From 6ff276c1a5ae0dd7703ff19e4b0a66754de65e9e Mon Sep 17 00:00:00 2001 From: jakobneef Date: Tue, 3 Dec 2024 17:23:31 +0000 Subject: [PATCH 01/13] Added PN stuff (not tested) --- Kernel/PN.wl | 2800 ++++++++++++++++++++++++++++++++++++++++++++ Kernel/Teukolsky.m | 1 + Kernel/Tools.wl | 277 +++++ 3 files changed, 3078 insertions(+) create mode 100644 Kernel/PN.wl create mode 100644 Kernel/Tools.wl diff --git a/Kernel/PN.wl b/Kernel/PN.wl new file mode 100644 index 0000000..218fce0 --- /dev/null +++ b/Kernel/PN.wl @@ -0,0 +1,2800 @@ +(* ::Package:: *) + +(* ::Input:: *) +(*SetOptions[EvaluationNotebook[],StyleDefinitions->$UserBaseDirectory<>"/SystemFiles/FrontEnd/StyleSheets/maTHEMEatica.nb"]*) + + +(* ::Section:: *) +(*Beginning Package*) + + +(* ::Subsection:: *) +(*Setting Context*) + + +BeginPackage["Teukolsky`PN`",{"Teukolsky`"}] + + +(* ::Subsection:: *) +(*Unprotecting*) + + +ClearAttributes[{TeukolskyRadialPN, TeukolskyRadialFunctionPN,TeukolskyPointParticleModePN}, {Protected, ReadProtected}]; + + +(* ::Section:: *) +(*Public *) + + +(* ::Subsection::Closed:: *) +(*MST Coefficients*) + + +(* ::Input:: *) +(*(*\[Nu]MST::usage="\[Nu]MST is representative of the \[Nu] coefficient in the MST solutions"*) +(*aMST::usage="aMST[\!\(\**) +(*StyleBox[\"n\",\nFontSlant->\"Italic\"]\)] is the \!\(\*SuperscriptBox[*) +(*StyleBox[\"n\",\nFontSlant->\"Italic\"], \(th\)]\) MST coefficient";*) +(*MSTCoefficients::usage="MSTCoefficients[\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]] gives the PN expanded MST coefficients aMST[n] for a given {\!\(\**) +(*StyleBox[\"\[ScriptS]\",\nFontSlant->\"Italic\"]\),\!\(\**) +(*StyleBox[\"\[ScriptL]\",\nFontSlant->\"Italic\"]\),\!\(\**) +(*StyleBox[\"\[ScriptM]\",\nFontSlant->\"Italic\"]\)} mode up to \!\(\*SuperscriptBox[\(\[Eta]\), \(\!\(\**) +(*StyleBox[\"order\[Eta]\",\nFontSlant->\"Italic\"]\)\)]\)."*) +(*KerrMSTSeries::usage="KerrMSTSeries[\!\(\**) +(*StyleBox[\"\[ScriptS]\",\nFontSlant->\"Italic\"]\),\!\(\**) +(*StyleBox[\"\[ScriptL]\",\nFontSlant->\"Italic\"]\),\!\(\**) +(*StyleBox[\"\[ScriptM]\",\nFontSlant->\"Italic\"]\),\!\(\**) +(*StyleBox[\"order\[Epsilon]\",\nFontSlant->\"Italic\"]\)] gives the PN expanded MST coefficients a[n] for a given {\!\(\**) +(*StyleBox[\"\[ScriptS]\",\nFontSlant->\"Italic\"]\), \!\(\**) +(*StyleBox[\"\[ScriptL]\",\nFontSlant->\"Italic\"]\), \!\(\**) +(*StyleBox[\"\[ScriptM]\",\nFontSlant->\"Italic\"]\)} mode up to \!\(\*SuperscriptBox[\(\[Epsilon]\), \(order\[Epsilon]\)]\). Where the relation to \[Eta] is given by \[Epsilon]=2 \[Omega] \!\(\*SuperscriptBox[\(\[Eta]\), \(3\)]\)."*)*) + + +(* ::Subsection::Closed:: *) +(*Spacetime replacements*) + + +(* ::Input:: *) +(*(*Kerr\[CapitalDelta]::usage="Kerr\[CapitalDelta][a,r] gives the Kerr \[CapitalDelta] \!\(\*SuperscriptBox[\(r\), \(2\)]\)-2r+\!\(\*SuperscriptBox[\(a\), \(2\)]\)"*)*) + + +(* ::Input:: *) +(*(*(*replsKerr::usage="a list of replacements for a Kerr spacetime."*) +(*replsSchwarzschild::usage="a list of replacements for Schwarzschild spacetime."*)*) +(*(*Schwarzschild::usage="Schwarzschild[\!\(\**) +(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)] will set a to 0 in \!\(\**) +(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)."*) +(**)*)*) + + +(* ::Subsection::Closed:: *) +(*Tools for Series*) + + +(* ::Subsubsection::Closed:: *) +(*General Tools for Series*) + + +(* ::Input:: *) +(*(*SeriesTake::usage="SeriesTake[series, n] takes the first n terms of series"*) +(*SeriesMinOrder::usage="SeriesMinOrder[series] gives the leading order of series"*) +(*SeriesMaxOrder::usage="SeriesMaxOrder[series] gives the first surpressed order of series"*) +(*SeriesLength::usage="SeriesLenght[series] gives the number of terms in series"*) +(*SeriesCollect::usage="SeriesCollect[expr, var, func] works like Collect but applied to each order individually. Crucially, unlike Collect it keeps the SeriesData structure."*) +(*SeriesTerms::usage="SeriesTerms[series, {x, \!\(\*SubscriptBox[\(x\), \(0\)]\), n}] works exactly like Series, with the difference that n gives the desired number of terms instead of a maximum order"*) +(*IgnoreExpansionParameter::usage="IgnoreExpansionParameter[series,x] sets all occurences of the expansion parameter in the series coefficients to x. If no value is entered x defaults to 1." *)*) + + +(* ::Subsubsection::Closed:: *) +(*Tools for PN Scalings*) + + +(* ::Input:: *) +(*(*PNScalings::usage="PNScalings[expr,params,var] applies the given powercounting scalings to the expression. E.g. PNScalings[\[Omega] r,{{\[Omega],3},{r,-2},\[Eta]]"*) +(*RemovePN::usage="PNScalings[expr,var] takes the Normal[] and sets var to 1"*) +(*Zero::usage="Zero[expr,vars] sets all vars in expr to 0"*) +(*One::usage="One[expr,vars] sets all vars in expr to 1"*)*) + + +(* ::Input:: *) +(*(*ExpandSpheroidals::usage="ExpandSpheroidal[expr,{param,order}] returns a all SpinWeightedSpheroidalHarmonicS in expr have been Series expanded around param->0 to order."*)*) + + +(* ::Input:: *) +(*(*CollectDerivatives::usage="CollectDerivatives[expr,f] works exactly like Collect[] but also collects for derivatives of f."*)*) + + +(* ::Subsection::Closed:: *) +(*Tools for Logs, Gammas, and PolyGammas*) + + +(* ::Input:: *) +(*(*ExpandLog::usage="ExpandLog[\!\(\**) +(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)] replaces all Logs in \!\(\**) +(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)\!\(\**) +(*StyleBox[\" \",\nFontSlant->\"Italic\"]\)with a PowerExpanded version"*) +(*ExpandGamma::usage="ExpandGamma[\!\(\**) +(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)] factors out all Integer facors out of the Gammas in \!\(\**) +(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\). E.g. Gamma[x+1]->x Gamma[x]"*) +(*ExpandPolyGamma::usage="ExpandPolyGamma[\!\(\**) +(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)] factors out all Integer facors out of the PolyGammas in \!\(\**) +(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\). E.g. PolyGamma[x+1]->\!\(\*FractionBox[\(1\), \(x\)]\) PolyGamma[x]"*) +(*PochhammerToGamma::usage="PochhammerToGamma[\!\(\**) +(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)] replaces all Pochhammer in \!\(\**) +(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\) with the respecive Gamma."*) +(*GammaToPochhammer::usage="PochhammerToGamma[\!\(\**) +(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\),\!\(\**) +(*StyleBox[\"n\",\nFontSlant->\"Italic\"]\)] replaces all Gamma in \!\(\**) +(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)\!\(\**) +(*StyleBox[\" \",\nFontSlant->\"Italic\"]\)that contain \!\(\**) +(*StyleBox[\"n\",\nFontSlant->\"Italic\"]\) with the respective Pochhammer[__,\!\(\**) +(*StyleBox[\"n\",\nFontSlant->\"Italic\"]\)]"*) +(**)*) + + +(* ::Subsection::Closed:: *) +(*Tools for DiracDelta*) + + +(* ::Input:: *) +(*(*ExpandDiracDelta::usage="ExpandDiracDelta[expr,r] applies identities for Dirac deltas and it's derivatives to expr."*)*) + + +(* ::Subsection::Closed:: *) +(*Amplitudes*) + + +(* ::Input:: *) +(*AAmplitude::usage="AAmplitude[\"+\"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]] gives the factor \!\(\*SubscriptBox[\(A\), \(+\)]\) from Sasaki Tagoshi Eq.(157). Likewise for [\"-\"]"*) +(*BAmplitude::usage="BAmplitude[\"trans\"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]] gives the factor \!\(\*SuperscriptBox[\(B\), \(trans\)]\) from Sasaki Tagoshi Eq.(167). Likewise for [\"inc\"]"*) +(*CAmplitude::usage="CAmplitude[\"trans\"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]] gives the factor \!\(\*SuperscriptBox[\(C\), \(trans\)]\) from Sasaki Tagoshi Eq.(170)."*) +(*\[ScriptCapitalK]Amplitude::usage="\[ScriptCapitalK]Amplitude[\"\[Nu]\"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]] gives the \!\(\*SuperscriptBox[\(\[ScriptCapitalK]\), \(\[Nu]\)]\) factor for \!\(\*SubscriptBox[\(R\), \(In\)]\). Likewise for \[ScriptCapitalK]Amplitude[\"-\[Nu]-1\"]. \[ScriptCapitalK]Amplitude[\"Ratio\"] gives the tidal response function \!\(\*FractionBox[SuperscriptBox[\(\[ScriptCapitalK]\), \(\(-\[Nu]\) - 1\)], SuperscriptBox[\(\[ScriptCapitalK]\), \(\[Nu]\)]]\)."*) + + +(* ::Input:: *) +(*(*TeukolskyAmplitudePN::usage="TeukolskyAmplitudePN[\"sol\"][\[ScriptS], \[ScriptL], \[ScriptM], a, \[Omega], {\[Eta], n}] gives the desired PN expanded amplitude. Options for sol are as follows: *) +(*\"A+\": Sasaki Tagoshi Eq.(157), *) +(*\"A-\": ST Eq.(158), *) +(*\"Btrans\": ST Eq.(167), *) +(*\"Binc\": ST Eq.(168) divided by \!\(\*SubscriptBox[\(\[ScriptCapitalK]\), \(\[Nu]\)]\), *) +(*\"Ctrans\": Eq.(170) ST, *) +(*\"\[ScriptCapitalK]\": , *) +(*\"\[ScriptCapitalK]\[Nu]\": , *) +(*\"\[ScriptCapitalK]-\[Nu]-1\": "*)*) + + +(* ::Subsection::Closed:: *) +(*Wronskian*) + + +(* ::Input:: *) +(*(*InvariantWronskian::usage="InvariantWronskian[\[ScriptS], \[ScriptL], \[ScriptM], a, \[Omega], {\[Eta], n}] gives the invariant Wronskian."*)*) + + +(* ::Subsection::Closed:: *) +(*Radial solutions*) + + +TeukolskyRadialPN::usage="TeukolskyRadialPN[\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{\[Eta],n}] gives the In and Up solution to the radial Teukolsky equation. {\[ScriptS],\[ScriptL],\[ScriptM]} specify the mode, a is the Kerr spin parameter, \[Omega] is the frequency, \[Eta] is the PN expansion parameter and n the number of terms." +TeukolskyRadialFunctionPN::usage="TeukolskyRadialFunctionPN[\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{\[Eta],n},\"sol\"] gives the specified solution to the radial Teukolsky equation. {\[ScriptS],\[ScriptL],\[ScriptM]} specify the mode, a is the Kerr spin parameter, \[Omega] is the frequency, \[Eta] is the PN expansion parameter and n the number of terms. Possible options for sol are In, Up, C\[Nu], and C-\[Nu]-1" + + +TeukolskyRadialFunctionPN::optx="`1` is not a valid boundary condition. Possible options are In, Up, C\[Nu], and C-\[Nu]-1"; +TeukolskyRadialFunctionPN::param\[ScriptS]="\[ScriptL]=`1` \[ScriptS]=`2`. \[ScriptL] has to be more or equal than |\[ScriptS]|"; +TeukolskyRadialFunctionPN::param\[ScriptM]="\[ScriptL]=`1` \[ScriptM]=`2`. \[ScriptL] has to be more or equal than |\[ScriptM]|"; +TeukolskyRadialFunctionPN::paramPN="varPN=`1`. The PN parameter has to be a Symbol. E.g. \[Eta] will work while \!\(\*SuperscriptBox[\(\[Eta]\), \(3\)]\) won't."; +TeukolskyRadialFunctionPN::paramaC="a=`1`. What do you even mean by a complex Kerr a???"; +TeukolskyRadialFunctionPN::parama="a=`1`. Numeric values for the Kerr parameter a have to be within [0,1]. It can however be left arbitrary."; +TeukolskyRadialFunctionPN::param\[Omega]="\[Omega]=`1`. Complex frequencies are not yet supported"; +TeukolskyRadialFunctionPN::paramorder="order=`1`. The given number of terms has to be an Integer"; + + +(* ::Subsection::Closed:: *) +(*Sourced things*) + + +(* ::Input:: *) +(*(*TeukolskySourceCircularOrbit::usage="TeukolskySource[\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{r,r\:2080}] gives an analytical expression for the Teukolsky point particle source for a given {\[ScriptS],\[ScriptL],\[ScriptM]} mode. "*)*) + + +TeukolskyPointParticleModePN::usage="TeukolskyPointParticleMode[\[ScriptS], \[ScriptL], \[ScriptM], orbit] produces a TeukolskyModePN representing a PN expanded analyitcal solution to the radial Teukolsky equation with a point particle source." + + +TeukolskyModePN::usage="aa" + + +TeukolskyPointParticleModePN::orbit="As of now TeukolskyPointParticleModePN only supports circular equatorial orbits, i.e., e=0 and x=1."; +TeukolskyPointParticleModePN::particle="TeukolskyPointParticleModePN cannot be evaluated at the particle."; + + +\[CapitalOmega]Kerr::usage="This is here as a quick fix. It is the particles orbital frequency, i.e., \!\(\*SqrtBox[FractionBox[\(1\), SuperscriptBox[SubscriptBox[\(r\), \(0\)], \(3\)]]]\) in Schwarzschild." + + +(* ::Subsection::Closed:: *) +(*Teukolsky Equation*) + + +(* ::Input:: *) +(*(*TeukolskyEquation::usage="TeukolskyEquation[\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{\[Eta],order},R[r]] gives the Teukolsky equation with for a given {\[ScriptS],\[ScriptL],\[ScriptM]} mode with included \[Eta] scalings. The {\[Eta],order} argument can be left out for a general expression."*)*) + + +(* ::Subsection::Closed:: *) +(*Developer options*) + + +(* ::Input:: *) +(*(*integrateDelta::usage="aala"*) +(*\[Delta]*) +(*\[Theta]*) +(**) +(*PNScalings*)*) + + +(* ::Input:: *) +(*(*z::usage="iuaeouia"*) +(*\[CapitalDelta]::usage="\[CapitalDelta][\!\(\**) +(*StyleBox[\"a\",\nFontSlant->\"Italic\"]\),\!\(\**) +(*StyleBox[\"M\",\nFontSlant->\"Italic\"]\),\!\(\**) +(*StyleBox[\"r\",\nFontSlant->\"Italic\"]\)] is the usual length scale of the Kerr spacetime"*)*) + + +(* ::Subsubsection::Closed:: *) +(*Post Newtonian Scalings*) + + +(* ::Input:: *) +(*(*replsPN="Replacements for PN scalings"*) +(*PNScalingsInternal::usage="PNScalingsInternal[\!\(\**) +(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)] reapplies the PN scalings to \!\(\**) +(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\). Should not be used on SeriesData objects. "*) +(*redo\[Eta]Repls::usage="iuaeuiae"*) +(*RemovePNInternal::usage="RemovePNInternal[expr] removes all PN scalings in the expression, i.e., taking the Normal and setting \[Eta]->1"*) +(*polyToSeries*) +(*IgnoreLog\[Eta]::usage="IgnoreLog\[Eta][\!\(\**) +(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)] sets all \[Eta] factors within \!\(\**) +(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\) to 1. This can be helpful when using SeriesCollect"*)*) + + +(* ::Subsubsection::Closed:: *) +(*Radial functions*) + + +(* ::Input:: *) +(*(*RPN::usage="RPN[\"In\"][\!\(\**) +(*StyleBox[\"\[ScriptS]\",\nFontSlant->\"Italic\"]\),\!\(\**) +(*StyleBox[\"\[ScriptL]\",\nFontSlant->\"Italic\"]\),\!\(\**) +(*StyleBox[\"\[ScriptM]\",\nFontSlant->\"Italic\"]\),\!\(\**) +(*StyleBox[\"a\",\nFontSlant->\"Italic\"]\),\!\(\**) +(*StyleBox[\"order\[Eta]\",\nFontSlant->\"Italic\"]\)] Gives the In solution to the radial Teukolsky equation. Analogously for \"Up\", \"C\", or \"secondTerm\". The inhomogeneous solution for a circular orbit can be obtained with \"CO\""*) +(*RPNF::usage="RPNF[\!\(\**) +(*StyleBox[\"\[ScriptS]\",\nFontSlant->\"Italic\"]\),\!\(\**) +(*StyleBox[\"\[ScriptL]\",\nFontSlant->\"Italic\"]\),\!\(\**) +(*StyleBox[\"\[ScriptM]\",\nFontSlant->\"Italic\"]\),\!\(\**) +(*StyleBox[\"a\",\nFontSlant->\"Italic\"]\),\!\(\**) +(*StyleBox[\"\[Omega]\",\nFontSlant->\"Italic\"]\),{\!\(\**) +(*StyleBox[\"\[Eta]\",\nFontSlant->\"Italic\"]\),order\[Eta]}] gives a function"*) +(*integrateDelta::usage="Function that integrates Dirac delta distributions. Used when getting the inhomogeneous solutions."*) +(*Normalization::usage="Normalization[\"In\"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]] gives a PN expanded normaliztion coefficient. If you divide RPN by it it will match the result of the toolkit"*)*) + + +(* ::Input:: *) +(*(*CCoefficient::usage="CCoefficient[\"In\"][\!\(\**) +(*StyleBox[\"\[ScriptS]\",\nFontSlant->\"Italic\"]\),\!\(\**) +(*StyleBox[\"\[ScriptL]\",\nFontSlant->\"Italic\"]\),\!\(\**) +(*StyleBox[\"\[ScriptM]\",\nFontSlant->\"Italic\"]\),\!\(\**) +(*StyleBox[\"a\",\nFontSlant->\"Italic\"]\),\!\(\**) +(*StyleBox[\"order\[Eta]\",\nFontSlant->\"Italic\"]\)] gives the \!\(\*SubscriptBox[\(c\), \(\(in\)\(\\\ \)\)]\)coefficient for the sourced solution R=\!\(\*SubscriptBox[\(c\), \(in\)]\) \!\(\*SubscriptBox[\(R\), \(in\)]\) + \!\(\*SubscriptBox[\(c\), \(up\)]\) \!\(\*SubscriptBox[\(R\), \(up\)]\). Likewise for [\"Up\"]"*) +(*InvariantWronskian::usage="InvariantWronskian[\!\(\**) +(*StyleBox[\"\[ScriptS]\",\nFontSlant->\"Italic\"]\),\!\(\**) +(*StyleBox[\"\[ScriptL]\",\nFontSlant->\"Italic\"]\),\!\(\**) +(*StyleBox[\"\[ScriptM]\",\nFontSlant->\"Italic\"]\),\!\(\**) +(*StyleBox[\"a\",\nFontSlant->\"Italic\"]\),\!\(\**) +(*StyleBox[\"order\[Eta]\",\nFontSlant->\"Italic\"]\)] gives the invariant Wronskian. C.f. Sasaki Tagoshi Eq.(23)"*) +(**)*) + + +(* ::Section:: *) +(*Private*) + + +Begin["`Private`"] + + +(* ::Subsection::Closed:: *) +(*Loading dependencies*) + + +(*< -s transform has factors \[Epsilon]^-2 for negative s *) +ClearAll[\[CapitalDelta]\[Nu]p2C]; +\[CapitalDelta]\[Nu]p2C[k_]:=\[CapitalDelta]\[Nu]p2C[k]=Sum[Expand[\[CapitalDelta]\[Nu]pC[i2]*\[CapitalDelta]\[Nu]pC[- i2 + k]], {i2, 0, k}]; +\[CapitalDelta]\[Nu]p3C[k_]:=\[CapitalDelta]\[Nu]p3C[k]=Sum[Expand[\[CapitalDelta]\[Nu]p2C[k - i3]*\[CapitalDelta]\[Nu]pC[i3]], {i3, 0, k}]; +\[CapitalDelta]\[Nu]p4C[k_]:=\[CapitalDelta]\[Nu]p4C[k]=Sum[Expand[\[CapitalDelta]\[Nu]p3C[k - i4]*\[CapitalDelta]\[Nu]pC[i4]], {i4, 0, k}]; +\[CapitalDelta]\[Nu]p5C[k_]:=\[CapitalDelta]\[Nu]p5C[k]=Sum[Expand[\[CapitalDelta]\[Nu]p4C[k - i5]*\[CapitalDelta]\[Nu]pC[i5]], {i5, 0, k}]; +\[CapitalDelta]\[Nu]p6C[k_]:=\[CapitalDelta]\[Nu]p6C[k]=Sum[Expand[\[CapitalDelta]\[Nu]p5C[k - i6]*\[CapitalDelta]\[Nu]pC[i6]], {i6, 0, k}]; + +\[Alpha]C[n_,k_]:=\[Alpha]C[n,k]=((l-2 l^2+n-4 l n-2 n^2) If[-4+k==0,1,0]+I (2 l^2+n (-1+2 n)+l (-1+4 n)) (-I m q+(1+l+n) \[Kappa]) If[-3+k==0,1,0]+(-2 l^2-n (-1+2 n)-l (-1+4 n)) (1+l+n+s)^2 If[-2+k==0,1,0]+I (2 l^2+n (-1+2 n)+l (-1+4 n)) (1+l+n+s)^2 (-I m q+(1+l+n) \[Kappa]) If[-1+k==0,1,0]+ +2 I \[Kappa] \[CapitalDelta]\[Nu]p3C[k-9]+(-3-8 l-8 n-4 s) \[CapitalDelta]\[Nu]p3C[k-8]+I (-I m q (3+8 l+8 n+4 s)+(3+20 l^2+20 n^2+6 s+2 s^2+4 n (5+4 s)+4 l (5+10 n+4 s)) \[Kappa]) \[CapitalDelta]\[Nu]p3C[k-7]- +2 \[CapitalDelta]\[Nu]p4C[k-10]+I (-2 I m q+(5+10 l+10 n+4 s) \[Kappa]) \[CapitalDelta]\[Nu]p4C[k-9]+2 I \[Kappa] \[CapitalDelta]\[Nu]p5C[k-11]-2\[CapitalDelta]\[Nu]p2C[k-8]+I (-2 I m q+\[Kappa]+6 l \[Kappa]+6 n \[Kappa]) \[CapitalDelta]\[Nu]p2C[k-7]+ +(-12 l^2-12 n^2-2 s (1+s)-3 n (3+4 s)-3 l (3+8 n+4 s))\[CapitalDelta]\[Nu]p2C[k-6]+I (-I m q (12 n^2+2 s (1+s)+3 n (3+4 s))+20 l^3 \[Kappa]+ +(-1+20 n^3+s^2+6 n^2 (5+4 s)+3 n (3+6 s+2 s^2)) \[Kappa]+6 l^2 (-2 I m q+(5+10 n+4 s) \[Kappa])+3 l (-I m q (3+8 n+4 s)+(3+20 n^2+6 s+2 s^2+4 n (5+4 s)) \[Kappa])) \[CapitalDelta]\[Nu]p2C[k-5]+ +(1-4 l-4 n) \[CapitalDelta]\[Nu]pC[-6+k]+I (-I m (-1+4 l+4 n) q+(-1+6 l^2+2 n+6 n^2+2 l (1+6 n)) \[Kappa]) \[CapitalDelta]\[Nu]pC[-5+k]+ +(-8 l^3-8 n^3-4 n s (1+s)+(1+s)^2-3 n^2 (3+4 s)-3 l^2 (3+8 n+4 s)-2 l (12 n^2+2 s (1+s)+3 n (3+4 s))) \[CapitalDelta]\[Nu]pC[-4+k]+ +I (1+l+n+s) (-I m q (-1+l+8 l^2+n+16 l n+8 n^2-s+4 l s+4 n s)+ +(-1+10 l^3+10 n^3-s+n (-1+2 s)+2 n^2 (5+3 s)+2 l^2 (5+15 n+3 s)+l (-1+30 n^2+2 s+4 n (5+3 s))) \[Kappa]) \[CapitalDelta]\[Nu]pC[-3+k]); + +\[Beta]C[n_,k_]:=\[Beta]C[n,k]=((-3+4 l^2+4 n+4 n^2+l (4+8 n)) If[-4+k==0,1,0]-m (-3+4 l^2+4 n+4 n^2+l (4+8 n)) q If[-3+k==0,1,0]+ +1/4 (3-4 l^2-4 n-4 n^2-l (4+8 n)) (l^2 (-8+q^2)+n (-8+q^2)+n^2 (-8+q^2)+l (1+2 n) (-8+q^2)-4 s^2) If[-2+k==0,1,0]- +m (-3+4 l^2+4 n+4 n^2+l (4+8 n)) q s^2 If[-1+k==0,1,0]+n (8 l^5+4 l^4 (5+9 n)+n (1+n)^2 (-3+4 n+4 n^2)+2 l^3 (5+36 n+32 n^2)+ +l^2 (-5+29 n+96 n^2+56 n^3)+l (-3-7 n+28 n^2+56 n^3+24 n^4)) If[k==0,1,0]-8 (1+2 l+2 n) Sum[\[CapitalDelta]\[Nu]p3C[k-6-i]*\[CapitalDelta]EC[s, l, m, i], {i, 0, k-6}]- +4 Sum[\[CapitalDelta]\[Nu]p4C[k-8-i]*\[CapitalDelta]EC[s, l, m, i], {i, 0, k - 8}]+(-1-24 l^2-24 n-24 n^2-24 l (1+2 n))Sum[\[CapitalDelta]\[Nu]p2C[k-4-i]*\[CapitalDelta]EC[s, l, m, i], {i, 0, k - 4}]- +2 (1+2 l+2 n) (-8+q^2) \[CapitalDelta]\[Nu]p3C[k-8]+(-2+64 l^3+36 n+120 n^2+80 n^3+32 l^2 (3+7 n)+4 l (7+56 n+60 n^2))\[CapitalDelta]\[Nu]p3C[k-6]+ +(8-q^2) \[CapitalDelta]\[Nu]p4C[k-10]+(9+56 l^2+60 n+60 n^2+8 l (7+15 n))\[CapitalDelta]\[Nu]p4C[k-8] +12 (1+2 l+2 n) \[CapitalDelta]\[Nu]p5C[k-10]+ +4 \[CapitalDelta]\[Nu]p6C[k-12]+(3-16 l^3-2 n-24 n^2-16 n^3-24 l^2 (1+2 n)-l (2+48 n+48 n^2)) Sum[\[CapitalDelta]EC[s, l, m, i]*\[CapitalDelta]\[Nu]pC[-2 - i + k], {i, 0, -2 + k}]+ +4 \[CapitalDelta]\[Nu]p2C[k-8]-4 m q \[CapitalDelta]\[Nu]p2C[k-7]+(2-q^2/4-6 l^2 (-8+q^2)-6 n (-8+q^2)-6 n^2 (-8+q^2)-6 l (1+2 n) (-8+q^2)+4 s^2) \[CapitalDelta]\[Nu]p2C[k-6]- +4 m q s^2 \[CapitalDelta]\[Nu]p2C[k-5]+(-3+36 l^4-6 n+54 n^2+120 n^3+60 n^4+24 l^3 (3+8 n)+l^2 (29+288 n+336 n^2)+l (-7+84 n+336 n^2+240 n^3)) \[CapitalDelta]\[Nu]p2C[k-4]+ +(-4 l^4-8 l^3 (1+2 n)-l^2 (1+24 n+24 n^2)-n (-3+n+8 n^2+4 n^3)-l (-3+2 n+24 n^2+16 n^3)) \[CapitalDelta]EC[s,l,m,k]+(4+8 l+8 n) \[CapitalDelta]\[Nu]pC[-6+k]- +4 m (1+2 l+2 n) q \[CapitalDelta]\[Nu]pC[-5+k]+1/4 (-1-2 l-2 n) (24-3 q^2+8 l^2 (-8+q^2)+8 n (-8+q^2)+8 n^2 (-8+q^2)+8 l (1+2 n) (-8+q^2)-16 s^2) \[CapitalDelta]\[Nu]pC[-4+k]- +4 m (1+2 l+2 n) q s^2 \[CapitalDelta]\[Nu]pC[-3+k]+(8 l^5+4 l^4 (5+18 n)+2 l^3 (5+72 n+96 n^2)+l^2 (-5+58 n+288 n^2+224 n^3)+6 n (-1-n+6 n^2+10 n^3+4 n^4)+ +l (-3-14 n+84 n^2+224 n^3+120 n^4)) \[CapitalDelta]\[Nu]pC[-2+k]); + +\[Gamma]C[n_,k_]:=\[Gamma]C[n,k]=((-3-2 l^2-5 n-2 n^2-l (5+4 n)) If[-4+k==0,1,0]-I (3+2 l^2+5 n+2 n^2+l (5+4 n)) (I m q+(l+n) \[Kappa]) If[-3+k==0,1,0]+ +(-3-2 l^2-5 n-2 n^2-l (5+4 n)) (l+n-s)^2 If[-2+k==0,1,0]-I (3+2 l^2+5 n+2 n^2+l (5+4 n)) (l+n-s)^2 (I m q+(l+n) \[Kappa]) If[-1+k==0,1,0]- +2 I \[Kappa] \[CapitalDelta]\[Nu]p3C[k-9]+(-5-8 l-8 n+4 s)\[CapitalDelta]\[Nu]p3C[k-8]-I (I m q (5+8 l+8 n-4 s)+(3+20 l^2+20 n+20 n^2+4 l (5+10 n-4 s)-10 s-16 n s+2 s^2) \[Kappa]) \[CapitalDelta]\[Nu]p3C[k-7]- +2 \[CapitalDelta]\[Nu]p4C[k-10]-I (2 I m q+(5+10 l+10 n-4 s) \[Kappa]) \[CapitalDelta]\[Nu]p4C[k-9]-2 I \[Kappa] \[CapitalDelta]\[Nu]p5C[k-11]-2 \[CapitalDelta]\[Nu]p2C[k-8]-I (2 I m q+(5+6 l+6 n) \[Kappa])\[CapitalDelta]\[Nu]p2C[k-7]+ +(-3-12 l^2-12 n^2-3 l (5+8 n-4 s)+10 s-2 s^2+3 n (-5+4 s))\[CapitalDelta]\[Nu]p2C[k-6]-I (I m q (3+12 l^2+12 n^2+3 l (5+8 n-4 s)-10 s+2 s^2-3 n (-5+4 s))+ +(20 l^3+20 n^3+6 l^2 (5+10 n-4 s)-6 n^2 (-5+4 s)+s (-6+5 s)+n (9-30 s+6 s^2)+l (9+60 n^2+n (60-48 s)-30 s+6 s^2)) \[Kappa]) \[CapitalDelta]\[Nu]p2C[k-5]+ +(-5-4 l-4 n) \[CapitalDelta]\[Nu]pC[-6+k]-I (I m (5+4 l+4 n) q+(3+6 l^2+10 n+6 n^2+2 l (5+6 n)) \[Kappa]) \[CapitalDelta]\[Nu]pC[-5+k]+ +(-8 l^3-8 n^3-3 l^2 (5+8 n-4 s)+(6-5 s) s+3 n^2 (-5+4 s)+n (-6+20 s-4 s^2)-2 l (3+12 n^2-10 s+2 s^2-3 n (-5+4 s))) \[CapitalDelta]\[Nu]pC[-4+k]- +I (l+n-s) (I m q (6+8 l^2+8 n^2+n (15-4 s)+l (15+16 n-4 s)-5 s)+ +(10 l^3+10 n^3+n (9-10 s)+l (9+30 n^2+n (40-12 s)-10 s)+n^2 (20-6 s)+l^2 (20+30 n-6 s)-3 s) \[Kappa]) \[CapitalDelta]\[Nu]pC[-3+k]); + +Do[\[CapitalDelta]\[Nu]pC[k]=0,{k,-12,-1}]; +\[Beta]C[-l,2]=\[Beta]C[-l,2]//Simplify; +\[Beta]C[-l,1]=\[Beta]C[-l,1]//Simplify; +\[Alpha]C[-l-1,2]=\[Alpha]C[-l-1,2]//Simplify; + +\[Kappa]Simplify[x_]:=Expand[x]/.\[Kappa]^2->(1-q^2); +\[CapitalDelta]\[Alpha]\[Beta]C[n_,k_]:=(\[Beta]C[-l,k]- \[Alpha]C[-l-1,k])//\[Kappa]Simplify; + +ProgressGrid:=Grid[ +Table[ +If[i==0,If[j<2||NumericQ[\[CapitalDelta]\[Nu]pC[j-2]/.{q->1/Sqrt[2],\[Kappa]->1/Sqrt[2]}],Item[j,Background->Green],Item[j,Background->Yellow]], +If[TrueQ[acSolved[i,j]],If[NumericQ[ac[i,j]/.{q->1/Sqrt[2],\[Kappa]->1/Sqrt[2]}],If[(ac[i,j]/.{q->1/Sqrt[2],\[Kappa]->1/Sqrt[2]})==0,Item[" ",Background->LightRed],Item[" ",Background->Red]],Item[" ",Background->Orange]] ,If[jBlue]]]],{i,ExpOrder+2,-ExpOrder-2,-1},{j,-1,ExpOrder+2}],ItemSize->All,Frame->All]; + +(* +FinalProgressGrid:=Grid[ +Table[ +If[i==0,If[j<2||NumericQ[\[CapitalDelta]\[Nu]pC[j-2]/.{q->1/Sqrt[2],\[Kappa]->1/Sqrt[2]}],Item[j,Background->Green],If[j==-1,Item["\[Nu]",Background->White],Item[j,Background->Yellow]]], +If[NumericQ[ac[i,j]/.{q->1/Sqrt[2],\[Kappa]->1/Sqrt[2]}],If[(ac[i,j]/.{q->1/Sqrt[2],\[Kappa]->1/Sqrt[2]})==0,Item[" ",Background->White],Item[" ",Background->Red]],If[jBlue]]]], +{i,ExpOrder+2,-ExpOrder-2,-1},{j,-1,ExpOrder+2}],ItemSize->All,Frame->All]; +*); + +FinalProgressGrid:=Grid[ +Table[ +If[i==0,If[j>=0 && NumericQ[\[CapitalDelta]\[Nu]pC[j-2]/.{q->1/Sqrt[2],\[Kappa]->1/Sqrt[2]}],Item[j,Background->Green],If[j==-1,Item["\[Nu]",Background->White],Item[j,Background->Yellow]]], +If[NumericQ[ac[i,j]/.{q->1/Sqrt[2],\[Kappa]->1/Sqrt[2]}],If[(ac[i,j]/.{q->1/Sqrt[2],\[Kappa]->1/Sqrt[2]})==0,Item[" ",Background->White],Item[" ",Background->Red]],If[jBlue]]]], +{i,ExpOrder+2,-ExpOrder-2,-1},{j,-1,ExpOrder+2}],ItemSize->All,Frame->All]; + +StructureGrid:=Grid[ +Table[ +If[i==0,Item[j,Background->Yellow], +If[jCyan],-4l,Item[" ",Background->Gray],_,Item[" ",Background->Blue]]]],{i,ExpOrder+2,-ExpOrder-2,-1},{j,-1,ExpOrder+2}],ItemSize->All,Frame->All]; + +<q \[Epsilon]/2)+O[\[Epsilon]]^(AngExpOrder+1)-l(l+1)+s(s+1)+ m q \[Epsilon]-(q \[Epsilon]/2)^2; +\[CapitalDelta]EC[s,l,m,0]=0; +Do[\[CapitalDelta]EC[s,l,m,i]=Coefficient[\[CapitalDelta]E[s,l,m],\[Epsilon],i],{i,1,AngExpOrder}]; + +ClearAll[aMST,ac,acq,eqnlist,\[CapitalDelta]\[Nu]p,\[CapitalDelta]\[Nu]pc]; + +If[s==-2, + aShift[l,n_]:=Which[n<=l-2,0,n==l-1,2,n==l,1, + n>l&&n<2l+1,0, + n>=2l+1,-2]; + eqShift[l,n_]:=Which[n<=l-3,0,n==l-2,4,n==l-1,4,n==l,5-Boole[l==2], + n>l&&n<2l+1,1, + n>=2l+1,-1];]; +If[s==-1, + aShift[l,n_]:=Which[n<=l-2,0,n==l-1,0,n==l,1, + n>l&&n<2l+1,0, + n>=2l+1,-2]; + eqShift[l,n_]:=Which[n<=l-3,0,n==l-2,0,n==l-1,3,n==l,4, + n>l&&n<2l+1,1, + n>=2l+1,-1];]; +If[s==0,aShift[l,n_]:=Which[n<2l+1,0,n>=2l+1,-2+Boole[l==0]]; + eqShift[l,n_]:=Which[n<=l-3,0,n==l-2,2,n==l-1,4,n==l,3, + n>l&&n<2l+1,1, + n>=2l+1,-1];]; + +aLeadingBehaviour[l,n_]=aLeadingBehaviour[l,n]=If[n>=0,n,-n+aShift[l,-n]]; + +aMST[0]=1; +ac[0,_]=0; +ac[0,0]=1; +Do[aMST[n]=Sum[ac[n,i]\[Epsilon]^i,{i,n,ExpOrder+1}],{n,1,ExpOrder+2}]; +Do[ac[n,i]=0,{n,1,ExpOrder+2},{i,0,n-1}]; +Do[ac[n,i]=If[EvenQ[i],Sum[acq[n,i,j]q^j,{j,0,i,2}]+I \[Kappa] Sum[acq[n,i,j]q^j,{j,1,i,2}],Sum[acq[n,i,j]q^j,{j,1,i,2}]+I \[Kappa] Sum[acq[n,i,j]q^j,{j,0,i,2}]],{n,1,ExpOrder+2},{i,n,ExpOrder+2}]; +Do[aMST[-n]=Sum[ac[-n,i]\[Epsilon]^i,{i,n+aShift[l,n],ExpOrder+1}],{n,1,ExpOrder+4}]; +Do[ac[-n,i]=0,{n,1,ExpOrder+4},{i,0,n+aShift[l,n]-1}]; +Do[ac[-n,i]=If[EvenQ[i],Sum[acq[-n,i,j]q^j,{j,0,i,2}]+I \[Kappa] Sum[acq[-n,i,j]q^j,{j,1,i,2}],Sum[acq[-n,i,j]q^j,{j,1,i,2}]+I \[Kappa] Sum[acq[-n,i,j]q^j,{j,0,i,2}]],{n,1,ExpOrder+4},{i,n+aShift[l,n],ExpOrder+3}]; +If[m == 0,Do[Do[acq[-n,n-2,i]=0;acqSolved[-n,n-2,i]=True,{i,0,n-2}];acSolved[-n,n-2]=True,{n,2l+1,ExpOrder+4}]]; + +\[CapitalDelta]\[Nu]p=Sum[\[CapitalDelta]\[Nu]pC[i] \[Epsilon]^i,{i,0,ExpOrder+2}]; +Do[\[CapitalDelta]\[Nu]pC[i]=If[EvenQ[i],Sum[\[CapitalDelta]\[Nu]pcq[i,j]q^j,{j,0,i,2}],Sum[\[CapitalDelta]\[Nu]pcq[i,j]q^j,{j,1,i,2}]],{i,0,ExpOrder+2}]; + +EqC[n_,k_]:=Sum[\[Alpha]C[n-1,k-i]ac[n,i]+\[Beta]C[n-1,k-i]ac[n-1,i]+\[Gamma]C[n-1,k-i]ac[n-2,i],{i,Min[Abs[n]+aShift[l,-n],Abs[n-1]+aShift[l,-(n-1)],Abs[n-2]+aShift[l,-(n-2)]],k}]; +\[CapitalDelta]EqC[n_,k_]:=Sum[\[Alpha]C[n,k-i]ac[n+1,i]+(\[Beta]C[n,k-i]- \[Alpha]C[n-1,k-i])ac[n,i]+(\[Gamma]C[n,k-i]- \[Beta]C[n-1,k-i])ac[n-1,i]+(-\[Gamma]C[n-1,k-i])ac[n-2,i],{i,Min[Abs[n+1]+aShift[l,-(n+1)],Abs[n]+aShift[l,-n],Abs[n-1]+aShift[l,-(n-1)],Abs[n-2]+aShift[l,-(n-2)]],k}]; + +EqCL[n_,p_]:=\[Kappa]Simplify[EqC[n,Min[Abs[n]-eqShift[l,-n]+If[n>1,0,2]+p,ExpOrder+eqShift[l,-n]+2]]]; +\[CapitalDelta]EqCL[n_,p_]:=\[Kappa]Simplify[\[CapitalDelta]EqC[n,Min[Abs[n]-eqShift[l,-n]+If[n>1,0,2]+p,ExpOrder+eqShift[l,-n]+2]]]; + +\[CapitalDelta]EqCTable[n_,k_]:=Table[{\[Alpha]C[n,k-i]ac[n+1,i],(\[Beta]C[n,k-i]- \[Alpha]C[n-1,k-i])ac[n,i]+(\[Gamma]C[n,k-i]- \[Beta]C[n-1,k-i])ac[n-1,i],(-\[Gamma]C[n-1,k-i])ac[n-2,i]},{i,Min[Abs[n+1]+aShift[l,-(n+1)],Abs[n]+aShift[l,-n],Abs[n-1]+aShift[l,-(n-1)],Abs[n-2]+aShift[l,-(n-2)]],k}]; + +Clear[Solveac]; +Solveac[i_?IntegerQ,j_?IntegerQ,Eq_,drop_?IntegerQ]:=Module[{tmpEq,tmp,tmpN,tmpD,shift,Verbose=False,acSolve=True}, +If[(i>0&&j(*+i*)>ExpOrder+Boole[l==0]+2Boole[s==-1&&l==1])||(i<0&&j(*-i*)>ExpOrder+2),acSolve=False;Return[]]; +(* remove commented terms if only want \[Nu] *) +If[Abs[j]<=ExpOrder+3&&acSolve,tmpEq=Drop[CoefficientList[Eq,q],drop]; +If[Verbose,Print[tmpEq,"\t",Dimensions[tmpEq][[1]]]]; +shift=0; +Do[ + tmpN=\[Kappa]Simplify[Coefficient[tmpEq[[k+1]],acq[i,j,k+shift],0]]; + tmpD=-\[Kappa]Simplify[Coefficient[tmpEq[[k+1]],acq[i,j,k+shift]]]; + If[Verbose,Print["i=",i,"\t j=",j, "\t k=",k,"\t tmpN:=",tmpN,"\t tmpD:=",tmpD]]; + If[tmpD==0,\[SZ]t["Attempted division by 0 in Solveac\n i=",i,"\t j=",j,"\t k+shift=",k+shift,"\t tmpEq=",Simplify[tmpEq]];Continue[]]; + tmp=tmpN/tmpD; + If[Verbose,Print["Eq:\t",tmpEq,"\t",tmpN,"\t",tmpD,"\t",tmp]]; + acq[i,j,k+shift]=\[Kappa]Simplify[tmp]; + acqSolved[i,j,k+shift]=True; + If[Verbose && acq[i,j,k+shift]==0,Print["Vanishing acq for i=",i,"\t j=",j,"\t k=",k]], +{k,0,j}]; +ac[i,j]=Collect[ac[i,j],\[Kappa]]; +acSolved[i,j]=True]; +If[Verbose,Print[ac[i,j]]]; +]; + +PrintCLSolveac[a_,b_,c_,d_]:=Print[a,"\t",b,"\t",CoefficientList[c,q],"\t",d]; + +Clear[Solve\[CapitalDelta]\[Nu]]; +Solve\[CapitalDelta]\[Nu][i_?IntegerQ,p_?IntegerQ, OptionsPattern[]]:=Module[{tmpEq,tmp,tmpN,tmpD,shift,Verbose=False}, +shift=If[EvenQ[i],0,1]; +tmpEq=Take[CoefficientList[EqCL[1,p],q],{shift+1,-1,2}]; +Do[ + tmpN=Coefficient[tmpEq[[k+1]],\[CapitalDelta]\[Nu]pcq[i,2k+shift],0]; + tmpD=-Coefficient[tmpEq[[k+1]],\[CapitalDelta]\[Nu]pcq[i,2k+shift]]; + If[tmpD==0,Print["Attempted division by 0 in Solve\[CapitalDelta]\[Nu] (i=",i,"\t 2k+shift=",2k+shift,")"],tmp=tmpN/tmpD]; + \[CapitalDelta]\[Nu]pcq[i,2k+shift]=\[Kappa]Simplify[tmp], +{k,0,Floor[i/2]}]; +If[Verbose,Print["i=",i,"\t \[CapitalDelta]\[Nu]pC[",i,"]=",\[CapitalDelta]\[Nu]pC[i]]]; +]; + + +Which[s==-2, + (*Print["s =-2 code"];*) + Which[l==2 && m!=0, + + Do[ + Do[If[n+p<=ExpOrder,Solveac[n,n+p,EqCL[n+1,p-1],0]],{n,1,ExpOrder}]; + Solve\[CapitalDelta]\[Nu][p,p-1], + {p,0,1}]; + p=1; + Solveac[-1,p+2,EqCL[0,p+4],0]; + Do[ + Do[If[n+p<=ExpOrder,Solveac[n,n+p,EqCL[n+1,p-1],0]],{n,1,ExpOrder}]; + Solve\[CapitalDelta]\[Nu][p,p-1]; + Solveac[-1,p+2,EqCL[0,p+4],0]; + If[p==2,Solveac[-2,3,EqCL[-1,5],1]]; + If[p2&&p<=ExpOrder, + Solveac[-5,p+1,EqCL[-4,p-2],0]; + Do[If[n+p-4<=ExpOrder+1,Solveac[-n,n+p-4,EqCL[-n+1,p-6],0]],{n,6,ExpOrder+2}]], + {p,2,ExpOrder}], + + l>2, + Do[ + Do[If[n+p<=ExpOrder,Solveac[n,n+p,EqCL[n+1,p-1],0]],{n,1,ExpOrder}]; + Do[Solveac[-n,n+p,EqCL[-n+1,p-1],0],{n,1,l-2}]; + Solve\[CapitalDelta]\[Nu][p,p-1], + {p,0,3}]; + + p=3; + If[m != 0,Solveac[-(l-1),l+1+(p-3),EqCL[-(l-2),(p-3)+5],0]]; + + Do[ + If[m == 0,Solveac[-(l+1),l+1+(p-4),EqCL[-(l-1),p+2],0],Solveac[-(l+1),l+1+(p-4),EqCL[-(l-1),p+1],1]]; + If[m == 0,Solveac[-l,l+1+(p-4),EqCL[-l,p+2],0],Solveac[-l,l+1+(p-4),(\[Beta]C[-l,2]- \[Alpha]C[-l-1,2])EqCL[-l+1,p+2]- \[Beta]C[-l,1] \[CapitalDelta]EqCL[-l,p+3],0]]; + If[m == 0,Solveac[-(l-1),l+1+(p-4),EqCL[-(l-2),p+1],0],Solveac[-(l-1),l+2+(p-4),EqCL[-(l-2),p+2],0]]; + + Do[If[n+p-4<=ExpOrder+2,Solveac[-n,n+(p-4),EqCL[-n+1,p-4],0]],{n,l+2,2l-1}]; + If[p>3+Boole[m==0]&&2l+p-4<=ExpOrder+1,Solveac[-2l-1,2l+(p-4)-1,EqCL[-2l,(p-4)],0]]; + If[p>3+Boole[m==0]&&2l+p-4<=ExpOrder+1,Solveac[-2l-2,2l+(p-4),EqCL[-2l-1,(p-4)-4],0]]; + If[2l+p-4<=ExpOrder+1,Solveac[-2l,2l+(p-4),EqCL[-2l+1,p-4],0]]; + Do[If[p>3+Boole[m==0]&&n-1+p-4<=ExpOrder,Solveac[-n-1,n-1+p-4,EqCL[-n,(p-4)-4],0]],{n,2l+2,ExpOrder+1}]; + Do[If[n+p<=ExpOrder,Solveac[n,n+p,EqCL[n+1,p-1],0]],{n,1,ExpOrder-p}]; + Do[If[n+p<=ExpOrder,Solveac[-n,n+p,EqCL[-n+1,p-1],0]],{n,1,l-2}]; + + If[p<=ExpOrder,Solve\[CapitalDelta]\[Nu][p,p-1]], + {p,4,ExpOrder+Boole[l==2]}]], + +s==-1, + (*Print["s =-1 code"];*) + Which[l==1 (* && m!=0 *), + (* Print["l = 1, m != 0 code"]; *) + + p=0; + Do[If[n+p<=ExpOrder+3,Solveac[n,n+p,EqCL[n+1,p-1],0]],{n,1,ExpOrder+1}]; + Solveac[-1,p+2,\[CapitalDelta]EqCL[-1,p+5],0]; + Solve\[CapitalDelta]\[Nu][p,p-1]; + + Do[ + Do[If[n+p<=ExpOrder+1,Solveac[n,n+p,EqCL[n+1,p-1],0]],{n,1,ExpOrder+1}]; + Solveac[-1,p+2,\[CapitalDelta]EqCL[-1,p+5],0]; + Solveac[-2,p+1,EqCL[0,p+4],0]; + Solve\[CapitalDelta]\[Nu][p,p-1]; + Solveac[-3,p+Boole[m==0],EqCL[-2,p-1+Boole[m==0]],0]; + Do[If[p+(n-4)+1+Boole[m==0]<=ExpOrder+1,Solveac[-n,p+(n-4)+1+Boole[m==0],EqCL[-n+1,p-5+Boole[m==0]],0]],{n,4,ExpOrder+2}], + {p,1,ExpOrder}], + + (* + l==1 && m==0, + (*Print["l = 1, m = 0 code"];*) + + Do[ + Do[If[n+p<=ExpOrder+2,Solveac[n,n+p,EqCL[n+1,p-1],0]],{n,1,ExpOrder}]; + If[p>0, + Solveac[-1,p+1,\[CapitalDelta]EqCL[-1,p+5],0]; + Solveac[-2,p+1,EqCL[-1,p+5],0]]; + If[p>1, + Solveac[-4,p+1,EqCL[-3,p-5],0]; + Solveac[-3,p,EqCL[-2,p-1],0]; + Do[Solveac[-n,p+n-3,EqCL[-n+1,p-5],0],{n,5,ExpOrder+2}]]; + Solve\[CapitalDelta]\[Nu][p,p+3], + {p,0,ExpOrder}], + *) + + l>1, + (* Print["|s| = 1, l > 1 code"]; *) + + Do[ + Do[If[n+p<=ExpOrder,Solveac[n,n+p,EqCL[n+1,p-1],0]],{n,1,ExpOrder}]; + Do[Solveac[-n,n+p,EqCL[-n+1,p-1],0],{n,1,l-1}]; + Solve\[CapitalDelta]\[Nu][p,p-1], + {p,0,4}]; + + p=4; + If[m==0,If[p+(l-3)<=ExpOrder+1,Solveac[-l,p+(l-3),EqCL[-l+1,p+1],0]]]; + Solveac[-l-1,p+(l-3),EqCL[-l,p+Boole[m==0]],1-Boole[m==0]]; + If[m==0,Do[If[n+(p-4)<=ExpOrder+1,Solveac[-n,n+(p-4),EqCL[-n+1,p-4],0]],{n,l+2,2l}]]; + + Do[ + Do[If[n+p<=ExpOrder,Solveac[n,n+p,EqCL[n+1,p-1],0]],{n,1,ExpOrder}]; + Do[Solveac[-n,n+p,EqCL[-n+1,p-1],0],{n,1,l-1}]; + If[p<= ExpOrder,Solve\[CapitalDelta]\[Nu][p,p-1]]; + If[m!=0, + (* m !=0 *) + Solveac[-l-1,p+(l-3),EqCL[-l+1,p],1] ; + Solveac[-l,p+(l-4),EqCL[-l+1,p]+q(EqCL[-l+1,p+1]-EqCL[-l,p+1]),0], + (* m==0 *) + Solveac[-l,p+(l-3),EqCL[-l+1,p+1],0]; + Solveac[-l-1,p+(l-3),EqCL[-l,p+Boole[m==0]],1-Boole[m==0]]]; + Do[If[n+(p-5)+Boole[m==0]<=ExpOrder+1,Solveac[-n,n+(p-5)+Boole[m==0],EqCL[-n+1,p-5+Boole[m==0]],0]],{n,l+2,2l}]; + If[p+2l-6+Boole[m==0]<=ExpOrder+1,Solveac[-2l-1,p+2l-6+Boole[m==0],EqCL[-2l,p-5+Boole[m==0]],0]]; + If[p+2l-5+Boole[m==0]<=ExpOrder+1,Solveac[-2l-2,p+2l-5+Boole[m==0],EqCL[-2l-1,p-9+Boole[m==0]],0]]; + Do[If[n-1+p-5+Boole[m==0]<=ExpOrder+1,Solveac[-n-1,n-1+p-5+Boole[m==0],EqCL[-n,p-9+Boole[m==0]],0]],{n,2l+2,ExpOrder+1}], + {p,5,ExpOrder+Max[4-l,0]}]; + ], + +s==0, + (*Print["s =0 code"];*) +Which[ + l==0, + (*Print["l = 0 code"];*) + p=0; + Solveac[1,1+p,EqCL[1+1,p+1],0]; + Do[If[n+p<=ExpOrder,Solveac[n,n+p,EqCL[n+1,p-1],0]],{n,2,ExpOrder-p}]; + Solveac[-2,1,EqCL[-1,-3],0]; + Solveac[-1,0,EqCL[0,5],0]; + (* \[CapitalDelta]\[Nu]pcq[0,0]=\[CapitalDelta]\[Nu]pcq[0,0]/.Solve[CoefficientList[EqCL[1,5],q]==0,\[CapitalDelta]\[Nu]pcq[0,0]][[1]] gives multiple solutions so hard code;*); + \[CapitalDelta]\[Nu]pcq[0,0]=-(7/6); + Do[Solveac[-n,n-1,EqCL[-n+1,-3],0],{n,3,ExpOrder+1}]; + + Do[ + Solveac[1,1+p,EqCL[1+1,p+1],0]; + Do[If[n+p<=ExpOrder,Solveac[n,n+p,EqCL[n+1,p-1],0]],{n,2,ExpOrder-p}]; + Solveac[-2,1+p,EqCL[-1,-3+p],0]; + Solveac[-1,p,EqCL[0,5+p],0]; + Solve\[CapitalDelta]\[Nu][p,p+5]; + Do[Solveac[-n,n-1+p,EqCL[-n+1,-3+p],0],{n,3,ExpOrder-p+2}], + {p,1,ExpOrder}], + l==1, + (*Print["l = 1 code"];*) + If[m==0&&l!=0,ac[-2l,2l]=0;acSolved[-2l,2l]=True]; + + Do[ + Do[If[n+p<=ExpOrder,Solveac[n,n+p,EqCL[n+1,p-1],0]],{n,1,ExpOrder-p}]; + Solveac[-(l),l+p,EqCL[-(l-1),p+5],0]; + Solve\[CapitalDelta]\[Nu][p,p+1]; + If[p>=Boole[m==0], + Solveac[-2l,2l+p,EqCL[-(2l-1),(p+4)],0]; + Solveac[-(2l+2),2l+p,EqCL[-(2l+1),p-4],0]; + Solveac[-(2l+1),2l-1+p,EqCL[-2l,p],0]; + Do[If[n+(p-2)<=ExpOrder+1,Solveac[-n-1,n+p-1,EqCL[-n,p-4],0]],{n,2l+2,ExpOrder-(p-1)}]], + {p,0,ExpOrder-1}], + l>=2, + (*Print["l >= 2 code"];*) + If[m==0&&l!=0,ac[-2l,2l]=0;acSolved[-2l,2l]=True]; + + Do[ + Do[If[n+p<=ExpOrder,Solveac[n,n+p,EqCL[n+1,p-1],0]],{n,1,ExpOrder}]; + Do[If[n+p<=ExpOrder,Solveac[-n,n+p,EqCL[-n+1,p-1],0]],{n,1,l-2}]; + Solve\[CapitalDelta]\[Nu][p,p-1]; + Solveac[-(l-1),l-1+p,EqCL[-(l-2),p+1],0]; + Solveac[-(l),l+p,EqCL[-(l-1),p+5],0]; + Solveac[-(l+1),l+1+p,EqCL[-l,p+4],0]; + Do[Solveac[-n,n+p,EqCL[-(n-1),p],0],{n,l+2,2l-1}]; + If[p>=2+Boole[m==0], + Solveac[-(2l+2),2l+(p-2),EqCL[-(2l+1),(p-2)-4],0]; + Solveac[-2l,2l+(p-2),EqCL[-(2l-1),(p-2)],0]; + Solveac[-(2l+1),2l-1+(p-2),EqCL[-2l,(p-2)],0]; + Do[If[p0,((n + \[Nu]MST + s)^2 + \[Epsilon]^2)/((n + \[Nu]MST- s)^2 + \[Epsilon]^2) flip[n-1],n==0,1,n<0,((n +1+ \[Nu]MST- s)^2 + \[Epsilon]^2)/((n +1+ \[Nu]MST+ s)^2 + \[Epsilon]^2) flip[n+1]]; +Do[aMST[n]=flip[n]aMST[n],{n,-ExpOrder-2,ExpOrder}]; +Print[Table[{n,Series[flip[n],{\[Epsilon],0,1}]},{n,-ExpOrder-2,ExpOrder}]]; +Do[ac[n,i]=Coefficient[aMST[n],\[Epsilon],i],{i,0,ExpOrder+2},{n,-ExpOrder-2,ExpOrder}] +]; +If[pos,ExpOrder-=2]; + +Print[FinalProgressGrid]; + +MST=Association[{\[Nu]->\[Nu]MST+O[\[Epsilon]]^(ExpOrder+1)}]; +MST=Append[MST,Table[a[n]->aMST[n]+If[n==0,0,O[\[Epsilon]]^(ExpOrder+1)],{n,-ExpOrder-2,ExpOrder}]] + +]*) + + +KerrMSTSeries[ss_,ll_,mm_,ExpOrder_]:=Module[{s=ss,l=ll,m=mm,\[CapitalDelta]\[Nu]pC,\[CapitalDelta]\[Nu]p2C,\[CapitalDelta]\[Nu]p3C,\[CapitalDelta]\[Nu]p4C,\[CapitalDelta]\[Nu]p5C,\[CapitalDelta]\[Nu]p6C,\[CapitalDelta]\[Nu]pcq,\[Alpha]C,\[Beta]C,\[Gamma]C,\[CapitalDelta]\[Alpha]\[Beta]C,\[Kappa]Simplify,aLeadingBehaviour,acSolved,acqSolved,aShift,eqShift,StructureGrid,AngExpOrder,\[CapitalDelta]E,\[CapitalDelta]EC,ProgressGrid,aMST,ac,acq,eqnlist,\[CapitalDelta]\[Nu]p,\[CapitalDelta]\[Nu]pc,EqC,\[CapitalDelta]EqC,EqCL,\[CapitalDelta]EqCL,\[CapitalDelta]EqCTable,Solveac,Solve\[CapitalDelta]\[Nu],i,j,k,n,p,\[Nu]MST,MST}, + + +ClearAll[\[CapitalDelta]\[Nu]p2C]; +\[CapitalDelta]\[Nu]p2C[k_]:=\[CapitalDelta]\[Nu]p2C[k]=Sum[Expand[\[CapitalDelta]\[Nu]pC[i2]*\[CapitalDelta]\[Nu]pC[- i2 + k]], {i2, 0, k}]; +\[CapitalDelta]\[Nu]p3C[k_]:=\[CapitalDelta]\[Nu]p3C[k]=Sum[Expand[\[CapitalDelta]\[Nu]p2C[k - i3]*\[CapitalDelta]\[Nu]pC[i3]], {i3, 0, k}]; +\[CapitalDelta]\[Nu]p4C[k_]:=\[CapitalDelta]\[Nu]p4C[k]=Sum[Expand[\[CapitalDelta]\[Nu]p3C[k - i4]*\[CapitalDelta]\[Nu]pC[i4]], {i4, 0, k}]; +\[CapitalDelta]\[Nu]p5C[k_]:=\[CapitalDelta]\[Nu]p5C[k]=Sum[Expand[\[CapitalDelta]\[Nu]p4C[k - i5]*\[CapitalDelta]\[Nu]pC[i5]], {i5, 0, k}]; +\[CapitalDelta]\[Nu]p6C[k_]:=\[CapitalDelta]\[Nu]p6C[k]=Sum[Expand[\[CapitalDelta]\[Nu]p5C[k - i6]*\[CapitalDelta]\[Nu]pC[i6]], {i6, 0, k}]; + +\[Alpha]C[n_,k_]:=\[Alpha]C[n,k]=((l-2 l^2+n-4 l n-2 n^2) If[-4+k==0,1,0]+I (2 l^2+n (-1+2 n)+l (-1+4 n)) (-I m q+(1+l+n) \[Kappa]) If[-3+k==0,1,0]+(-2 l^2-n (-1+2 n)-l (-1+4 n)) (1+l+n+s)^2 If[-2+k==0,1,0]+I (2 l^2+n (-1+2 n)+l (-1+4 n)) (1+l+n+s)^2 (-I m q+(1+l+n) \[Kappa]) If[-1+k==0,1,0]+ +2 I \[Kappa] \[CapitalDelta]\[Nu]p3C[k-9]+(-3-8 l-8 n-4 s) \[CapitalDelta]\[Nu]p3C[k-8]+I (-I m q (3+8 l+8 n+4 s)+(3+20 l^2+20 n^2+6 s+2 s^2+4 n (5+4 s)+4 l (5+10 n+4 s)) \[Kappa]) \[CapitalDelta]\[Nu]p3C[k-7]- +2 \[CapitalDelta]\[Nu]p4C[k-10]+I (-2 I m q+(5+10 l+10 n+4 s) \[Kappa]) \[CapitalDelta]\[Nu]p4C[k-9]+2 I \[Kappa] \[CapitalDelta]\[Nu]p5C[k-11]-2\[CapitalDelta]\[Nu]p2C[k-8]+I (-2 I m q+\[Kappa]+6 l \[Kappa]+6 n \[Kappa]) \[CapitalDelta]\[Nu]p2C[k-7]+ +(-12 l^2-12 n^2-2 s (1+s)-3 n (3+4 s)-3 l (3+8 n+4 s))\[CapitalDelta]\[Nu]p2C[k-6]+I (-I m q (12 n^2+2 s (1+s)+3 n (3+4 s))+20 l^3 \[Kappa]+ +(-1+20 n^3+s^2+6 n^2 (5+4 s)+3 n (3+6 s+2 s^2)) \[Kappa]+6 l^2 (-2 I m q+(5+10 n+4 s) \[Kappa])+3 l (-I m q (3+8 n+4 s)+(3+20 n^2+6 s+2 s^2+4 n (5+4 s)) \[Kappa])) \[CapitalDelta]\[Nu]p2C[k-5]+ +(1-4 l-4 n) \[CapitalDelta]\[Nu]pC[-6+k]+I (-I m (-1+4 l+4 n) q+(-1+6 l^2+2 n+6 n^2+2 l (1+6 n)) \[Kappa]) \[CapitalDelta]\[Nu]pC[-5+k]+ +(-8 l^3-8 n^3-4 n s (1+s)+(1+s)^2-3 n^2 (3+4 s)-3 l^2 (3+8 n+4 s)-2 l (12 n^2+2 s (1+s)+3 n (3+4 s))) \[CapitalDelta]\[Nu]pC[-4+k]+ +I (1+l+n+s) (-I m q (-1+l+8 l^2+n+16 l n+8 n^2-s+4 l s+4 n s)+ +(-1+10 l^3+10 n^3-s+n (-1+2 s)+2 n^2 (5+3 s)+2 l^2 (5+15 n+3 s)+l (-1+30 n^2+2 s+4 n (5+3 s))) \[Kappa]) \[CapitalDelta]\[Nu]pC[-3+k]); + +\[Beta]C[n_,k_]:=\[Beta]C[n,k]=((-3+4 l^2+4 n+4 n^2+l (4+8 n)) If[-4+k==0,1,0]-m (-3+4 l^2+4 n+4 n^2+l (4+8 n)) q If[-3+k==0,1,0]+ +1/4 (3-4 l^2-4 n-4 n^2-l (4+8 n)) (l^2 (-8+q^2)+n (-8+q^2)+n^2 (-8+q^2)+l (1+2 n) (-8+q^2)-4 s^2) If[-2+k==0,1,0]- +m (-3+4 l^2+4 n+4 n^2+l (4+8 n)) q s^2 If[-1+k==0,1,0]+n (8 l^5+4 l^4 (5+9 n)+n (1+n)^2 (-3+4 n+4 n^2)+2 l^3 (5+36 n+32 n^2)+ +l^2 (-5+29 n+96 n^2+56 n^3)+l (-3-7 n+28 n^2+56 n^3+24 n^4)) If[k==0,1,0]-8 (1+2 l+2 n) Sum[\[CapitalDelta]\[Nu]p3C[k-6-i]*\[CapitalDelta]EC[s, l, m, i], {i, 0, k-6}]- +4 Sum[\[CapitalDelta]\[Nu]p4C[k-8-i]*\[CapitalDelta]EC[s, l, m, i], {i, 0, k - 8}]+(-1-24 l^2-24 n-24 n^2-24 l (1+2 n))Sum[\[CapitalDelta]\[Nu]p2C[k-4-i]*\[CapitalDelta]EC[s, l, m, i], {i, 0, k - 4}]- +2 (1+2 l+2 n) (-8+q^2) \[CapitalDelta]\[Nu]p3C[k-8]+(-2+64 l^3+36 n+120 n^2+80 n^3+32 l^2 (3+7 n)+4 l (7+56 n+60 n^2))\[CapitalDelta]\[Nu]p3C[k-6]+ +(8-q^2) \[CapitalDelta]\[Nu]p4C[k-10]+(9+56 l^2+60 n+60 n^2+8 l (7+15 n))\[CapitalDelta]\[Nu]p4C[k-8] +12 (1+2 l+2 n) \[CapitalDelta]\[Nu]p5C[k-10]+ +4 \[CapitalDelta]\[Nu]p6C[k-12]+(3-16 l^3-2 n-24 n^2-16 n^3-24 l^2 (1+2 n)-l (2+48 n+48 n^2)) Sum[\[CapitalDelta]EC[s, l, m, i]*\[CapitalDelta]\[Nu]pC[-2 - i + k], {i, 0, -2 + k}]+ +4 \[CapitalDelta]\[Nu]p2C[k-8]-4 m q \[CapitalDelta]\[Nu]p2C[k-7]+(2-q^2/4-6 l^2 (-8+q^2)-6 n (-8+q^2)-6 n^2 (-8+q^2)-6 l (1+2 n) (-8+q^2)+4 s^2) \[CapitalDelta]\[Nu]p2C[k-6]- +4 m q s^2 \[CapitalDelta]\[Nu]p2C[k-5]+(-3+36 l^4-6 n+54 n^2+120 n^3+60 n^4+24 l^3 (3+8 n)+l^2 (29+288 n+336 n^2)+l (-7+84 n+336 n^2+240 n^3)) \[CapitalDelta]\[Nu]p2C[k-4]+ +(-4 l^4-8 l^3 (1+2 n)-l^2 (1+24 n+24 n^2)-n (-3+n+8 n^2+4 n^3)-l (-3+2 n+24 n^2+16 n^3)) \[CapitalDelta]EC[s,l,m,k]+(4+8 l+8 n) \[CapitalDelta]\[Nu]pC[-6+k]- +4 m (1+2 l+2 n) q \[CapitalDelta]\[Nu]pC[-5+k]+1/4 (-1-2 l-2 n) (24-3 q^2+8 l^2 (-8+q^2)+8 n (-8+q^2)+8 n^2 (-8+q^2)+8 l (1+2 n) (-8+q^2)-16 s^2) \[CapitalDelta]\[Nu]pC[-4+k]- +4 m (1+2 l+2 n) q s^2 \[CapitalDelta]\[Nu]pC[-3+k]+(8 l^5+4 l^4 (5+18 n)+2 l^3 (5+72 n+96 n^2)+l^2 (-5+58 n+288 n^2+224 n^3)+6 n (-1-n+6 n^2+10 n^3+4 n^4)+ +l (-3-14 n+84 n^2+224 n^3+120 n^4)) \[CapitalDelta]\[Nu]pC[-2+k]); + +\[Gamma]C[n_,k_]:=\[Gamma]C[n,k]=((-3-2 l^2-5 n-2 n^2-l (5+4 n)) If[-4+k==0,1,0]-I (3+2 l^2+5 n+2 n^2+l (5+4 n)) (I m q+(l+n) \[Kappa]) If[-3+k==0,1,0]+ +(-3-2 l^2-5 n-2 n^2-l (5+4 n)) (l+n-s)^2 If[-2+k==0,1,0]-I (3+2 l^2+5 n+2 n^2+l (5+4 n)) (l+n-s)^2 (I m q+(l+n) \[Kappa]) If[-1+k==0,1,0]- +2 I \[Kappa] \[CapitalDelta]\[Nu]p3C[k-9]+(-5-8 l-8 n+4 s)\[CapitalDelta]\[Nu]p3C[k-8]-I (I m q (5+8 l+8 n-4 s)+(3+20 l^2+20 n+20 n^2+4 l (5+10 n-4 s)-10 s-16 n s+2 s^2) \[Kappa]) \[CapitalDelta]\[Nu]p3C[k-7]- +2 \[CapitalDelta]\[Nu]p4C[k-10]-I (2 I m q+(5+10 l+10 n-4 s) \[Kappa]) \[CapitalDelta]\[Nu]p4C[k-9]-2 I \[Kappa] \[CapitalDelta]\[Nu]p5C[k-11]-2 \[CapitalDelta]\[Nu]p2C[k-8]-I (2 I m q+(5+6 l+6 n) \[Kappa])\[CapitalDelta]\[Nu]p2C[k-7]+ +(-3-12 l^2-12 n^2-3 l (5+8 n-4 s)+10 s-2 s^2+3 n (-5+4 s))\[CapitalDelta]\[Nu]p2C[k-6]-I (I m q (3+12 l^2+12 n^2+3 l (5+8 n-4 s)-10 s+2 s^2-3 n (-5+4 s))+ +(20 l^3+20 n^3+6 l^2 (5+10 n-4 s)-6 n^2 (-5+4 s)+s (-6+5 s)+n (9-30 s+6 s^2)+l (9+60 n^2+n (60-48 s)-30 s+6 s^2)) \[Kappa]) \[CapitalDelta]\[Nu]p2C[k-5]+ +(-5-4 l-4 n) \[CapitalDelta]\[Nu]pC[-6+k]-I (I m (5+4 l+4 n) q+(3+6 l^2+10 n+6 n^2+2 l (5+6 n)) \[Kappa]) \[CapitalDelta]\[Nu]pC[-5+k]+ +(-8 l^3-8 n^3-3 l^2 (5+8 n-4 s)+(6-5 s) s+3 n^2 (-5+4 s)+n (-6+20 s-4 s^2)-2 l (3+12 n^2-10 s+2 s^2-3 n (-5+4 s))) \[CapitalDelta]\[Nu]pC[-4+k]- +I (l+n-s) (I m q (6+8 l^2+8 n^2+n (15-4 s)+l (15+16 n-4 s)-5 s)+ +(10 l^3+10 n^3+n (9-10 s)+l (9+30 n^2+n (40-12 s)-10 s)+n^2 (20-6 s)+l^2 (20+30 n-6 s)-3 s) \[Kappa]) \[CapitalDelta]\[Nu]pC[-3+k]); + +Do[\[CapitalDelta]\[Nu]pC[k]=0,{k,-12,-1}]; +\[Beta]C[-l,2]=\[Beta]C[-l,2]//Simplify; +\[Beta]C[-l,1]=\[Beta]C[-l,1]//Simplify; +\[Alpha]C[-l-1,2]=\[Alpha]C[-l-1,2]//Simplify; + +\[Kappa]Simplify[x_]:=Expand[x]/.\[Kappa]^2->(1-q^2); +\[CapitalDelta]\[Alpha]\[Beta]C[n_,k_]:=(\[Beta]C[-l,k]- \[Alpha]C[-l-1,k])//\[Kappa]Simplify; + +ProgressGrid:=Grid[ +Table[ +If[i==0,If[j<2||NumericQ[\[CapitalDelta]\[Nu]pC[j-2]/.{q->1/Sqrt[2],\[Kappa]->1/Sqrt[2]}],Item[j,Background->Green],Item[j,Background->Yellow]], +If[TrueQ[acSolved[i,j]],If[NumericQ[ac[i,j]/.{q->1/Sqrt[2],\[Kappa]->1/Sqrt[2]}],If[(ac[i,j]/.{q->1/Sqrt[2],\[Kappa]->1/Sqrt[2]})==0,Item[" ",Background->Magenta],Item[" ",Background->Red]],Item[" ",Background->Orange]] ,If[jBlue]]]],{i,ExpOrder+2,-ExpOrder-2,-1},{j,-1,ExpOrder+2}],ItemSize->All,Frame->All]; +StructureGrid:=Grid[ +Table[ +If[i==0,Item[j,Background->Yellow], +If[jCyan],-4l,Item[" ",Background->Gray],_,Item[" ",Background->Blue]]]],{i,ExpOrder+2,-ExpOrder-2,-1},{j,-1,ExpOrder+2}],ItemSize->All,Frame->All]; + +<q \[Epsilon]/2)+O[\[Epsilon]]^(AngExpOrder+1)-l(l+1)+s(s+1)+ m q \[Epsilon]-(q \[Epsilon]/2)^2; +\[CapitalDelta]EC[s,l,m,0]=0; +Do[\[CapitalDelta]EC[s,l,m,i]=Coefficient[\[CapitalDelta]E[s,l,m],\[Epsilon],i],{i,1,AngExpOrder}]; + +ClearAll[aMST,ac,acq,eqnlist,\[CapitalDelta]\[Nu]p,\[CapitalDelta]\[Nu]pc]; + +If[s==-2, + aShift[l,n_]:=Which[n<=l-2,0,n==l-1,2,n==l,1, + n>l&&n<2l+1,0, + n>=2l+1,-2]; + eqShift[l,n_]:=Which[n<=l-3,0,n==l-2,4,n==l-1,4,n==l,5-Boole[l==2], + n>l&&n<2l+1,1, + n>=2l+1,-1];]; +If[s==-1, + aShift[l,n_]:=Which[n<=l-2,0,n==l-1,If[l==1||m==0,0,2],n==l,1+Boole[l>1&&m==0], + n>l&&n<2l+1,Boole[l>1&&m==0], + n>=2l+1,-2]; + eqShift[l_,n_]:=Which[n<=l-3,0,n==l-2,4,n==l-1,4,n==l,5, + n>l&&n<2l+1,1, + n>=2l+1,-1];]; +If[s==0,aShift[l,n_]:=Which[n<2l+1,0,n>=2l+1,-2+Boole[l==0]]; + (*aShift[l_,n_]:=Which[n<2l+1,0,n>=2l+1,-2];*) + eqShift[l,n_]:=Which[n<=l-3,0,n==l-2,2,n==l-1,4,n==l,3, + n>l&&n<2l+1,1, + n>=2l+1,-1]; + ]; + +aLeadingBehaviour[l,n_]=aLeadingBehaviour[l,n]=If[n>=0,n,-n+aShift[l,-n]]; + +(*aMST[0]=1; +ac[0,_]=0; +ac[0,0]=1; +Do[aMST[n]=Sum[ac[n,i]\[Epsilon]^i,{i,n,ExpOrder+1}],{n,1,ExpOrder+2}]; +Do[ac[n,i]=0,{n,1,ExpOrder+2},{i,0,n-1}]; +Do[ac[n,i]=If[EvenQ[i],Sum[acq[n,i,j]q^j,{j,0,i,2}]+I \[Kappa] Sum[acq[n,i,j]q^j,{j,1,i,2}],Sum[acq[n,i,j]q^j,{j,1,i,2}]+I \[Kappa] Sum[acq[n,i,j]q^j,{j,0,i,2}]],{n,1,ExpOrder+2},{i,n,ExpOrder+1}]; +Do[aMST[-n]=Sum[ac[-n,i]\[Epsilon]^i,{i,n+aShift[l,n],ExpOrder+1}],{n,1,ExpOrder+3}]; +Do[ac[-n,i]=0,{n,1,ExpOrder+3},{i,0,n+aShift[l,n]-1}]; +Do[ac[-n,i]=If[EvenQ[i],Sum[acq[-n,i,j]q^j,{j,0,i,2}]+I \[Kappa] Sum[acq[-n,i,j]q^j,{j,1,i,2}],Sum[acq[-n,i,j]q^j,{j,1,i,2}]+I \[Kappa] Sum[acq[-n,i,j]q^j,{j,0,i,2}]],{n,1,ExpOrder+2},{i,n+aShift[l,n],ExpOrder+1}]; +If[m == 0,Do[Do[acq[-n,n-2,i]=0;acqSolved[-n,n-2,i]=True,{i,0,n-2}];acSolved[-n,n-2]=True,{n,2l+1,ExpOrder+2}]]; +*) +aMST[0]=1; +ac[0,_]=0; +ac[0,0]=1; +Do[aMST[n]=Sum[ac[n,i]\[Epsilon]^i,{i,n,ExpOrder+1}],{n,1,ExpOrder+2}]; +Do[ac[n,i]=0,{n,1,ExpOrder+2},{i,0,n-1}]; +Do[ac[n,i]=If[EvenQ[i],Sum[acq[n,i,j]q^j,{j,0,i,2}]+I \[Kappa] Sum[acq[n,i,j]q^j,{j,1,i,2}],Sum[acq[n,i,j]q^j,{j,1,i,2}]+I \[Kappa] Sum[acq[n,i,j]q^j,{j,0,i,2}]],{n,1,ExpOrder+2},{i,n,ExpOrder+1}]; +Do[aMST[-n]=Sum[ac[-n,i]\[Epsilon]^i,{i,n+aShift[l,n],ExpOrder+1}],{n,1,ExpOrder+4}]; +Do[ac[-n,i]=0,{n,1,ExpOrder+4},{i,0,n+aShift[l,n]-1}]; +Do[ac[-n,i]=If[EvenQ[i],Sum[acq[-n,i,j]q^j,{j,0,i,2}]+I \[Kappa] Sum[acq[-n,i,j]q^j,{j,1,i,2}],Sum[acq[-n,i,j]q^j,{j,1,i,2}]+I \[Kappa] Sum[acq[-n,i,j]q^j,{j,0,i,2}]],{n,1,ExpOrder+4},{i,n+aShift[l,n],ExpOrder+1}]; +If[m == 0,Do[Do[acq[-n,n-2,i]=0;acqSolved[-n,n-2,i]=True,{i,0,n-2}];acSolved[-n,n-2]=True,{n,2l+1,ExpOrder+4}]]; +(*SetAttributes[aMST,NHoldAll]; +SetAttributes[ac,NHoldAll];*) + +\[CapitalDelta]\[Nu]p=Sum[\[CapitalDelta]\[Nu]pC[i] \[Epsilon]^i,{i,0,ExpOrder+1}]; +Do[\[CapitalDelta]\[Nu]pC[i]=If[EvenQ[i],Sum[\[CapitalDelta]\[Nu]pcq[i,j]q^j,{j,0,i,2}],Sum[\[CapitalDelta]\[Nu]pcq[i,j]q^j,{j,1,i,2}]],{i,0,ExpOrder+1}]; + +EqC[n_,k_]:=Sum[\[Alpha]C[n-1,k-i]ac[n,i]+\[Beta]C[n-1,k-i]ac[n-1,i]+\[Gamma]C[n-1,k-i]ac[n-2,i],{i,Min[Abs[n]+aShift[l,-n],Abs[n-1]+aShift[l,-(n-1)],Abs[n-2]+aShift[l,-(n-2)]],k}]; +\[CapitalDelta]EqC[n_,k_]:=Sum[\[Alpha]C[n,k-i]ac[n+1,i]+(\[Beta]C[n,k-i]- \[Alpha]C[n-1,k-i])ac[n,i]+(\[Gamma]C[n,k-i]- \[Beta]C[n-1,k-i])ac[n-1,i]+(-\[Gamma]C[n-1,k-i])ac[n-2,i],{i,Min[Abs[n+1]+aShift[l,-(n+1)],Abs[n]+aShift[l,-n],Abs[n-1]+aShift[l,-(n-1)],Abs[n-2]+aShift[l,-(n-2)]],k}]; + +EqCL[n_,p_]:=\[Kappa]Simplify[EqC[n,Min[Abs[n]-eqShift[l,-n]+If[n>1,0,2]+p,ExpOrder+eqShift[l,-n]+2]]]; +\[CapitalDelta]EqCL[n_,p_]:=\[Kappa]Simplify[\[CapitalDelta]EqC[n,Min[Abs[n]-eqShift[l,-n]+If[n>1,0,2]+p,ExpOrder+eqShift[l,-n]+2]]]; + +\[CapitalDelta]EqCTable[n_,k_]:=Table[{\[Alpha]C[n,k-i]ac[n+1,i],(\[Beta]C[n,k-i]- \[Alpha]C[n-1,k-i])ac[n,i]+(\[Gamma]C[n,k-i]- \[Beta]C[n-1,k-i])ac[n-1,i],(-\[Gamma]C[n-1,k-i])ac[n-2,i]},{i,Min[Abs[n+1]+aShift[l,-(n+1)],Abs[n]+aShift[l,-n],Abs[n-1]+aShift[l,-(n-1)],Abs[n-2]+aShift[l,-(n-2)]],k}]; + +Clear[Solveac]; +Solveac[i_?IntegerQ,j_?IntegerQ,Eq_,drop_?IntegerQ]:=Module[{tmpEq,tmp,tmpN,tmpD,shift,Verbose=False,acSolve=True}, +If[(i>0&&j(*+i*)>ExpOrder+Boole[l==0])||(i<0&&j(*-i*)>ExpOrder+2),acSolve=False;Return[]];(* remove commented terms if only want \[Nu] *) +If[Abs[j]<=ExpOrder+1&&acSolve,tmpEq=Drop[CoefficientList[Eq,q],drop]; +If[Verbose,Print[tmpEq,"\t",Dimensions[tmpEq][[1]]]]; +shift=0; +Do[ + tmpN=\[Kappa]Simplify[Coefficient[tmpEq[[k+1]],acq[i,j,k+shift],0]]; + tmpD=-\[Kappa]Simplify[Coefficient[tmpEq[[k+1]],acq[i,j,k+shift]]]; + If[Verbose,Print["i=",i,"\t j=",j, "\t k=",k,"\t tmpN:=",tmpN,"\t tmpD:=",tmpD]]; + If[tmpD==0,Print["Attempted division by 0 in Solveac\n i=",i,"\t j=",j,"\t k+shift=",k+shift,"\t tmpEq=",Simplify[tmpEq]];Continue[]]; + tmp=tmpN/tmpD; + If[Verbose,Print["Eq:\t",tmpEq,"\t",tmpN,"\t",tmpD,"\t",tmp]]; + acq[i,j,k+shift]=\[Kappa]Simplify[tmp]; + acqSolved[i,j,k+shift]=True; + If[Verbose && acq[i,j,k+shift]==0,Print["Vanishing acq for i=",i,"\t j=",j,"\t k=",k]], +{k,0,j}]; +ac[i,j]=Collect[ac[i,j],\[Kappa]]; +acSolved[i,j]=True]; +If[Verbose,Print[ac[i,j]]]; +]; + +Clear[Solve\[CapitalDelta]\[Nu]]; +Solve\[CapitalDelta]\[Nu][i_?IntegerQ,p_?IntegerQ, OptionsPattern[]]:=Module[{tmpEq,tmp,tmpN,tmpD,shift,Verbose=False}, +shift=If[EvenQ[i],0,1]; +tmpEq=Take[CoefficientList[EqCL[1,p],q],{shift+1,-1,2}]; +Do[ + tmpN=Coefficient[tmpEq[[k+1]],\[CapitalDelta]\[Nu]pcq[i,2k+shift],0]; + tmpD=-Coefficient[tmpEq[[k+1]],\[CapitalDelta]\[Nu]pcq[i,2k+shift]]; + If[tmpD==0,Print["Attempted division by 0 in Solve\[CapitalDelta]\[Nu] (i=",i,"\t 2k+shift=",2k+shift,")"],tmp=tmpN/tmpD]; + \[CapitalDelta]\[Nu]pcq[i,2k+shift]=\[Kappa]Simplify[tmp], +{k,0,Floor[i/2]}]; +If[Verbose,Print["i=",i,"\t \[CapitalDelta]\[Nu]pC[",i,"]=",\[CapitalDelta]\[Nu]pC[i]]]; +]; + + +Which[s==-2, + (*Print["s =-2 code"];*) + Which[l==2 && m!=0, + + p=0; + Do[If[n+p<=ExpOrder,Solveac[n,n+p,EqCL[n+1,p-1],0]],{n,1,ExpOrder}]; + Solve\[CapitalDelta]\[Nu][p,p-1]; + p=1; + Do[If[n+p<=ExpOrder,Solveac[n,n+p,EqCL[n+1,p-1],0]],{n,1,ExpOrder}]; + Solve\[CapitalDelta]\[Nu][p,p-1]; + Solveac[-1,p+2,EqCL[0,p+4],0]; + Do[ + Do[If[n+p<=ExpOrder,Solveac[n,n+p,EqCL[n+1,p-1],0]],{n,1,ExpOrder}]; + Solve\[CapitalDelta]\[Nu][p,p-1]; + Solveac[-1,p+2,EqCL[0,p+4],0]; + If[p==2,Solveac[-2,3,EqCL[-1,5],1]]; + Solveac[-2,p+2,(\[Beta]C[-2,2]- \[Alpha]C[-3,2])EqCL[-1,p+3]- \[Beta]C[-2,1] \[CapitalDelta]EqCL[-2,p+4],1]; + Solveac[-3,p+1,EqCL[-2,p+3],0]; + If[p2&&p2, + Do[ + Do[If[n+p<=ExpOrder,Solveac[n,n+p,EqCL[n+1,p-1],0]],{n,1,ExpOrder}]; + Do[Solveac[-n,n+p,EqCL[-n+1,p-1],0],{n,1,l-2}]; + Solve\[CapitalDelta]\[Nu][p,p-1], + {p,0,3}]; + + p=3; + If[m != 0,Solveac[-(l-1),l+1+(p-3),EqCL[-(l-2),(p-3)+5],0]]; + + Do[ + If[m == 0,Solveac[-(l+1),l+1+(p-4),EqCL[-(l-1),p+2],0],Solveac[-(l+1),l+1+(p-4),EqCL[-(l-1),p+1],1]]; + If[m == 0,Solveac[-l,l+1+(p-4),EqCL[-l,p+2],0],Solveac[-l,l+1+(p-4),(\[Beta]C[-l,2]- \[Alpha]C[-l-1,2])EqCL[-l+1,p+2]- \[Beta]C[-l,1] \[CapitalDelta]EqCL[-l,p+3],0]]; + If[m == 0,Solveac[-(l-1),l+1+(p-4),EqCL[-(l-2),p+1],0],Solveac[-(l-1),l+2+(p-4),EqCL[-(l-2),p+2],0]]; + + Do[If[n+p-4<=ExpOrder+2,Solveac[-n,n+(p-4),EqCL[-n+1,p-4],0]],{n,l+2,2l-1}]; + If[p>3+Boole[m==0]&&2l+p-4<=ExpOrder+1,Solveac[-2l-1,2l+(p-4)-1,EqCL[-2l,(p-4)],0]]; + If[p>3+Boole[m==0]&&2l+p-4<=ExpOrder+1,Solveac[-2l-2,2l+(p-4),EqCL[-2l-1,(p-4)-4],0]]; + If[2l+p-4<=ExpOrder+1,Solveac[-2l,2l+(p-4),EqCL[-2l+1,p-4],0]]; + Do[If[p>3+Boole[m==0]&&n-1+p-4<=ExpOrder,Solveac[-n-1,n-1+p-4,EqCL[-n,(p-4)-4],0]],{n,2l+2,ExpOrder+1}]; + Do[If[n+p<=ExpOrder,Solveac[n,n+p,EqCL[n+1,p-1],0]],{n,1,ExpOrder-p}]; + Do[If[n+p<=ExpOrder,Solveac[-n,n+p,EqCL[-n+1,p-1],0]],{n,1,l-2}]; + + If[p<=ExpOrder,Solve\[CapitalDelta]\[Nu][p,p-1]], + {p,4,ExpOrder+Boole[l==2]}]], + +s==-1, + (*Print["s =-1 code"];*) + Which[l==1 && m!=0, + (*Print["l = 1, m != 0 code"];*) + If[m==0,ac[-2l,2l]=0;acSolved[-2l,2l]=True]; + + Do[ + Do[If[n+p<=ExpOrder+2,Solveac[n,n+p,EqCL[n+1,p-1],0]],{n,1,ExpOrder}]; + Which[p>0&&p!=3, + Solveac[-1,p+1,\[CapitalDelta]EqCL[-1,p+5],0], + p==3, + Solveac[-1,p+1,FromDigits[Reverse[Join[Drop[CoefficientList[(\[Beta]C[-l,2]- \[Alpha]C[-l-1,2])EqCL[-l+1,p+4]- \[Beta]C[-l,1] \[CapitalDelta]EqCL[-l,p+5],q],1],Drop[CoefficientList[EqCL[-l+1,p+4],q],4]]],q],0]]; + Which[p>0&&p<3, + Solveac[-2,p+1,EqCL[-1,p+5],0], + p>3, + Solveac[-2,p,FromDigits[Reverse[Join[CoefficientList[(\[Beta]C[-l,2]- \[Alpha]C[-l-1,2])EqCL[-l+1,p+4]- \[Beta]C[-l,1] \[CapitalDelta]EqCL[-l,p+5],q],Drop[CoefficientList[EqCL[-l+1,p+4],q],4]]],q],0]]; + If[p>1, + Solveac[-4,p,EqCL[-3,p-6],0]; + Solveac[-3,p-1,EqCL[-2,p-2],0]; + Do[Solveac[-n,p+n-4,EqCL[-n+1,p-6],0],{n,5,ExpOrder+2}]]; + Solve\[CapitalDelta]\[Nu][p,p+3], + {p,0,ExpOrder+1}], + + l==1 && m==0, + (*Print["l = 1, m = 0 code"];*) + + Do[ + Do[If[n+p<=ExpOrder+2,Solveac[n,n+p,EqCL[n+1,p-1],0]],{n,1,ExpOrder}]; + If[p>0, + Solveac[-1,p+1,\[CapitalDelta]EqCL[-1,p+5],0]; + Solveac[-2,p+1,EqCL[-1,p+5],0]]; + If[p>1, + Solveac[-4,p+1,EqCL[-3,p-5],0]; + Solveac[-3,p,EqCL[-2,p-1],0]; + Do[Solveac[-n,p+n-3,EqCL[-n+1,p-5],0],{n,5,ExpOrder+2}]]; + Solve\[CapitalDelta]\[Nu][p,p+3], + {p,0,ExpOrder}], + + l>1, + (*Print["l > 1 code"];*) + Do[ + Do[If[n+p<=ExpOrder,Solveac[n,n+p,EqCL[n+1,p-1],0]],{n,1,ExpOrder}]; + Do[Solveac[-n,n+p,EqCL[-n+1,p-1],0],{n,1,l-2}]; + If[m==0,Solveac[-(l-1),(l-1)+p,EqCL[-(l-1)+1,p+3],0]]; + Solve\[CapitalDelta]\[Nu][p,p-1], + {p,0,3}]; + + Do[ + Do[If[n+p<=ExpOrder,Solveac[n,n+p,EqCL[n+1,p-1],0]],{n,1,ExpOrder}]; + Do[If[n+p<=ExpOrder,Solveac[-n,n+p,EqCL[-n+1,p-1],0]],{n,1,l-2}]; + If[m==0, + Solveac[-l,l+p-2,EqCL[-l+1,p+3],0]; + Solveac[-(l+1),l+1+p-3,EqCL[-l,p+3],0]; + Solveac[-(l-1),(l-1)+p,EqCL[-(l-1)+1,p+3],0], + Solveac[-(l+1),l+p-3,EqCL[-(l-1),p+1],1]; + Solveac[-l,l+1+(p-4),(\[Beta]C[-l,2]- \[Alpha]C[-l-1,2])EqCL[-l+1,p+2]- \[Beta]C[-l,1] \[CapitalDelta]EqCL[-l,p+3],0]; + Solveac[-(l-1),(l-1)+p-2,EqCL[-(l-2),p+1],0]]; + Do[If[True,Solveac[-n,n+(p-4)+Boole[m==0],EqCL[-n+1,p-4+Boole[m==0]],0]],{n,l+2,2l-1}]; + If[True,Solveac[-2l-1,2l+(p-4)-1+Boole[m==0],EqCL[-2l,(p-4)+Boole[m==0]],0]]; + If[2l-1+p-4<=ExpOrder+1,Solveac[-2l-2,2l+(p-4)+Boole[m==0],EqCL[-2l-1,(p-4)-4+Boole[m==0]],0]]; + If[True,Solveac[-2l,2l+(p-4)+Boole[m==0],EqCL[-2l+1,p-4+Boole[m==0]],0]]; + Do[If[True,Solveac[-n-1,n-1+p-4+Boole[m==0],EqCL[-n,(p-4)-4+Boole[m==0]],0]],{n,2l+2,ExpOrder-p+5}]; + If[p<=ExpOrder,Solve\[CapitalDelta]\[Nu][p,p-1]], + {p,4,ExpOrder+Boole[l==2]}]], + +s==0, + (*Print["s =0 code"];*) +Which[ + l==0, + (*Print["l = 0 code"];*) + p=0; + Solveac[1,1+p,EqCL[1+1,p+1],0]; + Do[If[n+p<=ExpOrder,Solveac[n,n+p,EqCL[n+1,p-1],0]],{n,2,ExpOrder-p}]; + Solveac[-2,1,EqCL[-1,-3],0]; + Solveac[-1,0,EqCL[0,5],0]; + (* \[CapitalDelta]\[Nu]pcq[0,0]=\[CapitalDelta]\[Nu]pcq[0,0]/.Solve[CoefficientList[EqCL[1,5],q]==0,\[CapitalDelta]\[Nu]pcq[0,0]][[1]] gives multiple solutions so hard code;*); + \[CapitalDelta]\[Nu]pcq[0,0]=-(7/6); + Do[Solveac[-n,n-1,EqCL[-n+1,-3],0],{n,3,ExpOrder+1}]; + + Do[ + Solveac[1,1+p,EqCL[1+1,p+1],0]; + Do[If[n+p<=ExpOrder,Solveac[n,n+p,EqCL[n+1,p-1],0]],{n,2,ExpOrder-p}]; + Solveac[-2,1+p,EqCL[-1,-3+p],0]; + Solveac[-1,p,EqCL[0,5+p],0]; + Solve\[CapitalDelta]\[Nu][p,p+5]; + Do[Solveac[-n,n-1+p,EqCL[-n+1,-3+p],0],{n,3,ExpOrder-p+2}], + {p,1,ExpOrder}], + l==1, + (*Print["l = 1 code"];*) + If[m==0&&l!=0,ac[-2l,2l]=0;acSolved[-2l,2l]=True]; + + Do[ + Do[If[n+p<=ExpOrder,Solveac[n,n+p,EqCL[n+1,p-1],0]],{n,1,ExpOrder-p}]; + Solveac[-(l),l+p,EqCL[-(l-1),p+5],0]; + Solve\[CapitalDelta]\[Nu][p,p+1]; + If[p>=Boole[m==0], + Solveac[-2l,2l+p,EqCL[-(2l-1),(p+4)],0]; + Solveac[-(2l+2),2l+p,EqCL[-(2l+1),p-4],0]; + Solveac[-(2l+1),2l-1+p,EqCL[-2l,p],0]; + Do[If[n+(p-2)<=ExpOrder+1,Solveac[-n-1,n+p-1,EqCL[-n,p-4],0]],{n,2l+2,ExpOrder-(p-1)}]], + {p,0,ExpOrder-1}], + l>=2, + (*Print["l >= 2 code"];*) + If[m==0&&l!=0,ac[-2l,2l]=0;acSolved[-2l,2l]=True]; + + Do[ + Do[If[n+p<=ExpOrder,Solveac[n,n+p,EqCL[n+1,p-1],0]],{n,1,ExpOrder}]; + Do[If[n+p<=ExpOrder,Solveac[-n,n+p,EqCL[-n+1,p-1],0]],{n,1,l-2}]; + Solve\[CapitalDelta]\[Nu][p,p-1]; + Solveac[-(l-1),l-1+p,EqCL[-(l-2),p+1],0]; + Solveac[-(l),l+p,EqCL[-(l-1),p+5],0]; + Solveac[-(l+1),l+1+p,EqCL[-l,p+4],0]; + Do[Solveac[-n,n+p,EqCL[-(n-1),p],0],{n,l+2,2l-1}]; + If[p>=2+Boole[m==0], + Solveac[-(2l+2),2l+(p-2),EqCL[-(2l+1),(p-2)-4],0]; + Solveac[-2l,2l+(p-2),EqCL[-(2l-1),(p-2)],0]; + Solveac[-(2l+1),2l-1+(p-2),EqCL[-2l,(p-2)],0]; + Do[If[p\[Nu]MST}]; +MST=Append[MST,Table[a[i]->aMST[i]+If[i==0,0,O[\[Epsilon]]^(ExpOrder+1)],{i,-ExpOrder-2,ExpOrder}]] + +] + + +(* ::Subsection:: *) +(*Definitions, replacements and auxiliary functions*) + + +(* ::Subsubsection::Closed:: *) +(*Assumptions *) + + +assumps={r>2,r0>2,a>=0,\[Eta]>0,\[Omega]>=0} + + +(* ::Subsubsection::Closed:: *) +(*MST Coefficients*) + + +(*replsMST[]:=Module[{aux,values,order\[Eta]=30}, +aux=Normal/@Block[{Print},KerrMSTSeries[-2,2,2,order\[Eta]/3//Ceiling]//.replsKerr]; +values=Replace[#,a_:>a+O[\[Eta]]^Ceiling[order\[Eta]+1,3]]&/@(Values@aux/.\[Epsilon]->2\[Omega]/.replsPN); +AssociationThread[Keys[aux]->values]];*) +(* +replsMST[\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,order\[Eta]_]:=Module[{aux,values,res,\[Nu]Value}, +aux=Normal/@Block[{Print},KerrMSTSeries[\[ScriptS],\[ScriptL],\[ScriptM],order\[Eta]/3+1//Ceiling]//.replsKerr/.q->a]; +\[Nu]Value=Replace[#,a_:>a+O[\[Eta]]^Ceiling[order\[Eta]+1,3]]&@(aux[\[Nu]]/.\[Epsilon]->2\[Omega]/.replsPN); +values=Replace[#,a_:>a+O[\[Eta]]^Ceiling[order\[Eta]+1,3]]&/@(Values@KeyDrop[#,\[Nu]]&@aux/.\[Epsilon]->2\[Omega]/.replsPN); +values=Insert[values,\[Nu]Value,1]; +res=AssociationThread[Keys[aux]->values]];*) + +replsMST[\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,order\[Eta]_]:=Module[{aux,values,res,\[Nu]Value}, +aux=Normal/@Block[{Print},KerrMSTSeries[\[ScriptS],\[ScriptL],\[ScriptM],order\[Eta]/3+1//Ceiling]//.replsKerr/.q->a]; +\[Nu]Value=Replace[#,a_:>a+O[\[Eta]]^(order\[Eta]+1)]&@(aux[\[Nu]]/.\[Epsilon]->2\[Omega]/.replsPN); +values=Replace[#,a_:>a+O[\[Eta]]^(order\[Eta]+1)]&/@(Values@KeyDrop[#,\[Nu]]&@aux/.\[Epsilon]->2\[Omega]/.replsPN); +values=Insert[values,\[Nu]Value,1]; +res=AssociationThread[Keys[aux]->values]]; + + +MSTCoefficientsInternal[\[ScriptS]_Integer,\[ScriptL]_Integer,\[ScriptM]_,aKerr_,order\[Eta]_Integer]:=Block[{aux,values,keys}, +aux=replsMST[\[ScriptS],\[ScriptL],\[ScriptM],order\[Eta]]; +values=Values[aux]/.a->aKerr; +keys=Keys[aux]/.{a[n_]:>aMST[n],\[Nu]->\[Nu]MST}; +keys->values//Thread//Association +] + + +MSTCoefficients[\[ScriptS]_Integer,\[ScriptL]_Integer,\[ScriptM]_,aKerr_,\[Omega]Var_,{\[Eta]Var_,order\[Eta]_Integer}]:=MSTCoefficientsInternal[\[ScriptS],\[ScriptL],\[ScriptM],aKerr,order\[Eta]]/.{\[Omega]->\[Omega]Var,\[Eta]->\[Eta]Var} + + +(* ::Subsubsection::Closed:: *) +(*Spacetime replacements*) + + +(*x[r_]:=((1+\[Kappa])-r)/(2 \[Kappa]);*) +z[r_]:=\[Epsilon] \[Kappa] (1-((1+\[Kappa])-r)/(2 \[Kappa])); +replsKerr={\[Tau]->(\[Epsilon]-\[ScriptM] a)/\[Kappa],\[Kappa]->Sqrt[1-a^2]}; +replsSchwarzschild={a->0,\[Kappa]->1,\[Lambda]->\[ScriptL](\[ScriptL]+1)-\[ScriptS](\[ScriptS]+1),\[Tau]->\[Epsilon]}; +Schwarzschild=#/.replsSchwarzschild&; +Kerr\[CapitalDelta][a_,r_]:=\[CapitalDelta][a,1,r]; + + +(* ::Subsubsection::Closed:: *) +(*Post Newtonian Scalings*) + + +replsPN={r->r \[Eta]^-2,r0->r0 \[Eta]^-2,\[Omega]->\[Omega] \[Eta]^3,\[CapitalOmega]Kerr->\[CapitalOmega]Kerr \[Eta]^3}; +RemovePNInternal=Normal[#]/.\[Eta]->1& +RemovePN[expr_,\[Eta]_Symbol]:=Normal[expr]/.\[Eta]->1 + + +PNScalingsInternal[expr_]:=expr/.\[Eta]->1/.replsPN +PNScalingsInternal[series_SeriesData]:=Module[{aux,termOrder}, +termOrder=series//SeriesLength; +aux=series//RemovePNInternal; +aux//PNScalingsInternal//SeriesTerms[#,{\[Eta],0,termOrder}]& +] + + +Options[PNScalings]={"IgnoreHarmonics"->False} + + +PNScalings[expr_,arguments_List,\[Eta]_Symbol,OptionsPattern[]]:=Module[{aux,check,repls,replsOpt}, +check=MatchQ[#,{a_Symbol,b_Integer}]&/@{{\[Omega],3},{r,2}}//Union; +If[!check,Return[$Failed]]; +repls=(#[[1]]->#[[1]]\[Eta]^#[[2]]&)/@arguments; +replsOpt=If[OptionValue["IgnoreHarmonics"],{ SpinWeightedSpheroidalHarmonicS[a___]:>( SpinWeightedSpheroidalHarmonicS[a]/.{\[Eta]->1})},{}]; +expr/.\[Eta]->1/.repls/.replsOpt +] + +PNScalings[series_SeriesData,arguments_List,\[Eta]_Symbol,opt:OptionsPattern[]]:=Module[{aux,termOrder}, +termOrder=series//SeriesLength; +aux=series//RemovePN[#,\[Eta]]&; +aux//PNScalings[#,arguments,\[Eta],opt]&//SeriesTerms[#,{\[Eta],0,termOrder}]& +] + + +IgnoreExpansionParameter[series_SeriesData,symbol_:1]:=Module[{aux,param,newList}, +param=series[[1]]; +newList=series[[3]]/.param->symbol; +ReplacePart[series,3->newList] +] + + +Zero[expr_,var_Symbol]:=Module[{aux,repls}, +repls=var->0; +expr/.repls] +Zero[expr_,vars_List]:=Module[{aux,repls}, +repls=vars->0//Thread; +expr/.repls] +One[expr_,var_Symbol]:=Module[{aux,repls}, +repls=var->1; +expr/.repls] +One[expr_,vars_List]:=Module[{aux,repls}, +repls=vars->1//Thread; +expr/.repls] + + +CollectDerivatives[expr_,func_,extra_:Identity]:=Module[{aux,list}, +list={func[__],Derivative[__][func][__]}; +Collect[expr,list,extra] +]; +CollectDerivatives[expr_,funcs_List,extra_:Identity]:=Module[{aux,list}, +list={#[__],Derivative[__][#][__]}&/@funcs//Flatten; +Collect[expr,list,extra] +]; + + +ExpandSpheroidals[expr_/;MatchQ[expr,SpinWeightedSpheroidalHarmonicS[\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,\[Gamma]_][\[Theta]_,\[Phi]_]],{\[Eta]_,n_}]:=Module[{aux,\[ScriptS],\[ScriptL],\[ScriptM],\[Gamma],\[Theta],\[Phi],gam,n\[Gamma]}, +{\[ScriptS],\[ScriptL],\[ScriptM],\[Gamma],\[Theta],\[Phi]}=expr/.{SpinWeightedSpheroidalHarmonicS[s_,l_,m_,gamma_][\[CurlyTheta]_,\[CurlyPhi]_]:>{s,l,m,gamma,\[CurlyTheta],\[CurlyPhi]}}; +n\[Gamma]=\[Gamma]//Exponent[#,\[Eta]]&; +aux=SpinWeightedSpheroidalHarmonicS[\[ScriptS],\[ScriptL],\[ScriptM],gam][\[Theta],\[Phi]]//Series[#,{gam,0,n/n\[Gamma]//Ceiling}]&; +aux=Normal[aux]/.gam->\[Gamma]; +aux//Series[#,{\[Eta],0,n}]& +] +ExpandSpheroidals[expr_/;MatchQ[expr,Derivative[__][SpinWeightedSpheroidalHarmonicS[\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,\[Gamma]_]][\[Theta]_,\[Phi]_]],{\[Eta]_,n_}]:=Module[{aux,d\[Theta],d\[Phi],\[ScriptS],\[ScriptL],\[ScriptM],\[Gamma],\[Theta],\[Phi],gam,n\[Gamma],aux\[Theta],aux\[Phi]}, +{d\[Theta],d\[Phi],\[ScriptS],\[ScriptL],\[ScriptM],\[Gamma],\[Theta],\[Phi]}=expr/.{Derivative[d\[CurlyTheta]_,d\[CurlyPhi]_][SpinWeightedSpheroidalHarmonicS[s_,l_,m_,gamma_]][\[CurlyTheta]_,\[CurlyPhi]_]:>{d\[CurlyTheta],d\[CurlyPhi],s,l,m,gamma,\[CurlyTheta],\[CurlyPhi]}}; +aux=SpinWeightedSpheroidalHarmonicS[\[ScriptS],\[ScriptL],\[ScriptM],\[Gamma]][aux\[Theta],aux\[Phi]]//ExpandSpheroidals[#,{\[Eta],n}]&; +aux=aux//D[#,{aux\[Theta],d\[Theta]}]&//D[#,{aux\[Phi],d\[Phi]}]&; +aux=aux/.{aux\[Theta]->\[Theta],aux\[Phi]->\[Phi]} +] +ExpandSpheroidals[expr_Plus,{\[Eta]_,n_}]:=ExpandSpheroidals[#,{\[Eta],n}]&/@expr; +ExpandSpheroidals[expr_Times,{\[Eta]_,n_}]:=ExpandSpheroidals[#,{\[Eta],n}]&/@expr; +ExpandSpheroidals[expr_,{\[Eta]_,n_}]:=expr; + + +(* ::Subsubsection::Closed:: *) +(*Tools for Series*) + + +SeriesMinOrder[series_SeriesData]:=Block[{}, +series[[4]] +] +SeriesMinOrder[1]=0; +Attributes[SeriesMinOrder]={Listable}; + +SeriesMaxOrder[series_SeriesData]:=Block[{}, +series[[5]] +] +Attributes[SeriesMaxOrder]={Listable}; + +SeriesLength[series_SeriesData]:=Block[{}, +SeriesMaxOrder[series]-SeriesMinOrder[series] +] +Attributes[SeriesLenght]={Listable}; + + + +SeriesCollect[series_SeriesData,var__,func_:Identity]:=Collect[#,var,func]&/@series; + + +SeriesTake[series_SeriesData,order_Integer:1]:=Block[{aux}, +series(1+O[series[[1]]]^(order)) +] +SeriesTake[series_SeriesData,0]:=Block[{aux,minOrder}, +minOrder=series//SeriesMinOrder; +O[series[[1]]]^(minOrder) +] +SeriesTake[series_O,order_Integer:1]:=Block[{aux,minOrder}, +series +] +SeriesTake[expr_/;MatchQ[expr,Times[__,_SeriesData]],order_Integer]:=Block[{aux,factor,series}, +factor=expr/.Times[a__,b_SeriesData]:>a; +series=expr/.Times[a__,b_SeriesData]:>b; +factor SeriesTake[series,order] +] +Attributes[SeriesTake]={Listable}; + + + +SeriesTerms[expr_,{x_,x0_,termOrder_}]:=Module[{aux,minOrder}, +minOrder=Series[expr,x->x0]//SeriesMinOrder; +Series[expr,{x,x0,minOrder+termOrder-1}] +]; +SeriesTerms[expr___]:=Series[expr] + + +polyToSeries[poly_,x_:\[Eta],x\:2080_:0]:=Block[{aux,maxPower}, +maxPower=poly//Exponent[#,x]&//Ceiling; +poly+O[(x-x\:2080)]^(maxPower+1)] +polyToSeries[0,x_:\[Eta],x\:2080_:0]:=Block[{aux,maxPower}, +0 +] + + +(* ::Subsubsection:: *) +(*Tools for Logs, Gammas, and PolyGammas*) + + +IgnoreLog\[Eta][expr_]:=expr/.Log[x_]/;!FreeQ[x,\[Eta]]:>Log[x/.\[Eta]->1]; +ExpandLog[expr_]:=(expr/.Log[a_]:>PowerExpand[Log[a]]); +ExpandGamma[expr_]:=(expr/. Gamma[n_Integer+x_]:>(\!\( +\*UnderoverscriptBox[\(\[Product]\), \(i = 0\), \(n - 1\)]\((x + i)\)\))(\!\( +\*UnderoverscriptBox[\(\[Product]\), \(i = n\), \(-1\)] +\*SuperscriptBox[\((x + i)\), \(-1\)]\)) Gamma[x]); +ExpandPolyGamma[expr_]:=(expr/.PolyGamma[m_Integer,n_Integer+x_]:>(-1)^m (m!)(\!\( +\*UnderoverscriptBox[\(\[Sum]\), \(i = 0\), \(n - 1\)] +\*FractionBox[\(1\), +SuperscriptBox[\((x + i)\), \(m + 1\)]]\)-\!\( +\*UnderoverscriptBox[\(\[Sum]\), \(i = n\), \(-1\)] +\*FractionBox[\(1\), +SuperscriptBox[\((x + i)\), \(m + 1\)]]\))+PolyGamma[m,x]); +PochhammerToGamma[expr_]:=(expr/.Pochhammer[x_,n_]:>Gamma[x+n]/Gamma[x]); +GammaToPochhammer[expr_,n_]:=expr/.(Gamma[x_+sign_. n]:>Pochhammer[x,sign n]Gamma[x]); + + +(* ::Subsubsection::Closed:: *) +(*Tools for DiracDeltas*) + + +ExpandDiracDelta[expr_,x_]/;MatchQ[expr,__ DiracDelta[a__]/;!FreeQ[a,x]]:=Module[{aux,f,f0,arg,repls,sign,flip}, +f=expr/.{f_ DiracDelta[arg_]:>f}; +arg=expr/.{f_ DiracDelta[arg_]:>arg}; +sign=arg//Coefficient[#,x]&; +flip=If[sign==-1,-1,1]; +repls=Solve[arg==0,x]//Flatten; +f0=f/.repls; +f0 DiracDelta[flip arg] +] + +ExpandDiracDelta[expr_,x_]/;MatchQ[expr,__ Derivative[__][DiracDelta][a__]/;!FreeQ[a,x]]:=Module[{aux,ret,f,n,f0,df0,arg,repls,coeff}, +f=expr/.{f_ Derivative[n_][DiracDelta][arg_]/;!FreeQ[arg,x]:>f}; +n=expr/.{f_ Derivative[n_][DiracDelta][arg_]/;!FreeQ[arg,x]:>n}; +arg=expr/.{f_ Derivative[n_][DiracDelta][arg_]/;!FreeQ[arg,x]:>arg}; +coeff=arg//Coefficient[#,x]&; +repls=Solve[arg==0,x]//Flatten; +f0[i_]:=D[f,{x,i}]/.repls; +ret=\!\( +\*UnderoverscriptBox[\(\[Sum]\), \(i = 0\), \(n\)]\( +\*SuperscriptBox[\((\(-1\))\), \(i\)] +\*SuperscriptBox[\(( +\*FractionBox[\(1\), \(coeff\)])\), \(i\)]\ Binomial[n, i]\ f0[i]\ \(\(Derivative[n - i]\)[DiracDelta]\)[arg]\)\); +ret +] + +ExpandDiracDelta[expr_,x_]/;!FreeQ[expr,DiracDelta[a_]/;!FreeQ[a,x]]:=Module[{aux,ret}, +aux=expr//Collect[#,{DiracDelta[__],Derivative[__][DiracDelta][__]}]&; +ExpandDiracDelta[#,x]&/@aux +] + +ExpandDiracDelta[expr_Plus,x_]:=(ExpandDiracDelta[#,x]&/@expr); +ExpandDiracDelta[expr_,x_]:=expr; + + +(* ::Subsection::Closed:: *) +(*Point particle source*) + + +(* ::Subsubsection::Closed:: *) +(*Interface*) + + +Options[TeukolskySourceCircularOrbit]={"InvariantWronskianForm"->False}; + + +TeukolskySourceCircularOrbit[\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,aVar_,{rVar_,r0Var_},opt:OptionsPattern[]]:=Module[{aux}, +TeukolskySource[\[ScriptS],\[ScriptL],\[ScriptM],aVar,{rVar,r0Var},opt] +] + + +(* ::Subsubsection::Closed:: *) +(*\[ScriptS] = -2*) + + +Options[TeukolskySource]={"Form"->"Default"}; + + +TeukolskySource[-2,\[ScriptL]_,\[ScriptM]_,a_,{r_,r0_},OptionsPattern[]] :=Assuming[{r0>0,r>0,1>a>=0}, + Module[{\[ScriptS]=-2,aux,auxFactor, \[ScriptCapitalE], \[ScriptCapitalL], \[Theta]0, \[CapitalDelta],\[CapitalDelta]1d,\[CapitalDelta]2d, Kt, \[CapitalUpsilon]t, \[Omega],\[CapitalOmega], SH,S0, dS0, d2S0, L1, L2, L2S, L2p, L1Sp, L1L2S, rcomp, invFactor0,invFactor1,invFactor2,\[Theta]comp, \[Rho], \[Rho]bar, \[CapitalSigma], Ann0, Anmbar0, Anmbar1, Ambarmbar0, Ambarmbar1, Ambarmbar2, Cnnp1p1, Cnmbarp1p1, Cmbarmbarp1p1,ret}, +\[ScriptCapitalE]=(a+(-2+r0) Sqrt[r0])/Sqrt[2 a r0^(3/2)+(-3+r0) r0^2]; +\[ScriptCapitalL]=(a^2-2 a Sqrt[r0]+r0^2)/(Sqrt[2 a+(-3+r0) Sqrt[r0]] r0^(3/4)); + + \[CapitalUpsilon]t = (r0^(5/4) (a+r0^(3/2)))/Sqrt[2 a+(-3+r0) Sqrt[r0]]; +\[Omega]=\[ScriptM] \[CapitalOmega]Kerr; +(*\[CapitalOmega]=1/Sqrt[r0^3];*) + \[Theta]0 = \[Pi]/2; + + \[CapitalDelta] = Kerr\[CapitalDelta][a,r0]; + \[CapitalDelta]1d=D[\[CapitalDelta],r0]; + \[CapitalDelta]2d=D[\[CapitalDelta],{r0,2}]; + + Kt=(r0^2+a^2)\[Omega]-\[ScriptM] a; + SH=SpinWeightedSpheroidalHarmonicS[\[ScriptS],\[ScriptL],\[ScriptM],a \[Omega]]; + S0 = SH[\[Theta]0, 0]; + dS0 = Derivative[1,0][SH][\[Theta]0, 0]; + d2S0 = Derivative[2,0][SH][\[Theta]0, 0]; + L1 = -\[ScriptM]/Sin[\[Theta]0] + a \[Omega] Sin[\[Theta]0] + Cos[\[Theta]0]/Sin[\[Theta]0]; + L2 = -\[ScriptM]/Sin[\[Theta]0] + a \[Omega] Sin[\[Theta]0] + 2 Cos[\[Theta]0]/Sin[\[Theta]0]; + L2S = dS0 + L2 S0; + L2p = \[ScriptM] Cos[\[Theta]0]/Sin[\[Theta]0]^2 + a \[Omega] Cos[\[Theta]0] - 2/Sin[\[Theta]0]^2; + L1Sp = d2S0 + L1 dS0; + L1L2S = L1Sp + L2p S0 + L2 dS0 + L1 L2 S0; + + \[Rho] = -1/(r0 - I a Cos[\[Theta]0]); + \[Rho]bar = -1/(r0 + I a Cos[\[Theta]0]); + \[CapitalSigma] = 1/(\[Rho] \[Rho]bar); + + Ann0 = -\[Rho]^(-2) \[Rho]bar^(-1) (Sqrt[2] \[CapitalDelta])^(-2) (\[Rho]^(-1) L1L2S + 3 I a Sin[\[Theta]0] L1 S0 + 3 I a Cos[\[Theta]0] S0 + 2 I a Sin[\[Theta]0] dS0 - I a Sin[\[Theta]0] L2 S0 ); + Anmbar0 = \[Rho]^(-3) (Sqrt[2]\[CapitalDelta])^(-1) ( (\[Rho] + \[Rho]bar - I Kt/\[CapitalDelta]) L2S + (\[Rho] - \[Rho]bar) a Sin[\[Theta]0] Kt/\[CapitalDelta] S0 ); + Anmbar1 = -\[Rho]^(-3) (Sqrt[2]\[CapitalDelta])^(-1) ( L2S + I (\[Rho] - \[Rho]bar) a Sin[\[Theta]0] S0 ); + Ambarmbar0 = (Kt^2 S0 \[Rho]bar)/(4 \[CapitalDelta]^2 \[Rho]^3)+(I Kt S0 (1-r0+\[CapitalDelta] \[Rho]) \[Rho]bar)/(2 \[CapitalDelta]^2 \[Rho]^3)+(I r0 S0 \[Rho]bar \[Omega])/(2 \[CapitalDelta] \[Rho]^3); + Ambarmbar1 = -\[Rho]^(-3) \[Rho]bar S0/2 ( I Kt/\[CapitalDelta] - \[Rho] ); + Ambarmbar2 = -\[Rho]^(-3) \[Rho]bar S0/4; + + rcomp = (\[ScriptCapitalE](r0^2+a^2) - a \[ScriptCapitalL])/(2\[CapitalSigma]); + \[Theta]comp = \[Rho] (I Sin[\[Theta]0](a \[ScriptCapitalE] - \[ScriptCapitalL]/Sin[\[Theta]0]^2))/Sqrt[2]; + + {Cnnp1p1, Cnmbarp1p1, Cmbarmbarp1p1} = {rcomp^2, rcomp \[Theta]comp, \[Theta]comp^2}; + + aux =(-8Pi)/\[CapitalUpsilon]t ((Ann0*Cnnp1p1 + Anmbar0*Cnmbarp1p1 + Ambarmbar0*Cmbarmbarp1p1) DiracDelta[r-r0]+(Anmbar1*Cnmbarp1p1 + Ambarmbar1*Cmbarmbarp1p1) DiracDelta'[r-r0]+(Ambarmbar2*Cmbarmbarp1p1) DiracDelta''[r-r0]); +auxFactor=Switch[OptionValue["Form"],"Default",Kerr\[CapitalDelta][a,r]^-\[ScriptS],"InvariantWronskian",1]; +If[auxFactor//MatchQ[#,_Switch]&,Return[$Failed]]; +aux=auxFactor aux//ExpandDiracDelta[#,r]&//Collect[#,{DiracDelta[__],Derivative[__][DiracDelta][__]},Simplify]&; +ret=aux; +ret +]] + + +(* ::Subsubsection::Closed:: *) +(*\[ScriptS] = -1*) + + +TeukolskySource[-1,\[ScriptL]_,\[ScriptM]_,a_,{r_,r0_},OptionsPattern[]] :=Assuming[{r0>0,r>0,1>a>=0}, + Module[{aux,auxFactor,\[ScriptS]=-1, \[ScriptCapitalE], \[ScriptCapitalL], \[CapitalDelta], Kt, \[CapitalUpsilon]t,SH,\[Omega],\[CapitalOmega],\[Theta]0,S0,dS0,L1,\[Rho],\[Rho]bar,\[CapitalSigma],An0,Ambar0,Ambar1,rcomp,\[Theta]comp,Cnp1,Cmbarp1,ret}, +\[ScriptCapitalE]=(a+(-2+r0) Sqrt[r0])/Sqrt[2 a r0^(3/2)+(-3+r0) r0^2]; +\[ScriptCapitalL]=(a^2-2 a Sqrt[r0]+r0^2)/(Sqrt[2 a+(-3+r0) Sqrt[r0]] r0^(3/4)); + \[CapitalUpsilon]t = (r0^(5/4) (a+r0^(3/2)))/Sqrt[2 a+(-3+r0) Sqrt[r0]]; +\[CapitalDelta] = r0^2-2r0+a^2; +Kt=(r0^2+a^2)\[Omega]-\[ScriptM] a; +\[Omega]=\[ScriptM] \[CapitalOmega]Kerr; +(*\[CapitalOmega]=1/Sqrt[r0^3];*) +SH= SpinWeightedSpheroidalHarmonicS[\[ScriptS],\[ScriptL],\[ScriptM],a \[Omega]]; +S0 = SH[\[Theta]0, 0]; +dS0 = Derivative[1,0][SH][\[Theta]0, 0]; +L1 = -\[ScriptM]/Sin[\[Theta]0] + a \[Omega] Sin[\[Theta]0] + Cos[\[Theta]0]/Sin[\[Theta]0]; +\[Rho] = -1/(r0 - I a Cos[\[Theta]0]); +\[Rho]bar = -1/(r0 + I a Cos[\[Theta]0]); +\[CapitalSigma] = 1/(\[Rho] \[Rho]bar); +\[Theta]0 =\[Pi]/2; + +An0 =-((dS0+L1 S0+I a S0 \[Rho] Sin[\[Theta]0])/(2Sqrt[2] \[CapitalDelta] \[Rho]^2 \[Rho]bar)); +Ambar0 =(S0 (-((I Kt)/\[CapitalDelta])+\[Rho]))/(4 \[Rho]^2); +Ambar1 =-(S0/(4 \[Rho]^2)); + +rcomp = (\[ScriptCapitalE](r0^2+a^2) - a \[ScriptCapitalL] )/(2\[CapitalSigma]); +\[Theta]comp = \[Rho] (I Sin[\[Theta]0](a \[ScriptCapitalE] - \[ScriptCapitalL]/Sin[\[Theta]0]^2))/Sqrt[2]; +Cnp1=rcomp; +Cmbarp1=\[Theta]comp; + +aux=(-8Pi)/\[CapitalUpsilon]t (-(An0*Cnp1 + Ambar0*Cmbarp1) DiracDelta[r-r0]-(Ambar1*Cmbarp1) DiracDelta'[r-r0]); +auxFactor=Switch[OptionValue["Form"],"Default",Kerr\[CapitalDelta][a,r]^-\[ScriptS],"InvariantWronskian",1]; +If[auxFactor//MatchQ[#,_Switch]&,Return[$Failed]]; +aux=auxFactor aux//ExpandDiracDelta[#,r]&//Collect[#,{DiracDelta[__],Derivative[__][DiracDelta][__]},Simplify]&; +ret=aux; +ret + ]] + + +(* ::Subsubsection::Closed:: *) +(*\[ScriptS] = 0*) + + +TeukolskySource[0,\[ScriptL]_,\[ScriptM]_,a_,{r_,r0_},OptionsPattern[]] :=Assuming[{r0>0,r>0,1>a>=0}, + Module[{aux,auxFactor,\[ScriptS]=0, \[Theta]0, \[Omega],\[CapitalOmega], \[CapitalUpsilon]t, S,SH,ret,\[CapitalDelta]}, +\[CapitalUpsilon]t =(r0^(5/4) (a+r0^(3/2)))/Sqrt[2 a+(-3+r0) Sqrt[r0]]; +\[Theta]0 = \[Pi]/2; +\[CapitalDelta] = r0^2-2r0+a^2; +\[Omega]=\[ScriptM] \[CapitalOmega]Kerr; +(*\[CapitalOmega]=1/Sqrt[r0^3];*) +SH= SpinWeightedSpheroidalHarmonicS[\[ScriptS],\[ScriptL],\[ScriptM],a \[Omega]]; +S = SH[\[Pi]/2,0]; +aux = -((4 \[Pi])/\[CapitalUpsilon]t) r0^2 S DiracDelta[r-r0]; +auxFactor=Switch[OptionValue["Form"],"Default",Kerr\[CapitalDelta][a,r]^-\[ScriptS],"InvariantWronskian",1]; +If[auxFactor//MatchQ[#,_Switch]&,Return[$Failed]]; +aux=auxFactor aux//ExpandDiracDelta[#,r]&//Collect[#,{DiracDelta[__],Derivative[__][DiracDelta][__]},Simplify]&; +ret=aux; +ret +]] + + +(*TeukolskySource[0,\[ScriptL]_,\[ScriptM]_,order\[Eta]_:"exact",OptionsPattern[]] := + Block[{s=0,l=\[ScriptL],m=\[ScriptM],a, p,r0, \[Theta]0, \[Omega],\[Gamma], \[CapitalUpsilon]t, S,SH,ret,aux,\[CapitalDelta],invFactor0}, +p = r0; +\[CapitalUpsilon]t =(p^(5/4) (a+p^(3/2)))/Sqrt[2 a+(-3+p) Sqrt[p]]; +r0 = p; +\[Theta]0 = \[Pi]/2; +SH= SpinWeightedSpheroidalHarmonicS[s,l,m,\[Gamma]]; +S = SH[\[Pi]/2,0]; +\[CapitalDelta] = r0^2-2r0+a^2; +invFactor0=If[OptionValue["InvariantWronskianForm"],1,\[CapitalDelta]^-1]; +If[order\[Eta]!="exact", +S=S//Series[#,{\[Gamma],0,order\[Eta]/3//Ceiling}]&//Normal; +]; +S=S/.\[Gamma]->a \[Omega]; +If[order\[Eta]!="exact", +ret = -((4 \[Pi])/\[CapitalUpsilon]t) r0^2 invFactor0 S \[Delta][r-r0]/.replsPN//Series[#,{\[Eta],0,order\[Eta]}]&, +(*else*) +ret= -((4 \[Pi])/\[CapitalUpsilon]t) r0^2 invFactor0 S \[Delta][r-r0] +]; +ret +]*) + + +(* ::Subsubsection::Closed:: *) +(*s = +2*) + + +(* ::Input:: *) +(*(*TeukolskySource[2,\[ScriptL]_,\[ScriptM]_,OptionsPattern[]] := *) +(* Block[{s=2,l=\[ScriptL], m=\[ScriptM], a, p, \[ScriptCapitalE], \[ScriptCapitalL], \[CapitalUpsilon]t,SH, \[Omega], r0, \[Theta]0, \[CapitalDelta],d\[CapitalDelta],d2\[CapitalDelta], Kt, S0, dS0, d2S0, \[Delta]L\[Dagger]1, \[Delta]L\[Dagger]2, d\[Delta]L\[Dagger]2, \[Rho], \[Rho]bar, d\[Rho]over\[Rho], d2\[Rho]over\[Rho], u\[Theta]0,invFactor0,invFactor1,invFactor2, rcomp, \[Theta]comp, All0, Alm0, Alm1, Amm0, Amm1, Amm2, Cllp1p1, Clmp1p1, Cmmp1p1, Cllp1m1, Clmp1m1,ret,A,B,C},*) +(* a = \[ScriptA];*) +(* p=r0;*) +(* \[ScriptCapitalE]=(a+(-2+p) Sqrt[p])/Sqrt[2 a p^(3/2)+(-3+p) p^2];*) +(*\[ScriptCapitalL]=(a^2-2 a Sqrt[p]+p^2)/(Sqrt[2 a+(-3+p) Sqrt[p]] p^(3/4));*) +(* \[CapitalUpsilon]t = (p^(5/4) (a+p^(3/2)))/Sqrt[2 a+(-3+p) Sqrt[p]];*) +(* \[Theta]0 = \[Pi]/2;*) +(* *) +(* \[CapitalDelta] = r0^2 + a^2 - 2 r0;*) +(*d\[CapitalDelta]=2 (-1+r0);*) +(*d2\[CapitalDelta]=2;*) +(* Kt = (r0^2 + a^2) \[Omega] - m a;*) +(* *) +(* invFactor0=If[OptionValue["InvariantWronskianForm"],1,\[CapitalDelta]^-3+3 d\[CapitalDelta]/\[CapitalDelta]^4-3 (-4 d\[CapitalDelta]^2+\[CapitalDelta] d2\[CapitalDelta])/\[CapitalDelta]^5];*) +(* invFactor1=If[OptionValue["InvariantWronskianForm"],1,(6 d\[CapitalDelta])/\[CapitalDelta]^4+\[CapitalDelta]^-3];*) +(* invFactor2=If[OptionValue["InvariantWronskianForm"],1,\[CapitalDelta]^-3];*) +(* *) +(* \[Rho] = -1/(r0 - I a Cos[\[Theta]0]);*) +(* \[Rho]bar = -1/(r0 + I a Cos[\[Theta]0]);*) +(**) +(* SH=SpinWeightedSpheroidalHarmonicS[s,l,m,a \[Omega]];*) +(* S0 = SH[\[Theta]0, 0];*) +(* dS0 = Derivative[1,0][SH][\[Theta]0, 0];*) +(* d2S0 = Derivative[2,0][SH][\[Theta]0, 0];*) +(* \[Delta]L\[Dagger]1 = m/Sin[\[Theta]0] - a \[Omega] Sin[\[Theta]0] + Cos[\[Theta]0]/Sin[\[Theta]0];*) +(* \[Delta]L\[Dagger]2 = m/Sin[\[Theta]0] - a \[Omega] Sin[\[Theta]0] + 2 Cos[\[Theta]0]/Sin[\[Theta]0];*) +(* d\[Delta]L\[Dagger]2 = -m Cos[\[Theta]0]/Sin[\[Theta]0]^2 - a \[Omega] Cos[\[Theta]0] - 2/Sin[\[Theta]0]^2;*) +(* d\[Rho]over\[Rho] = I a \[Rho] Sin[\[Theta]0];*) +(* d2\[Rho]over\[Rho] = I a \[Rho](Cos[\[Theta]0] + 2 Sin[\[Theta]0] d\[Rho]over\[Rho]);*) +(* *) +(* All0 = -(1/2) \[Rho]^(-1) \[Rho]bar (d2S0 + (\[Delta]L\[Dagger]1 + \[Delta]L\[Dagger]2 + 2 d\[Rho]over\[Rho])dS0 + (d\[Delta]L\[Dagger]2 + \[Delta]L\[Dagger]1 \[Delta]L\[Dagger]2 - 6 d\[Rho]over\[Rho]^2 + 3 d2\[Rho]over\[Rho] + (3 \[Delta]L\[Dagger]1 - \[Delta]L\[Dagger]2)d\[Rho]over\[Rho]) S0);*) +(* Alm0 = (2/Sqrt[2]) \[Rho]^(-1) ( -(\[Rho] + \[Rho]bar + I Kt/\[CapitalDelta]) (dS0 + \[Delta]L\[Dagger]2 S0) + (\[Rho] - \[Rho]bar) a Sin[\[Theta]0] Kt/\[CapitalDelta] S0 );*) +(* Alm1 = (2/Sqrt[2]) \[Rho]^(-1) ( (dS0 + \[Delta]L\[Dagger]2 S0) + I (\[Rho] - \[Rho]bar) a Sin[\[Theta]0] S0 );*) +(* Amm0 = (Kt^2 S0)/(\[CapitalDelta]^2 \[Rho] \[Rho]bar)+(2 I Kt S0 (-1+r0-\[CapitalDelta] \[Rho]))/(\[CapitalDelta]^2 \[Rho] \[Rho]bar)-(2 I r0 S0 \[Omega])/(\[CapitalDelta] \[Rho] \[Rho]bar);*) +(* Amm1 = 2 \[Rho]^(-1) \[Rho]bar^(-1) S0 ( I Kt/\[CapitalDelta] + \[Rho] );*) +(* Amm2 = -\[Rho]^(-1) \[Rho]bar^(-1) S0;*) +(**) +(* rcomp = (\[ScriptCapitalE](r0^2+a^2) - a \[ScriptCapitalL])/(\[CapitalDelta]);*) +(* \[Theta]comp = -\[Rho]bar (I Sin[\[Theta]0](a \[ScriptCapitalE] - \[ScriptCapitalL]/Sin[\[Theta]0]^2))/Sqrt[2];*) +(* *) +(* {Cllp1p1,Clmp1p1,Cmmp1p1} = {rcomp^2, rcomp \[Theta]comp, \[Theta]comp^2};*) +(*A=(All0*Cllp1p1 + Alm0*Clmp1p1 + Amm0*Cmmp1p1);*) +(*B=(Alm1*Clmp1p1 + Amm1*Cmmp1p1);*) +(*C=Amm2*Cmmp1p1;*) +(* *) +(* ret =-((8Pi)/\[CapitalUpsilon]t)(invFactor0(\[CapitalDelta]^2 A -2\[CapitalDelta] d\[CapitalDelta] B+2(d\[CapitalDelta]^2+\[CapitalDelta] d2\[CapitalDelta])C )\[Delta][r-r0]+ invFactor1(\[CapitalDelta]^2 B-4 \[CapitalDelta] d\[CapitalDelta] C) \[Delta]'[r-r0] +invFactor2 \[CapitalDelta]^2 C \[Delta]''[r-r0]);*) +(*ret//Simplify*) +(*]*)*) + + +TeukolskySource[2,\[ScriptL]_,\[ScriptM]_,a_,{r_,r0_},OptionsPattern[]] :=Assuming[{r0>0,r>0,1>a>=0}, + Module[{aux,auxFactor,\[ScriptS]=2, \[ScriptCapitalE], \[ScriptCapitalL], \[CapitalUpsilon]t,SH, \[Omega], \[CapitalOmega], \[Theta]0, \[CapitalDelta],d\[CapitalDelta],d2\[CapitalDelta], Kt, S0, dS0, d2S0, \[Delta]L\[Dagger]1, \[Delta]L\[Dagger]2, d\[Delta]L\[Dagger]2, \[Rho], \[Rho]bar, d\[Rho]over\[Rho], d2\[Rho]over\[Rho], u\[Theta]0, rcomp, \[Theta]comp, All0, Alm0, Alm1, Amm0, Amm1, Amm2, Cllp1p1, Clmp1p1, Cmmp1p1, Cllp1m1, Clmp1m1,ret,A,B,C}, + \[ScriptCapitalE]=(a+(-2+r0) Sqrt[r0])/Sqrt[2 a r0^(3/2)+(-3+r0) r0^2]; +\[ScriptCapitalL]=(a^2-2 a Sqrt[r0]+r0^2)/(Sqrt[2 a+(-3+r0) Sqrt[r0]] r0^(3/4)); + \[CapitalUpsilon]t = (r0^(5/4) (a+r0^(3/2)))/Sqrt[2 a+(-3+r0) Sqrt[r0]]; + \[Theta]0 = \[Pi]/2; + +\[Omega]=\[ScriptM] \[CapitalOmega]Kerr; +(*\[CapitalOmega]=1/Sqrt[r0^3];*) + + \[CapitalDelta] = r0^2 + a^2 - 2 r0; +d\[CapitalDelta]=2 (-1+r0); +d2\[CapitalDelta]=2; + Kt = (r0^2 + a^2) \[Omega] - \[ScriptM] a; + + \[Rho] = -1/(r0 - I a Cos[\[Theta]0]); + \[Rho]bar = -1/(r0 + I a Cos[\[Theta]0]); + + SH=SpinWeightedSpheroidalHarmonicS[\[ScriptS],\[ScriptL],\[ScriptM],a \[Omega]]; + S0 = SH[\[Theta]0, 0]; + dS0 = Derivative[1,0][SH][\[Theta]0, 0]; + d2S0 = Derivative[2,0][SH][\[Theta]0, 0]; + \[Delta]L\[Dagger]1 = \[ScriptM]/Sin[\[Theta]0] - a \[Omega] Sin[\[Theta]0] + Cos[\[Theta]0]/Sin[\[Theta]0]; + \[Delta]L\[Dagger]2 = \[ScriptM]/Sin[\[Theta]0] - a \[Omega] Sin[\[Theta]0] + 2 Cos[\[Theta]0]/Sin[\[Theta]0]; + d\[Delta]L\[Dagger]2 = -\[ScriptM] Cos[\[Theta]0]/Sin[\[Theta]0]^2 - a \[Omega] Cos[\[Theta]0] - 2/Sin[\[Theta]0]^2; + d\[Rho]over\[Rho] = I a \[Rho] Sin[\[Theta]0]; + d2\[Rho]over\[Rho] = I a \[Rho](Cos[\[Theta]0] + 2 Sin[\[Theta]0] d\[Rho]over\[Rho]); + + All0 = -(1/2) \[Rho]^(-1) \[Rho]bar (d2S0 + (\[Delta]L\[Dagger]1 + \[Delta]L\[Dagger]2 + 2 d\[Rho]over\[Rho])dS0 + (d\[Delta]L\[Dagger]2 + \[Delta]L\[Dagger]1 \[Delta]L\[Dagger]2 - 6 d\[Rho]over\[Rho]^2 + 3 d2\[Rho]over\[Rho] + (3 \[Delta]L\[Dagger]1 - \[Delta]L\[Dagger]2)d\[Rho]over\[Rho]) S0); + Alm0 = (2/Sqrt[2]) \[Rho]^(-1) ( -(\[Rho] + \[Rho]bar + I Kt/\[CapitalDelta]) (dS0 + \[Delta]L\[Dagger]2 S0) + (\[Rho] - \[Rho]bar) a Sin[\[Theta]0] Kt/\[CapitalDelta] S0 ); + Alm1 = (2/Sqrt[2]) \[Rho]^(-1) ( (dS0 + \[Delta]L\[Dagger]2 S0) + I (\[Rho] - \[Rho]bar) a Sin[\[Theta]0] S0 ); + Amm0 = (Kt^2 S0)/(\[CapitalDelta]^2 \[Rho] \[Rho]bar)+(2 I Kt S0 (-1+r0-\[CapitalDelta] \[Rho]))/(\[CapitalDelta]^2 \[Rho] \[Rho]bar)-(2 I r0 S0 \[Omega])/(\[CapitalDelta] \[Rho] \[Rho]bar); + Amm1 = 2 \[Rho]^(-1) \[Rho]bar^(-1) S0 ( I Kt/\[CapitalDelta] + \[Rho] ); + Amm2 = -\[Rho]^(-1) \[Rho]bar^(-1) S0; + + rcomp = (\[ScriptCapitalE](r0^2+a^2) - a \[ScriptCapitalL])/(\[CapitalDelta]); + \[Theta]comp = -\[Rho]bar (I Sin[\[Theta]0](a \[ScriptCapitalE] - \[ScriptCapitalL]/Sin[\[Theta]0]^2))/Sqrt[2]; + + {Cllp1p1,Clmp1p1,Cmmp1p1} = {rcomp^2, rcomp \[Theta]comp, \[Theta]comp^2}; +A=(All0*Cllp1p1 + Alm0*Clmp1p1 + Amm0*Cmmp1p1); +B=(Alm1*Clmp1p1 + Amm1*Cmmp1p1); +C=Amm2*Cmmp1p1; + + aux =-((8Pi)/\[CapitalUpsilon]t)(A DiracDelta[r-r0]+ B DiracDelta'[r-r0] +C DiracDelta''[r-r0]); +auxFactor=Switch[OptionValue["Form"],"Default",1,"InvariantWronskian",Kerr\[CapitalDelta][a,r]^\[ScriptS]]; +If[auxFactor//MatchQ[#,_Switch]&,Return[$Failed]]; +aux=auxFactor aux//ExpandDiracDelta[#,r]&//Collect[#,{DiracDelta[__],Derivative[__][DiracDelta][__]},Simplify]&; +ret=aux; +ret +]] + + +(* ::Subsection::Closed:: *) +(*Teukolsky Equation*) + + +rstar[a_,M_,r_]:=Block[{rp=M (1+Sqrt[1-(a/M)^2]),rm=M (1-Sqrt[1-(a/M)^2])},r+((2 M rp) Log[(r-rp)/(2 M)])/(rp-rm)-((2 M rm) Log[(r-rm)/(2 M)])/(rp-rm)]; +s\[Lambda]\[ScriptL]\[ScriptM][0,\[ScriptL]_,\[ScriptM]_,\[Omega]_,a_]:=SpheroidalEigenvalue[\[ScriptL],\[ScriptM],I a \[Omega]]-2 \[ScriptM] a \[Omega]; +s\[Lambda]\[ScriptL]\[ScriptM][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,\[Omega]_,0]:=\[ScriptL](\[ScriptL]+1)-\[ScriptS](\[ScriptS]+1); +s\[Lambda]\[ScriptL]\[ScriptM][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,\[Omega]_,a_]/;\[ScriptS]!=0&&a!=0:=(Print["Unsupported: \[ScriptS]=",\[ScriptS]," a=",a];Abort[];) +\[CapitalDelta][a_,M_,r_]:=r^2-2 M r+a^2; +ClearAll[equation] +equation[\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,\[Omega]_,a_,M_,r_]:=\[CapitalDelta][a,M,r]^-\[ScriptS] D[(\[CapitalDelta][a,M,r]^(\[ScriptS]+1) D[R[r],r]),r]+(1/\[CapitalDelta][a,M,r] ((r^2+a^2)^2 \[Omega]^2-4 a M r \[Omega] \[ScriptM]+a^2 \[ScriptM]^2+2 I a (r-M) \[ScriptM] \[ScriptS]-2 I M (r^2-a^2) \[Omega] \[ScriptS])+2 I r \[Omega] \[ScriptS]-SpinWeightedSpheroidalEigenvalue[\[ScriptS],\[ScriptL],\[ScriptM],a \[Omega]]-2 a \[ScriptM] \[Omega]) R[r] +equation[\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,\[Omega]_,a_,M_,r_,order\[Eta]_]:=\[CapitalDelta][a,M,r]^-\[ScriptS] D[(\[CapitalDelta][a,M,r]^(\[ScriptS]+1) D[R[r],r]),r]+(1/\[CapitalDelta][a,M,r] ((r^2+a^2)^2 \[Omega]^2-4 a M r \[Omega] \[ScriptM]+a^2 \[ScriptM]^2+2 I a (r-M) \[ScriptM] \[ScriptS]-2 I M (r^2-a^2) \[Omega] \[ScriptS])+2 I r \[Omega] \[ScriptS]-eigenValue[\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]-2 a \[ScriptM] \[Omega]) R[r] + + +\[Lambda][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=Block[{aux,order\[Epsilon],\[Gamma],\[ScriptA]=a,\[Omega]}, +order\[Epsilon]=order\[Eta]/3//Ceiling; +aux=SpinWeightedSpheroidalEigenvalue[\[ScriptS],\[ScriptL],\[ScriptM],\[Gamma]]; +aux=aux//Series[#,{\[Gamma],0,order\[Epsilon]}]&//Normal; +aux/.\[Gamma]->\[ScriptA] \[Omega]/.replsPN//Series[#,{\[Eta],0,order\[Eta]}]& +] + + +TeukolskyEquation[\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_] := Collect[equation[\[ScriptS], \[ScriptL], \[ScriptM], \[Omega], a, 1, r]/.R->(R[# \[Eta]^2]&)/.replsPN,Derivative[__][R][__],Simplify]; +TeukolskyEquation[\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_] := Collect[equation[\[ScriptS], \[ScriptL], \[ScriptM], \[Omega], a, 1, r,order\[Eta]]/.R->(R[# \[Eta]^2]&)/.replsPN/.eigenValue->\[Lambda],{R[__],Derivative[__][R][__]},Simplify]; + + +TeukolskyEquation[\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]Var_,{\[Eta]Var_,order_},RVar_[rvar_]]:=Module[{aux}, +aux=SeriesCollect[equation[\[ScriptS], \[ScriptL], \[ScriptM], \[Omega], a, 1, r,order]/.R->(R[# \[Eta]^2]&)/.replsPN/.eigenValue->\[Lambda],{R[__],Derivative[__][R][__]},Simplify]; +aux/.{\[Eta]->\[Eta]Var,\[Omega]->\[Omega]Var,R->RVar,r->rvar}] + + +TeukolskyEquation[\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]Var_,RVar_[rvar_]]:=Module[{aux}, +aux=Collect[equation[\[ScriptS], \[ScriptL], \[ScriptM], \[Omega], a, 1, r,order]/.eigenValue[___]->SpinWeightedSpheroidalEigenvalue[\[ScriptS],\[ScriptL],\[ScriptM],a \[Omega]],{R[__],Derivative[__][R][__]},Simplify]; +aux/.{\[Omega]->\[Omega]Var,R->RVar,r->rvar}] + + +teukolsky[r_] := Collect[equation[-2, 2, \[ScriptM], \[Omega], 0, 1, r],Derivative[__][R][__],Simplify]; +teukolsky[\[ScriptS]_,\[ScriptL]_] := Collect[equation[\[ScriptS], \[ScriptL], 0, \[Omega], 0, 1, r],Derivative[__][R][__],Simplify]; +teukolsky[\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,order\[Eta]_] := Collect[equation[\[ScriptS], \[ScriptL], \[ScriptM], \[Omega], \[ScriptA], 1, r,order\[Eta]]/.eigenValue->\[Lambda],Derivative[__][R][__],Simplify]; + + +(* ::Subsection::Closed:: *) +(*integrateDelta*) + + +integrateDelta[expr_Plus,dx_,toInf_?BooleanQ]:=integrateDelta[#,dx,toInf]&/@expr; +integrateDelta[expr_,dx_,toInf_?BooleanQ]:=Module[{collected,sign,border}, + sign=If[toInf,-1,1]; + border=If[toInf,\[Infinity],0]; + (*collected=Collect[expr,{DiracDelta[__],HeavisideTheta[__],Derivative[__][DiracDelta][__]}];*) + collected=Expand@expr; + If[MatchQ[collected,_Plus], + integrateDelta[#,dx,toInf]&@collected, + (*else*) + sign(#-Quiet[Check[(#/.{dx->dx^sign}/.{dx->0}), \:2665 Normal@Series[#,dx->border]]])&@\[Integral]expr \[DifferentialD]dx + ] +] + +integrateDelta[expr_/;MatchQ[expr,__ \[Delta][__]],dx_,toInf_?BooleanQ]:=Module[{repl,sign,border,arg,flip}, + sign=If[toInf,-1,1]; + border=If[toInf,\[Infinity],0]; + arg=expr/.{__ \[Delta][arg_]:>arg}; + flip=Coefficient[arg,dx]; + repl={a_ \[Delta][arg_]:> (a/.Flatten@Solve[arg==0,dx])\[Theta][flip sign arg]}; + expr/.repl +] +integrateDelta[expr_/;MatchQ[expr,__ Derivative[__][\[Delta]][__]],dx_,toInf_?BooleanQ]:=Module[{repl,sign,arg,flip}, + sign=If[toInf,-1,1]; + arg=expr/.__ Derivative[__][\[Delta]][arg_]:>arg; + flip=Coefficient[arg,dx]; + (*To keep the boundary terms just uncomment the first term in the line below*) + repl={(a_ Derivative[n_][\[Delta]][arg_]):>(*( \!\( +\*UnderoverscriptBox[\(\[Sum]\), \(i = 0\), \(n - 1\)]\ \(sign\ flip +\*SuperscriptBox[\((\(-flip\))\), \(i\)]\ D[a, {dx, i}]\ \(\(Derivative[n - i]\)[\[Theta]]\)[arg]\)\))*)+(-flip)^n (D[a,{dx,n}]/.Flatten@Solve[arg==0,dx])\[Theta][flip sign arg] }; +expr/.repl +] + +integrateDelta[expr_/;MatchQ[expr,__ \[Theta][__]],dx_,toInf_?BooleanQ]:=Module[{repl,sign,border,arg,basePiece,x\:2080}, + sign=If[toInf,-1,1]; + border=If[toInf,\[Infinity],0]; + arg=expr/.__ \[Theta][arg_]:>arg; + x\:2080=Flatten@Solve[arg==0,dx]; + basePiece=Negative[Coefficient[arg,dx] sign]; + repl={a_ \[Theta][arg_]:>((sign \[Theta][arg](#-(#/.x\:2080))-If[basePiece, sign (Quiet[Check[(Simplify[#/.{dx->dx^sign}]/.{dx->0}), \:2665 Normal@Series[#,dx->border]]]-(#/.x\:2080)),0])&@\[Integral]a \[DifferentialD]dx)}; + expr/.repl +] + + + +\[Theta]'[arg_]:=\[Delta][arg] +Derivative[n_][\[Theta]][arg_]:=Derivative[n-1][\[Delta]][arg]; +\[Theta][\[Eta]^-2 a_]:=\[Theta][a]; +\[Delta][\[Eta]^-2 a_]:=\[Eta]^2 \[Delta][a]; +\[Delta]'[\[Eta]^-2 a_]:=\[Eta]^2 \[Delta]'[a]; +\[Delta]''[\[Eta]^-2 a_]:=\[Eta]^2 \[Delta]''[a]; + + +(* ::Subsection::Closed:: *) +(*Amplitudes*) + + +(* ::Subsubsection::Closed:: *) +(*A Amplitudes*) + + +(* ::Text:: *) +(*These are the amplitudes \!\(\*SubscriptBox[\(A\), \(\[PlusMinus]\)]\) from Sasaki Tagoshi Eq.(157-158)*) + + +AAmplitude["+"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=Module[{aux,\[CurlyEpsilon],\[Kappa],\[CurlyEpsilon]p,\[Tau],nMax,nMin}, +\[CurlyEpsilon]=2 \[Omega]; +\[Kappa]=Sqrt[1-a^2]; +\[CurlyEpsilon]p=(\[CurlyEpsilon]+\[Tau])/2; +\[Tau]=(-a \[ScriptM]+\[CurlyEpsilon])/\[Kappa]; +nMax=order\[Eta]/3//Ceiling; +nMin=-(order\[Eta]/3+2)//Floor; +aux=E^(-(\[Pi]/2)\[CurlyEpsilon]) E^(\[Pi]/2 I(\[Nu]MST+1+\[ScriptS])) 2^(-1+\[ScriptS]-I \[CurlyEpsilon]) Gamma[\[Nu]MST+1-\[ScriptS]+I \[CurlyEpsilon]]/Gamma[\[Nu]MST+1+\[ScriptS]-I \[CurlyEpsilon]] \!\( +\*UnderoverscriptBox[\(\[Sum]\), \(n = nMin\), \(nMax\)]\(aMST[n]\)\)//PNScalingsInternal; +aux/.MSTCoefficientsInternal[\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]+3]//SeriesTake[#,order\[Eta]]&//IgnoreExpansionParameter +] + + +AAmplitude["-"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=Module[{aux,\[CurlyEpsilon],\[Kappa],\[CurlyEpsilon]p,\[Tau],nMax,nMin}, +\[CurlyEpsilon]=2 \[Omega]; +\[Kappa]=Sqrt[1-a^2]; +\[CurlyEpsilon]p=(\[CurlyEpsilon]+\[Tau])/2; +\[Tau]=(-a \[ScriptM]+\[CurlyEpsilon])/\[Kappa]; +nMax=order\[Eta]/3//Ceiling; +nMin=-(order\[Eta]/3+2)//Floor; +aux=2^(-1-\[ScriptS]+I \[CurlyEpsilon]) E^(-(\[Pi]/2)I(\[Nu]MST+1+\[ScriptS])) E^(-(\[Pi]/2)\[CurlyEpsilon]) \!\( +\*UnderoverscriptBox[\(\[Sum]\), \(n = nMin\), \(nMax\)]\( +\*SuperscriptBox[\((\(-1\))\), \(n\)] +\*FractionBox[\(Pochhammer[\[Nu]MST + 1 + \[ScriptS] - I\ \[CurlyEpsilon], n]\), \(Pochhammer[\[Nu]MST + 1 - \[ScriptS] + I\ \[CurlyEpsilon], n]\)] aMST[n]\)\)//PNScalingsInternal; +aux/.MSTCoefficientsInternal[\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]+3]//SeriesTake[#,order\[Eta]]&//IgnoreExpansionParameter +] + + +(* ::Subsubsection::Closed:: *) +(*B Amplitudes*) + + +(* ::Text:: *) +(*These are the amplitudes Subscript[B, trans] and Subscript[B, inc] from Sasaki Tagoshi Eq.(167-169) TODO: Add Subscript[B, ref]*) + + +Options[BAmplitude]={"Normalization"->"SFPN"} + + +BAmplitude["Inc",OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=Module[{aux,\[CurlyEpsilon],\[Kappa],\[CurlyEpsilon]p,\[Tau],\[ScriptCapitalK]1,\[ScriptCapitalK]2,A}, +\[CurlyEpsilon]=2 \[Omega]; +\[Kappa]=Sqrt[1-a^2]; +\[CurlyEpsilon]p=(\[CurlyEpsilon]+\[Tau])/2; +\[Tau]=(-a \[ScriptM]+\[CurlyEpsilon])/\[Kappa]; +aux=\[Omega]^-1 (\[ScriptCapitalK]1 -I E^(-I \[Pi] \[Nu]MST) Sin[\[Pi](\[Nu]MST-\[ScriptS]+I \[CurlyEpsilon])]/Sin[\[Pi](\[Nu]MST+\[ScriptS]-I \[CurlyEpsilon])] \[ScriptCapitalK]2) A E^(-I(\[CurlyEpsilon] Log[\[CurlyEpsilon]]-(1-\[Kappa])/2 \[CurlyEpsilon]))//PNScalingsInternal; +aux=aux/.MSTCoefficientsInternal[\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]+3]//IgnoreExpansionParameter; +\[ScriptCapitalK]1=Switch[OptionValue["Normalization"], + "SFPN",1, + "SasakiTagoshi",\[ScriptCapitalK]Amplitude["\[Nu]"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]//ExpandPolyGamma//ExpandGamma//SeriesCollect[#,PolyGamma[__,__]]& +]; +\[ScriptCapitalK]2=Switch[OptionValue["Normalization"], + "SFPN",\[ScriptCapitalK]Amplitude["Ratio"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]//ExpandPolyGamma//ExpandGamma//SeriesCollect[#,PolyGamma[__,__]]&, + "SasakiTagoshi",\[ScriptCapitalK]Amplitude["-\[Nu]-1"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]//ExpandPolyGamma//ExpandGamma//SeriesCollect[#,PolyGamma[__,__]]& +]; +A=AAmplitude["+"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]//SeriesCollect[#,Log[__]]&//IgnoreExpansionParameter; +aux//SeriesTake[#,order\[Eta]]&//IgnoreExpansionParameter +] + + +BAmplitude["Inc","Normalization"->"Toolkit"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=BAmplitude["Inc"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]/BAmplitude["Trans"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]; + + +BAmplitude["Trans","Normalization"->"Toolkit"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=1 (1+O[\[Eta]] \[Eta]^(order\[Eta]-1)); + + +BAmplitude["Trans",OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=Module[{aux,\[CurlyEpsilon],\[Kappa],\[CurlyEpsilon]p,\[Tau],\[ScriptCapitalK],nMin,nMax}, +\[CurlyEpsilon]=2 \[Omega]; +\[Kappa]=Sqrt[1-a^2]; +\[CurlyEpsilon]p=(\[CurlyEpsilon]+\[Tau])/2; +\[Tau]=(-a \[ScriptM]+\[CurlyEpsilon])/\[Kappa]; +nMax=order\[Eta]/3//Ceiling; +nMin=-(order\[Eta]/3+2)//Floor; +\[ScriptCapitalK]=Switch[OptionValue["Normalization"], + "SFPN",\[ScriptCapitalK]Amplitude["\[Nu]"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]//ExpandPolyGamma//ExpandGamma//SeriesCollect[#,PolyGamma[__,__]]&, + "SasakiTagoshi",1 +]; +aux=((\[CurlyEpsilon] \[Kappa])/\[Omega])^(2\[ScriptS]) E^(I \[Kappa] \[CurlyEpsilon]p(1+(2 Log[\[Kappa]])/(1+\[Kappa]))) \!\( +\*UnderoverscriptBox[\(\[Sum]\), \(n = nMin\), \(nMax\)]\(aMST[n]\)\)//PNScalingsInternal; +aux=aux/.MSTCoefficientsInternal[\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]+3]//SeriesTake[#,order\[Eta]]&//IgnoreExpansionParameter; +aux=aux \[ScriptCapitalK]^-1; +aux//IgnoreExpansionParameter +] + + +(* ::Subsubsection::Closed:: *) +(*C Amplitude*) + + +(* ::Text:: *) +(*These is the amplitudes Subscript[C, trans] from Sasaki Tagoshi Eq.(170) TODO: Add Subscript[C, ref] and Subscript[C, inc]*) + + +Options[CAmplitude]={"Normalization"->"SFPN"} + + +CAmplitude["Trans","Normalization"->"Toolkit"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=1 (1+O[\[Eta]] \[Eta]^(order\[Eta]-1)); + + +CAmplitude["Trans",OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=Module[{aux,\[CurlyEpsilon],\[Kappa],\[CurlyEpsilon]p,\[Tau],\[ScriptCapitalK],A}, +\[CurlyEpsilon]=2 \[Omega]; +\[Kappa]=Sqrt[1-a^2]; +\[CurlyEpsilon]p=(\[CurlyEpsilon]+\[Tau])/2; +\[Tau]=(-a \[ScriptM]+\[CurlyEpsilon])/\[Kappa]; +aux=\[Omega]^(-1-2\[ScriptS]) A E^(I (\[CurlyEpsilon] Log[\[CurlyEpsilon]]-(1-\[Kappa])/2 \[CurlyEpsilon]))//PNScalingsInternal; +aux=aux/.MSTCoefficientsInternal[\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]+3]; +A=AAmplitude["-"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]; +aux//SeriesTake[#,order\[Eta]]&//IgnoreExpansionParameter +] + + +(* ::Subsubsection::Closed:: *) +(*\[ScriptCapitalK] Amplitude*) + + +(* ::Text:: *) +(*These is the amplitudes \[ScriptCapitalK]^\[Nu]and \[ScriptCapitalK]^(-\[Nu]-1) from Sasaki Tagoshi Eq.(165)*) + + +Options[\[ScriptCapitalK]Amplitude]={"PochhammerForm"->True} + + +\[ScriptCapitalK]Amplitude["\[Nu]",OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_,OptionsPattern[]]:=Module[{\[ScriptR]=0,ret,\[CapitalGamma]=Gamma,PH=Pochhammer,coeff,sumUp,sumUpPHCoeff,sumUpPH,sumDown,\[CurlyEpsilon]p,\[Tau],\[CurlyEpsilon],\[Kappa]=Sqrt[1-a^2],nMax,nMin,jump,jumpCount,repls,repls\[Nu]}, +\[CurlyEpsilon]=2 \[Omega]; +\[CurlyEpsilon]p=(\[CurlyEpsilon]+\[Tau])/2; +\[Tau]=(-a \[ScriptM]+\[CurlyEpsilon])/\[Kappa]; +jump[1]=2+2\[ScriptL]+\[ScriptR]<=Abs[n]; +jump[2]=1+\[ScriptS]+\[ScriptL]<=n; +jump[3]=1-\[ScriptS]+\[ScriptL]<=n; +jumpCount=1+(jump[#]&/@Range[3]//Boole//Total); +jumpCount=1; +repls=MSTCoefficientsInternal[\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]+3]; +repls\[Nu]=<|\[Nu]MST->(repls[\[Nu]MST]//SeriesTake[#,order\[Eta]-Floor[Abs[n]/3]+3jumpCount]&)|>; +coeff=E^(I \[CurlyEpsilon] \[Kappa] ) (\[CurlyEpsilon] \[Kappa])^(\[ScriptS]-\[Nu]MST-\[ScriptR]) 2^-\[Nu]MST I^-\[ScriptR] (\[CapitalGamma][1-\[ScriptS]-2I \[CurlyEpsilon]p]\[CapitalGamma][\[ScriptR]+2 \[Nu]MST+2])/\[CapitalGamma][\[ScriptR]+\[Nu]MST+1-\[ScriptS]+I \[CurlyEpsilon]] If[\[ScriptR]==0&&OptionValue["PochhammerForm"],1,1/(\[CapitalGamma][\[ScriptR]+\[Nu]MST+1+\[ScriptS]+I \[CurlyEpsilon]] \[CapitalGamma][\[ScriptR]+\[Nu]MST+1+I \[Tau]])]//IgnoreExpansionParameter; +nMax=Ceiling[order\[Eta]/3]; +nMin=Ceiling[-(order\[Eta]/3)-2]; +(*sumUp=\!\( +\*UnderoverscriptBox[\(\[Sum]\), \(n = \[ScriptR]\), \(nMax\)]\( +\*FractionBox[ +SuperscriptBox[\((\(-1\))\), \(n\)], \(\((n - \[ScriptR])\)!\)]\[CapitalGamma][n + \[ScriptR] + 2 \[Nu]MST + 1] +\*FractionBox[\(\[CapitalGamma][n + \[Nu]MST + 1 + \[ScriptS] + I\ \[CurlyEpsilon]]\[CapitalGamma][n + \[Nu]MST + 1 + I\ \[Tau]]\), \(\[CapitalGamma][n + \[Nu]MST + 1 - \[ScriptS] - I\ \[CurlyEpsilon]]\[CapitalGamma][n + \[Nu]MST + 1 - I\ \[Tau]]\)]aMST[n]\)\)/.repls;*) +sumUpPH=\!\( +\*UnderoverscriptBox[\(\[Sum]\), \(n = \[ScriptR]\), \(nMax\)]\(\((\( +\*FractionBox[\( +\*SuperscriptBox[\((\(-1\))\), \(n\)]\ PH[1 + \[ScriptS] + I\ \[CurlyEpsilon] + \[Nu]MST, n]\ PH[1 + \[ScriptR] + 2\ \[Nu]MST, n]\ PH[1 + \[Nu]MST + I\ \[Tau], n]\), \(\(\((n - \[ScriptR])\)!\)\ PH[1 - \[ScriptS] - I\ \[CurlyEpsilon] + \[Nu]MST, n]\ PH[1 + \[Nu]MST - I\ \[Tau], n]\)] /. replsPN\) /. repls\[Nu])\) \((aMST[n]\ /. repls)\)\)\)//IgnoreExpansionParameter; +sumUpPHCoeff= Gamma[1+\[ScriptR]+2 \[Nu]MST] /(Gamma[1-\[ScriptS]-I \[CurlyEpsilon]+\[Nu]MST] Gamma[1+\[Nu]MST-I \[Tau]]) If[\[ScriptR]==0,1,Gamma[1+\[ScriptS]+I \[CurlyEpsilon]+\[Nu]MST] Gamma[1+\[Nu]MST+I \[Tau]]]//IgnoreExpansionParameter; +coeff=(coeff If[OptionValue["PochhammerForm"],sumUpPHCoeff,1])/.replsPN/.repls//IgnoreExpansionParameter; + +sumDown=\!\( +\*UnderoverscriptBox[\(\[Sum]\), \(n = nMin\), \(\[ScriptR]\)]\(\((\( +\*FractionBox[ +SuperscriptBox[\((\(-1\))\), \(n\)], \(\(\((\[ScriptR] - n)\)!\) PH[\[ScriptR] + 2\ \[Nu]MST + 2, n]\)] +\*FractionBox[\(PH[\[Nu]MST + 1 + \[ScriptS] - I\ \[CurlyEpsilon], n]\), \(PH[\[Nu]MST + 1 - \[ScriptS] + I\ \[CurlyEpsilon], n]\)] /. replsPN\) /. repls\[Nu])\) \((aMST[n] /. repls)\)\)\)//IgnoreExpansionParameter; +ret=coeff/sumDown If[OptionValue["PochhammerForm"],sumUpPH,sumUp]//IgnoreExpansionParameter; +ret//SeriesTake[#,order\[Eta]]& +] + + +\[ScriptCapitalK]Amplitude["C-\[Nu]-1",OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_,OptionsPattern[]]:=Module[{\[ScriptR]=0,ret,\[CapitalGamma]=Gamma,PH=Pochhammer,coeff,sumUp,sumUpPHCoeff,sumUpPH,sumDown,\[CurlyEpsilon]p,\[Tau],\[CurlyEpsilon],\[Kappa]=Sqrt[1-a^2],nMax,nMin,jump,jumpCount,repls,repls\[Nu]}, +\[CurlyEpsilon]=2 \[Omega]; +\[CurlyEpsilon]p=(\[CurlyEpsilon]+\[Tau])/2; +\[Tau]=(-a \[ScriptM]+\[CurlyEpsilon])/\[Kappa]; +jump[1]=2+2\[ScriptL]+\[ScriptR]<=Abs[n]; +jump[2]=1+\[ScriptS]+\[ScriptL]<=n; +jump[3]=1-\[ScriptS]+\[ScriptL]<=n; +jumpCount=1+(jump[#]&/@Range[3]//Boole//Total); +repls=MSTCoefficientsInternal[\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]+3]; +repls\[Nu]=<|\[Nu]MST->(repls[\[Nu]MST]//SeriesTake[#,order\[Eta]-Floor[n/3]+3jumpCount]&)|>; +coeff=I^-\[ScriptR] 2^(1+\[Nu]MST) E^(I \[CurlyEpsilon] \[Kappa]) (\[CurlyEpsilon] \[Kappa])^(1-\[ScriptR]+\[ScriptS]+\[Nu]MST) ( \[CapitalGamma][1-\[ScriptS]-2 I \[CurlyEpsilon]p] \[CapitalGamma][\[ScriptR]-2 \[Nu]MST])/\[CapitalGamma][\[ScriptR]-\[ScriptS]+I \[CurlyEpsilon]-\[Nu]MST] If[\[ScriptR]==0&&OptionValue["PochhammerForm"],1,1/(\[CapitalGamma][\[ScriptR]-\[Nu]MST+I \[Tau]]\[CapitalGamma][\[ScriptR]+\[ScriptS]+I \[CurlyEpsilon]-\[Nu]MST] )]//IgnoreExpansionParameter; +nMax=Ceiling[order\[Eta]/3]; +nMin=Ceiling[-(order\[Eta]/3)-2]; +sumUp=\!\( +\*UnderoverscriptBox[\(\[Sum]\), \(n = nMin\), \(-\[ScriptR]\)]\( +\*FractionBox[\( +\*SuperscriptBox[\((\(-1\))\), \(-n\)]\ \ \[CapitalGamma][\(-1\) - n + \[ScriptR] - 2\ \[Nu]MST]\ \[CapitalGamma][\(-n\) + \[ScriptS] + I\ \[CurlyEpsilon] - \[Nu]MST]\ \[CapitalGamma][\(-n\) - \[Nu]MST + I\ \[Tau]]\), \(\(\((\(-n\) - \[ScriptR])\)!\)\ \[CapitalGamma][\(-n\) - \[ScriptS] - I\ \[CurlyEpsilon] - \[Nu]MST]\ \[CapitalGamma][\(-n\) - \[Nu]MST - I\ \[Tau]]\)] aMST[n]\)\)/.repls//IgnoreExpansionParameter; +sumUpPH=\!\( +\*UnderoverscriptBox[\(\[Sum]\), \(n = nMin\), \(-\[ScriptR]\)]\(\((\( +\*FractionBox[\( +\*SuperscriptBox[\((\(-1\))\), \(-n\)]\ PH[\(-1\) + \[ScriptR] - 2\ \[Nu]MST, \(-n\)]\ PH[\[ScriptS] + I\ \[CurlyEpsilon] - \[Nu]MST, \(-n\)]\ PH[\(-\[Nu]MST\) + I\ \[Tau], \(-n\)]\), \(\(\((\(-n\) - \[ScriptR])\)!\)\ PH[\(-\[ScriptS]\) - I\ \[CurlyEpsilon] - \[Nu]MST, \(-n\)]\ PH[\(-\[Nu]MST\) - I\ \[Tau], \(-n\)]\)] /. replsPN\) /. repls\[Nu])\) \((aMST[n]\ /. repls)\)\)\)//IgnoreExpansionParameter; +sumUpPHCoeff=Gamma[-1+\[ScriptR]-2 \[Nu]MST] /(Gamma[-\[ScriptS]-I \[CurlyEpsilon]-\[Nu]MST] Gamma[-\[Nu]MST-I \[Tau]]) If[\[ScriptR]==0,1,Gamma[\[ScriptS]+I \[CurlyEpsilon]-\[Nu]MST] Gamma[-\[Nu]MST+I \[Tau]]]//IgnoreExpansionParameter; +coeff=(coeff If[OptionValue["PochhammerForm"],sumUpPHCoeff,1])/.replsPN/.repls; + +sumDown=\!\( +\*UnderoverscriptBox[\(\[Sum]\), \(n = \(-\[ScriptR]\)\), \(nMax\)]\(\((\( +\*FractionBox[\( +\*SuperscriptBox[\((\(-1\))\), \(-n\)]\ PH[\[ScriptS] - I\ \[CurlyEpsilon] - \[Nu]MST, \(-n\)]\), \(\(\((n + \[ScriptR])\)!\)\ PH[\[ScriptR] - 2\ \[Nu]MST, \(-n\)]\ PH[\(-\[ScriptS]\) + I\ \[CurlyEpsilon] - \[Nu]MST, \(-n\)]\)] /. replsPN\) /. repls\[Nu])\) \((aMST[n] /. repls)\)\)\)//IgnoreExpansionParameter; +ret=coeff/sumDown If[OptionValue["PochhammerForm"],sumUpPHCoeff sumUpPH,sumUp]//IgnoreExpansionParameter; +ret//SeriesTake[#,order\[Eta]]& +] + + +\[ScriptCapitalK]Amplitude["Ratio"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_,OptionsPattern[]]:=Module[{\[ScriptR]=0,ret,\[CapitalGamma]=Gamma,PH=Pochhammer,coeff,sumUp,sumUp2,sumUpPHCoeff,sumUpPH,sumDown,sumDown2,\[CurlyEpsilon]p,\[Tau],\[CurlyEpsilon],\[Kappa]=Sqrt[1-a^2],nMax,nMin,jump,jumpCount,repls,repls\[Nu]}, +\[CurlyEpsilon]=2 \[Omega]; +\[CurlyEpsilon]p=(\[CurlyEpsilon]+\[Tau])/2; +\[Tau]=(-a \[ScriptM]+\[CurlyEpsilon])/\[Kappa]; +jumpCount=1; +repls=MSTCoefficientsInternal[\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]+6]; +repls\[Nu]=<|\[Nu]MST->(repls[\[Nu]MST]//SeriesTake[#,order\[Eta]-Floor[n/3]+6jumpCount]&)|>; +coeff=2^(1+2 \[Nu]MST) \[CurlyEpsilon] \[Kappa] (\[CurlyEpsilon] \[Kappa])^(2 \[Nu]MST) (\[CapitalGamma][\[ScriptR]-2 \[Nu]MST] \[CapitalGamma][1+\[ScriptR]-\[ScriptS]+I \[CurlyEpsilon]+\[Nu]MST])/(\[CapitalGamma][\[ScriptR]-\[ScriptS]+I \[CurlyEpsilon]-\[Nu]MST] \[CapitalGamma][2+\[ScriptR]+2 \[Nu]MST]) If[\[ScriptR]==0,1,(\[CapitalGamma][\[ScriptR]+\[Nu]MST+1+\[ScriptS]+I \[CurlyEpsilon]] \[CapitalGamma][\[ScriptR]+\[Nu]MST+1+I \[Tau]])/(\[CapitalGamma][\[ScriptR]-\[Nu]MST+I \[Tau]]\[CapitalGamma][\[ScriptR]+\[ScriptS]+I \[CurlyEpsilon]-\[Nu]MST] )]//IgnoreExpansionParameter; +nMax=Ceiling[order\[Eta]/3]; +nMin=Ceiling[-(order\[Eta]/3)-2]; +sumUpPH=\!\( +\*UnderoverscriptBox[\(\[Sum]\), \(n = nMin\), \(-\[ScriptR]\)]\(\((\( +\*FractionBox[\( +\*SuperscriptBox[\((\(-1\))\), \(-n\)]\ PH[\(-1\) + \[ScriptR] - 2\ \[Nu]MST, \(-n\)]\ PH[\[ScriptS] + I\ \[CurlyEpsilon] - \[Nu]MST, \(-n\)]\ PH[\(-\[Nu]MST\) + I\ \[Tau], \(-n\)]\), \(\(\((\(-n\) - \[ScriptR])\)!\)\ PH[\(-\[ScriptS]\) - I\ \[CurlyEpsilon] - \[Nu]MST, \(-n\)]\ PH[\(-\[Nu]MST\) - I\ \[Tau], \(-n\)]\)] /. replsPN\) /. repls\[Nu])\) \((aMST[n]\ /. repls)\)\)\)//IgnoreExpansionParameter; +sumUp2=\!\( +\*UnderoverscriptBox[\(\[Sum]\), \(n = nMin\), \(\[ScriptR]\)]\(\((\( +\*FractionBox[ +SuperscriptBox[\((\(-1\))\), \(n\)], \(\(\((\[ScriptR] - n)\)!\) PH[\[ScriptR] + 2\ \[Nu]MST + 2, n]\)] +\*FractionBox[\(PH[\[Nu]MST + 1 + \[ScriptS] - I\ \[CurlyEpsilon], n]\), \(PH[\[Nu]MST + 1 - \[ScriptS] + I\ \[CurlyEpsilon], n]\)] /. replsPN\) /. repls\[Nu])\) \((aMST[n] /. repls)\)\)\)//IgnoreExpansionParameter; +sumUpPHCoeff=(Gamma[-1+\[ScriptR]-2 \[Nu]MST] Gamma[1-\[ScriptS]-I \[CurlyEpsilon]+\[Nu]MST] Gamma[1+\[Nu]MST-I \[Tau]])/(Gamma[-\[ScriptS]-I \[CurlyEpsilon]-\[Nu]MST] Gamma[1+\[ScriptR]+2 \[Nu]MST] Gamma[-\[Nu]MST-I \[Tau]]) If[\[ScriptR]==0,1,(Gamma[\[ScriptS]+I \[CurlyEpsilon]-\[Nu]MST] Gamma[-\[Nu]MST+I \[Tau]])/(Gamma[1+\[ScriptS]+I \[CurlyEpsilon]+\[Nu]MST] Gamma[1+\[Nu]MST+I \[Tau]])]//IgnoreExpansionParameter; +coeff=(coeff sumUpPHCoeff)/.replsPN/.repls//IgnoreExpansionParameter; + +sumDown=\!\( +\*UnderoverscriptBox[\(\[Sum]\), \(n = \(-\[ScriptR]\)\), \(nMax\)]\(\((\( +\*FractionBox[\( +\*SuperscriptBox[\((\(-1\))\), \(-n\)]\ PH[\[ScriptS] - I\ \[CurlyEpsilon] - \[Nu]MST, \(-n\)]\), \(\(\((n + \[ScriptR])\)!\)\ PH[\[ScriptR] - 2\ \[Nu]MST, \(-n\)]\ PH[\(-\[ScriptS]\) + I\ \[CurlyEpsilon] - \[Nu]MST, \(-n\)]\)] /. replsPN\) /. repls\[Nu])\) \((aMST[n] /. repls)\)\)\)//IgnoreExpansionParameter; +sumDown2=\!\( +\*UnderoverscriptBox[\(\[Sum]\), \(n = \[ScriptR]\), \(nMax\)]\(\((\( +\*FractionBox[\( +\*SuperscriptBox[\((\(-1\))\), \(n\)]\ PH[1 + \[ScriptS] + I\ \[CurlyEpsilon] + \[Nu]MST, n]\ PH[1 + \[ScriptR] + 2\ \[Nu]MST, n]\ PH[1 + \[Nu]MST + I\ \[Tau], n]\), \(\(\((n - \[ScriptR])\)!\)\ PH[1 - \[ScriptS] - I\ \[CurlyEpsilon] + \[Nu]MST, n]\ PH[1 + \[Nu]MST - I\ \[Tau], n]\)] /. replsPN\) /. repls\[Nu])\) \((aMST[n]\ /. repls)\)\)\)//IgnoreExpansionParameter; +ret=coeff sumUp2/sumDown sumUpPH/sumDown2//IgnoreExpansionParameter; +ret//SeriesTake[#,order\[Eta]]& +] + + +(* ::Subsubsection::Closed:: *) +(*Interface*) + + +Options[TeukolskyAmplitudePN]={"Normalization"->"SFPN"} + + +TeukolskyAmplitudePN["A+",opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]Var_,{\[Eta]Var_,order\[Eta]_}] :=AAmplitude["+"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]/.{\[Omega]->\[Omega]Var,\[Eta]->\[Eta]Var}; +TeukolskyAmplitudePN["A-",opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]Var_,{\[Eta]Var_,order\[Eta]_}] :=AAmplitude["-"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]/.{\[Omega]->\[Omega]Var,\[Eta]->\[Eta]Var}; + + +TeukolskyAmplitudePN["Binc",opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]Var_,{\[Eta]Var_,order\[Eta]_}] :=BAmplitude["Inc",opt][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]/.{\[Omega]->\[Omega]Var,\[Eta]->\[Eta]Var}; +TeukolskyAmplitudePN["Btrans",opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]Var_,{\[Eta]Var_,order\[Eta]_}] :=BAmplitude["Trans",opt][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]/.{\[Omega]->\[Omega]Var,\[Eta]->\[Eta]Var}; + + +TeukolskyAmplitudePN["Ctrans",opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]Var_,{\[Eta]Var_,order\[Eta]_}] :=CAmplitude["Trans",opt][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]/.{\[Omega]->\[Omega]Var,\[Eta]->\[Eta]Var}; + + +TeukolskyAmplitudePN["\[ScriptCapitalK]\[Nu]",opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]Var_,{\[Eta]Var_,order\[Eta]_}] :=\[ScriptCapitalK]Amplitude["\[Nu]"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]/.{\[Omega]->\[Omega]Var,\[Eta]->\[Eta]Var}; +TeukolskyAmplitudePN["\[ScriptCapitalK]-\[Nu]-1",opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]Var_,{\[Eta]Var_,order\[Eta]_}] :=\[ScriptCapitalK]Amplitude["C-\[Nu]-1"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]/.{\[Omega]->\[Omega]Var,\[Eta]->\[Eta]Var}; +TeukolskyAmplitudePN["\[ScriptCapitalK]",opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]Var_,{\[Eta]Var_,order\[Eta]_}] :=\[ScriptCapitalK]Amplitude["Ratio"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]/.{\[Omega]->\[Omega]Var,\[Eta]->\[Eta]Var}; + + +(* ::Subsection::Closed:: *) +(*Wronskian*) + + +(* ::Subsubsection::Closed:: *) +(*Invariant Wronskian*) + + +InvariantWronskian[\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]Var_,{\[Eta]Var_,order\[Eta]_}]:=Module[{aux,Rup,Rin,B,C,ret}, +C=CAmplitude["Trans"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]//ExpandPolyGamma//ExpandGamma; +B=BAmplitude["Inc"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]//ExpandPolyGamma//ExpandGamma; +aux=PNScalingsInternal[2 I \[Omega]]B C//ExpandLog//SeriesCollect[#,Log[__]]&; +ret=aux/.{\[Omega]->\[Omega]Var,\[Eta]->\[Eta]Var}; +ret +] + + +(* ::Subsection:: *) +(*Subscript[R, In]*) + + +(* ::Text:: *) +(*To construct Subscript[R, In] we follow Eq.166 in Sasaki Tagoshi ( https://doi.org/10.12942/lrr-2003-6 ). Subscript[R, in]=Subscript[R, C]^\[Nu]+Subscript[\[ScriptCapitalK], -\[Nu]-1]/Subscript[\[ScriptCapitalK], \[Nu]] Subscript[R, C]^(-\[Nu]-1). Notice that we have divided out a factor of Subscript[\[ScriptCapitalK], \[Nu]], which is allowed since Subscript[\[ScriptCapitalK], \[Nu]] does not depend on r. This is helpful as the second term now dies off drastically with the increase of \[ScriptL]. \[ScriptCapitalK]^\[Nu] ~\[Omega]^-\[ScriptL]. \[ScriptCapitalK]^(-\[Nu]-1) ~\[Omega]^\[ScriptL] (Schwarzschild?)*) + + +(* ::Subsubsection::Closed:: *) +(*Definitions (depreciated)*) + + +(* ::Text:: *) +(*We start by typing up the definitions given in Sasaki Tagoshi Eq.162 - Eq.165. Notice that we use the symmetry Subscript[a, n]^(-\[Nu]-1)=Subscript[a, -n]^\[Nu] for the MST coefficients. One key challenge is figuring out where to truncate the infinite sums for a given scaling in \[Eta] most efficiently. This will be solved later in tableOverNJ[]. *) + + +(*c["In"][z_]:=E^(-I z) 2^\[Nu] z^\[Nu] (z-\[Epsilon] \[Kappa])^-s (1-(\[Epsilon] \[Kappa])/z)^(-I \[Epsilon]p); +\[Epsilon]p=(\[Epsilon]+\[Tau])/2; +\[ConstantC]D[\[Nu],n_,j_]:=((-1)^n (2 I)^(n+j) Gamma[n+\[Nu]+1-s+I \[Epsilon]] Pochhammer[\[Nu]+1+s-I \[Epsilon],n] Pochhammer[n+\[Nu]+1-s+I \[Epsilon],j] a[n])/(Gamma[2 n+2 \[Nu]+2] Pochhammer[\[Nu]+1-s+I \[Epsilon],n] Pochhammer[2 n+2 \[Nu]+2,j] j!); +\[ConstantC]D[-\[Nu]-1,n_,j_]:=((-1)^n (2 I)^(n+j) Gamma[n+\[Nu]+1-s+I \[Epsilon]] Pochhammer[\[Nu]+1+s-I \[Epsilon],n] Pochhammer[n+\[Nu]+1-s+I \[Epsilon],j] a[-n])/(Gamma[2 n+2 \[Nu]+2] Pochhammer[\[Nu]+1-s+I \[Epsilon],n] Pochhammer[2 n+2 \[Nu]+2,j] j!)/. \[Nu]->-\[Nu]-1; +(*\[ScriptCapitalK][\[Nu],order\[Eta]_]:=((E^(I \[Epsilon] \[Kappa]) (2 \[Epsilon] \[Kappa])^(s-\[Nu]-rInt) 2^-s I^rInt Gamma[1-s-2 I \[Epsilon]p] Gamma[rInt+2 \[Nu]+2]) \!\( +\*UnderoverscriptBox[\(\[Sum]\), \(n = rInt\), \(Ceiling[ +\*FractionBox[\(order\[Eta]\), \(3\)]]\)] +\*FractionBox[\( +\*SuperscriptBox[\((\(-1\))\), \(n\)]\ Gamma[n + rInt + 2\ \[Nu] + 1]\ Gamma[n + \[Nu] + 1 + s + I\ \[Epsilon]]\ Gamma[n + \[Nu] + 1 + I\ \[Tau]]\ a[n]\), \(\(\((n - rInt)\)!\)\ Gamma[n + \[Nu] + 1 - s - I\ \[Epsilon]]\ Gamma[n + \[Nu] + 1 - I\ \[Tau]]\)]\))/((Gamma[rInt+\[Nu]+1-s+I \[Epsilon]] Gamma[rInt+\[Nu]+1+I \[Tau]] Gamma[rInt+\[Nu]+1+s+I \[Epsilon]]) \!\( +\*UnderoverscriptBox[\(\[Sum]\), \(n = Floor[\(- +\*FractionBox[\(1\), \(3\)]\)\ \((order\[Eta] + 6)\)]\), \(rInt\)] +\*FractionBox[\( +\*SuperscriptBox[\((\(-1\))\), \(n\)]\ Pochhammer[\[Nu] + 1 + s - I\ \[Epsilon], n]\ a[n]\), \(\((\(\((rInt - n)\)!\)\ Pochhammer[rInt + 2\ \[Nu] + 2, n])\)\ Pochhammer[\[Nu] + 1 - s + I\ \[Epsilon], n]\)]\))/. rInt->Ceiling[order\[Eta]/3]; +\[ScriptCapitalK][-\[Nu]-1,order\[Eta]_]:=((E^(I \[Epsilon] \[Kappa]) (2 \[Epsilon] \[Kappa])^(s-\[Nu]-rInt) 2^-s I^rInt Gamma[1-s-2 I \[Epsilon]p] Gamma[rInt+2 \[Nu]+2]) \!\( +\*UnderoverscriptBox[\(\[Sum]\), \(n = rInt\), \(Ceiling[ +\*FractionBox[\(order\[Eta] + 6\), \(3\)]]\)] +\*FractionBox[\( +\*SuperscriptBox[\((\(-1\))\), \(n\)]\ Gamma[n + rInt + 2\ \[Nu] + 1]\ Gamma[n + \[Nu] + 1 + s + I\ \[Epsilon]]\ Gamma[n + \[Nu] + 1 + I\ \[Tau]]\ a[\(-n\)]\), \(\(\((n - rInt)\)!\)\ Gamma[n + \[Nu] + 1 - s - I\ \[Epsilon]]\ Gamma[n + \[Nu] + 1 - I\ \[Tau]]\)]\))/((Gamma[rInt+\[Nu]+1-s+I \[Epsilon]] Gamma[rInt+\[Nu]+1+I \[Tau]] Gamma[rInt+\[Nu]+1+s+I \[Epsilon]]) \!\( +\*UnderoverscriptBox[\(\[Sum]\), \(n = Floor[\(- +\*FractionBox[\(order\[Eta]\), \(3\)]\)]\), \(rInt\)] +\*FractionBox[\( +\*SuperscriptBox[\((\(-1\))\), \(n\)]\ Pochhammer[\[Nu] + 1 + s - I\ \[Epsilon], n]\ a[\(-n\)]\), \(\((\(\((rInt - n)\)!\)\ Pochhammer[rInt + 2\ \[Nu] + 2, n])\)\ Pochhammer[\[Nu] + 1 - s + I\ \[Epsilon], n]\)]\))/. {rInt->0,\[Nu]->-\[Nu]-1};*)*) + + +(* ::Subsubsection::Closed:: *) +(*Alternative Definitions*) + + +z[a_,r_]:=(-1+Sqrt[1-a^2]+r) \[Omega]//PNScalingsInternal; + + +c["In"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,z_]:=Module[{aux,\[CurlyEpsilon]p,\[Tau],\[CurlyEpsilon],\[Kappa]}, +\[CurlyEpsilon]=2 \[Omega]; +\[CurlyEpsilon]p=(\[CurlyEpsilon]+\[Tau])/2; +\[Kappa]=Sqrt[1-a^2]; +\[Tau]=(-a \[ScriptM]+\[CurlyEpsilon])/\[Kappa]; +aux=E^(-I z) 2^\[Nu]MST z^\[Nu]MST (z-\[CurlyEpsilon] \[Kappa])^-\[ScriptS] (1-(\[CurlyEpsilon] \[Kappa])/z)^(-I \[CurlyEpsilon]p); +aux//PNScalingsInternal] + + +\[ConstantC]D["\[Nu]"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,n_,j_]:=Module[{aux,\[CapitalGamma]=Gamma,PH=Pochhammer,\[CurlyEpsilon]p,\[Tau],\[CurlyEpsilon],\[Kappa]}, +\[CurlyEpsilon]=2 \[Omega]; +\[CurlyEpsilon]p=(\[CurlyEpsilon]+\[Tau])/2; +\[Kappa]=Sqrt[1-a^2]; +\[Tau]=(-a \[ScriptM]+\[CurlyEpsilon])/\[Kappa]; +aux=((-1)^n (2 I)^(n+j) \[CapitalGamma][n+\[Nu]MST+1-\[ScriptS]+I \[CurlyEpsilon]] PH[\[Nu]MST+1+\[ScriptS]-I \[CurlyEpsilon],n] PH[n+\[Nu]MST+1-\[ScriptS]+I \[CurlyEpsilon],j] aMST[n])/(\[CapitalGamma][2 n+2 \[Nu]MST+2] PH[\[Nu]MST+1-\[ScriptS]+I \[CurlyEpsilon],n] PH[2 n+2 \[Nu]MST+2,j] j!); +aux//PNScalingsInternal +]; + + +\[ConstantC]D["C-\[Nu]-1"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,n_,j_]:=Module[{aux,\[CapitalGamma]=Gamma,PH=Pochhammer,\[CurlyEpsilon]p,\[Tau],\[CurlyEpsilon],\[Kappa]}, +\[CurlyEpsilon]=2 \[Omega]; +\[CurlyEpsilon]p=(\[CurlyEpsilon]+\[Tau])/2; +\[Kappa]=Sqrt[1-a^2]; +\[Tau]=(-a \[ScriptM]+\[CurlyEpsilon])/\[Kappa]; +aux=((-1)^n (2 I)^(j+n) PH[\[ScriptS]-I \[CurlyEpsilon]-\[Nu]MST,n] PH[n-\[ScriptS]+I \[CurlyEpsilon]-\[Nu]MST,j] \[CapitalGamma][n-\[ScriptS]+I \[CurlyEpsilon]-\[Nu]MST])/(j! PH[2 n-2 \[Nu]MST,j] PH[-\[ScriptS]+I \[CurlyEpsilon]-\[Nu]MST,n] \[CapitalGamma][2 n-2 \[Nu]MST]) aMST[-n]; +aux//PNScalingsInternal +]; + + +(* ::Subsubsection::Closed:: *) +(*Constructing \!\(\*SubsuperscriptBox[\(R\), \(C\), \(\[Nu]\)]\)*) + + +(* ::Text:: *) +(*To construct Subscript[R, C]^\[Nu] we follow the first instance of Eq.162, i.e., Subscript[R, C]^\[Nu]=coeff \!\( *) +(*\*UnderoverscriptBox[\(\[Sum]\), \(n = \(-\[Infinity]\)\), \(\[Infinity]\)]\( *) +(*\*UnderoverscriptBox[\(\[Sum]\), \(j = 0\), \(\[Infinity]\)]\[ConstantC]D[\[Nu], n, j]\ *) +(*\*SuperscriptBox[\(z[r]\), \(n + j\)]\)\). To execute the sums we use tableOverNJ[] which constructs a table of all needed terms. You can find some plots illustrating the variables nMin, nMax and firstRegularj in the Hyperlink["Plots section",{NotebookObject["c6ff3b79-aebc-4585-b130-dd1a930b511b", "9e23625d-8944-48d8-a536-bca7efa5e050"], "\[Nu]Plots"}]. These are essential in telling you where to truncate the sums. *) + + +(* ::Input:: *) +(*(*tableOverNJ["C\[Nu]"][\[ScriptL]_,term_,repls_,order\[Eta]_]:=Block[{table,aux,finalj,firstRegularj,firstRegular\[Eta],leading\[Eta]OrderTerm,leading\[Eta]Order,goal\[Eta]Order,replsAux,replsLeading,auxOrder,nMin,nMax},*) +(*replsLeading=SeriesTake[#,7]&/@repls;*) +(*firstRegularj=(Abs[2+2\[ScriptL]+2 n]+1);*) +(*leading\[Eta]Order=(term/.j->#/.n->0/.replsLeading//SeriesMinOrder)&/@{0,firstRegularj}//Min;*) +(*goal\[Eta]Order=order\[Eta]+leading\[Eta]Order;*) +(*nMin=-(goal\[Eta]Order/2)//Ceiling;*) +(*nMax=goal\[Eta]Order/4//Floor;*) +(*table=Table[*) +(*firstRegular\[Eta]=(term/.j->firstRegularj/.replsLeading//SeriesMinOrder);*) +(*finalj=firstRegularj+(goal\[Eta]Order-firstRegular\[Eta]);*) +(*Table[*) +(*leading\[Eta]OrderTerm=term/.replsLeading//SeriesMinOrder;*) +(*If[leading\[Eta]OrderTerm<=goal\[Eta]Order,*) +(*auxOrder=goal\[Eta]Order-leading\[Eta]OrderTerm+If[n<0,6,0]//Max[#,7]&;*) +(*replsAux=SeriesTake[#,auxOrder]&/@repls;*) +(*aux=term/.replsAux;*) +(*,(*else*)*) +(*aux=O[\[Eta]]^(goal\[Eta]Order+1);*) +(*];*) +(*aux*) +(*,{j,0,finalj}]*) +(*,{n,nMin,nMax}]//Simplify;*) +(*table]*) +(**) +(*RPN["C\[Nu]"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]Var_,aKerr_,order\[Eta]_]:=Block[{s=\[ScriptS],l=\[ScriptL],m=\[ScriptM]Var,\[ScriptM]=\[ScriptM]Var,a=aKerr,\[ScriptA]=aKerr,aux,repls,replsAux,replsCoeff,replsLeading,coeff,term,status,table},*) +(*Monitor[status="getting repls";*) +(*repls=replsMST[\[ScriptS],\[ScriptL],\[ScriptM],Max[Ceiling[order\[Eta],3]+4,7]];*) +(*replsCoeff=SeriesTake[#,Max[order\[Eta],7]]&/@repls;*) +(*status="coefficient";*) +(*coeff=(c["In"][z[r]]//.replsKerr/.\[Epsilon]->2\[Omega]/.replsPN//Simplify)/.replsCoeff;*) +(*status="term";*) +(*term=\[ConstantC]D[\[Nu],n,j] z[r]^(n+j)//.replsKerr/.\[Epsilon]->2\[Omega]/.replsPN;*) +(*status="tableing";*) +(*table=tableOverNJ["C\[Nu]"][\[ScriptL],term,repls,order\[Eta]];*) +(*status="summing";*) +(*table=table//Flatten//Total;*) +(*status="assembling";*) +(*coeff table//SeriesTake[#,order\[Eta]]&*) +(*,{status,n,j}]]*)*) + + +RPN["C\[Nu]"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=Module[{aux,repls,replsAux,replsCoeff,replsLeading,coeff,term,status,table,ret,nMin}, +nMin=Ceiling[(order\[Eta]+7)/3]; +repls=MSTCoefficientsInternal[\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]+nMin+1]; +replsCoeff=SeriesTake[#,Max[order\[Eta],7]]&/@repls; +coeff=c["In"][\[ScriptS],\[ScriptL],\[ScriptM],a,z[a,r]]/.replsCoeff; +term=\[ConstantC]D["\[Nu]"][\[ScriptS],\[ScriptL],\[ScriptM],a,n,j] z[a,r]^(n+j); +table=tableOverNJ["C\[Nu]"][\[ScriptL],term,repls,order\[Eta]]; +table=table//Flatten//Total; +ret=coeff table//SeriesTake[#,order\[Eta]]&; +ret +] + + +tableOverNJ["C\[Nu]"][\[ScriptL]_,term_,repls_,order\[Eta]_]:=Module[{table,aux,finalj,firstRegularj,firstRegular\[Eta],leading\[Eta]OrderTerm,leading\[Eta]Order,goal\[Eta]Order,replsAux,replsLeading,auxOrder,nMin,nMax},replsLeading=(SeriesTake[#1,7]&)/@repls;firstRegularj=Abs[2+2 \[ScriptL]+2 n]+1;leading\[Eta]Order=(term/. j->#1&/@{0,firstRegularj})/. n->0/. replsLeading//SeriesMinOrder//Min;goal\[Eta]Order=order\[Eta]+leading\[Eta]Order; +nMin=Ceiling[-((goal\[Eta]Order+3)/2)]; +nMax=Floor[goal\[Eta]Order/4]; +table=Simplify[Table[firstRegular\[Eta]=SeriesMinOrder[term/. j->firstRegularj/. replsLeading]; +finalj=firstRegularj+(goal\[Eta]Order-firstRegular\[Eta]); +Table[leading\[Eta]OrderTerm=SeriesMinOrder[term/. replsLeading]; +If[leading\[Eta]OrderTerm<=goal\[Eta]Order, +auxOrder=(Max[#1,7]&)[goal\[Eta]Order-leading\[Eta]OrderTerm+If[n<0,7,0]]; +replsAux=(SeriesTake[#1,auxOrder]&)/@repls; +aux=term/.replsAux; +,(*else*) +aux=O[\[Eta]]^(goal\[Eta]Order+1); +]; +aux,{j,0,finalj}] +,{n,nMin,nMax}]];table] + + +(* ::Subsubsection::Closed:: *) +(*Constructing \!\(\*SubsuperscriptBox[\(R\), \(C\), \(\(-\[Nu]\) - 1\)]\) *) + + +(* ::Text:: *) +(*The construction of the second term is very similar to Subscript[R, C] just with different scalings and coefficients. Again Plots can be found in the Hyperlink["Plots section",{NotebookObject["c6ff3b79-aebc-4585-b130-dd1a930b511b", "9e23625d-8944-48d8-a536-bca7efa5e050"], "2ndPlots"}]*) + + +(* ::Input:: *) +(*(*RPN["2ndTerm"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]Var_,aKerr_,order\[Eta]_]:=Block[{s=\[ScriptS],l=\[ScriptL],m=\[ScriptM]Var,\[ScriptM]=\[ScriptM]Var,\[ScriptA]=aKerr,a=aKerr,aux,ret,repls,repls\[ScriptCapitalK],replsLeading,coeff,term,status,table,factor\[ScriptCapitalK],replsCut},*) +(*Monitor[status="getting repls";*) +(*repls=replsMST[\[ScriptS],\[ScriptL],\[ScriptM],Max[Ceiling[order\[Eta],3]+10,7]];*) +(* status="\[ScriptCapitalK] factor";*) +(*replsCut=SeriesTake[#,Max[order\[Eta]+1,7]]&/@repls;*) +(*repls\[ScriptCapitalK]=SeriesTake[#,Max[order\[Eta]+7,7]]&/@repls;*) +(*(*EchoTiming[factor\[ScriptCapitalK]=(\[ScriptCapitalK][-\[Nu]-1,order\[Eta](*-2 \[ScriptL]*)]/\[ScriptCapitalK][\[Nu],order\[Eta](*-2 \[ScriptL]*)]//.replsKerr/.\[Epsilon]->2\[Omega]/.replsPN)/.repls\[ScriptCapitalK];*) +(*factor\[ScriptCapitalK]=factor\[ScriptCapitalK]//Collect[#,{\[Eta],Gamma[__]},Simplify]&//polyToSeries;*) +(*,"\[ScriptCapitalK] factor"];*)*) +(*factor\[ScriptCapitalK]=\[ScriptCapitalK]Amplitude["Ratio"][\[ScriptS],\[ScriptL],\[ScriptM]Var,aKerr,order\[Eta]]//ExpandGamma//ExpandPolyGamma//SeriesCollect[#,{Gamma[__],PolyGamma[__,__]},Simplify]&;*) +(* status="coeff";*) +(*coeff=(c["In"][z[r]] /.\[Nu]->-\[Nu]-1//.replsKerr/.\[Epsilon]->2\[Omega]/.replsPN//Simplify);*) +(*coeff=coeff/.replsCut//Collect[#,{\[Eta],Log[__]}]&//Simplify//polyToSeries;*) +(* status="table";*) +(*term=\[ConstantC]D[-\[Nu]-1,n,j] z[r]^(n+j)//.replsKerr/.\[Epsilon]->2\[Omega]/.replsPN;*) +(*table=tableOverNJ["2ndTerm"][\[ScriptL],term,repls,order\[Eta]];*) +(* status="summing";*) +(*table=table//Flatten//Total//Simplify;*) +(* status="assembling";*) +(*ret=factor\[ScriptCapitalK] coeff table;*) +(* status="cutting the result";*) +(**) +(*(*EchoTiming[ret=ret//Collect[#,{\[Eta],Gamma[__]},Simplify]&//polyToSeries,"collecting"];*) +(*Print[ret];*)*) +(*ret=ret//SeriesTake[#,order\[Eta]]&;*) +(*ret*) +(*,{status,n,j}]]*)*) + + +tableOverNJ["C-\[Nu]-1"][\[ScriptL]_,term_,repls_,order\[Eta]_]:=Block[{table,aux,finalj,firstRegularj,firstRegular\[Eta],leading\[Eta]OrderTerm,leading\[Eta]Order,goal\[Eta]Order,replsAux,replsLeading,auxOrder,nMin,nMax}, +replsLeading=repls//SeriesTake[#1,7]&; +firstRegularj=Abs[-2 \[ScriptL]+2 n]+1; +leading\[Eta]Order=Min[(SeriesMinOrder[term/. j->#1/. n->0/. replsLeading]&)/@{0,firstRegularj}]; +goal\[Eta]Order=order\[Eta]+leading\[Eta]Order; +nMin=(Min[#1,0]&)[Ceiling[(3-goal\[Eta]Order)/2]]; +nMax=Floor[(6+goal\[Eta]Order)/4]; +table=Table[firstRegular\[Eta]=SeriesMinOrder[term/. j->firstRegularj/. replsLeading]; +finalj=firstRegularj+(goal\[Eta]Order-firstRegular\[Eta]); +Table[leading\[Eta]OrderTerm=SeriesMinOrder[term/. replsLeading]; +If[leading\[Eta]OrderTerm<=goal\[Eta]Order,auxOrder=(Max[#1,7]&)[goal\[Eta]Order-leading\[Eta]OrderTerm+6]; +replsAux=(SeriesTake[#1,auxOrder]&)/@repls; +aux=term/. replsAux; +,aux=O[\[Eta]]^(goal\[Eta]Order+1);]; +aux,{j,0,finalj}],{n,nMin,nMax}]; +table] + + +RPN["C-\[Nu]-1"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=Module[{aux,ret,repls,repls\[ScriptCapitalK],replsLeading,coeff,term,table,factor\[ScriptCapitalK],replsCut}, +repls=MSTCoefficientsInternal[\[ScriptS],\[ScriptL],\[ScriptM],a,Max[Ceiling[order\[Eta],3]+10,7]]; +replsCut=repls//SeriesTake[#,Max[order\[Eta]+1,7]]&; +coeff=c["In"][\[ScriptS],\[ScriptL],\[ScriptM],a,z[a,r]]/. \[Nu]MST->-\[Nu]MST-1/.replsCut//SeriesCollect[#,Log[__],Simplify]&; +term=\[ConstantC]D["C-\[Nu]-1"][\[ScriptS],\[ScriptL],\[ScriptM],a,n,j] z[a,r]^(n+j)//PNScalingsInternal; +table=tableOverNJ["C-\[Nu]-1"][\[ScriptL],term,repls,order\[Eta]]; +table=Simplify[Total[Flatten[table]]]; +ret=coeff table; +ret=(SeriesTake[#1,order\[Eta]]&)[ret]; +ret +] + + +(* ::Input:: *) +(*(*RPN["2ndTerm"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]Var_,aKerr_,order\[Eta]_]:=Block[{s=\[ScriptS],l=\[ScriptL],m=\[ScriptM]Var,\[ScriptM]=\[ScriptM]Var,\[ScriptA]=aKerr,a=aKerr,aux,ret,repls,repls\[ScriptCapitalK],replsLeading,coeff,term,status,table,factor\[ScriptCapitalK],replsCut},*) +(*Monitor[status="getting repls";*) +(*repls=replsMST[\[ScriptS],\[ScriptL],\[ScriptM],Max[Ceiling[order\[Eta],3]+10,7]];*) +(* status="\[ScriptCapitalK] factor";*) +(*replsCut=SeriesTake[#,Max[order\[Eta]+1,7]]&/@repls;*) +(*repls\[ScriptCapitalK]=SeriesTake[#,Max[order\[Eta]+7,7]]&/@repls;*) +(*factor\[ScriptCapitalK]=(\[ScriptCapitalK]Amplitude[-\[Nu]-1][\[ScriptS],\[ScriptL],\[ScriptM]Var,aKerr,order\[Eta](*-2 \[ScriptL]*)]/\[ScriptCapitalK]Amplitude[\[Nu]][\[ScriptS],\[ScriptL],\[ScriptM]Var,aKerr,order\[Eta](*-2 \[ScriptL]*)])//ExpandGamma//ExpandPolyGamma;*) +(*factor\[ScriptCapitalK]=factor\[ScriptCapitalK]//Collect[#,{\[Eta],Gamma[__]},Simplify]&//polyToSeries;*) +(* status="coeff";*) +(*coeff=(c["In"][z[r]] /.\[Nu]->-\[Nu]-1//.replsKerr/.\[Epsilon]->2\[Omega]/.replsPN//Simplify);*) +(*coeff=coeff/.replsCut//Collect[#,{\[Eta],Log[__]}]&//Simplify//polyToSeries;*) +(* status="table";*) +(*term=\[ConstantC]D[-\[Nu]-1,n,j] z[r]^(n+j)//.replsKerr/.\[Epsilon]->2\[Omega]/.replsPN;*) +(*table=tableOverNJ["2ndTerm"][\[ScriptL],term,repls,order\[Eta]];*) +(* status="summing";*) +(*table=table//Flatten//Total//Simplify;*) +(* status="assembling";*) +(*ret=factor\[ScriptCapitalK] coeff table;*) +(* status="cutting the result";*) +(**) +(*(*EchoTiming[ret=ret//Collect[#,{\[Eta],Gamma[__]},Simplify]&//polyToSeries,"collecting"];*) +(*Print[ret];*)*) +(*ret=ret//SeriesTake[#,order\[Eta]]&;*) +(*ret*) +(*,{status,n,j}]]*)*) + + +(* ::Subsubsection:: *) +(*Constructing Subscript[R, In]*) + + +(* ::Text:: *) +(*Subscript[R, in] is now simply the sum of the two terms. A visualisation of the gap can be found in the Hyperlink["Plots section",{NotebookObject["c6ff3b79-aebc-4585-b130-dd1a930b511b", "9e23625d-8944-48d8-a536-bca7efa5e050"], "gapPlot"}]*) + + +InGap[\[ScriptL]_,\[ScriptM]aKerr_]:=4\[ScriptL]-1; +InGap[0,\[ScriptM]aKerr_]:=0 +InGap[0,0]=0; +InGap[\[ScriptL]_,0]:=4\[ScriptL]+2; + + +Options[RPN]={"Normalization"->"SFPN"} + + +RPN["In",OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=Module[{aux,ret,gap,\[ScriptCapitalK],secondTerm,secondR,normalization}, +gap=InGap[\[ScriptL],\[ScriptM] a]; +\[ScriptCapitalK]=If[order\[Eta]>=gap,\[ScriptCapitalK]Amplitude["Ratio"][\[ScriptS],\[ScriptL],\[ScriptM],a,Max[order\[Eta]-gap,2]]//ExpandGamma//ExpandPolyGamma//SeriesCollect[#,PolyGamma[__,__]]&,1]; +secondR=If[order\[Eta]>=gap,RPN["C-\[Nu]-1"][\[ScriptS],\[ScriptL],\[ScriptM],a,Max[order\[Eta]-gap,2]],0]; +secondTerm=\[ScriptCapitalK] secondR; +aux=RPN["C\[Nu]"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]+secondTerm; +normalization=Switch[OptionValue["Normalization"], + "Default",1, + "SasakiTagoshi",\[ScriptCapitalK]Amplitude["\[Nu]"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]], + "UnitTransmission",\[ScriptCapitalK]Amplitude["\[Nu]"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]/BAmplitude["Trans"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]] +]; +ret=normalization aux; +ret//IgnoreExpansionParameter +] + + +RPN["In"][0,0,0,aKerr_,1]:=49/81+O[\[Eta]] +RPN["In"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,aKerr_,1]:=(2^\[ScriptL] (-\[ScriptS]+\[ScriptL])!)/(2\[ScriptL]+1)! (r \[Omega] \[Eta])^(-\[ScriptS]+\[ScriptL]) +O[\[Eta]] \[Eta]^(-\[ScriptS]+\[ScriptL]) + +RPN["C\[Nu]"][0,0,0,aKerr_,1]:=7/9+O[\[Eta]] +RPN["C\[Nu]"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,aKerr_,1]:=RPN["In"][\[ScriptS],\[ScriptL],\[ScriptM],aKerr,1] + + +RPN["In"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,aKerr_,0]:=O[\[Eta]] \[Eta]^(-\[ScriptS]+\[ScriptL]-1) +RPN["C\[Nu]"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,aKerr_,0]:=O[\[Eta]] \[Eta]^(-\[ScriptS]+\[ScriptL]-1) + + +(* ::Subsection:: *) +(*Subscript[R, Up]*) + + +(* ::Text:: *) +(*To construct Subscript[R, up] we follow Eq.159 in Sasaki Tagoshi ( https://doi.org/10.12942/lrr-2003-6 ) where \[CapitalPsi] is identical to HypergeometricU[]..*) + + +(* ::Subsubsection::Closed:: *) +(*Definitions (depreciated)*) + + +(* ::Text:: *) +(*We expand the Hypergeometrics in Kummer functions (c.f. Eq.13.2.42 in the Digital Library of Mathematical Functions https://dlmf.nist.gov/13.2#E42 ) and then expand the Kummer functions in Pochhammers ( c.f. https://dlmf.nist.gov/13.2#E2 ). The result of this are the individual elements over which we will table again, just as we did for Subscript[R, in].*) + + +(*(*z[r_]:=\[Omega](r-(1-\[Kappa]))*) + +(*kummerM[a_,b_,z_,order_]:=\!\( +\*UnderoverscriptBox[\(\[Sum]\), \(i = 0\), \(order\)]\( +\*FractionBox[\(Pochhammer[a, i]\), \(Pochhammer[b, i]\)] +\*FractionBox[\(1\), \(i!\)] +\*SuperscriptBox[\(z\), \(i\)]\)\); +UtoM[order_]:=(HypergeometricU[a_,b_,z_]:> +Gamma[1-b]/Gamma[a-b+1]kummerM[a,b,z,order]+Gamma[b-1]/Gamma[a]z^(1-b) kummerM[a-b+1,2-b,z,order]); + +term["up"][n_,z_,order_]:=a[n](2 \[ImaginaryI] z)^nPochhammer[1+s+\[Nu]-\[ImaginaryI] \[Epsilon],n]/Pochhammer[1+s+\[Nu]+\[ImaginaryI] \[Epsilon]-2s,n]HypergeometricU[n+1+s+\[Nu]-\[ImaginaryI] \[Epsilon],2n+2+2\[Nu],-2\[ImaginaryI] z]/.UtoM[order]; +R["up"][z_,order_]:=c["up"][z](*Gamma[\[Nu]-s+1+\[ImaginaryI] \[Epsilon]]/Gamma[\[Nu]+s+1-\[ImaginaryI] \[Epsilon]]*)\!\( +\*UnderoverscriptBox[\(\[Sum]\), \(n = \(-\((order + 2)\)\)\), \(order\)]\(\(term["\"]\)[n, z, order]\)\)*) +(*c["up"][z_]:=2^\[Nu]\[ExponentialE]^(-\[Pi] \[Epsilon])\[ExponentialE]^(-\[ImaginaryI] \[Pi](\[Nu]+1+s))\[ExponentialE]^(\[ImaginaryI] z)z^(\[Nu]+\[ImaginaryI](\[Epsilon]+\[Tau])/2)/(z-\[Epsilon] \[Kappa])^(s+\[ImaginaryI] (\[Epsilon]+\[Tau])/2);*) +c["Up"][z_]:=2^\[Nu] E^(-\[Pi] \[Epsilon]) E^(-I \[Pi](\[Nu]+1+s)) E^(I z) (z-\[Epsilon] \[Kappa])^-s z^\[Nu] (1-(\[Epsilon] \[Kappa])/z)^(-I \[Epsilon]p) +element["Up"][n_,j_,z_]:=2^n (I z)^n Pochhammer[1+s-I \[Epsilon]+\[Nu],n]/Pochhammer[1-s+I \[Epsilon]+\[Nu],n] ((2^(-1+j-2 n-2 \[Nu]) (-I z)^(-1+j-2 n-2 \[Nu]) Gamma[1+2 n+2 \[Nu]] Pochhammer[-n+s-I \[Epsilon]-\[Nu],j])/(j! Gamma[1+n+s-I \[Epsilon]+\[Nu]] Pochhammer[-2 n-2 \[Nu],j])+(2^j (-I z)^j Gamma[-1-2 n-2 \[Nu]] Pochhammer[1+n+s-I \[Epsilon]+\[Nu],j])/(j! Gamma[-n+s-I \[Epsilon]-\[Nu]] Pochhammer[2+2 n+2 \[Nu],j]))*) + + +(* ::Subsubsection::Closed:: *) +(*Constructions (depreciated)*) + + +(* ::Text:: *) +(*The construction of Subscript[R, up] is very much like the one for Subscript[R, C]. To execute the sums we use tableOverNJ[] which constructs a table of all needed terms. Again you can find some plots illustrating the variables nMin, nMax and firstRegularj in the Hyperlink["plot section",{NotebookObject["c6ff3b79-aebc-4585-b130-dd1a930b511b", "9e23625d-8944-48d8-a536-bca7efa5e050"], "plotsUp"}]. These are essential in telling you where to truncate the sums. *) + + +(*tableOverNJ["Up"][\[ScriptL]_,term_,repls_,order\[Eta]_]:=Block[{table,tableNeg,tablePos,aux,aux2,finalj,firstRegularj,firstRegular\[Eta],leading\[Eta]OrderTerm,leading\[Eta]Order,goal\[Eta]Order,aux\[Eta],replsAux,replsLeading,auxOrder,nMin,nMax,status}, + status="tableing setup"; +replsLeading=SeriesTake[#,7]&/@repls; +firstRegularj=Min[Abs[2+2\[ScriptL]+2 n]+1,Abs[-2 n-2 \[ScriptL]]+1]; +leading\[Eta]Order=(term/.j->#/.n->0/.replsLeading//SeriesMinOrder)&/@{0,firstRegularj}//Min; +goal\[Eta]Order=order\[Eta]+leading\[Eta]Order; +nMin=Min[1/4 (-1-goal\[Eta]Order-2 \[ScriptL]),1/2 (-3-goal\[Eta]Order)]//Ceiling; +nMax=1/2 (1+goal\[Eta]Order+2 \[ScriptL])//Floor; +table=Table[ +status="getting finalj"; +firstRegular\[Eta]=(term/.j->firstRegularj/.replsLeading//SeriesMinOrder); +finalj=firstRegularj+(goal\[Eta]Order-firstRegular\[Eta]); +Table[ +status="j table"; +leading\[Eta]OrderTerm=term/.replsLeading//SeriesMinOrder; +If[leading\[Eta]OrderTerm<=goal\[Eta]Order, +status="entered doing something"; +auxOrder=goal\[Eta]Order-leading\[Eta]OrderTerm+6//Max[#,7]&; +replsAux=SeriesTake[#,auxOrder]&/@repls; +status="doing something"; +aux=term/.replsAux;, +(*else*) +status="entered doing nothing"; +aux=Quiet[O[\[Eta]]^(goal\[Eta]Order+1)]; +]; +aux +,{j,0,finalj}] +,{n,nMin,nMax}]; +table] + +RPN["Up"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]Var_,aKerr_,order\[Eta]_]:=Block[{s=\[ScriptS],l=\[ScriptL],m=\[ScriptM]Var,\[ScriptM]=\[ScriptM]Var,\[ScriptA]=aKerr,a=aKerr,aux,aux2,repls,replsCoeff,coeff,table,tableNeg,tablePos,status,list,term}, +Monitor[ + status="repls"; +repls=replsMST[\[ScriptS],\[ScriptL],\[ScriptM],Max[Ceiling[order\[Eta],3]+9,7]]; + status="coeff"; +replsCoeff=SeriesTake[#,Max[order\[Eta],7]]&/@repls; +coeff=(c["Up"][z[r]]//.replsKerr/.\[Epsilon]->2\[Omega]/.replsPN//Simplify)/.replsCoeff; + status="constructing term"; +term=element["Up"][n,j,z[r]]a[n]//.replsKerr/.\[Epsilon]->2\[Omega]/.replsPN; + status="tableing"; +table=tableOverNJ["Up"][\[ScriptL],term,repls,order\[Eta]]; + status="flattening"; +table=Total@Flatten@table; + status="assembling"; +coeff table//SeriesTake[#,order\[Eta]]& +,{status,n,j}]] +*) + + +(* ::Subsubsection:: *) +(*Constructing Subscript[R, up] from Subscript[R, C]*) + + +(* ::Text:: *) +(*We turn to Throwe's bachelor thesis where Eq.(B.7) gives Subscript[R, up] in terms of \!\(\*SubsuperscriptBox[\(R\), \(C\), \(\[Nu]\)]\) and \!\(\*SubsuperscriptBox[\(R\), \(C\), \(\(-\[Nu]\) - 1\)]\).*) + + +RPN["Up",OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=Module[{aux,ret,\[CurlyEpsilon],\[CurlyEpsilon]p,\[Kappa],\[Tau],C1,C2,repls,term1,term2,coeff,normalization}, +\[CurlyEpsilon]=2 \[Omega]; +\[CurlyEpsilon]p=(\[CurlyEpsilon]+\[Tau])/2; +\[Kappa]=Sqrt[1-a^2]; +\[Tau]=(-a \[ScriptM]+\[CurlyEpsilon])/\[Kappa]; +repls=MSTCoefficientsInternal[\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]+7]; +coeff=(-I E^(-\[Pi] \[CurlyEpsilon]-I \[Pi] \[ScriptS])Sin[\[Pi](\[Nu]MST+\[ScriptS]-I \[CurlyEpsilon])]/Sin[2\[Pi] \[Nu]MST]); +(*coeff=1;*) +C1=PNScalingsInternal[coeff]/.repls; +C2=PNScalingsInternal[coeff I E^(-I \[Pi] \[Nu]MST) Sin[\[Pi](\[Nu]MST-\[ScriptS]+I \[CurlyEpsilon])]/ Sin[\[Pi](\[Nu]MST+\[ScriptS]-I \[CurlyEpsilon])]]/.repls; +term1=C1 RPN["C-\[Nu]-1"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]; +term2=C2 RPN["C\[Nu]"][\[ScriptS],\[ScriptL],\[ScriptM],a,Max[order\[Eta]-2\[ScriptL]+2,0]]; +aux=term1+term2; +normalization=Switch[OptionValue["Normalization"], + "Default",1, + "SasakiTagoshi",1, + "UnitTransmission",1/CAmplitude["Trans"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]] +]; +aux=aux//SeriesTake[#,order\[Eta]]&; +ret=aux normalization; +ret//IgnoreExpansionParameter +] + + +RPN["Up"][0,0,0,aKerr_,1]:=-((7 I)/(18 r \[Omega] \[Eta]))+O[\[Eta]] \[Eta]^-1 +RPN["Up"][\[ScriptS]_/;\[ScriptS]<=0,\[ScriptL]_,\[ScriptM]Var_,aKerr_,1]:=((-1)^(\[ScriptS]+1) 2^-\[ScriptL] \[ScriptL] ( r \[Eta] \[Omega])^(-\[ScriptL]-\[ScriptS]-1) (2 \[ScriptL]-1)!)/(\[ScriptL]+\[ScriptS])!+O[\[Eta]] \[Eta]^(-\[ScriptS]-\[ScriptL]-1) + + +RPN["Up"][\[ScriptS]_/;\[ScriptS]<=0,\[ScriptL]_,\[ScriptM]Var_,aKerr_,0]:=O[\[Eta]] \[Eta]^(-\[ScriptS]-\[ScriptL]-2) +RPN["Up"][0,0,0,aKerr_,0]:=O[\[Eta]]^-1 + + +(* ::Subsubsection:: *) +(*Normalization*) + + +(* ::Input:: *) +(*Normalization["Up"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,aKerr_,order\[Eta]_]:=Block[{s=\[ScriptS],l=\[ScriptL],m=\[ScriptM],a=aKerr,maxn,minn,ret},*) +(*maxn=order\[Eta]/3//Ceiling;*) +(*minn=-(order\[Eta]/3+2)//Floor;*) +(*ret=2^(-1-s+4 I \[Eta]^3 \[Omega]) E^(-(1/2) I \[Pi] (1+s+\[Nu]MST)+I (-1+Sqrt[1-a^2]) \[Eta]^3 \[Omega]-\[Pi] \[Eta]^3 \[Omega]) (\[Eta]^3 \[Omega])^(-1-2 s+2 I \[Eta]^3 \[Omega]) \!\( *) +(*\*UnderoverscriptBox[\(\[Sum]\), \(n = minn\), \(maxn\)]\(aMST[n]\)\)/.MSTCoefficientsInternal[\[ScriptS],\[ScriptL],\[ScriptM],aKerr,order\[Eta]]*) +(*]*) + + +(* ::Subsection:: *) +(*Outputting Radial solutions as functions*) + + +Options[RPNF]={"Normalization"->"Default","Simplify"->True} + + +RPNF[sol_,opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]Var_,{\[Eta]Var_/;MatchQ[\[Eta]Var,_Symbol],order_}]:=Block[{aux,ret}, +aux=RPN[sol,"Normalization"->OptionValue["Normalization"]][\[ScriptS],\[ScriptL],\[ScriptM],a,order]/.{\[Eta]->\[Eta]Var,\[Omega]->\[Omega]Var}; +If[OptionValue["Simplify"],aux=aux//Simplify]; +ret=Function[r,Evaluate[aux]]; +ret +] + + +(* ::Subsection::Closed:: *) +(*Positive spins *) + + +(* ::Subsubsection::Closed:: *) +(*Teukolsky-Starobinsky identities*) + + +\[ScriptCapitalL]\[Dagger][n_][x_]:=D[x,\[Theta]]-\[ScriptM] Csc[\[Theta]]x+n Cot[\[Theta]]x; +\[ScriptCapitalL][n_][x_]:=D[x,\[Theta]]+\[ScriptM] Csc[\[Theta]]x+n Cot[\[Theta]]x; +\[ScriptCapitalD][n_][x_]:=D[x,r]-( I((r^2+\[ScriptA]^2)\[Omega]-\[ScriptM] \[ScriptA]))/\[CapitalDelta][\[ScriptA],1,r] x+2n (r-M)/\[CapitalDelta][\[ScriptA],1,r]; +\[ScriptCapitalD]\[Dagger][n_][x_]:=D[x,r]+(I ((r^2+\[ScriptA]^2)\[Omega]-\[ScriptM] \[ScriptA]))/\[CapitalDelta][\[ScriptA],1,r] x+2n (r-M)/\[CapitalDelta][\[ScriptA],1,r]; + + +RPN["In",opt:OptionsPattern[]][2,\[ScriptL]_,\[ScriptM]Var_,aVar_,order\[Eta]_]:=Block[{aux,\[ScriptA]=aVar,a=aVar,m=\[ScriptM]Var,\[ScriptM]=\[ScriptM]Var}, +aux=RPN["In",opt][-2,\[ScriptL],\[ScriptM],a,order\[Eta]]; +aux=\[ScriptCapitalD][0]@\[ScriptCapitalD][0]@\[ScriptCapitalD][0]@\[ScriptCapitalD][0]@aux; +aux//Simplify//PNScalings[#,{{\[Omega],3},{r,-2}},\[Eta]]&//Simplify//SeriesTake[#,order\[Eta]]&] + +RPN["Up",opt:OptionsPattern[]][2,\[ScriptL]_,\[ScriptM]Var_,aVar_,order\[Eta]_]:=Block[{aux,\[ScriptA]=aVar,a=aVar,m=\[ScriptM]Var,\[ScriptM]=\[ScriptM]Var}, +aux=RPN["Up",opt][-2,\[ScriptL],\[ScriptM],a,order\[Eta]+1]; +aux=\[ScriptCapitalD][0]@\[ScriptCapitalD][0]@\[ScriptCapitalD][0]@\[ScriptCapitalD][0]@aux; +aux//Simplify//PNScalings[#,{{\[Omega],3},{r,-2}},\[Eta]]&//Simplify//SeriesTake[#,order\[Eta]]&] + +RPN["In",opt:OptionsPattern[]][1,\[ScriptL]_,\[ScriptM]Var_,aVar_,order\[Eta]_]:=Block[{aux,\[ScriptA]=aVar,a=aVar,m=\[ScriptM]Var,\[ScriptM]=\[ScriptM]Var}, +aux=RPN["In",opt][-1,\[ScriptL],\[ScriptM],a,order\[Eta]]; +aux=\[ScriptCapitalD][0]@\[ScriptCapitalD][0][aux]; +aux//Simplify//PNScalings[#,{{\[Omega],3},{r,-2}},\[Eta]]&//Simplify//SeriesTake[#,order\[Eta]]&] +RPN["Up",opt:OptionsPattern[]][1,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=Block[{aux,\[ScriptA]=a}, +aux=RPN["Up",opt][-1,\[ScriptL],\[ScriptM],a,order\[Eta]+1]; +aux=\[ScriptCapitalD][0]@\[ScriptCapitalD][0][aux]; +aux//Simplify//redo\[Eta]Repls//Simplify//SeriesTake[#,order\[Eta]]&] + + +(* ::Subsection::Closed:: *) +(*Inhomogeneous solution (depreciated)*) + + +(* ::Text:: *) +(*This part is now handled directly by TeukolskyPointParticleMode*) + + +(* ::Subsubsection::Closed:: *) +(*Coefficients*) + + +(* ::Input:: *) +(*CCoefficient["Up"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,aKerr_,order\[Eta]_]:=Block[{aux,W,S,a=aKerr,R,RF,auxF,list,ret},*) +(*Assuming[{assumps},W=InvariantWronskian[\[ScriptS],\[ScriptL],\[ScriptM],aKerr,order\[Eta]];*) +(*S=TeukolskySource[\[ScriptS],\[ScriptL],\[ScriptM],"InvariantWronskianForm"->True]//Simplify;*) +(*R=RPN["In"][\[ScriptS],\[ScriptL],\[ScriptM],aKerr,order\[Eta]];*) +(*RF=r|->Evaluate[R];*) +(*aux=integrateDelta[auxF[r] S,r,False]//RemovePNInternal;*) +(*aux=aux//Collect[#,{auxF[__],Derivative[__][auxF][__]},Collect[#,{SpinWeightedSpheroidalHarmonicS[__][__],Derivative[__][SpinWeightedSpheroidalHarmonicS[__]][__]},Simplify]&]&;*) +(*aux=PNScalingsInternal[aux]/.{auxF->(auxF[# \[Eta]^2]&),\[Theta]->(\[Theta][# \[Eta]^2]&),SpinWeightedSpheroidalHarmonicS[args___]:>(SpinWeightedSpheroidalHarmonicS[args]/. {\[Eta]->1})};*) +(*aux=aux/.auxF->RF;*) +(*ret=aux/W;*) +(*ret*) +(*]]*) + + +(* ::Input:: *) +(*CCoefficient["In"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,aKerr_,order\[Eta]_]:=Block[{aux,W,S,a=aKerr,R,RF,auxF,list,ret},*) +(*Assuming[{assumps},W=InvariantWronskian[\[ScriptS],\[ScriptL],\[ScriptM],aKerr,order\[Eta]];*) +(*S=TeukolskySource[\[ScriptS],\[ScriptL],\[ScriptM],"InvariantWronskianForm"->True]//Simplify;*) +(*R=RPN["Up"][\[ScriptS],\[ScriptL],\[ScriptM],aKerr,order\[Eta]];*) +(*RF=r|->Evaluate[R];*) +(*aux=integrateDelta[auxF[r] S,r,True]//RemovePNInternal;*) +(*aux=aux//Collect[#,{auxF[__],Derivative[__][auxF][__]},Collect[#,{SpinWeightedSpheroidalHarmonicS[__][__],Derivative[__][SpinWeightedSpheroidalHarmonicS[__]][__]},Simplify]&]&;*) +(*aux=PNScalingsInternal[aux]/.{auxF->(auxF[# \[Eta]^2]&),\[Theta]->(\[Theta][# \[Eta]^2]&),SpinWeightedSpheroidalHarmonicS[args___]:>(SpinWeightedSpheroidalHarmonicS[args]/. {\[Eta]->1})};*) +(*aux=aux/.auxF->RF;*) +(*ret=aux/W;*) +(*ret*) +(*]]*) + + +(* ::Subsubsection::Closed:: *) +(*Sourced solution *) + + +(* ::Input:: *) +(*RPN["CO"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,aKerr_,order\[Eta]_]:=Assuming[{assumps},Block[{s=\[ScriptS],l=\[ScriptL],m=\[ScriptM],\[ScriptA]=aKerr,a=aKerr,Rup,RupF,Rin,RinF,wronskian,integrandUp,integrandIn,cIn,cUp,aux,auxF},*) +(*EchoTiming[Rup=RPN["Up"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]//Simplify;*) +(*RupF[r_]:=Evaluate[Rup],"Rup"];*) +(*EchoTiming[Rin=RPN["In"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]//Simplify;*) +(*RinF[r_]:=Evaluate[Rin],"Rin"];*) +(*EchoTiming[wronskian=\[CapitalDelta][\[ScriptA],1,r \[Eta]^-2]^(\[ScriptS]+1) (Rin D[Rup,r]-D[Rin,r] Rup)//Simplify[#,Assumptions->r>2]&,"wronskian"];*) +(*EchoTiming[cIn=1/ wronskian integrateDelta[auxF[r]Simplify[TeukolskySource[\[ScriptS],\[ScriptL],\[ScriptM],"InvariantWronskianForm"->True]],r,True]//Normal//Simplify,"integrating cIn"];*) +(*EchoTiming[cIn=cIn/.auxF->RupF//redo\[Eta]Repls,"subbing cIn"];*) +(*EchoTiming[cUp=1/ wronskian integrateDelta[auxF[r]Simplify[TeukolskySource[\[ScriptS],\[ScriptL],\[ScriptM],"InvariantWronskianForm"->True]],r,False]//Normal//Simplify,"integrating cUp"];*) +(*EchoTiming[cUp=cUp/.auxF->RinF//redo\[Eta]Repls,"subbing cUp"];*) +(*EchoTiming[aux=cIn Rin+cUp Rup,"assembling"]//Simplify[#,Assumptions->{\[Eta]>0,r0>2,r>2}]&*) +(*]]*) + + +(* ::Input:: *) +(*RPN["CO"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=Assuming[{assumps},*) +(*Module[{aux,Rin,Rup,wronskian,integrandUp,integrandIn,cIn,cUp,auxF,sourceCoeffs},*) +(*aux=TeukolskyRadialPN[\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{\[Eta],order\[Eta]}];*) +(*Rin=aux["In"]["RadialFunction"];*) +(*Rup=aux["Up"]["RadialFunction"];*) +(*wronskian=EchoTiming[wronskian=(Simplify[#1,Assumptions->r>2]&)[\[CapitalDelta][a,1,r/\[Eta]^2]^(\[ScriptS]+1) (Rin[r] Rup'[r]-Rin'[r] Rup[r])],"wronskian"];*) +(*EchoTiming[sourceCoeffs=TeukolskySourceCircularOrbit[\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{r,r0}]//Coefficient[#,{DiracDelta[r-r0],Derivative[1][DiracDelta][r-r0],Derivative[2][DiracDelta][r-r0]}]&,"coeffs"];*) +(*EchoTiming[sourceCoeffs=redo\[Eta]Repls[sourceCoeffs]//Simplify,"PNScalings"];*) +(*cIn=integrateDelta[f[r]#,r,True]&/@{\[Delta][r-r0],\[Delta]'[r-r0],\[Delta]''[r-r0]};*) +(*cUp=integrateDelta[f[r]#,r,False]&/@{\[Delta][r-r0],\[Delta]'[r-r0],\[Delta]''[r-r0]};*) +(*EchoTiming[cIn=cIn/.f->Rin,"subbing In"];*) +(*EchoTiming[cUp=cUp/.f->Rup,"subbing Up"];*) +(*EchoTiming[cIn=sourceCoeffs cIn//Total,"total In"];*) +(*EchoTiming[cUp=sourceCoeffs cUp//Total,"total Up"];*) +(*(cIn Rin[r])/wronskian+(cUp Rup[r])/wronskian//Simplify*) +(*]]*) + + +(* ::Subsection::Closed:: *) +(*Checking input is correct*) + + +PossibleSols={"In","Up","C\[Nu]","C-\[Nu]-1"} + + +CheckInput[sol_,\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]_,{varPN_,order_}]:=Module[{aux}, +(*Checking boundary conditions*) +If[!MemberQ[PossibleSols,sol],Message[TeukolskyRadialFunctionPN::optx,sol];Abort[];]; +(*Checking modes*) +If[\[ScriptL]1||a<0,Message[TeukolskyRadialFunctionPN::parama,a];Abort[];]; +(*Checking \[Omega] is real*) +If[MatchQ[\[Omega],_Complex],Message[TeukolskyRadialFunctionPN::param\[Omega],\[Omega]];Abort[];]; +(*Checking PN parameter is a symbol*) +If[!MatchQ[varPN,_Symbol],Message[TeukolskyRadialFunctionPN::parama,varPN];Abort[];]; +(*Checking order is an integer*) +If[!MatchQ[order,_Integer],Message[TeukolskyRadialFunctionPN::paramorder,order];Abort[];]; +] + + +(* ::Subsection:: *) +(*TeukolskyRadialFunctionPN*) + + +(* ::Subsubsection::Closed:: *) +(*Icons*) + + +icons = <| + "In" -> Graphics[{ + Line[{{0,1/2},{1/2,1},{1,1/2},{1/2,0},{0,1/2}}], + Line[{{3/4,1/4},{1/2,1/2}}], + {Arrowheads[0.2],Arrow[Line[{{1/2,1/2},{1/4,3/4}}]]}, + {Arrowheads[0.2],Arrow[Line[{{1/2,1/2},{3/4,3/4}}]]}}, + Background -> White, + ImageSize -> Dynamic[{Automatic, 3.5 CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[Magnification]}]], + "Up" -> Graphics[{ + Line[{{0,1/2},{1/2,1},{1,1/2},{1/2,0},{0,1/2}}], + Line[{{1/4,1/4},{1/2,1/2}}], + {Arrowheads[0.2],Arrow[Line[{{1/2,1/2},{1/4,3/4}}]]}, + {Arrowheads[0.2],Arrow[Line[{{1/2,1/2},{3/4,3/4}}]]}}, + Background -> White, + ImageSize -> Dynamic[{Automatic, 3.5 CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[Magnification]}]], +"CO"->Graphics[{ + Black,Disk[{0,0},{1,1},{\[Pi]-.1,2\[Pi]+.1}],Red,Thick,Circle[{0,0},{2,.5}],Black,Disk[{0,0},{1,1},{0,\[Pi]}]}, + Background->White, + ImageSize -> Dynamic[{Automatic, 3.5 CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[Magnification]}]] +|>; + + +(* ::Subsubsection:: *) +(*TeukolskyRadialPN*) + + +Options[RadialAssociation]={"Normalization"->"Default", "Amplitudes"->False, "Simplify"->True} +Options[TeukolskyRadialPN]={"Normalization"->"Default", "Amplitudes"->False, "Simplify"->True} + + +RadialAssociation[sol_String,opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]_,{varPN_,order_}]/;MemberQ[PossibleSols,sol]:=Module[{aux,ret,R,BC,lead,termCount,normalization,amplitudes,trans}, +CheckInput[sol,\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,order}]; +R=RPNF[sol,"Normalization"->OptionValue["Normalization"],"Simplify"->OptionValue["Simplify"]][\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,order}]; +BC=sol; +lead=RPNF[sol,"Normalization"->OptionValue["Normalization"],"Simplify"->OptionValue["Simplify"]][\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,1}]; +termCount=R[r]//SeriesLength; +normalization=OptionValue["Normalization"]; +trans=If[OptionValue["Amplitudes"],TeukolskyAmplitudePN[Switch[sol,"In","Btrans","Up","Ctrans"],"Normalization"->OptionValue["Normalization"]][\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,order}],"Not Computed"]; +amplitudes=<|"Transmission"->trans|>; +ret=<|"\[ScriptS]"->\[ScriptS],"\[ScriptL]"->\[ScriptL],"\[ScriptM]"->\[ScriptM],"a"->a,"PN"->{varPN,order},"RadialFunction"->R,"BoundaryCondition"->BC,"LeadingOrder"->lead,"TermCount"->termCount,"Normalization"->normalization,"Amplitudes"->amplitudes,"Simplified"->OptionValue["Simplify"]|>; +ret +] + + +TeukolskyRadialPN[\[ScriptS]_, \[ScriptL]_, \[ScriptM]_, a_, \[Omega]_,{varPN_,order_},opt:OptionsPattern[]]:=Module[{aux,Rin,Rup,assocUp,assocIn,retIn,retUp}, +assocIn=RadialAssociation["In",opt][\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,order}]; +retIn=TeukolskyRadialFunctionPN[\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,order},assocIn]; +assocUp=RadialAssociation["Up",opt][\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,order}]; +retUp=TeukolskyRadialFunctionPN[\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,order},assocUp]; +<|"In"->retIn,"Up"->retUp|> +] + + +(* ::Subsubsection:: *) +(*TeukolskyRadialFunctionPN*) + + +Options[TeukolskyRadialFunctionPN]={"Normalization"->"Default"} + + +TeukolskyRadialFunctionPN /: + MakeBoxes[trf:TeukolskyRadialFunctionPN[s_, l_, m_, a_, \[Omega]_,{varPN_,order_},assoc_], form:(StandardForm|TraditionalForm)] := + Module[{summary, extended}, + CheckInput[assoc["BoundaryCondition"],s,l,m,a,\[Omega],{varPN,order}]; + summary = {Row[{BoxForm`SummaryItem[{"s: ", s}], " ", + BoxForm`SummaryItem[{"l: ", l}], " ", + BoxForm`SummaryItem[{"m: ", m}], " ", + BoxForm`SummaryItem[{"a: ", a}], " ", + BoxForm`SummaryItem[{"\[Omega]: ", \[Omega]}]," ", + BoxForm`SummaryItem[{"PN parameter: ", varPN}]," ", + BoxForm`SummaryItem[{"PN order: ", N[(order-1)/2]"PN"}] +}], + BoxForm`SummaryItem[{"Boundary Condition: ", assoc["BoundaryCondition"]}]}; + extended = {BoxForm`SummaryItem[{"Leading order: ",assoc["LeadingOrder"]["r"]}], + BoxForm`SummaryItem[{"Normalization: ",assoc["Normalization"]}], + BoxForm`SummaryItem[{"Simplified: ",assoc["Simplified"]}], + BoxForm`SummaryItem[{"Number of terms: ",assoc["TermCount"]}]}; + + BoxForm`ArrangeSummaryBox[ + TeukolskyRadialFunctionPN, + trf, + Lookup[icons, assoc["BoundaryCondition"], None], + summary, + extended, + form] +]; + + +TeukolskyRadialFunctionPN[\[ScriptS]_, \[ScriptL]_, \[ScriptM]_, a_, \[Omega]_,{varPN_,order_},sol_String,opt:OptionsPattern[]]:=Module[{aux,assoc,ret}, +assoc=RadialAssociation[sol,opt][\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,order}]; +ret=TeukolskyRadialFunctionPN[\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,order},assoc]; +ret +] + + +(* ::Subsubsection:: *) +(*TeukolskyPointParticleModePN*) + + +Options[RadialSourcedAssociation]={"Normalization"->"Default"} + + +RadialSourcedAssociation["CO",opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,r0_,{varPN_,order_}]:=Assuming[{varPN>0,r0>0,r>0,1>a>=0,\[ScriptA]>=0},Module[{aux,ret,Rin,dRin,ddRin,Rup,dRup,ddRup,wronskian,source,sourceCoeffs,cUp,cIn,deltaCoeff,innerF,outerF,radialF}, +CheckInput["Up",\[ScriptS],\[ScriptL],\[ScriptM],a,\[ScriptM]/Sqrt[r0^3],{varPN,order}]; +aux=TeukolskyRadialPN[\[ScriptS],\[ScriptL],\[ScriptM],a,If[\[ScriptM]!=0,\[ScriptM],Style["0",Red]]\[CapitalOmega]Kerr,{varPN,order},"Normalization"->OptionValue["Normalization"]]; +Rin=aux["In"]["RadialFunction"]; +Rup=aux["Up"]["RadialFunction"]; +dRup=Rup'; +dRin=Rin'; +ddRup=dRup'; +ddRin=dRin'; +wronskian=(Simplify[#1,Assumptions->r>2]&)[Kerr\[CapitalDelta][a,r/varPN^2]^(\[ScriptS]+1) varPN^2 (Rin[r] dRup[r]-dRin[r] Rup[r])]; +source=TeukolskySourceCircularOrbit[\[ScriptS],\[ScriptL],\[ScriptM],a,{#,r0},"Form"->"InvariantWronskian"]&; +sourceCoeffs=source[r]//Coefficient[#,{DiracDelta[r-r0],Derivative[1][DiracDelta][r-r0],Derivative[2][DiracDelta][r-r0]}]&; +sourceCoeffs=Collect[#,{SpinWeightedSpheroidalHarmonicS[__][__],Derivative[__][SpinWeightedSpheroidalHarmonicS[__]][__]},Simplify]&/@sourceCoeffs; +sourceCoeffs=sourceCoeffs//PNScalings[#,{{r0,-2},{\[CapitalOmega]Kerr,3}},varPN,"IgnoreHarmonics"->True]&//Simplify; +cIn=1/wronskian Total[sourceCoeffs {Rup[r0],-varPN^2dRup[r0],varPN^4 ddRup[r0]}]//SeriesTake[#,order]&; +cUp=1/wronskian Total[sourceCoeffs {Rin[r0],-varPN^2dRin[r0],varPN^4 ddRin[r0]}]//SeriesTake[#,order]&; +deltaCoeff=Coefficient[source[r],Derivative[2][DiracDelta][r-r0]]/Kerr\[CapitalDelta][a,r0]; +deltaCoeff=If[deltaCoeff===0,0,deltaCoeff//PNScalings[#,{{r0,-2},{\[CapitalOmega]Kerr,3}},varPN,"IgnoreHarmonics"->True]&//SeriesTerms[#,{varPN,0,order}]&]; +innerF=cIn Rin[#]&; +outerF=cUp Rup[#]&; +radialF=innerF[#] HeavisideTheta[r0-#] + outerF[#] HeavisideTheta[#-r0]+deltaCoeff DiracDelta[#-r0]&; +ret=<|"s"->\[ScriptS],"l"->\[ScriptL],"m"->\[ScriptM],"a"->a,"r0"->r0,"PN"->{varPN,order},"RadialFunction"->radialF,("ExtendedHomogeneous"->"\[ScriptCapitalI]")->innerF,("ExtendedHomogeneous"->"\[ScriptCapitalH]")->outerF,"\[Delta]"->deltaCoeff,"cUp"->cUp,"cIn"->cIn,"Wronskian"->wronskian,"Source"->source,"In"->aux["In"],"Up"->aux["Up"]|>; +ret +] +] + + +(* ::Input:: *) +(*(*RadialSourcedAssociation["CO"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,r0Var_,{varPN_,order_}]/;NumericQ[a]:=Module[{aux,keys,values},*) +(*aux=RadialSourcedAssociation["CO"][\[ScriptS],\[ScriptL],\[ScriptM],\[ScriptA],r0Var,{varPN,order}];*) +(*keys=aux//Keys;*) +(*values=Values[aux]/.\[ScriptA]->a;*) +(*AssociationThread[keys,values]*) +(*]*) +(*RadialSourcedAssociation["CO"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,r0Var_,{varPN_,order_}]/;NumericQ[r0Var]:=RadialSourcedAssociation["CO"][\[ScriptS],\[ScriptL],\[ScriptM],a,r0,{varPN,order}]/.r0->r0Var;*)*) + + +Options[TeukolskyPointParticleModePN]={"Normalization"->"Default"} + + +TeukolskyModePN /: + MakeBoxes[trf:TeukolskyModePN[s_, l_, m_, a_,r0_,{varPN_,order_},assoc_], form:(StandardForm|TraditionalForm)] := + Module[{summary, extended}, +(* CheckInput[assoc["BoundaryCondition"],s,l,m,a,\[Omega],{varPN,order}];*) + summary = {Row[{BoxForm`SummaryItem[{"s: ", s}], " ", + BoxForm`SummaryItem[{"l: ", l}], " ", + BoxForm`SummaryItem[{"m: ", m}], " ", + BoxForm`SummaryItem[{"a: ", a}], " ", + BoxForm`SummaryItem[{"\[Omega]: ", m \[CapitalOmega]Kerr}]," ", + BoxForm`SummaryItem[{"\!\(\*SubscriptBox[\(r\), \(0\)]\): ", r0}]," ", + BoxForm`SummaryItem[{"PN parameter: ", varPN}]," ", + BoxForm`SummaryItem[{"PN order: ", N[(order-1)/2]"PN"}] +}], + BoxForm`SummaryItem[{"Orbit: ", "Circular Equatorial"}]}; + extended = {}; + + BoxForm`ArrangeSummaryBox[ + TeukolskyModePN, + trf, + Lookup[icons, "CO", None], + summary, + extended, + form] +]; + + +TeukolskyPointParticleModePN[\[ScriptS]_, \[ScriptL]_, \[ScriptM]_,orbit_KerrGeodesics`KerrGeoOrbit`KerrGeoOrbitFunction,{varPN_,order_},opt:OptionsPattern[]]:=Module[{aux,assoc,ret,a,r0Var,eccentricity,inclination}, +{a,r0Var,eccentricity,inclination}=orbit[#]&/@{"a","p","e","Inclination"}; +If[!(eccentricity===0),Message[TeukolskyPointParticleModePN::orbit];Abort[];]; +If[!(inclination===1),Message[TeukolskyPointParticleModePN::orbit];Abort[];]; +assoc=RadialSourcedAssociation["CO",opt][\[ScriptS],\[ScriptL],\[ScriptM],a,r0Var,{varPN,order}]; +ret=TeukolskyModePN[\[ScriptS],\[ScriptL],\[ScriptM],a,r0Var,{varPN,order},assoc]; +ret +] + + +(* ::Subsubsection:: *) +(*Accessing functions and keys*) + + +TeukolskyRadialFunctionPN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_][y_String]:= + assoc[y]; + + +TeukolskyRadialFunctionPN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_][r_Symbol] :=assoc["RadialFunction"][r] +TeukolskyRadialFunctionPN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_][r_/;NumericQ[r]] :=assoc["RadialFunction"][r] + + +Derivative[n_Integer][trf_TeukolskyRadialFunctionPN][r_Symbol]:=trf[[6,1]]^(2 n) Derivative[n][trf["RadialFunction"]][r] + + +Keys[trfpn_TeukolskyRadialFunctionPN]^:=trfpn[[-1]]//Keys + + +TeukolskyModePN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_][y:(_String|("ExtendedHomogeneous"->"\[ScriptCapitalH]")|("ExtendedHomogeneous"->"\[ScriptCapitalI]"))]:= + assoc[y]; + + +TeukolskyModePN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_][r0_] :=Message[TeukolskyModePN::particle]Abort[]; +TeukolskyModePN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_][r_Symbol] :=assoc["RadialFunction"][r] +TeukolskyModePN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_][r_/;NumericQ[r]] :=assoc["RadialFunction"][r] + + +Keys[trfpn_TeukolskyModePN]^:=trfpn[[-1]]//Keys + + +Derivative[n_Integer][tppm_TeukolskyModePN][r_Symbol]:=tppm[[6,1]]^(2 n) Derivative[n][tppm["RadialFunction"]][r] + + +(* ::Section:: *) +(*Ending Package*) + + +(* ::Subsection:: *) +(*Protecting*) + + +SetAttributes[{TeukolskyRadialPN, TeukolskyRadialFunctionPN, TeukolskyPointParticleModePN}, {Protected, ReadProtected}]; + + +(* ::Subsection:: *) +(*Ending*) + + +End[] +EndPackage[] diff --git a/Kernel/Teukolsky.m b/Kernel/Teukolsky.m index 1cbb184..5cf3894 100644 --- a/Kernel/Teukolsky.m +++ b/Kernel/Teukolsky.m @@ -31,3 +31,4 @@ Get["Teukolsky`TeukolskyMode`"]; Get["Teukolsky`NumericalIntegration`"]; Get["Teukolsky`ConvolveSource`"]; +Get["Teukolsky`PN`"]; diff --git a/Kernel/Tools.wl b/Kernel/Tools.wl new file mode 100644 index 0000000..97bb8ec --- /dev/null +++ b/Kernel/Tools.wl @@ -0,0 +1,277 @@ +(* ::Package:: *) + +(* ::Input:: *) +(*SetOptions[EvaluationNotebook[],StyleDefinitions->$UserBaseDirectory<>"/SystemFiles/FrontEnd/StyleSheets/maTHEMEatica.nb"]*) + + +(* ::Section:: *) +(*Beginning*) + + +(* ::Subsection:: *) +(*Beginning*) + + +BeginPackage["Teukolsky`PN`Tools`",{"Teukolsky`","Teukolsky`PN`"}] + + +(* ::Subsection:: *) +(*Unprotecting*) + + +ClearAttributes[{\[Nu]MST, aMST,MSTCoefficients}, {Protected, ReadProtected}]; + + +ClearAttributes[{SeriesTake, SeriesMinOrder,SeriesMaxOrder,SeriesLength,SeriesCollect,SeriesTerms,IgnoreExpansionParameter}, {Protected, ReadProtected}]; + + +ClearAttributes[{PNScalings, RemovePN,Zero,One}, {Protected, ReadProtected}]; + + +ClearAttributes[{ExpandLog, ExpandGamma,ExpandPolyGamma,PochhammerToGamma,GammaToPochhammer,ExpandDiracDelta,ExpandSpheroidals,CollectDerivatives}, {Protected, ReadProtected}]; + + +ClearAttributes[{TeukolskyAmplitudePN, InvariantWronskian,TeukolskySourceCircularOrbit,TeukolskyEquation}, {Protected, ReadProtected}]; + + +(* ::Section:: *) +(*Public*) + + +(* ::Subsection::Closed:: *) +(*MST Coefficients*) + + +\[Nu]MST::usage="\[Nu]MST is representative of the \[Nu] coefficient in the MST solutions" +aMST::usage="aMST[\!\(\* +StyleBox[\"n\",\nFontSlant->\"Italic\"]\)] is the \!\(\*SuperscriptBox[ +StyleBox[\"n\",\nFontSlant->\"Italic\"], \(th\)]\) MST coefficient"; +MSTCoefficients::usage="MSTCoefficients[\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]] gives the PN expanded MST coefficients aMST[n] for a given {\[ScriptS],\[ScriptL],\[ScriptM]} mode up to \[Eta]^order\[Eta]." +(*KerrMSTSeries::usage="KerrMSTSeries[\!\(\* +StyleBox[\"\[ScriptS]\",\nFontSlant->\"Italic\"]\),\!\(\* +StyleBox[\"\[ScriptL]\",\nFontSlant->\"Italic\"]\),\!\(\* +StyleBox[\"\[ScriptM]\",\nFontSlant->\"Italic\"]\),\!\(\* +StyleBox[\"order\[Epsilon]\",\nFontSlant->\"Italic\"]\)] gives the PN expanded MST coefficients a[n] for a given {\!\(\* +StyleBox[\"\[ScriptS]\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"\[ScriptL]\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"\[ScriptM]\",\nFontSlant->\"Italic\"]\)} mode up to \!\(\*SuperscriptBox[\(\[Epsilon]\), \(order\[Epsilon]\)]\). Where the relation to \[Eta] is given by \[Epsilon]=2 \[Omega] \!\(\*SuperscriptBox[\(\[Eta]\), \(3\)]\)."*) + + +(* ::Subsection::Closed:: *) +(*General Tools for Series*) + + +SeriesTake::usage="SeriesTake[series, n] takes the first n terms of series" +SeriesMinOrder::usage="SeriesMinOrder[series] gives the leading order of series" +SeriesMaxOrder::usage="SeriesMaxOrder[series] gives the first surpressed order of series" +SeriesLength::usage="SeriesLenght[series] gives the number of terms in series" +SeriesCollect::usage="SeriesCollect[expr, var, func] works like Collect but applied to each order individually. Crucially, unlike Collect it keeps the SeriesData structure." +SeriesTerms::usage="SeriesTerms[series, {x, x0, n}] works exactly like Series, with the difference that n gives the desired number of terms instead of a maximum order" +IgnoreExpansionParameter::usage="IgnoreExpansionParameter[series,x] sets all occurences of the expansion parameter in the series coefficients to x. If no value is entered x defaults to 1." + + +(* ::Subsection::Closed:: *) +(*Tools for PN Scalings*) + + +PNScalings::usage="PNScalings[expr,params,var] applies the given powercounting scalings to the expression. E.g. PNScalings[\[Omega] r,{{\[Omega],3},{r,-2},\[Eta]]" +RemovePN::usage="PNScalings[expr,var] takes the Normal[] and sets var to 1" +Zero::usage="Zero[expr,vars] sets all vars in expr to 0" +One::usage="One[expr,vars] sets all vars in expr to 1" + + +(* ::Subsection:: *) +(*Tools for Logs, Gammas, and PolyGammas*) + + +ExpandLog::usage="ExpandLog[expr] replaces all Logs in expr with a PowerExpanded version" +ExpandGamma::usage="ExpandGamma[expr] factors out all Integer facors out of the Gammas in expr. E.g. Gamma[x+1]->x Gamma[x]" +ExpandPolyGamma::usage="ExpandPolyGamma[expr] factors out all Integer facors out of the PolyGammas in expr. E.g. PolyGamma[x+1]->\!\(\*FractionBox[\(1\), \(x\)]\) PolyGamma[x]" +PochhammerToGamma::usage="PochhammerToGamma[expr] replaces all Pochhammer in expr with the respecive Gamma." +GammaToPochhammer::usage="PochhammerToGamma[expr,n] replaces all Gamma in expr that contain n with the respective Pochhammer[__,n]" + + + +(* ::Subsection:: *) +(*Tools for DiracDelta *) + + +ExpandDiracDelta::usage="ExpandDiracDelta[expr,r] applies identities for Dirac deltas and it's derivatives to expr." + + +(* ::Subsection:: *) +(*Tools for SpinWeightedSpheroidalHarmonics *) + + +ExpandSpheroidals::usage="ExpandSpheroidal[expr,{param,order}] returns a all SpinWeightedSpheroidalHarmonicS in expr have been Series expanded around param->0 to order." + + +(* ::Subsection:: *) +(*Other Tools*) + + +CollectDerivatives::usage="CollectDerivatives[expr,f] works exactly like Collect[] but also collects for derivatives of f." + + +(* ::Subsection:: *) +(*Amplitudes*) + + +TeukolskyAmplitudePN::usage="TeukolskyAmplitudePN[\"sol\"][\[ScriptS], \[ScriptL], \[ScriptM], a, \[Omega], {\[Eta], n}] gives the desired PN expanded amplitude. Options for sol are as follows: +\"A+\": Sasaki Tagoshi Eq.(157), +\"A-\": ST Eq.(158), +\"Btrans\": ST Eq.(167), +\"Binc\": ST Eq.(168) divided by \!\(\*SubscriptBox[\(\[ScriptCapitalK]\), \(\[Nu]\)]\), +\"Ctrans\": Eq.(170) ST, +\"\[ScriptCapitalK]\": , +\"\[ScriptCapitalK]\[Nu]\": , +\"\[ScriptCapitalK]-\[Nu]-1\": " + + +(* ::Subsection:: *) +(*Wronskian*) + + +InvariantWronskian::usage="InvariantWronskian[\[ScriptS], \[ScriptL], \[ScriptM], a, \[Omega], {\[Eta], n}] gives the invariant Wronskian." + + +(* ::Subsection:: *) +(*Source*) + + +TeukolskySourceCircularOrbit::usage="TeukolskySource[\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{r,r\:2080}] gives an analytical expression for the Teukolsky point particle source for a given {\[ScriptS],\[ScriptL],\[ScriptM]} mode. " + + +(* ::Subsection:: *) +(*Teukolsky Equation*) + + +TeukolskyEquation::usage="TeukolskyEquation[\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{\[Eta],order},R[r]] gives the Teukolsky equation with for a given {\[ScriptS],\[ScriptL],\[ScriptM]} mode with included \[Eta] scalings. The {\[Eta],order} argument can be left out for a general expression." + + +(* ::Section::Closed:: *) +(*Private*) + + +Begin["Private`"] + + +(* ::Subsection::Closed:: *) +(*MST Coefficients*) + + +MSTCoefficients=Teukolsky`PN`Private`MSTCoefficients + + +(* ::Subsection:: *) +(*General Tools for Series*) + + +SeriesTake=Teukolsky`PN`Private`SeriesTake +SeriesMinOrder=Teukolsky`PN`Private`SeriesMinOrder +SeriesMaxOrder=Teukolsky`PN`Private`SeriesMaxOrder +SeriesLength=Teukolsky`PN`Private`SeriesLength +SeriesCollect=Teukolsky`PN`Private`SeriesCollect +SeriesTerms=Teukolsky`PN`Private`SeriesTerms +IgnoreExpansionParameter=Teukolsky`PN`Private`IgnoreExpansionParameter + + +(* ::Subsection::Closed:: *) +(*Tools for PN Scalings*) + + +PNScalings=Teukolsky`PN`Private`PNScalings +RemovePN=Teukolsky`PN`Private`RemovePN +Zero=Teukolsky`PN`Private`Zero +One=Teukolsky`PN`Private`One + + +(* ::Subsection::Closed:: *) +(*Tools for Logs, Gammas, and PolyGammas*) + + +ExpandLog=Teukolsky`PN`Private`ExpandLog +ExpandGamma=Teukolsky`PN`Private`ExpandGamma +ExpandPolyGamma=Teukolsky`PN`Private`ExpandPolyGamma +PochhammerToGamma=Teukolsky`PN`Private`PochhammerToGamma +GammaToPochhammer=Teukolsky`PN`Private`GammaToPochhammer + + +(* ::Subsection::Closed:: *) +(*Tools for DiracDelta *) + + +ExpandDiracDelta=Teukolsky`PN`Private`ExpandDiracDelta + + +(* ::Subsection::Closed:: *) +(*Tools for SpinWeightedSpheroidalHarmonics *) + + +ExpandSpheroidals=Teukolsky`PN`Private`ExpandSpheroidals + + +(* ::Subsection::Closed:: *) +(*Other Tools*) + + +CollectDerivatives=Teukolsky`PN`Private`CollectDerivatives + + +(* ::Subsection::Closed:: *) +(*Amplitudes*) + + +TeukolskyAmplitudePN=Teukolsky`PN`Private`TeukolskyAmplitudePN + + +(* ::Subsection::Closed:: *) +(*Wronskian*) + + +InvariantWronskian=Teukolsky`PN`Private`InvariantWronskian + + +(* ::Subsection::Closed:: *) +(*Source*) + + +TeukolskySourceCircularOrbit=Teukolsky`PN`Private`TeukolskySourceCircularOrbit + + +(* ::Subsection::Closed:: *) +(*TeukolskyEquation*) + + +TeukolskyEquation=Teukolsky`PN`Private`TeukolskyEquation + + +(* ::Section:: *) +(*Ending Package*) + + +(* ::Subsection:: *) +(*Protecting*) + + +SetAttributes[{\[Nu]MST, aMST,MSTCoefficients}, {Protected, ReadProtected}]; + + +SetAttributes[{SeriesTake, SeriesMinOrder,SeriesMaxOrder,SeriesLength,SeriesCollect,SeriesTerms,IgnoreExpansionParameter}, {Protected, ReadProtected}]; + + +SetAttributes[{PNScalings, RemovePN,Zero,One}, {Protected, ReadProtected}]; + + +SetAttributes[{ExpandLog, ExpandGamma,ExpandPolyGamma,PochhammerToGamma,GammaToPochhammer,ExpandDiracDelta,ExpandSpheroidals,CollectDerivatives}, {Protected, ReadProtected}]; + + +SetAttributes[{TeukolskyAmplitudePN, InvariantWronskian,TeukolskySourceCircularOrbit,TeukolskyEquation}, {Protected, ReadProtected}]; + + +(* ::Subsection:: *) +(*Ending*) + + +End[] +EndPackage[] From f0cbb6c0e376afc2491d7ab07b7ee9e999b4f2bf Mon Sep 17 00:00:00 2001 From: jakobneef Date: Wed, 11 Dec 2024 14:38:10 +0000 Subject: [PATCH 02/13] fixed bug in SeriesMinOrder --- .../Symbols/TeukolskyPointParticleModePN.nb | 10258 ++++++++++++++++ .../Symbols/TeukolskyRadialPN.nb | 3694 ++++++ Kernel/PN.wl | 41 +- 3 files changed, 13973 insertions(+), 20 deletions(-) create mode 100644 Documentation/English/ReferencePages/Symbols/TeukolskyPointParticleModePN.nb create mode 100644 Documentation/English/ReferencePages/Symbols/TeukolskyRadialPN.nb diff --git a/Documentation/English/ReferencePages/Symbols/TeukolskyPointParticleModePN.nb b/Documentation/English/ReferencePages/Symbols/TeukolskyPointParticleModePN.nb new file mode 100644 index 0000000..bc07c7c --- /dev/null +++ b/Documentation/English/ReferencePages/Symbols/TeukolskyPointParticleModePN.nb @@ -0,0 +1,10258 @@ +(* Content-type: application/vnd.wolfram.mathematica *) + +(*** Wolfram Notebook File ***) +(* http://www.wolfram.com/nb *) + +(* CreatedBy='Mathematica 14.0' *) + +(*CacheID: 234*) +(* Internal cache information: +NotebookFileLineBreakTest +NotebookFileLineBreakTest +NotebookDataPosition[ 158, 7] +NotebookDataLength[ 444837, 10250] +NotebookOptionsPosition[ 431924, 9976] +NotebookOutlinePosition[ 432708, 10002] +CellTagsIndexPosition[ 432627, 9997] +WindowFrame->Normal*) + +(* Beginning of Notebook Content *) +Notebook[{ + +Cell[CellGroupData[{ +Cell["TeukolskyPointParticleModePN", "ObjectName", + CellID->1397816838,ExpressionUUID->"22d23bd2-f676-4b1f-ba41-7ba131ecf226"], + +Cell[TextData[{ + Cell[" ", "ModInfo",ExpressionUUID->"ab67f0ed-259b-4fb5-bdee-b429755bf293"], + "TeukolskyPointParticleModePN[\[ScriptS], ", + Cell[BoxData[ + StyleBox["\[ScriptL]", "TI"]], "InlineFormula",ExpressionUUID-> + "be37f3ff-85ce-4b96-9b80-7f8fb32b89ce"], + ", ", + Cell[BoxData[ + StyleBox["\[ScriptM]", "TI"]], "InlineFormula",ExpressionUUID-> + "d4f97d34-1aec-4e1d-a2ab-d9c41177094e"], + ", orbit, { \[Eta], n}] produces a TeukolskyModePN representing a PN \ +expanded analytical solution to the radial Teukolsky equation with a point \ +particle source." +}], "Usage", + CellChangeTimes->{{3.942221938113145*^9, 3.9422219491809483`*^9}, { + 3.942307934869256*^9, 3.942307958753386*^9}}, + CellID->502739332,ExpressionUUID->"7fc170f0-fb0a-4785-bb5a-972c73accea2"], + +Cell["\<\ +This Function computes a TeukolskyModePN[] for a given mode and orbit. As of \ +now it only supports circular equatorial orbits in Kerr. \ +\>", "Notes", + CellChangeTimes->{{3.942308248878138*^9, 3.9423082558379707`*^9}, { + 3.942308287433778*^9, 3.942308330953247*^9}}, + CellID->308730800,ExpressionUUID->"2006ab6f-1bbf-439f-918e-919e9054ec76"], + +Cell[TextData[{ + "The main output is accessed by the Key \"RadialFuncition\" which gives ", + Cell[BoxData[ + RowBox[{ + SubscriptBox["R", "\[ScriptL]\[ScriptM]\[Omega]"], "=", " ", + RowBox[{ + RowBox[{ + SubscriptBox["c", "in"], " ", + SubscriptBox["R", "In"]}], "+", + RowBox[{ + SubscriptBox["c", "up"], " ", + SubscriptBox["R", "In"]}]}]}]], "InlineFormula", + FormatType->StandardForm,ExpressionUUID-> + "5f6cb3b1-e152-49f4-8768-a1cf725872b8"], + " where ", + Cell[BoxData[ + RowBox[{ + SubscriptBox["c", "In"], "=", + RowBox[{ + SubsuperscriptBox["\[Integral]", "r", "\[Infinity]"], + RowBox[{ + FractionBox[ + RowBox[{ + SubscriptBox["R", "Up"], "S"}], "Wronskian"], + RowBox[{"\[DifferentialD]", "r"}]}]}]}]], "InlineFormula", + FormatType->StandardForm,ExpressionUUID-> + "42fe0dd7-b2af-4d80-9b6d-7f1648eb4d21"], + " and ", + Cell[BoxData[ + RowBox[{ + SubscriptBox["c", "Up"], "=", + RowBox[{ + SubsuperscriptBox["\[Integral]", "0", "r"], + RowBox[{ + FractionBox[ + RowBox[{ + SubscriptBox["R", "In"], "S"}], "Wronskian"], + RowBox[{"\[DifferentialD]", "r"}]}]}]}]], "InlineFormula", + FormatType->StandardForm,ExpressionUUID-> + "8e15cf07-4e97-4cca-9829-df51585d360f"], + ". 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Teukolsky`PN`TeukolskyPointParticleModePN[-2, 2, + 2, $CellContext`a, $CellContext`r0, {$CellContext`\[Eta], 3}, <| + "\[ScriptS]" -> -2, "\[ScriptL]" -> 2, "\[ScriptM]" -> 2, + "a" -> $CellContext`a, "r0" -> $CellContext`r0, + "PN" -> {$CellContext`\[Eta], 3}, + "RadialFunction" -> ( + Teukolsky`PN`Private`innerF$104659[#] + HeavisideTheta[$CellContext`r0 - #] + + Teukolsky`PN`Private`outerF$104659[#] + HeavisideTheta[# - $CellContext`r0] + + Teukolsky`PN`Private`deltaCoeff$104659 + DiracDelta[# - $CellContext`r0]& ), + "Inner" -> (Teukolsky`PN`Private`cIn$104659 + Teukolsky`PN`Private`Rin$104659[#]& ), + "Outer" -> (Teukolsky`PN`Private`cUp$104659 + Teukolsky`PN`Private`Rup$104659[#]& ), "\[Delta]" -> + SeriesData[$CellContext`\[Eta], + 0, {-Pi $CellContext`r0^(-3) + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, + 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0], 0, 2 $CellContext`a + Pi $CellContext`r0^Rational[-7, 2] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, + 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0], 0, + Pi (Rational[-3, 2] $CellContext`r0^Rational[-17, 4] - ( + 2 $CellContext`r0^Rational[-7, 2] + $CellContext`a^2 $CellContext`r0^ + Rational[-7, 2]) $CellContext`r0^Rational[-3, 4]) $CellContext`r0^ + Rational[1, 4] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, + 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0], 0, 9 $CellContext`a + Pi $CellContext`r0^Rational[-9, 2] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, + 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0], 0, + Pi (Rational[-27, 8] $CellContext`r0^Rational[-21, 4] - + 2 $CellContext`a^2 $CellContext`r0^Rational[-21, 4] + + Rational[-3, 2] ( + 2 $CellContext`r0^ + Rational[-7, 2] + $CellContext`a^2 $CellContext`r0^ + Rational[-7, 2]) $CellContext`r0^Rational[-7, 4] - ( + 4 $CellContext`a^2 $CellContext`r0^Rational[-9, 2] + ( + 4 $CellContext`r0^(-2) - $CellContext`a^2 $CellContext`r0^(-2)) \ +$CellContext`r0^Rational[-5, 2]) $CellContext`r0^ + Rational[-3, 4]) $CellContext`r0^Rational[1, 4] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, + 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0], 0, + Pi (Rational[75, 4] $CellContext`a $CellContext`r0^ + Rational[-23, 4] + $CellContext`a ( + 2 $CellContext`r0^ + Rational[-7, 2] + $CellContext`a^2 $CellContext`r0^ + Rational[-7, 2]) $CellContext`r0^ + Rational[-9, + 4] - ((-2) $CellContext`a $CellContext`r0^(-5) - $CellContext`a^3 \ +$CellContext`r0^(-5) - + 2 $CellContext`a ( + 4 $CellContext`r0^(-2) - $CellContext`a^2 $CellContext`r0^(-2)) \ +$CellContext`r0^(-3)) $CellContext`r0^Rational[-3, 4]) $CellContext`r0^ + Rational[1, 4] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, + 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0], 0, + Pi $CellContext`r0^ + Rational[ + 1, 4] ((-14) $CellContext`a^2 $CellContext`r0^Rational[-25, 4] + + Rational[-1, 6] (Rational[405, 8] $CellContext`r0^(-3) + + 9 $CellContext`a^2 $CellContext`r0^(-3)) $CellContext`r0^ + Rational[-13, 4] + + Rational[-27, 8] ( + 2 $CellContext`r0^ + Rational[-7, 2] + $CellContext`a^2 $CellContext`r0^ + Rational[-7, 2]) $CellContext`r0^Rational[-11, 4] + + Rational[-3, 2] ( + 4 $CellContext`a^2 $CellContext`r0^Rational[-9, 2] + ( + 4 $CellContext`r0^(-2) - $CellContext`a^2 $CellContext`r0^(-2)) \ +$CellContext`r0^Rational[-5, 2]) $CellContext`r0^ + Rational[-7, 4] - $CellContext`r0^Rational[-3, 4] ( + 4 $CellContext`a^2 $CellContext`r0^ + Rational[-11, 2] + $CellContext`a^2 ( + 4 $CellContext`r0^(-2) - $CellContext`a^2 $CellContext`r0^(-2)) \ +$CellContext`r0^ + Rational[-7, 2] + ($CellContext`a^2 $CellContext`r0^ + Rational[-13, 2] + (8 $CellContext`r0^(-3) - + 4 $CellContext`a^2 $CellContext`r0^(-3)) $CellContext`r0^ + Rational[-7, 2]) $CellContext`r0)) + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, + 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0], 0, + Pi (Rational[135, 4] $CellContext`a $CellContext`r0^Rational[-27, 4] + + Rational[1, 3] $CellContext`a (Rational[405, 8] $CellContext`r0^(-3) + + 9 $CellContext`a^2 $CellContext`r0^(-3)) $CellContext`r0^ + Rational[-15, 4] + + Rational[9, 2] $CellContext`a ( + 2 $CellContext`r0^ + Rational[-7, 2] + $CellContext`a^2 $CellContext`r0^ + Rational[-7, 2]) $CellContext`r0^ + Rational[-13, 4] + $CellContext`a ( + 4 $CellContext`a^2 $CellContext`r0^Rational[-9, 2] + ( + 4 $CellContext`r0^(-2) - $CellContext`a^2 $CellContext`r0^(-2)) \ +$CellContext`r0^Rational[-5, 2]) $CellContext`r0^Rational[-9, 4] + + Rational[-3, + 2] ((-2) $CellContext`a $CellContext`r0^(-5) - $CellContext`a^3 \ +$CellContext`r0^(-5) - + 2 $CellContext`a ( + 4 $CellContext`r0^(-2) - $CellContext`a^2 $CellContext`r0^(-2)) \ +$CellContext`r0^(-3)) $CellContext`r0^ + Rational[-7, + 4] - ((-2) $CellContext`a^3 $CellContext`r0^(-6) - $CellContext`a \ +(4 $CellContext`r0^(-2) - $CellContext`a^2 $CellContext`r0^(-2)) \ +$CellContext`r0^(-4) - + 2 $CellContext`a ($CellContext`a^2 $CellContext`r0^ + Rational[-13, 2] + (8 $CellContext`r0^(-3) - + 4 $CellContext`a^2 $CellContext`r0^(-3)) $CellContext`r0^ + Rational[-7, 2]) $CellContext`r0^Rational[1, 2]) $CellContext`r0^ + Rational[-3, 4]) $CellContext`r0^Rational[1, 4] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, + 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0], 0, + Pi $CellContext`r0^Rational[1, 4] ( + Rational[-225, 4] $CellContext`a^2 $CellContext`r0^Rational[-29, 4] + + Rational[-1, 8] ( + Rational[117, 2] $CellContext`a^2 $CellContext`r0^(-4) + + Rational[7, 2] (Rational[405, 8] $CellContext`r0^(-3) + + 9 $CellContext`a^2 $CellContext`r0^(-3))/$CellContext`r0) \ +$CellContext`r0^Rational[-13, 4] + + Rational[-27, 8] ( + 4 $CellContext`a^2 $CellContext`r0^Rational[-9, 2] + ( + 4 $CellContext`r0^(-2) - $CellContext`a^2 $CellContext`r0^(-2)) \ +$CellContext`r0^Rational[-5, 2]) $CellContext`r0^ + Rational[-11, + 4] + $CellContext`a ((-2) $CellContext`a $CellContext`r0^(-5) - \ +$CellContext`a^3 $CellContext`r0^(-5) - + 2 $CellContext`a ( + 4 $CellContext`r0^(-2) - $CellContext`a^2 $CellContext`r0^(-2)) \ +$CellContext`r0^(-3)) $CellContext`r0^Rational[-9, 4] + + Rational[-1, 6] ( + 2 $CellContext`r0^ + Rational[-7, 2] + $CellContext`a^2 $CellContext`r0^ + Rational[-7, 2]) (Rational[405, 8] $CellContext`r0^(-3) + + 9 $CellContext`a^2 $CellContext`r0^(-3)) $CellContext`r0^ + Rational[-3, 4] - $CellContext`r0^ + Rational[-3, + 4] ($CellContext`a^2 ($CellContext`a^2 $CellContext`r0^ + Rational[-13, 2] + (8 $CellContext`r0^(-3) - + 4 $CellContext`a^2 $CellContext`r0^(-3)) $CellContext`r0^ + Rational[-7, 2]) + + 2 $CellContext`a^2 ( + 4 $CellContext`r0^(-2) - $CellContext`a^2 $CellContext`r0^(-2)) \ +$CellContext`r0^Rational[-9, 2] + ( + 2 $CellContext`a^2 $CellContext`r0^Rational[-15, 2] + ( + 16 $CellContext`r0^(-4) - + 12 $CellContext`a^2 $CellContext`r0^(-4) + $CellContext`a^4 \ +$CellContext`r0^(-4)) $CellContext`r0^Rational[-7, 2]) $CellContext`r0) + + Rational[-3, 2] $CellContext`r0^Rational[-7, 4] ( + 4 $CellContext`a^2 $CellContext`r0^ + Rational[-11, 2] + $CellContext`a^2 ( + 4 $CellContext`r0^(-2) - $CellContext`a^2 $CellContext`r0^(-2)) \ +$CellContext`r0^ + Rational[-7, 2] + ($CellContext`a^2 $CellContext`r0^ + Rational[-13, 2] + (8 $CellContext`r0^(-3) - + 4 $CellContext`a^2 $CellContext`r0^(-3)) $CellContext`r0^ + Rational[-7, 2]) $CellContext`r0)) + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, + 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0]}, 12, 29, 2], "cUp" -> + SeriesData[$CellContext`\[Eta], 0, {Complex[0, + Rational[-2, 15]] Pi $CellContext`r0 + Teukolsky`PN`\[CapitalOmega]Kerr ( + 2 $CellContext`r0 + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]), 0, Complex[0, + Rational[-1, 96]] + Teukolsky`PN`\[CapitalOmega]Kerr^(-3) ( + Rational[512, 5] Pi $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr^4 ( + Complex[0, -2] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + + Rational[64, 5] Pi $CellContext`r0 + Teukolsky`PN`\[CapitalOmega]Kerr^4 ( + Complex[0, 8] $CellContext`r0^Rational[1, 2] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + + Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + Complex[0, + Rational[256, 15]] Pi $CellContext`r0^2 + Teukolsky`PN`\[CapitalOmega]Kerr^5 ( + 2 $CellContext`r0 + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0])), 0, Complex[0, + Rational[-1, 96]] + Teukolsky`PN`\[CapitalOmega]Kerr^(-3) ( + Rational[-256, 5] Pi $CellContext`r0 + Teukolsky`PN`\[CapitalOmega]Kerr^4 + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + $CellContext`r0^Rational[-1, 2] ( + Rational[-512, 5] $CellContext`a Pi $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr^4 + Complex[0, + Rational[512, 3]] Pi $CellContext`r0^3 + Teukolsky`PN`\[CapitalOmega]Kerr^5) ( + Complex[0, -2] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + Complex[0, + Rational[256, 15]] Pi $CellContext`r0^2 + Teukolsky`PN`\[CapitalOmega]Kerr^5 ( + Complex[0, 8] $CellContext`r0^Rational[1, 2] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + + Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + + Rational[64, 5] Pi $CellContext`r0 + Teukolsky`PN`\[CapitalOmega]Kerr^4 ((-4) ( + Complex[0, 1] $CellContext`a + + 2 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr) + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + + 2 (Complex[0, 1] $CellContext`a + + 2 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr) + Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + ( + Rational[96, 5] Pi Teukolsky`PN`\[CapitalOmega]Kerr^4 + + Pi $CellContext`r0^(-3) ( + Rational[256, 21] $CellContext`r0^6 + Teukolsky`PN`\[CapitalOmega]Kerr^6 + + Rational[ + 1, 5] ((-64) $CellContext`r0^3 + Teukolsky`PN`\[CapitalOmega]Kerr^3 ( + 2 Teukolsky`PN`\[CapitalOmega]Kerr + + 2 (1 - $CellContext`a^2)^Rational[1, 2] + Teukolsky`PN`\[CapitalOmega]Kerr) + + 4 $CellContext`r0^2 + Teukolsky`PN`\[CapitalOmega]Kerr^2 ( + Complex[0, -32] $CellContext`a $CellContext`r0 + Teukolsky`PN`\[CapitalOmega]Kerr^2 + + 32 (-1 + (1 - $CellContext`a^2)^ + Rational[1, 2]) $CellContext`r0 + Teukolsky`PN`\[CapitalOmega]Kerr^2 - 32 $CellContext`r0^4 + Teukolsky`PN`\[CapitalOmega]Kerr^4)))) ( + 2 $CellContext`r0 + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]))}, -2, 4, 2], "cIn" -> + SeriesData[$CellContext`\[Eta], 0, { + Rational[-1, 64] Pi $CellContext`r0^(-4) + Teukolsky`PN`\[CapitalOmega]Kerr^(-4) ( + 2 $CellContext`r0 + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]), 0, Complex[0, + Rational[-1, 96]] + Teukolsky`PN`\[CapitalOmega]Kerr^(-3) ( + Complex[0, 3] Pi $CellContext`r0^Rational[-7, 2] + Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ( + Complex[0, -2] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + Complex[0, + Rational[-3, 2]] Pi $CellContext`r0^(-4) + Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ( + Complex[0, 8] $CellContext`r0^Rational[1, 2] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + + Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) - 3 + Pi $CellContext`r0^(-3) ( + 2 $CellContext`r0 + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0])), 0, Complex[0, + Rational[-1, 96]] + Teukolsky`PN`\[CapitalOmega]Kerr^(-3) ( + Complex[0, 6] Pi $CellContext`r0^(-4) + Teukolsky`PN`\[CapitalOmega]Kerr^(-1) + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + + Complex[0, -3] $CellContext`a Pi $CellContext`r0^(-4) + Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ( + Complex[0, -2] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) - 3 + Pi $CellContext`r0^(-3) ( + Complex[0, 8] $CellContext`r0^Rational[1, 2] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + + Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + Complex[0, + Rational[-3, 2]] Pi $CellContext`r0^(-4) + Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ((-4) ( + Complex[0, 1] $CellContext`a + + 2 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr) + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + + 2 (Complex[0, 1] $CellContext`a + + 2 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr) + Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + (Complex[0, + Rational[-9, 4]] Pi $CellContext`r0^(-5)/ + Teukolsky`PN`\[CapitalOmega]Kerr + + Pi $CellContext`r0^(-3) (Complex[0, + Rational[-336, 107]] $CellContext`r0^4 + Teukolsky`PN`\[CapitalOmega]Kerr^3 + + Rational[105, 428] + Teukolsky`PN`\[CapitalOmega]Kerr^(-1) (Complex[0, + Rational[-107, 35]] $CellContext`r0^(-2) (2 + + Complex[0, -4] $CellContext`a - + 10 (1 - $CellContext`a^2)^Rational[1, 2] - + 4 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2) + + Rational[2, 105] $CellContext`r0^(-2) (1070 $CellContext`a + + Complex[0, -1605] (1 - $CellContext`a^2)^Rational[1, 2] + + Complex[0, 672] $CellContext`r0^6 + Teukolsky`PN`\[CapitalOmega]Kerr^4)))) ( + 2 $CellContext`r0 + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]))}, -12, -6, 2], "Wronskian" -> + SeriesData[$CellContext`\[Eta], 0, { + Complex[0, 96] Teukolsky`PN`\[CapitalOmega]Kerr^3}, 9, 12, 1], + "Source" -> ( + Teukolsky`PN`Private`TeukolskySourceCircularOrbit[-2, 2, + 2, $CellContext`a, {#, $CellContext`r0}, "Form" -> + "InvariantWronskian"]& ), "In" -> + Teukolsky`PN`TeukolskyRadialFunctionPN[-2, 2, 2, $CellContext`a, 2 + Teukolsky`PN`\[CapitalOmega]Kerr, {$CellContext`\[Eta], 3}, <| + "\[ScriptS]" -> -2, "\[ScriptL]" -> 2, "\[ScriptM]" -> 2, + "a" -> $CellContext`a, "PN" -> {$CellContext`\[Eta], 3}, + "RadialFunction" -> Function[Teukolsky`PN`Private`r, + SeriesData[$CellContext`\[Eta], 0, { + Rational[64, 5] Teukolsky`PN`Private`r^4 + Teukolsky`PN`\[CapitalOmega]Kerr^4, Complex[0, + Rational[256, 15]] Teukolsky`PN`Private`r^5 + Teukolsky`PN`\[CapitalOmega]Kerr^5, + Rational[256, 21] Teukolsky`PN`Private`r^6 + Teukolsky`PN`\[CapitalOmega]Kerr^6 + + Rational[ + 1, 5] ((-64) Teukolsky`PN`Private`r^3 + Teukolsky`PN`\[CapitalOmega]Kerr^3 ( + 2 Teukolsky`PN`\[CapitalOmega]Kerr + + 2 (1 - $CellContext`a^2)^Rational[1, 2] + Teukolsky`PN`\[CapitalOmega]Kerr) + + 4 Teukolsky`PN`Private`r^2 + Teukolsky`PN`\[CapitalOmega]Kerr^2 ( + Complex[0, -32] $CellContext`a Teukolsky`PN`Private`r + Teukolsky`PN`\[CapitalOmega]Kerr^2 + + 32 (-1 + (1 - $CellContext`a^2)^Rational[1, 2]) + Teukolsky`PN`Private`r Teukolsky`PN`\[CapitalOmega]Kerr^2 - + 32 Teukolsky`PN`Private`r^4 + Teukolsky`PN`\[CapitalOmega]Kerr^4))}, 4, 7, 1]], + "BoundaryCondition" -> "In", "LeadingOrder" -> + Function[Teukolsky`PN`Private`r, + SeriesData[$CellContext`\[Eta], 0, { + Rational[64, 5] Teukolsky`PN`Private`r^4 + Teukolsky`PN`\[CapitalOmega]Kerr^4}, 4, 5, 1]], "TermCount" -> 3, + "Normalization" -> "SFPN", + "Amplitudes" -> <|"Transmission" -> "Not Computed"|>|>], "Up" -> + Teukolsky`PN`TeukolskyRadialFunctionPN[-2, 2, 2, $CellContext`a, 2 + Teukolsky`PN`\[CapitalOmega]Kerr, {$CellContext`\[Eta], 3}, <| + "\[ScriptS]" -> -2, "\[ScriptL]" -> 2, "\[ScriptM]" -> 2, + "a" -> $CellContext`a, "PN" -> {$CellContext`\[Eta], 3}, + "RadialFunction" -> Function[Teukolsky`PN`Private`r, + SeriesData[$CellContext`\[Eta], 0, {Complex[0, + Rational[-3, 2]] Teukolsky`PN`Private`r^(-1)/ + Teukolsky`PN`\[CapitalOmega]Kerr, -3, Complex[0, + Rational[-336, 107]] Teukolsky`PN`Private`r^4 + Teukolsky`PN`\[CapitalOmega]Kerr^3 + + Rational[105, 428] + Teukolsky`PN`\[CapitalOmega]Kerr^(-1) (Complex[0, + Rational[-107, 35]] + Teukolsky`PN`Private`r^(-2) (2 + Complex[0, -4] $CellContext`a - + 10 (1 - $CellContext`a^2)^Rational[1, 2] - 4 + Teukolsky`PN`Private`r^3 Teukolsky`PN`\[CapitalOmega]Kerr^2) + + Rational[2, 105] + Teukolsky`PN`Private`r^(-2) (1070 $CellContext`a + + Complex[0, -1605] (1 - $CellContext`a^2)^Rational[1, 2] + + Complex[0, 672] Teukolsky`PN`Private`r^6 + Teukolsky`PN`\[CapitalOmega]Kerr^4))}, -1, 2, 1]], + "BoundaryCondition" -> "Up", "LeadingOrder" -> + Function[Teukolsky`PN`Private`r, + SeriesData[$CellContext`\[Eta], 0, {Complex[0, + Rational[-3, 2]] Teukolsky`PN`Private`r^(-1)/ + Teukolsky`PN`\[CapitalOmega]Kerr}, -1, 0, 1]], "TermCount" -> 3, + "Normalization" -> "SFPN", + "Amplitudes" -> <|"Transmission" -> "Not Computed"|>|>]|>], + Editable->False, + SelectWithContents->True, + Selectable->False]], "Output", + CellChangeTimes->{{3.94222200433076*^9, 3.942222016566121*^9}, + 3.942222158382213*^9}, + CellLabel->"Out[10]=", + CellID->856291874,ExpressionUUID->"eaa4202c-5951-44c8-9216-204e8c654020"] +}, Open ]], + +Cell["The main use is now to use it as a Function[] ", "ExampleText", + CellChangeTimes->{{3.9422225202895*^9, 3.9422225372103558`*^9}}, + CellID->334635788,ExpressionUUID->"864cd375-bb7b-437b-8f73-52b511703b31"], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{ + RowBox[{ + RowBox[{"mode", 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((-42) \ +$CellContext`r0 + + 84 $CellContext`r^3 $CellContext`r0 Teukolsky`PN`\[CapitalOmega]Kerr^2 - + 112 $CellContext`r^2 $CellContext`r0^Rational[1, 2] + Teukolsky`PN`\[CapitalOmega]Kerr (-1 + $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr) + $CellContext`r (119 - + 56 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr + + 44 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2)) + HeavisideTheta[$CellContext`r - $CellContext`r0] + $CellContext`r^5 ( + 168 $CellContext`r0 + + 44 $CellContext`r^3 $CellContext`r0 Teukolsky`PN`\[CapitalOmega]Kerr^2 - + 112 $CellContext`r^2 $CellContext`r0^Rational[1, 2] + Teukolsky`PN`\[CapitalOmega]Kerr (-1 + $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr) + + 7 $CellContext`r (-13 - 8 $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr + + 12 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2)) + HeavisideTheta[-$CellContext`r + $CellContext`r0])}, -4, 2, 2], + Editable->False]], 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"\<\"\[ScriptCapitalH]\"\>"}], ",", "\<\"\[Delta]\"\>", + ",", "\<\"cUp\"\>", ",", "\<\"cIn\"\>", ",", "\<\"Wronskian\"\>", + ",", "\<\"Source\"\>", ",", "\<\"In\"\>", ",", "\<\"Up\"\>"}], + "}"}]], "Output", + CellChangeTimes->{3.942222499430403*^9, 3.9423091918116503`*^9}, + CellLabel->"Out[6]=", + CellID->1555125986,ExpressionUUID->"ad781b88-77aa-4552-9e30-1cfb824066f2"] +}, Open ]], + +Cell["\<\ +To see an explanation of all the Keys see the examples in the Scope section\ +\>", "ExampleText", + CellChangeTimes->{{3.942308786470313*^9, 3.942308810476489*^9}}, + CellID->2123700404,ExpressionUUID->"3c147753-d453-491c-89ae-32da365681c5"] +}, Open ]], + +Cell[CellGroupData[{ + +Cell[TextData[{ + "More Examples", + Cell[BoxData[ + TemplateBox[{"MoreExamples", + Cell[ + BoxData[ + FrameBox[ + Cell["Extended examples in standardized sections.", "MoreInfoText"], + BaseStyle -> "IFrameBox"]], "MoreInfoTextOuter"]}, + "MoreInfoOpenerButtonTemplate"]],ExpressionUUID-> + "3225729b-cd20-4c0c-bfcb-657b7b9021e1"] +}], "ExtendedExamplesSection", + CellTags->"ExtendedExamples", + CellID->822723576,ExpressionUUID->"4f0e9116-1c64-4883-8f53-e21686f7b049"], + +Cell[CellGroupData[{ + +Cell[BoxData[ + InterpretationBox[Cell[ + "Scope", "ExampleSection",ExpressionUUID-> + "ac0dec70-1428-454e-9f36-4bf89982db6b"], + $Line = 0; Null]], "ExampleSection", + CellID->1512661770,ExpressionUUID->"57c3a8c5-4f1c-4c78-9e9e-2b52d024a06f"], + +Cell["\<\ +Here we want to display the possible Keys that can be used to query a \ +TeukolskyModePN\ +\>", "ExampleText", + CellChangeTimes->{{3.942308774836884*^9, 3.942308779609502*^9}, { + 3.942308822367017*^9, 3.942308846368334*^9}}, + CellID->214174106,ExpressionUUID->"01d5f544-7788-4a34-bfce-b0fe3f48d890"], + +Cell[BoxData[{ + RowBox[{ + RowBox[{"orbit", "=", + RowBox[{"KerrGeoOrbit", "[", + RowBox[{"a", ",", "r0", ",", "0", ",", "1"}], "]"}]}], + ";"}], "\[IndentingNewLine]", + RowBox[{ + RowBox[{"mode", "=", + 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"\<\"Source\"\>", ",", "\<\"In\"\>", ",", "\<\"Up\"\>"}], + "}"}]], "Output", + CellChangeTimes->{{3.9423088493082743`*^9, 3.942308873691544*^9}, + 3.942308960743535*^9}, + CellLabel->"Out[5]=", + CellID->324726175,ExpressionUUID->"7831d538-ec44-46d3-a6aa-199c366976bf"] +}, Open ]], + +Cell["\<\ +\"s\", \"l\", and \"m\" give the mode values. \"a\" and \"r0\" give the Kerr \ +spin parameter and the particles radius respectively. \"PN\" gives the \ +expansion parameter and the number of terms. \ +\>", "ExampleText", + CellChangeTimes->{{3.9423091876035557`*^9, 3.942309300576494*^9}, { + 3.94230935001857*^9, 3.942309413828618*^9}}, + CellID->1057982257,ExpressionUUID->"0721f743-8a90-4335-9c9c-cc3d54b19b1b"], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{ + RowBox[{ + RowBox[{"mode", "[", "#", "]"}], "&"}], "/@", + RowBox[{"{", + RowBox[{"\"\\"", ",", "\"\\"", ",", "\"\\""}], + "}"}]}]], "Input", + CellChangeTimes->{{3.942309303978402*^9, 3.942309329641718*^9}}, + 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SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + + Rational[8, 45] $CellContext`r^5 + Teukolsky`PN`\[CapitalOmega]Kerr^2 ( + Complex[0, 3] Pi $CellContext`r0^Rational[-7, 2] + Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ( + Complex[0, -2] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + Complex[0, + Rational[-3, 2]] Pi $CellContext`r0^(-4) + Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ( + Complex[0, 8] 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Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]))), 0, + HeavisideTheta[-100 + $CellContext`r0] ( + Rational[400000, 21] + Pi $CellContext`r0^(-4) (42 + Complex[0, 21] $CellContext`a + + 11000000 Teukolsky`PN`\[CapitalOmega]Kerr^2) ( + 2 $CellContext`r0 + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + + Rational[16000000000, 9] + Teukolsky`PN`\[CapitalOmega]Kerr^2 ( + Complex[0, 3] Pi $CellContext`r0^Rational[-7, 2] + Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ( + Complex[0, -2] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + Complex[0, + Rational[-3, 2]] Pi $CellContext`r0^(-4) + Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ( + Complex[0, 8] $CellContext`r0^Rational[1, 2] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + + Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) - 3 + Pi $CellContext`r0^(-3) ( + 2 $CellContext`r0 + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0])) + Complex[0, + Rational[-40000000, 3]] + Teukolsky`PN`\[CapitalOmega]Kerr ( + Complex[0, 6] Pi $CellContext`r0^(-4) + Teukolsky`PN`\[CapitalOmega]Kerr^(-1) + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + + Complex[0, -3] $CellContext`a Pi $CellContext`r0^(-4) + Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ( + Complex[0, -2] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) - 3 + Pi $CellContext`r0^(-3) ( + Complex[0, 8] $CellContext`r0^Rational[1, 2] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + + Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + Complex[0, + Rational[-3, 2]] Pi $CellContext`r0^(-4) + Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ((-4) ( + Complex[0, 1] $CellContext`a + + 2 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr) + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + + 2 (Complex[0, 1] $CellContext`a + + 2 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr) + Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + (Complex[0, + Rational[-9, 4]] Pi $CellContext`r0^(-5)/ + Teukolsky`PN`\[CapitalOmega]Kerr + + Rational[1, 2] Pi $CellContext`r0^(-5) + Teukolsky`PN`\[CapitalOmega]Kerr^(-1) (Complex[0, -3] + + 4 $CellContext`a + + Complex[0, 6] $CellContext`r0^3 + Teukolsky`PN`\[CapitalOmega]Kerr^2)) ( + 2 $CellContext`r0 + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]))) + + HeavisideTheta[100 - $CellContext`r0] (Complex[0, + Rational[-1, 150000]] + Pi $CellContext`r0 (Complex[0, -3] + 4 $CellContext`a + + Complex[0, 6000000] Teukolsky`PN`\[CapitalOmega]Kerr^2) ( + 2 $CellContext`r0 + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + Complex[0, + Rational[1, 32]] + Teukolsky`PN`\[CapitalOmega]Kerr^(-3) ( + Rational[512, 5] Pi $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr^4 ( + Complex[0, -2] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + + Rational[64, 5] Pi $CellContext`r0 + Teukolsky`PN`\[CapitalOmega]Kerr^4 ( + Complex[0, 8] $CellContext`r0^Rational[1, 2] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + + Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + Complex[0, + Rational[256, 15]] Pi $CellContext`r0^2 + Teukolsky`PN`\[CapitalOmega]Kerr^5 ( + 2 $CellContext`r0 + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0])) + + Rational[-1, 6400] + Teukolsky`PN`\[CapitalOmega]Kerr^(-4) ( + Rational[-256, 5] Pi $CellContext`r0 + Teukolsky`PN`\[CapitalOmega]Kerr^4 + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + $CellContext`r0^Rational[-1, 2] ( + Rational[-512, 5] $CellContext`a Pi $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr^4 + Complex[0, + Rational[512, 3]] Pi $CellContext`r0^3 + Teukolsky`PN`\[CapitalOmega]Kerr^5) ( + Complex[0, -2] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + Complex[0, + Rational[256, 15]] Pi $CellContext`r0^2 + Teukolsky`PN`\[CapitalOmega]Kerr^5 ( + Complex[0, 8] $CellContext`r0^Rational[1, 2] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + + Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + + Rational[64, 5] Pi $CellContext`r0 + Teukolsky`PN`\[CapitalOmega]Kerr^4 ((-4) ( + Complex[0, 1] $CellContext`a + + 2 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr) + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + + 2 (Complex[0, 1] $CellContext`a + + 2 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr) + Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + ( + Rational[96, 5] Pi Teukolsky`PN`\[CapitalOmega]Kerr^4 + 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looks weird but is \ +operational. This is due to the fact that it is a Function[] object \ +internally that will not be evaluated unless given an explicit variable.\ +\>", "ExampleText", + CellChangeTimes->{{3.9423096373480053`*^9, 3.9423097501991253`*^9}}, + CellID->1605161360,ExpressionUUID->"9e0f866e-55a7-49f8-aaae-dce309c4b4ad"], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"rafu", "=", + RowBox[{"mode", "[", "\"\\"", "]"}]}]], "Input", + CellChangeTimes->{{3.9423097521435757`*^9, 3.942309762792602*^9}}, + CellLabel->"In[19]:=", + CellID->566631531,ExpressionUUID->"2ae0a031-bb99-4836-b23b-42ba66156178"], + +Cell[BoxData[ + RowBox[{ + RowBox[{ + RowBox[{ + RowBox[{"Teukolsky`PN`Private`innerF$7304", "[", "#1", "]"}], " ", + RowBox[{"HeavisideTheta", "[", + RowBox[{"r0", "-", "#1"}], "]"}]}], "+", + RowBox[{ + RowBox[{"Teukolsky`PN`Private`outerF$7304", "[", "#1", "]"}], " ", + RowBox[{"HeavisideTheta", "[", + RowBox[{"#1", "-", "r0"}], "]"}]}], "+", + RowBox[{"Teukolsky`PN`Private`deltaCoeff$7304", " ", + 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SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + + Rational[64, 5] Pi $CellContext`r0 + Teukolsky`PN`\[CapitalOmega]Kerr^4 ( + Complex[0, 8] $CellContext`r0^Rational[1, 2] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + + Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + Complex[0, + Rational[256, 15]] Pi $CellContext`r0^2 + Teukolsky`PN`\[CapitalOmega]Kerr^5 ( + 2 $CellContext`r0 + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]))), 0, + HeavisideTheta[-$CellContext`r + $CellContext`r0] ( + Rational[2, 105] + Pi $CellContext`r^3 $CellContext`r0^(-4) (42 + + Complex[0, 21] $CellContext`a + + 11 $CellContext`r^3 Teukolsky`PN`\[CapitalOmega]Kerr^2) ( + 2 $CellContext`r0 + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + + Rational[8, 45] $CellContext`r^5 + Teukolsky`PN`\[CapitalOmega]Kerr^2 ( + Complex[0, 3] Pi $CellContext`r0^Rational[-7, 2] + Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ( + Complex[0, -2] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + Complex[0, + Rational[-3, 2]] Pi $CellContext`r0^(-4) + Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ( + Complex[0, 8] $CellContext`r0^Rational[1, 2] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + + Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) - 3 + Pi $CellContext`r0^(-3) ( + 2 $CellContext`r0 + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0])) + Complex[0, + Rational[-2, 15]] $CellContext`r^4 + Teukolsky`PN`\[CapitalOmega]Kerr ( + Complex[0, 6] Pi $CellContext`r0^(-4) + Teukolsky`PN`\[CapitalOmega]Kerr^(-1) + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + + Complex[0, -3] $CellContext`a Pi $CellContext`r0^(-4) + Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ( + Complex[0, -2] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) - 3 + Pi $CellContext`r0^(-3) ( + Complex[0, 8] $CellContext`r0^Rational[1, 2] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + + Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + Complex[0, + Rational[-3, 2]] Pi $CellContext`r0^(-4) + Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ((-4) ( + Complex[0, 1] $CellContext`a + + 2 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr) + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + + 2 (Complex[0, 1] $CellContext`a + + 2 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr) + Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + (Complex[0, + Rational[-9, 4]] Pi $CellContext`r0^(-5)/ + Teukolsky`PN`\[CapitalOmega]Kerr + + Rational[1, 2] Pi $CellContext`r0^(-5) + Teukolsky`PN`\[CapitalOmega]Kerr^(-1) (Complex[0, -3] + + 4 $CellContext`a + + Complex[0, 6] $CellContext`r0^3 + Teukolsky`PN`\[CapitalOmega]Kerr^2)) ( + 2 $CellContext`r0 + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]))) + + HeavisideTheta[$CellContext`r - $CellContext`r0] (Complex[0, + Rational[-1, 15]] + Pi $CellContext`r^(-2) $CellContext`r0 (Complex[0, -3] + + 4 $CellContext`a + + Complex[0, 6] $CellContext`r^3 Teukolsky`PN`\[CapitalOmega]Kerr^2) ( + 2 $CellContext`r0 + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + Complex[0, + Rational[1, 32]] + Teukolsky`PN`\[CapitalOmega]Kerr^(-3) ( + Rational[512, 5] Pi $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr^4 ( + Complex[0, -2] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + + Rational[64, 5] Pi $CellContext`r0 + Teukolsky`PN`\[CapitalOmega]Kerr^4 ( + Complex[0, 8] $CellContext`r0^Rational[1, 2] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + + Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + Complex[0, + Rational[256, 15]] Pi $CellContext`r0^2 + Teukolsky`PN`\[CapitalOmega]Kerr^5 ( + 2 $CellContext`r0 + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0])) + + Rational[-1, 64] $CellContext`r^(-1) + Teukolsky`PN`\[CapitalOmega]Kerr^(-4) ( + Rational[-256, 5] Pi $CellContext`r0 + Teukolsky`PN`\[CapitalOmega]Kerr^4 + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + $CellContext`r0^Rational[-1, 2] ( + Rational[-512, 5] $CellContext`a Pi $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr^4 + Complex[0, + Rational[512, 3]] Pi $CellContext`r0^3 + Teukolsky`PN`\[CapitalOmega]Kerr^5) ( + Complex[0, -2] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + Complex[0, + Rational[256, 15]] Pi $CellContext`r0^2 + Teukolsky`PN`\[CapitalOmega]Kerr^5 ( + Complex[0, 8] $CellContext`r0^Rational[1, 2] + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + + Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + + Rational[64, 5] Pi $CellContext`r0 + Teukolsky`PN`\[CapitalOmega]Kerr^4 ((-4) ( + Complex[0, 1] $CellContext`a + + 2 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr) + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + + 2 (Complex[0, 1] $CellContext`a + + 2 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr) + Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0]) + ( + Rational[96, 5] Pi Teukolsky`PN`\[CapitalOmega]Kerr^4 + + Rational[-128, 105] Pi + Teukolsky`PN`\[CapitalOmega]Kerr^4 (42 + + Complex[0, 21] $CellContext`a + + 11 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2)) ( + 2 $CellContext`r0 + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ + Rational[1, 2] Pi, 0])))}, -4, 2, 2], + Editable->False]], "Output", + CellChangeTimes->{{3.942309768469756*^9, 3.9423097739478903`*^9}}, + CellLabel->"Out[21]=", + CellID->1199434827,ExpressionUUID->"2664a597-b3c1-49ec-9041-f7337c595d57"] +}, Closed]], + +Cell[TextData[{ + "\"ExtendedHomogeneous\" \[Rule] \"\[ScriptCapitalI]\" gives the extended \ +homogenous solution on the infinity side of the particle, i.e., ", + Cell[BoxData[ + RowBox[{ + SubscriptBox["Z", "Up"], + SubscriptBox["R", "Up"]}]], "InlineFormula", + FormatType->StandardForm,ExpressionUUID-> + "bd82a563-c1b2-46df-b6cf-67a7533a06ec"], + " where ", + Cell[BoxData[ + RowBox[{ + RowBox[{ + SubscriptBox["Z", "Up"], + RowBox[{"HeavisideTheta", "[", + RowBox[{"r", "-", "r0"}], "]"}]}], "=", + SubscriptBox["c", "Up"]}]], "InlineFormula", + FormatType->StandardForm,ExpressionUUID-> + "06fddb09-77ce-41ce-a0a0-0ab5fbaf010c"], + ". Likewise for \"ExtendedHomogeneous\" \[Rule] \"\[ScriptCapitalH]\" and In\ +\[LeftRightArrow]Up. (Note that the output is again a Function and thus has \ +the same weird behaviour as \"RadialFunction\" )" +}], "ExampleText", + CellChangeTimes->{{3.942309863097804*^9, 3.942310041198903*^9}, { + 3.9423101107986603`*^9, 3.942310177163575*^9}}, + CellID->715202156,ExpressionUUID->"09af8ff7-a58d-437b-8e6e-0c719590edf6"], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{ + RowBox[{"mode", "[", + RowBox[{ + "\"\\"", "\[Rule]", "\"\<\[ScriptCapitalI]\>\""}], + "]"}], "[", "r", "]"}]], "Input", + CellChangeTimes->{{3.942310046450639*^9, 3.942310057727206*^9}}, + CellLabel->"In[23]:=", + CellID->1645563088,ExpressionUUID->"2e15e418-5e71-4c97-ac20-def3b9046005"], + +Cell[BoxData[ + InterpretationBox[ + RowBox[{ + RowBox[{"-", + RowBox[{ + FractionBox["1", + RowBox[{"5", " ", + SuperscriptBox["r0", "4"], " ", + SuperscriptBox["\[Eta]", "2"]}]], + RowBox[{"\[Pi]", " ", + SuperscriptBox["r", "4"], " ", + RowBox[{"(", + RowBox[{ + RowBox[{"2", " ", "r0", " ", + RowBox[{ + RowBox[{"SpinWeightedSpheroidalHarmonicS", "[", + RowBox[{ + RowBox[{"-", "2"}], ",", "2", ",", "2", ",", + RowBox[{"2", " ", "a", " ", "\[CapitalOmega]Kerr"}]}], "]"}], "[", + RowBox[{ + FractionBox["\[Pi]", "2"], ",", "0"}], "]"}]}], "-", + RowBox[{"4", " ", "r0", " ", + RowBox[{ + SuperscriptBox[ + RowBox[{"SpinWeightedSpheroidalHarmonicS", "[", + RowBox[{ + RowBox[{"-", "2"}], ",", "2", ",", "2", ",", + RowBox[{"2", " ", "a", " ", "\[CapitalOmega]Kerr"}]}], "]"}], + TagBox[ + RowBox[{"(", + RowBox[{"1", ",", "0"}], ")"}], + Derivative], + MultilineFunction->None], "[", + RowBox[{ + FractionBox["\[Pi]", "2"], ",", "0"}], "]"}]}], "+", + RowBox[{"r0", " ", + RowBox[{ + SuperscriptBox[ + RowBox[{"SpinWeightedSpheroidalHarmonicS", "[", + RowBox[{ + RowBox[{"-", "2"}], ",", "2", ",", "2", ",", + RowBox[{"2", " ", "a", " ", "\[CapitalOmega]Kerr"}]}], "]"}], + TagBox[ + RowBox[{"(", + RowBox[{"2", ",", "0"}], ")"}], + Derivative], + MultilineFunction->None], "[", + RowBox[{ + 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It gives the solution to the \ +Teukolsky equation. 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Complex frequencies are TeukolskyRadialFunctionPN::paramorder="order=`1`. The given number of terms has to be an Integer"; -(* ::Subsection::Closed:: *) +(* ::Subsection:: *) (*Sourced things*) @@ -197,7 +197,7 @@ TeukolskyRadialFunctionPN::paramorder="order=`1`. The given number of terms has (*(*TeukolskySourceCircularOrbit::usage="TeukolskySource[\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{r,r\:2080}] gives an analytical expression for the Teukolsky point particle source for a given {\[ScriptS],\[ScriptL],\[ScriptM]} mode. "*)*) -TeukolskyPointParticleModePN::usage="TeukolskyPointParticleMode[\[ScriptS], \[ScriptL], \[ScriptM], orbit] produces a TeukolskyModePN representing a PN expanded analyitcal solution to the radial Teukolsky equation with a point particle source." +TeukolskyPointParticleModePN::usage="TeukolskyPointParticleModePN[s, l, m, orbit, {\[Eta], n}] produces a TeukolskyModePN representing a PN expanded analyitcal solution to the radial Teukolsky equation with a point particle source. s, l, and m specify the mode and need to be Integers. orbit needs to be a KerrGeoOrbitFunction (computed with KerrGeoOrbit[]). {\[Eta], n} specify the PN information. \[Eta] needs to be a symbol, while n is an integer specifying the amount of terms (including the Newtonian order), i.e., n=2 PNorder+1. " TeukolskyModePN::usage="aa" @@ -1205,18 +1205,18 @@ ExpandSpheroidals[expr_Times,{\[Eta]_,n_}]:=ExpandSpheroidals[#,{\[Eta],n}]&/@ex ExpandSpheroidals[expr_,{\[Eta]_,n_}]:=expr; -(* ::Subsubsection::Closed:: *) +(* ::Subsubsection:: *) (*Tools for Series*) SeriesMinOrder[series_SeriesData]:=Block[{}, -series[[4]] +series[[4]]/series[[6]] ] SeriesMinOrder[1]=0; Attributes[SeriesMinOrder]={Listable}; SeriesMaxOrder[series_SeriesData]:=Block[{}, -series[[5]] +series[[5]]/series[[6]] ] Attributes[SeriesMaxOrder]={Listable}; @@ -1253,7 +1253,8 @@ SeriesTerms[expr_,{x_,x0_,termOrder_}]:=Module[{aux,minOrder}, minOrder=Series[expr,x->x0]//SeriesMinOrder; Series[expr,{x,x0,minOrder+termOrder-1}] ]; -SeriesTerms[expr___]:=Series[expr] +SeriesTerms[expr___]:=Module[{aux}, +Series[expr]] polyToSeries[poly_,x_:\[Eta],x\:2080_:0]:=Block[{aux,maxPower}, @@ -1264,7 +1265,7 @@ polyToSeries[0,x_:\[Eta],x\:2080_:0]:=Block[{aux,maxPower}, ] -(* ::Subsubsection:: *) +(* ::Subsubsection::Closed:: *) (*Tools for Logs, Gammas, and PolyGammas*) @@ -1964,7 +1965,7 @@ ret ] -(* ::Subsection:: *) +(* ::Subsection::Closed:: *) (*Subscript[R, In]*) @@ -2220,7 +2221,7 @@ ret (*,{status,n,j}]]*)*) -(* ::Subsubsection:: *) +(* ::Subsubsection::Closed:: *) (*Constructing Subscript[R, In]*) @@ -2264,7 +2265,7 @@ RPN["In"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,aKerr_,0]:=O[\[Eta]] \[Eta]^(-\[Sc RPN["C\[Nu]"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,aKerr_,0]:=O[\[Eta]] \[Eta]^(-\[ScriptS]+\[ScriptL]-1) -(* ::Subsection:: *) +(* ::Subsection::Closed:: *) (*Subscript[R, Up]*) @@ -2355,7 +2356,7 @@ coeff table//SeriesTake[#,order\[Eta]]& *) -(* ::Subsubsection:: *) +(* ::Subsubsection::Closed:: *) (*Constructing Subscript[R, up] from Subscript[R, C]*) @@ -2395,7 +2396,7 @@ RPN["Up"][\[ScriptS]_/;\[ScriptS]<=0,\[ScriptL]_,\[ScriptM]Var_,aKerr_,0]:=O[\[E RPN["Up"][0,0,0,aKerr_,0]:=O[\[Eta]]^-1 -(* ::Subsubsection:: *) +(* ::Subsubsection::Closed:: *) (*Normalization*) @@ -2408,7 +2409,7 @@ RPN["Up"][0,0,0,aKerr_,0]:=O[\[Eta]]^-1 (*]*) -(* ::Subsection:: *) +(* ::Subsection::Closed:: *) (*Outputting Radial solutions as functions*) @@ -2427,7 +2428,7 @@ ret (*Positive spins *) -(* ::Subsubsection::Closed:: *) +(* ::Subsubsection:: *) (*Teukolsky-Starobinsky identities*) @@ -2592,7 +2593,7 @@ icons = <| |>; -(* ::Subsubsection:: *) +(* ::Subsubsection::Closed:: *) (*TeukolskyRadialPN*) @@ -2609,7 +2610,7 @@ termCount=R[r]//SeriesLength; normalization=OptionValue["Normalization"]; trans=If[OptionValue["Amplitudes"],TeukolskyAmplitudePN[Switch[sol,"In","Btrans","Up","Ctrans"],"Normalization"->OptionValue["Normalization"]][\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,order}],"Not Computed"]; amplitudes=<|"Transmission"->trans|>; -ret=<|"\[ScriptS]"->\[ScriptS],"\[ScriptL]"->\[ScriptL],"\[ScriptM]"->\[ScriptM],"a"->a,"PN"->{varPN,order},"RadialFunction"->R,"BoundaryCondition"->BC,"LeadingOrder"->lead,"TermCount"->termCount,"Normalization"->normalization,"Amplitudes"->amplitudes,"Simplified"->OptionValue["Simplify"]|>; +ret=<|"s"->\[ScriptS],"l"->\[ScriptL],"m"->\[ScriptM],"a"->a,"PN"->{varPN,order},"RadialFunction"->R,"BoundaryCondition"->BC,"LeadingOrder"->lead,"TermCount"->termCount,"Normalization"->normalization,"Amplitudes"->amplitudes,"Simplified"->OptionValue["Simplify"]|>; ret ] @@ -2623,7 +2624,7 @@ retUp=TeukolskyRadialFunctionPN[\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{var ] -(* ::Subsubsection:: *) +(* ::Subsubsection::Closed:: *) (*TeukolskyRadialFunctionPN*) @@ -2689,7 +2690,7 @@ sourceCoeffs=sourceCoeffs//PNScalings[#,{{r0,-2},{\[CapitalOmega]Kerr,3}},varPN, cIn=1/wronskian Total[sourceCoeffs {Rup[r0],-varPN^2dRup[r0],varPN^4 ddRup[r0]}]//SeriesTake[#,order]&; cUp=1/wronskian Total[sourceCoeffs {Rin[r0],-varPN^2dRin[r0],varPN^4 ddRin[r0]}]//SeriesTake[#,order]&; deltaCoeff=Coefficient[source[r],Derivative[2][DiracDelta][r-r0]]/Kerr\[CapitalDelta][a,r0]; -deltaCoeff=If[deltaCoeff===0,0,deltaCoeff//PNScalings[#,{{r0,-2},{\[CapitalOmega]Kerr,3}},varPN,"IgnoreHarmonics"->True]&//SeriesTerms[#,{varPN,0,order}]&]; +deltaCoeff=Assuming[{varPN>0},If[deltaCoeff===0,0,deltaCoeff//PNScalings[#,{{r0,-2},{\[CapitalOmega]Kerr,3}},varPN,"IgnoreHarmonics"->True]&//SeriesTerms[#,{varPN,0,order}]&]]; innerF=cIn Rin[#]&; outerF=cUp Rup[#]&; radialF=innerF[#] HeavisideTheta[r0-#] + outerF[#] HeavisideTheta[#-r0]+deltaCoeff DiracDelta[#-r0]&; @@ -2748,7 +2749,7 @@ ret ] -(* ::Subsubsection:: *) +(* ::Subsubsection::Closed:: *) (*Accessing functions and keys*) From 9050fbc6f60c8f943f70a46f1db04a2aec220d9c Mon Sep 17 00:00:00 2001 From: jakobneef Date: Mon, 20 Jan 2025 14:24:02 +0000 Subject: [PATCH 03/13] added documentation --- .../Symbols/TeukolskyPointParticleModePN.nb | 10336 +++------------- .../Symbols/TeukolskyRadialPN.nb | 4509 ++++--- Kernel/PN.wl | 173 +- Kernel/Tools.wl | 17 +- 4 files changed, 5273 insertions(+), 9762 deletions(-) diff --git a/Documentation/English/ReferencePages/Symbols/TeukolskyPointParticleModePN.nb b/Documentation/English/ReferencePages/Symbols/TeukolskyPointParticleModePN.nb index bc07c7c..ef5a17a 100644 --- a/Documentation/English/ReferencePages/Symbols/TeukolskyPointParticleModePN.nb +++ b/Documentation/English/ReferencePages/Symbols/TeukolskyPointParticleModePN.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 444837, 10250] -NotebookOptionsPosition[ 431924, 9976] -NotebookOutlinePosition[ 432708, 10002] -CellTagsIndexPosition[ 432627, 9997] +NotebookDataLength[ 163844, 3940] +NotebookOptionsPosition[ 150332, 3654] +NotebookOutlinePosition[ 151116, 3680] +CellTagsIndexPosition[ 151035, 3675] WindowFrame->Normal*) (* Beginning of Notebook Content *) @@ -137,10 +137,10 @@ Cell[TextData[{ "83d1757e-96d2-44c4-8211-de196c886561"], DynamicModuleBox[{$CellContext`nbobj$$ = NotebookObject[ "4cdc53be-4474-4e7d-8aa9-8c4b6af9a759", - "fbc5a385-f115-442f-9b9a-793c79ee4e26"], $CellContext`cellobj$$ = + "f8b797ce-49f5-4b23-9cd8-fb5daf2e0891"], $CellContext`cellobj$$ = CellObject[ "bfafe1ae-4382-474b-97c7-755897deb3ff", - "6872b1be-545b-42e8-8e8f-1810e701ea37"]}, + "d085d55a-1770-4305-89b3-1c7e08d2a004"]}, TemplateBox[{ GraphicsBox[{{ Thickness[0.06], @@ -261,8 +261,9 @@ Cell[BoxData[ $Line = 0; Null]], "PrimaryExamplesSection", CellID->1901548751,ExpressionUUID->"792f10b2-ef4c-49af-8e46-73c9a141bdad"], -Cell["We can start by computing a mode", "ExampleText", - CellChangeTimes->{{3.94222250523709*^9, 3.9422225119624577`*^9}}, +Cell["We can start by computing a mode ", "ExampleText", + CellChangeTimes->{{3.94222250523709*^9, 3.942222511962458*^9}, { + 3.946104731057811*^9, 3.946104755323127*^9}, 3.946104817320264*^9}, CellID->1665245063,ExpressionUUID->"95b73ce6-6d47-4075-b7ba-e26748338cb4"], Cell[CellGroupData[{ @@ -279,15 +280,16 @@ Cell[BoxData[{ RowBox[{"-", "2"}], ",", "2", ",", "2", ",", "orbit", ",", RowBox[{"{", RowBox[{"\[Eta]", ",", "3"}], "}"}]}], "]"}]}]}], "Input", - CellChangeTimes->{{3.9422219565829277`*^9, 3.94222199381381*^9}, { - 3.942222147391078*^9, 3.942222148633972*^9}}, - CellLabel->"In[9]:=", + CellChangeTimes->{{3.942221956582928*^9, 3.94222199381381*^9}, { + 3.942222147391078*^9, 3.942222148633972*^9}, 3.946104722586904*^9, + 3.946104814370336*^9}, + CellLabel->"In[1]:=", CellID->103538757,ExpressionUUID->"3ed8b034-7711-4c9d-b4d6-76f204a012e2"], Cell[BoxData[ InterpretationBox[ RowBox[{ - TagBox["TeukolskyPointParticleModePN", + TagBox["TeukolskyModePN", "SummaryHead"], "[", DynamicModuleBox[{Typeset`open$$ = False, Typeset`embedState$$ = "Ready"}, @@ -440,22 +442,19 @@ Cell[BoxData[ Dynamic[Typeset`open$$], ImageSize -> Automatic]}, "SummaryPanel"], DynamicModuleValues:>{}], "]"}], - Teukolsky`PN`TeukolskyPointParticleModePN[-2, 2, + Teukolsky`PN`TeukolskyModePN[-2, 2, 2, $CellContext`a, $CellContext`r0, {$CellContext`\[Eta], 3}, <| - "\[ScriptS]" -> -2, "\[ScriptL]" -> 2, "\[ScriptM]" -> 2, - "a" -> $CellContext`a, "r0" -> $CellContext`r0, - "PN" -> {$CellContext`\[Eta], 3}, + "s" -> -2, "l" -> 2, "m" -> 2, "a" -> $CellContext`a, + "r0" -> $CellContext`r0, "PN" -> {$CellContext`\[Eta], 3}, "RadialFunction" -> ( - Teukolsky`PN`Private`innerF$104659[#] - HeavisideTheta[$CellContext`r0 - #] + - Teukolsky`PN`Private`outerF$104659[#] - HeavisideTheta[# - $CellContext`r0] + - Teukolsky`PN`Private`deltaCoeff$104659 - DiracDelta[# - $CellContext`r0]& ), - "Inner" -> (Teukolsky`PN`Private`cIn$104659 - Teukolsky`PN`Private`Rin$104659[#]& ), - "Outer" -> (Teukolsky`PN`Private`cUp$104659 - Teukolsky`PN`Private`Rup$104659[#]& ), "\[Delta]" -> + Teukolsky`PN`Private`innerF$4169[#] HeavisideTheta[$CellContext`r0 - #] + + Teukolsky`PN`Private`outerF$4169[#] HeavisideTheta[# - $CellContext`r0] + + Teukolsky`PN`Private`deltaCoeff$4169 + DiracDelta[# - $CellContext`r0]& ), ("ExtendedHomogeneous" -> + "\[ScriptCapitalI]") -> (Teukolsky`PN`Private`cIn$4169 + Teukolsky`PN`Private`Rin$4169[#]& ), ("ExtendedHomogeneous" -> + "\[ScriptCapitalH]") -> (Teukolsky`PN`Private`cUp$4169 + Teukolsky`PN`Private`Rup$4169[#]& ), "\[Delta]" -> SeriesData[$CellContext`\[Eta], 0, {-Pi $CellContext`r0^(-3) SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, @@ -464,394 +463,119 @@ Cell[BoxData[ Pi $CellContext`r0^Rational[-7, 2] SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0], 0, - Pi (Rational[-3, 2] $CellContext`r0^Rational[-17, 4] - ( - 2 $CellContext`r0^Rational[-7, 2] + $CellContext`a^2 $CellContext`r0^ - Rational[-7, 2]) $CellContext`r0^Rational[-3, 4]) $CellContext`r0^ - Rational[1, 4] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, - 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0], 0, 9 $CellContext`a - Pi $CellContext`r0^Rational[-9, 2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, - 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0], 0, - Pi (Rational[-27, 8] $CellContext`r0^Rational[-21, 4] - - 2 $CellContext`a^2 $CellContext`r0^Rational[-21, 4] + - Rational[-3, 2] ( - 2 $CellContext`r0^ - Rational[-7, 2] + $CellContext`a^2 $CellContext`r0^ - Rational[-7, 2]) $CellContext`r0^Rational[-7, 4] - ( - 4 $CellContext`a^2 $CellContext`r0^Rational[-9, 2] + ( - 4 $CellContext`r0^(-2) - $CellContext`a^2 $CellContext`r0^(-2)) \ -$CellContext`r0^Rational[-5, 2]) $CellContext`r0^ - Rational[-3, 4]) $CellContext`r0^Rational[1, 4] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, - 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0], 0, - Pi (Rational[75, 4] $CellContext`a $CellContext`r0^ - Rational[-23, 4] + $CellContext`a ( - 2 $CellContext`r0^ - Rational[-7, 2] + $CellContext`a^2 $CellContext`r0^ - Rational[-7, 2]) $CellContext`r0^ - Rational[-9, - 4] - ((-2) $CellContext`a $CellContext`r0^(-5) - $CellContext`a^3 \ -$CellContext`r0^(-5) - - 2 $CellContext`a ( - 4 $CellContext`r0^(-2) - $CellContext`a^2 $CellContext`r0^(-2)) \ -$CellContext`r0^(-3)) $CellContext`r0^Rational[-3, 4]) $CellContext`r0^ - Rational[1, 4] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, - 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0], 0, - Pi $CellContext`r0^ - Rational[ - 1, 4] ((-14) $CellContext`a^2 $CellContext`r0^Rational[-25, 4] + - Rational[-1, 6] (Rational[405, 8] $CellContext`r0^(-3) + - 9 $CellContext`a^2 $CellContext`r0^(-3)) $CellContext`r0^ - Rational[-13, 4] + - Rational[-27, 8] ( - 2 $CellContext`r0^ - Rational[-7, 2] + $CellContext`a^2 $CellContext`r0^ - Rational[-7, 2]) $CellContext`r0^Rational[-11, 4] + - Rational[-3, 2] ( - 4 $CellContext`a^2 $CellContext`r0^Rational[-9, 2] + ( - 4 $CellContext`r0^(-2) - $CellContext`a^2 $CellContext`r0^(-2)) \ -$CellContext`r0^Rational[-5, 2]) $CellContext`r0^ - Rational[-7, 4] - $CellContext`r0^Rational[-3, 4] ( - 4 $CellContext`a^2 $CellContext`r0^ - Rational[-11, 2] + $CellContext`a^2 ( - 4 $CellContext`r0^(-2) - $CellContext`a^2 $CellContext`r0^(-2)) \ -$CellContext`r0^ - Rational[-7, 2] + ($CellContext`a^2 $CellContext`r0^ - Rational[-13, 2] + (8 $CellContext`r0^(-3) - - 4 $CellContext`a^2 $CellContext`r0^(-3)) $CellContext`r0^ - Rational[-7, 2]) $CellContext`r0)) - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, - 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0], 0, - Pi (Rational[135, 4] $CellContext`a $CellContext`r0^Rational[-27, 4] + - Rational[1, 3] $CellContext`a (Rational[405, 8] $CellContext`r0^(-3) + - 9 $CellContext`a^2 $CellContext`r0^(-3)) $CellContext`r0^ - Rational[-15, 4] + - Rational[9, 2] $CellContext`a ( - 2 $CellContext`r0^ - Rational[-7, 2] + $CellContext`a^2 $CellContext`r0^ - Rational[-7, 2]) $CellContext`r0^ - Rational[-13, 4] + $CellContext`a ( - 4 $CellContext`a^2 $CellContext`r0^Rational[-9, 2] + ( - 4 $CellContext`r0^(-2) - $CellContext`a^2 $CellContext`r0^(-2)) \ -$CellContext`r0^Rational[-5, 2]) $CellContext`r0^Rational[-9, 4] + - Rational[-3, - 2] ((-2) $CellContext`a $CellContext`r0^(-5) - $CellContext`a^3 \ -$CellContext`r0^(-5) - - 2 $CellContext`a ( - 4 $CellContext`r0^(-2) - $CellContext`a^2 $CellContext`r0^(-2)) \ -$CellContext`r0^(-3)) $CellContext`r0^ - Rational[-7, - 4] - ((-2) $CellContext`a^3 $CellContext`r0^(-6) - $CellContext`a \ -(4 $CellContext`r0^(-2) - $CellContext`a^2 $CellContext`r0^(-2)) \ -$CellContext`r0^(-4) - - 2 $CellContext`a ($CellContext`a^2 $CellContext`r0^ - Rational[-13, 2] + (8 $CellContext`r0^(-3) - - 4 $CellContext`a^2 $CellContext`r0^(-3)) $CellContext`r0^ - Rational[-7, 2]) $CellContext`r0^Rational[1, 2]) $CellContext`r0^ - Rational[-3, 4]) $CellContext`r0^Rational[1, 4] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, - 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0], 0, - Pi $CellContext`r0^Rational[1, 4] ( - Rational[-225, 4] $CellContext`a^2 $CellContext`r0^Rational[-29, 4] + - Rational[-1, 8] ( - Rational[117, 2] $CellContext`a^2 $CellContext`r0^(-4) + - Rational[7, 2] (Rational[405, 8] $CellContext`r0^(-3) + - 9 $CellContext`a^2 $CellContext`r0^(-3))/$CellContext`r0) \ -$CellContext`r0^Rational[-13, 4] + - Rational[-27, 8] ( - 4 $CellContext`a^2 $CellContext`r0^Rational[-9, 2] + ( - 4 $CellContext`r0^(-2) - $CellContext`a^2 $CellContext`r0^(-2)) \ -$CellContext`r0^Rational[-5, 2]) $CellContext`r0^ - Rational[-11, - 4] + $CellContext`a ((-2) $CellContext`a $CellContext`r0^(-5) - \ -$CellContext`a^3 $CellContext`r0^(-5) - - 2 $CellContext`a ( - 4 $CellContext`r0^(-2) - $CellContext`a^2 $CellContext`r0^(-2)) \ -$CellContext`r0^(-3)) $CellContext`r0^Rational[-9, 4] + - Rational[-1, 6] ( - 2 $CellContext`r0^ - Rational[-7, 2] + $CellContext`a^2 $CellContext`r0^ - Rational[-7, 2]) (Rational[405, 8] $CellContext`r0^(-3) + - 9 $CellContext`a^2 $CellContext`r0^(-3)) $CellContext`r0^ - Rational[-3, 4] - $CellContext`r0^ - Rational[-3, - 4] ($CellContext`a^2 ($CellContext`a^2 $CellContext`r0^ - Rational[-13, 2] + (8 $CellContext`r0^(-3) - - 4 $CellContext`a^2 $CellContext`r0^(-3)) $CellContext`r0^ - Rational[-7, 2]) + - 2 $CellContext`a^2 ( - 4 $CellContext`r0^(-2) - $CellContext`a^2 $CellContext`r0^(-2)) \ -$CellContext`r0^Rational[-9, 2] + ( - 2 $CellContext`a^2 $CellContext`r0^Rational[-15, 2] + ( - 16 $CellContext`r0^(-4) - - 12 $CellContext`a^2 $CellContext`r0^(-4) + $CellContext`a^4 \ -$CellContext`r0^(-4)) $CellContext`r0^Rational[-7, 2]) $CellContext`r0) + - Rational[-3, 2] $CellContext`r0^Rational[-7, 4] ( - 4 $CellContext`a^2 $CellContext`r0^ - Rational[-11, 2] + $CellContext`a^2 ( - 4 $CellContext`r0^(-2) - $CellContext`a^2 $CellContext`r0^(-2)) \ -$CellContext`r0^ - Rational[-7, 2] + ($CellContext`a^2 $CellContext`r0^ - Rational[-13, 2] + (8 $CellContext`r0^(-3) - - 4 $CellContext`a^2 $CellContext`r0^(-3)) $CellContext`r0^ - Rational[-7, 2]) $CellContext`r0)) + Rational[1, 2] Pi, 0], 0, Rational[-1, 2] (7 + 2 $CellContext`a^2) + Pi $CellContext`r0^(-4) SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0]}, 12, 29, 2], "cUp" -> - SeriesData[$CellContext`\[Eta], 0, {Complex[0, - Rational[-2, 15]] Pi $CellContext`r0 - Teukolsky`PN`\[CapitalOmega]Kerr ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]), 0, Complex[0, - Rational[-1, 96]] - Teukolsky`PN`\[CapitalOmega]Kerr^(-3) ( - Rational[512, 5] Pi $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr^4 ( - Complex[0, -2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + - Rational[64, 5] Pi $CellContext`r0 - Teukolsky`PN`\[CapitalOmega]Kerr^4 ( - Complex[0, 8] $CellContext`r0^Rational[1, 2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ + Rational[1, 2] Pi, 0]}, 12, 17, 2], + "Amplitudes" -> <| + "\[ScriptCapitalI]" -> SeriesData[$CellContext`\[Eta], 0, {Complex[0, + Rational[-2, 15]] Pi $CellContext`r0^2 + Teukolsky`PN`\[CapitalOmega]Kerr ( + 2 SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] - 4 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ 2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + Complex[0, - Rational[256, 15]] Pi $CellContext`r0^2 - Teukolsky`PN`\[CapitalOmega]Kerr^5 ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ + Rational[1, 2] Pi, 0] + Derivative[2, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ 2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ + Rational[1, 2] Pi, 0]), 0, Rational[8, 45] + Pi $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr ( + 2 (-3 + $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr) + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + (3 - 4 $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr) Derivative[1, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ 2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0])), 0, Complex[0, - Rational[-1, 96]] - Teukolsky`PN`\[CapitalOmega]Kerr^(-3) ( - Rational[-256, 5] Pi $CellContext`r0 - Teukolsky`PN`\[CapitalOmega]Kerr^4 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + $CellContext`r0^Rational[-1, 2] ( - Rational[-512, 5] $CellContext`a Pi $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr^4 + Complex[0, - Rational[512, 3]] Pi $CellContext`r0^3 - Teukolsky`PN`\[CapitalOmega]Kerr^5) ( - Complex[0, -2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ + Rational[1, 2] Pi, 0] + $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr Derivative[2, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ 2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + Complex[0, - Rational[256, 15]] Pi $CellContext`r0^2 - Teukolsky`PN`\[CapitalOmega]Kerr^5 ( - Complex[0, 8] $CellContext`r0^Rational[1, 2] + Rational[1, 2] Pi, 0]), 0, Complex[0, + Rational[1, 315]] Pi $CellContext`r0 + Teukolsky`PN`\[CapitalOmega]Kerr ((378 + + Complex[0, -336] $CellContext`a - + 336 $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr + + 88 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2) SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ Rational[1, 2] Pi, 0] + - Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ + 4 (-105 + Complex[0, -21] $CellContext`a + + 42 $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr - 44 $CellContext`r0^3 + Teukolsky`PN`\[CapitalOmega]Kerr^2) Derivative[1, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ 2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + - Rational[64, 5] Pi $CellContext`r0 - Teukolsky`PN`\[CapitalOmega]Kerr^4 ((-4) ( - Complex[0, 1] $CellContext`a + - 2 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr) - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - 2 (Complex[0, 1] $CellContext`a + - 2 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr) - Derivative[1, 0][ + Rational[1, 2] Pi, 0] + (105 + Complex[0, 84] $CellContext`a + + 44 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2) + Derivative[2, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ 2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + ( - Rational[96, 5] Pi Teukolsky`PN`\[CapitalOmega]Kerr^4 + - Pi $CellContext`r0^(-3) ( - Rational[256, 21] $CellContext`r0^6 - Teukolsky`PN`\[CapitalOmega]Kerr^6 + - Rational[ - 1, 5] ((-64) $CellContext`r0^3 - Teukolsky`PN`\[CapitalOmega]Kerr^3 ( - 2 Teukolsky`PN`\[CapitalOmega]Kerr + - 2 (1 - $CellContext`a^2)^Rational[1, 2] - Teukolsky`PN`\[CapitalOmega]Kerr) + - 4 $CellContext`r0^2 - Teukolsky`PN`\[CapitalOmega]Kerr^2 ( - Complex[0, -32] $CellContext`a $CellContext`r0 - Teukolsky`PN`\[CapitalOmega]Kerr^2 + - 32 (-1 + (1 - $CellContext`a^2)^ - Rational[1, 2]) $CellContext`r0 - Teukolsky`PN`\[CapitalOmega]Kerr^2 - 32 $CellContext`r0^4 - Teukolsky`PN`\[CapitalOmega]Kerr^4)))) ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ + Rational[1, 2] Pi, 0])}, -2, 4, 2], "\[ScriptCapitalH]" -> + SeriesData[$CellContext`\[Eta], 0, { + Rational[-1, 64] Pi $CellContext`r0^(-3) + Teukolsky`PN`\[CapitalOmega]Kerr^(-4) ( + 2 SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] - 4 Derivative[1, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ 2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ + Rational[1, 2] Pi, 0] + Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ 2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]))}, -2, 4, 2], "cIn" -> - SeriesData[$CellContext`\[Eta], 0, { - Rational[-1, 64] Pi $CellContext`r0^(-4) - Teukolsky`PN`\[CapitalOmega]Kerr^(-4) ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]), 0, Complex[0, - Rational[-1, 96]] - Teukolsky`PN`\[CapitalOmega]Kerr^(-3) ( - Complex[0, 3] Pi $CellContext`r0^Rational[-7, 2] - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ( - Complex[0, -2] + Rational[1, 2] Pi, 0]), 0, Complex[0, + Rational[1, 32]] Pi $CellContext`r0^Rational[-7, 2] + Teukolsky`PN`\[CapitalOmega]Kerr^(-4) ( + 2 (-3 + $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr) SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ + Rational[1, 2] Pi, 0] + (3 - 4 $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr) Derivative[1, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ 2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + Complex[0, - Rational[-3, 2]] Pi $CellContext`r0^(-4) - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ( - Complex[0, 8] $CellContext`r0^Rational[1, 2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ + Rational[1, 2] Pi, 0] + $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr Derivative[2, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ 2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) - 3 - Pi $CellContext`r0^(-3) ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0])), 0, Complex[0, - Rational[-1, 96]] - Teukolsky`PN`\[CapitalOmega]Kerr^(-3) ( - Complex[0, 6] Pi $CellContext`r0^(-4) - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - Complex[0, -3] $CellContext`a Pi $CellContext`r0^(-4) - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ( - Complex[0, -2] + Rational[1, 2] Pi, 0]), 0, Rational[1, 384] + Pi $CellContext`r0^(-4) + Teukolsky`PN`\[CapitalOmega]Kerr^(-4) ((-6 + + Complex[0, 32] $CellContext`a - 48 $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr + + 24 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2) SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) - 3 - Pi $CellContext`r0^(-3) ( - Complex[0, 8] $CellContext`r0^Rational[1, 2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + Complex[0, - Rational[-3, 2]] Pi $CellContext`r0^(-4) - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ((-4) ( - Complex[0, 1] $CellContext`a + - 2 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr) - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ Rational[1, 2] Pi, 0] + - 2 (Complex[0, 1] $CellContext`a + - 2 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr) - Derivative[1, 0][ + 4 (15 + Complex[0, 2] $CellContext`a + + 6 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr - + 12 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2) + Derivative[1, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ 2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + (Complex[0, - Rational[-9, 4]] Pi $CellContext`r0^(-5)/ - Teukolsky`PN`\[CapitalOmega]Kerr + - Pi $CellContext`r0^(-3) (Complex[0, - Rational[-336, 107]] $CellContext`r0^4 - Teukolsky`PN`\[CapitalOmega]Kerr^3 + - Rational[105, 428] - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) (Complex[0, - Rational[-107, 35]] $CellContext`r0^(-2) (2 + - Complex[0, -4] $CellContext`a - - 10 (1 - $CellContext`a^2)^Rational[1, 2] - - 4 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2) + - Rational[2, 105] $CellContext`r0^(-2) (1070 $CellContext`a + - Complex[0, -1605] (1 - $CellContext`a^2)^Rational[1, 2] + - Complex[0, 672] $CellContext`r0^6 - Teukolsky`PN`\[CapitalOmega]Kerr^4)))) ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ + Rational[1, 2] Pi, 0] + (-15 + Complex[0, -8] $CellContext`a + + 12 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2) + Derivative[2, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ 2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]))}, -12, -6, 2], "Wronskian" -> + Rational[1, 2] Pi, 0])}, -12, -6, 2]|>, "Wronskian" -> SeriesData[$CellContext`\[Eta], 0, { Complex[0, 96] Teukolsky`PN`\[CapitalOmega]Kerr^3}, 9, 12, 1], "Source" -> ( @@ -860,72 +584,56 @@ $CellContext`r0^ "InvariantWronskian"]& ), "In" -> Teukolsky`PN`TeukolskyRadialFunctionPN[-2, 2, 2, $CellContext`a, 2 Teukolsky`PN`\[CapitalOmega]Kerr, {$CellContext`\[Eta], 3}, <| - "\[ScriptS]" -> -2, "\[ScriptL]" -> 2, "\[ScriptM]" -> 2, - "a" -> $CellContext`a, "PN" -> {$CellContext`\[Eta], 3}, - "RadialFunction" -> Function[Teukolsky`PN`Private`r, + "s" -> -2, "l" -> 2, "m" -> 2, "a" -> $CellContext`a, + "PN" -> {$CellContext`\[Eta], 3}, "RadialFunction" -> + Function[Teukolsky`PN`Private`r, SeriesData[$CellContext`\[Eta], 0, { Rational[64, 5] Teukolsky`PN`Private`r^4 Teukolsky`PN`\[CapitalOmega]Kerr^4, Complex[0, Rational[256, 15]] Teukolsky`PN`Private`r^5 - Teukolsky`PN`\[CapitalOmega]Kerr^5, - Rational[256, 21] Teukolsky`PN`Private`r^6 - Teukolsky`PN`\[CapitalOmega]Kerr^6 + - Rational[ - 1, 5] ((-64) Teukolsky`PN`Private`r^3 - Teukolsky`PN`\[CapitalOmega]Kerr^3 ( - 2 Teukolsky`PN`\[CapitalOmega]Kerr + - 2 (1 - $CellContext`a^2)^Rational[1, 2] - Teukolsky`PN`\[CapitalOmega]Kerr) + - 4 Teukolsky`PN`Private`r^2 - Teukolsky`PN`\[CapitalOmega]Kerr^2 ( - Complex[0, -32] $CellContext`a Teukolsky`PN`Private`r - Teukolsky`PN`\[CapitalOmega]Kerr^2 + - 32 (-1 + (1 - $CellContext`a^2)^Rational[1, 2]) - Teukolsky`PN`Private`r Teukolsky`PN`\[CapitalOmega]Kerr^2 - - 32 Teukolsky`PN`Private`r^4 - Teukolsky`PN`\[CapitalOmega]Kerr^4))}, 4, 7, 1]], - "BoundaryCondition" -> "In", "LeadingOrder" -> + Teukolsky`PN`\[CapitalOmega]Kerr^5, Rational[-128, 105] + Teukolsky`PN`Private`r^3 + Teukolsky`PN`\[CapitalOmega]Kerr^4 (42 + + Complex[0, 21] $CellContext`a + + 11 Teukolsky`PN`Private`r^3 Teukolsky`PN`\[CapitalOmega]Kerr^2)}, + 4, 7, 1]], "BoundaryCondition" -> "In", "LeadingOrder" -> Function[Teukolsky`PN`Private`r, SeriesData[$CellContext`\[Eta], 0, { Rational[64, 5] Teukolsky`PN`Private`r^4 Teukolsky`PN`\[CapitalOmega]Kerr^4}, 4, 5, 1]], "TermCount" -> 3, - "Normalization" -> "SFPN", - "Amplitudes" -> <|"Transmission" -> "Not Computed"|>|>], "Up" -> + "Normalization" -> "Default", + "Amplitudes" -> <|"Transmission" -> "Not Computed"|>, "Simplified" -> + True|>], "Up" -> Teukolsky`PN`TeukolskyRadialFunctionPN[-2, 2, 2, $CellContext`a, 2 Teukolsky`PN`\[CapitalOmega]Kerr, {$CellContext`\[Eta], 3}, <| - "\[ScriptS]" -> -2, "\[ScriptL]" -> 2, "\[ScriptM]" -> 2, - "a" -> $CellContext`a, "PN" -> {$CellContext`\[Eta], 3}, - "RadialFunction" -> Function[Teukolsky`PN`Private`r, + "s" -> -2, "l" -> 2, "m" -> 2, "a" -> $CellContext`a, + "PN" -> {$CellContext`\[Eta], 3}, "RadialFunction" -> + Function[Teukolsky`PN`Private`r, SeriesData[$CellContext`\[Eta], 0, {Complex[0, Rational[-3, 2]] Teukolsky`PN`Private`r^(-1)/ - Teukolsky`PN`\[CapitalOmega]Kerr, -3, Complex[0, - Rational[-336, 107]] Teukolsky`PN`Private`r^4 - Teukolsky`PN`\[CapitalOmega]Kerr^3 + - Rational[105, 428] - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) (Complex[0, - Rational[-107, 35]] - Teukolsky`PN`Private`r^(-2) (2 + Complex[0, -4] $CellContext`a - - 10 (1 - $CellContext`a^2)^Rational[1, 2] - 4 - Teukolsky`PN`Private`r^3 Teukolsky`PN`\[CapitalOmega]Kerr^2) + - Rational[2, 105] - Teukolsky`PN`Private`r^(-2) (1070 $CellContext`a + - Complex[0, -1605] (1 - $CellContext`a^2)^Rational[1, 2] + - Complex[0, 672] Teukolsky`PN`Private`r^6 - Teukolsky`PN`\[CapitalOmega]Kerr^4))}, -1, 2, 1]], + Teukolsky`PN`\[CapitalOmega]Kerr, -3, Rational[1, 2] + Teukolsky`PN`Private`r^(-2) + Teukolsky`PN`\[CapitalOmega]Kerr^(-1) (Complex[0, -3] + + 4 $CellContext`a + + Complex[0, 6] Teukolsky`PN`Private`r^3 + Teukolsky`PN`\[CapitalOmega]Kerr^2)}, -1, 2, 1]], "BoundaryCondition" -> "Up", "LeadingOrder" -> Function[Teukolsky`PN`Private`r, SeriesData[$CellContext`\[Eta], 0, {Complex[0, Rational[-3, 2]] Teukolsky`PN`Private`r^(-1)/ Teukolsky`PN`\[CapitalOmega]Kerr}, -1, 0, 1]], "TermCount" -> 3, - "Normalization" -> "SFPN", - "Amplitudes" -> <|"Transmission" -> "Not Computed"|>|>]|>], + "Normalization" -> "Default", + "Amplitudes" -> <|"Transmission" -> "Not Computed"|>, "Simplified" -> + True|>], "Simplified" -> True|>], Editable->False, SelectWithContents->True, Selectable->False]], "Output", CellChangeTimes->{{3.94222200433076*^9, 3.942222016566121*^9}, - 3.942222158382213*^9}, - CellLabel->"Out[10]=", - CellID->856291874,ExpressionUUID->"eaa4202c-5951-44c8-9216-204e8c654020"] + 3.942222158382213*^9, 3.946104634610979*^9, 3.9461047796987343`*^9, + 3.9461048377087297`*^9, 3.94611005563437*^9, 3.946113940645184*^9, + 3.9461141370507*^9}, + CellLabel->"Out[2]=", + CellID->331545821,ExpressionUUID->"7ad3ffb0-7f8c-4ad8-96c9-086bd974b68c"] }, Open ]], Cell["The main use is now to use it as a 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SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, - 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]), 0, - HeavisideTheta[-$CellContext`r + $CellContext`r0] (Complex[0, - Rational[-4, 15]] Pi $CellContext`r^5 $CellContext`r0^(-4) - Teukolsky`PN`\[CapitalOmega]Kerr ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + Complex[0, - Rational[-2, 15]] $CellContext`r^4 - Teukolsky`PN`\[CapitalOmega]Kerr ( - Complex[0, 3] Pi $CellContext`r0^Rational[-7, 2] - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ( - Complex[0, -2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + Complex[0, - Rational[-3, 2]] Pi $CellContext`r0^(-4) - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ( - Complex[0, 8] $CellContext`r0^Rational[1, 2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) - 3 - Pi $CellContext`r0^(-3) ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ + Rational[1, 2] Pi, 0], 0, Rational[-1, 2] (7 + 2 $CellContext`a^2) + Pi $CellContext`r0^(-4) + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, + 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0]}, 12, 17, 2], + "Amplitudes" -> <| + "\[ScriptCapitalI]" -> SeriesData[$CellContext`\[Eta], 0, {Complex[0, + Rational[-2, 15]] Pi $CellContext`r0^2 + Teukolsky`PN`\[CapitalOmega]Kerr ( + 2 SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] - 4 Derivative[1, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ 2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]))) + - HeavisideTheta[$CellContext`r - $CellContext`r0] (Complex[0, - Rational[2, 5]] Pi $CellContext`r0 - Teukolsky`PN`\[CapitalOmega]Kerr ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + - Rational[-1, 64] $CellContext`r^(-1) - Teukolsky`PN`\[CapitalOmega]Kerr^(-4) ( - Rational[512, 5] Pi $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr^4 ( - Complex[0, -2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + - Rational[64, 5] Pi $CellContext`r0 - Teukolsky`PN`\[CapitalOmega]Kerr^4 ( - Complex[0, 8] $CellContext`r0^Rational[1, 2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + Complex[0, - Rational[256, 15]] Pi $CellContext`r0^2 - Teukolsky`PN`\[CapitalOmega]Kerr^5 ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ + Rational[1, 2] Pi, 0] + Derivative[2, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ 2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ + Rational[1, 2] Pi, 0]), 0, Rational[8, 45] + Pi $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr ( + 2 (-3 + $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr) + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] + (3 - 4 $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr) Derivative[1, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ 2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]))), 0, - HeavisideTheta[-$CellContext`r + $CellContext`r0] ( - Rational[2, 105] - Pi $CellContext`r^3 $CellContext`r0^(-4) (42 + - Complex[0, 21] $CellContext`a + - 11 $CellContext`r^3 Teukolsky`PN`\[CapitalOmega]Kerr^2) ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + - Rational[8, 45] $CellContext`r^5 - Teukolsky`PN`\[CapitalOmega]Kerr^2 ( - Complex[0, 3] Pi $CellContext`r0^Rational[-7, 2] - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ( - Complex[0, -2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ + Rational[1, 2] Pi, 0] + $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr Derivative[2, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ 2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + Complex[0, - Rational[-3, 2]] Pi $CellContext`r0^(-4) - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ( - Complex[0, 8] $CellContext`r0^Rational[1, 2] + Rational[1, 2] Pi, 0]), 0, Complex[0, + Rational[1, 315]] Pi $CellContext`r0 + Teukolsky`PN`\[CapitalOmega]Kerr ((378 + + Complex[0, -336] $CellContext`a - + 336 $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr + + 88 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2) SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ Rational[1, 2] Pi, 0] + - Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ + 4 (-105 + Complex[0, -21] $CellContext`a + + 42 $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr - 44 $CellContext`r0^3 + Teukolsky`PN`\[CapitalOmega]Kerr^2) Derivative[1, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ 2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) - 3 - Pi $CellContext`r0^(-3) ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0])) + Complex[0, - Rational[-2, 15]] $CellContext`r^4 - Teukolsky`PN`\[CapitalOmega]Kerr ( - Complex[0, 6] Pi $CellContext`r0^(-4) - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - Complex[0, -3] $CellContext`a Pi $CellContext`r0^(-4) - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ( - Complex[0, -2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ + Rational[1, 2] Pi, 0] + (105 + Complex[0, 84] $CellContext`a + + 44 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2) + Derivative[2, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ 2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) - 3 - Pi $CellContext`r0^(-3) ( - Complex[0, 8] $CellContext`r0^Rational[1, 2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ + Rational[1, 2] Pi, 0])}, -2, 4, 2], "\[ScriptCapitalH]" -> + SeriesData[$CellContext`\[Eta], 0, { + Rational[-1, 64] Pi $CellContext`r0^(-3) + Teukolsky`PN`\[CapitalOmega]Kerr^(-4) ( + 2 SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + Rational[1, 2] Pi, 0] - 4 Derivative[1, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ 2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + Complex[0, - Rational[-3, 2]] Pi $CellContext`r0^(-4) - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ((-4) ( - Complex[0, 1] $CellContext`a + - 2 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr) - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - 2 (Complex[0, 1] $CellContext`a + - 2 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr) - Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + (Complex[0, - Rational[-9, 4]] Pi $CellContext`r0^(-5)/ - Teukolsky`PN`\[CapitalOmega]Kerr + - Rational[1, 2] Pi $CellContext`r0^(-5) - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) (Complex[0, -3] + - 4 $CellContext`a + - Complex[0, 6] $CellContext`r0^3 - Teukolsky`PN`\[CapitalOmega]Kerr^2)) ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ + Rational[1, 2] Pi, 0] + Derivative[2, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ 2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]))) + - HeavisideTheta[$CellContext`r - $CellContext`r0] (Complex[0, - Rational[-1, 15]] - Pi $CellContext`r^(-2) $CellContext`r0 (Complex[0, -3] + - 4 $CellContext`a + - Complex[0, 6] $CellContext`r^3 Teukolsky`PN`\[CapitalOmega]Kerr^2) ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + Complex[0, - Rational[1, 32]] - Teukolsky`PN`\[CapitalOmega]Kerr^(-3) ( - Rational[512, 5] Pi $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr^4 ( - Complex[0, -2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + - Rational[64, 5] Pi $CellContext`r0 - Teukolsky`PN`\[CapitalOmega]Kerr^4 ( - Complex[0, 8] $CellContext`r0^Rational[1, 2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + Complex[0, - Rational[256, 15]] Pi $CellContext`r0^2 - Teukolsky`PN`\[CapitalOmega]Kerr^5 ( - 2 $CellContext`r0 + Rational[1, 2] Pi, 0]), 0, Complex[0, + Rational[1, 32]] Pi $CellContext`r0^Rational[-7, 2] + Teukolsky`PN`\[CapitalOmega]Kerr^(-4) ( + 2 (-3 + $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr) SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ + Rational[1, 2] Pi, 0] + (3 - 4 $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr) Derivative[1, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ 2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0])) + - Rational[-1, 64] $CellContext`r^(-1) - Teukolsky`PN`\[CapitalOmega]Kerr^(-4) ( - Rational[-256, 5] Pi $CellContext`r0 - Teukolsky`PN`\[CapitalOmega]Kerr^4 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + $CellContext`r0^Rational[-1, 2] ( - Rational[-512, 5] $CellContext`a Pi $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr^4 + Complex[0, - Rational[512, 3]] Pi $CellContext`r0^3 - Teukolsky`PN`\[CapitalOmega]Kerr^5) ( - Complex[0, -2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ + Rational[1, 2] Pi, 0] + $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr Derivative[2, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ 2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + Complex[0, - Rational[256, 15]] Pi $CellContext`r0^2 - Teukolsky`PN`\[CapitalOmega]Kerr^5 ( - Complex[0, 8] $CellContext`r0^Rational[1, 2] + Rational[1, 2] Pi, 0]), 0, Rational[1, 384] + Pi $CellContext`r0^(-4) + Teukolsky`PN`\[CapitalOmega]Kerr^(-4) ((-6 + + Complex[0, 32] $CellContext`a - 48 $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr + + 24 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2) SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ Rational[1, 2] Pi, 0] + - Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ + 4 (15 + Complex[0, 2] $CellContext`a + + 6 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr - + 12 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2) + Derivative[1, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ 2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + - Rational[64, 5] Pi $CellContext`r0 - Teukolsky`PN`\[CapitalOmega]Kerr^4 ((-4) ( - Complex[0, 1] $CellContext`a + - 2 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr) - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - 2 (Complex[0, 1] $CellContext`a + - 2 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr) - Derivative[1, 0][ + Rational[1, 2] Pi, 0] + (-15 + Complex[0, -8] $CellContext`a + + 12 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2) + Derivative[2, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ 2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + ( - Rational[96, 5] Pi Teukolsky`PN`\[CapitalOmega]Kerr^4 + - Rational[-128, 105] Pi + Rational[1, 2] Pi, 0])}, -12, -6, 2]|>, "Wronskian" -> + SeriesData[$CellContext`\[Eta], 0, { + Complex[0, 96] Teukolsky`PN`\[CapitalOmega]Kerr^3}, 9, 12, 1], + "Source" -> ( + Teukolsky`PN`Private`TeukolskySourceCircularOrbit[-2, 2, + 2, $CellContext`a, {#, $CellContext`r0}, "Form" -> + "InvariantWronskian"]& ), "In" -> + Teukolsky`PN`TeukolskyRadialFunctionPN[-2, 2, 2, $CellContext`a, 2 + Teukolsky`PN`\[CapitalOmega]Kerr, {$CellContext`\[Eta], 3}, <| + "s" -> -2, "l" -> 2, "m" -> 2, "a" -> $CellContext`a, + "PN" -> {$CellContext`\[Eta], 3}, "RadialFunction" -> + Function[Teukolsky`PN`Private`r, + SeriesData[$CellContext`\[Eta], 0, { + Rational[64, 5] Teukolsky`PN`Private`r^4 + Teukolsky`PN`\[CapitalOmega]Kerr^4, Complex[0, + Rational[256, 15]] Teukolsky`PN`Private`r^5 + Teukolsky`PN`\[CapitalOmega]Kerr^5, Rational[-128, 105] + Teukolsky`PN`Private`r^3 Teukolsky`PN`\[CapitalOmega]Kerr^4 (42 + Complex[0, 21] $CellContext`a + - 11 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2)) ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0])))}, -4, 2, 2], - Editable->False]], "Output", - CellChangeTimes->{{3.942309478663083*^9, 3.94230948651221*^9}}, - CellLabel->"Out[13]=", - CellID->502689445,ExpressionUUID->"8b8522de-2b9f-4b2a-a34a-b91b5560eb50"] -}, Closed]], + 11 Teukolsky`PN`Private`r^3 Teukolsky`PN`\[CapitalOmega]Kerr^2)}, + 4, 7, 1]], "BoundaryCondition" -> "In", "LeadingOrder" -> + Function[Teukolsky`PN`Private`r, + SeriesData[$CellContext`\[Eta], 0, { + Rational[64, 5] Teukolsky`PN`Private`r^4 + Teukolsky`PN`\[CapitalOmega]Kerr^4}, 4, 5, 1]], "TermCount" -> 3, + "Normalization" -> "Default", + "Amplitudes" -> <|"Transmission" -> "Not Computed"|>, "Simplified" -> + True|>], "Up" -> + Teukolsky`PN`TeukolskyRadialFunctionPN[-2, 2, 2, $CellContext`a, 2 + Teukolsky`PN`\[CapitalOmega]Kerr, {$CellContext`\[Eta], 3}, <| + "s" -> -2, "l" -> 2, "m" -> 2, "a" -> $CellContext`a, + "PN" -> {$CellContext`\[Eta], 3}, "RadialFunction" -> + Function[Teukolsky`PN`Private`r, + SeriesData[$CellContext`\[Eta], 0, {Complex[0, + Rational[-3, 2]] Teukolsky`PN`Private`r^(-1)/ + Teukolsky`PN`\[CapitalOmega]Kerr, -3, Rational[1, 2] + Teukolsky`PN`Private`r^(-2) + Teukolsky`PN`\[CapitalOmega]Kerr^(-1) (Complex[0, -3] + + 4 $CellContext`a + + Complex[0, 6] Teukolsky`PN`Private`r^3 + Teukolsky`PN`\[CapitalOmega]Kerr^2)}, -1, 2, 1]], + "BoundaryCondition" -> "Up", "LeadingOrder" -> + Function[Teukolsky`PN`Private`r, + SeriesData[$CellContext`\[Eta], 0, {Complex[0, + Rational[-3, 2]] Teukolsky`PN`Private`r^(-1)/ + Teukolsky`PN`\[CapitalOmega]Kerr}, -1, 0, 1]], "TermCount" -> 3, + "Normalization" -> "Default", + "Amplitudes" -> <|"Transmission" -> "Not Computed"|>, "Simplified" -> + True|>], "Simplified" -> True|>], + Editable->False, + SelectWithContents->True, + Selectable->False]], "Output", + CellChangeTimes->{3.946103993320241*^9, 3.946104644969741*^9, + 3.946104789864366*^9, 3.946104847705056*^9, 3.9461100653902884`*^9, + 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"ExampleSection",ExpressionUUID-> + "ac0dec70-1428-454e-9f36-4bf89982db6b"], + $Line = 0; Null]], "ExampleSection", + CellID->1512661770,ExpressionUUID->"57c3a8c5-4f1c-4c78-9e9e-2b52d024a06f"], + +Cell["\<\ +Here we want to display the possible Keys that can be used to query a \ +TeukolskyModePN (we set a=0 here to improve the readability of the output)\ +\>", "ExampleText", + CellChangeTimes->{{3.942308774836884*^9, 3.942308779609502*^9}, { + 3.942308822367017*^9, 3.942308846368334*^9}, {3.946104822516281*^9, + 3.946104822735633*^9}}, + CellID->214174106,ExpressionUUID->"01d5f544-7788-4a34-bfce-b0fe3f48d890"], + +Cell[BoxData[{ + RowBox[{ + RowBox[{"orbit", "=", + RowBox[{"KerrGeoOrbit", "[", + RowBox[{"0", ",", "r0", ",", "0", ",", "1"}], "]"}]}], + ";"}], "\[IndentingNewLine]", + RowBox[{ + RowBox[{"mode", "=", + RowBox[{"TeukolskyPointParticleModePN", "[", + RowBox[{ + RowBox[{"-", "2"}], ",", "2", ",", "2", ",", "orbit", ",", + RowBox[{"{", + RowBox[{"\[Eta]", ",", "3"}], "}"}]}], 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2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, - 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]), 0, - HeavisideTheta[-100 + $CellContext`r0] (Complex[0, - Rational[-8000000000, 3]] Pi $CellContext`r0^(-4) - Teukolsky`PN`\[CapitalOmega]Kerr ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + Complex[0, - Rational[-40000000, 3]] - Teukolsky`PN`\[CapitalOmega]Kerr ( - Complex[0, 3] Pi $CellContext`r0^Rational[-7, 2] - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ( - Complex[0, -2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + Complex[0, - Rational[-3, 2]] Pi $CellContext`r0^(-4) - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ( - Complex[0, 8] $CellContext`r0^Rational[1, 2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) - 3 - Pi $CellContext`r0^(-3) ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]))) + - HeavisideTheta[100 - $CellContext`r0] (Complex[0, - Rational[2, 5]] Pi $CellContext`r0 - Teukolsky`PN`\[CapitalOmega]Kerr ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + - Rational[-1, 6400] - Teukolsky`PN`\[CapitalOmega]Kerr^(-4) ( - Rational[512, 5] Pi $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr^4 ( - Complex[0, -2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + - Rational[64, 5] Pi $CellContext`r0 - Teukolsky`PN`\[CapitalOmega]Kerr^4 ( - Complex[0, 8] $CellContext`r0^Rational[1, 2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + Complex[0, - Rational[256, 15]] Pi $CellContext`r0^2 - Teukolsky`PN`\[CapitalOmega]Kerr^5 ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]))), 0, - HeavisideTheta[-100 + $CellContext`r0] ( - Rational[400000, 21] - Pi $CellContext`r0^(-4) (42 + Complex[0, 21] $CellContext`a + - 11000000 Teukolsky`PN`\[CapitalOmega]Kerr^2) ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + - Rational[16000000000, 9] - Teukolsky`PN`\[CapitalOmega]Kerr^2 ( - Complex[0, 3] Pi $CellContext`r0^Rational[-7, 2] - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ( - Complex[0, -2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + Complex[0, - Rational[-3, 2]] Pi $CellContext`r0^(-4) - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ( - Complex[0, 8] $CellContext`r0^Rational[1, 2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) - 3 - Pi $CellContext`r0^(-3) ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0])) + Complex[0, - Rational[-40000000, 3]] - Teukolsky`PN`\[CapitalOmega]Kerr ( - Complex[0, 6] Pi $CellContext`r0^(-4) - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - Complex[0, -3] $CellContext`a Pi $CellContext`r0^(-4) - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ( - Complex[0, -2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) - 3 - Pi $CellContext`r0^(-3) ( - Complex[0, 8] $CellContext`r0^Rational[1, 2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + Complex[0, - Rational[-3, 2]] Pi $CellContext`r0^(-4) - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ((-4) ( - Complex[0, 1] $CellContext`a + - 2 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr) - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - 2 (Complex[0, 1] $CellContext`a + - 2 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr) - Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + (Complex[0, - Rational[-9, 4]] Pi $CellContext`r0^(-5)/ - Teukolsky`PN`\[CapitalOmega]Kerr + - Rational[1, 2] Pi $CellContext`r0^(-5) - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) (Complex[0, -3] + - 4 $CellContext`a + - Complex[0, 6] $CellContext`r0^3 - Teukolsky`PN`\[CapitalOmega]Kerr^2)) ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]))) + - HeavisideTheta[100 - $CellContext`r0] (Complex[0, - Rational[-1, 150000]] - Pi $CellContext`r0 (Complex[0, -3] + 4 $CellContext`a + - Complex[0, 6000000] Teukolsky`PN`\[CapitalOmega]Kerr^2) ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + Complex[0, - Rational[1, 32]] - Teukolsky`PN`\[CapitalOmega]Kerr^(-3) ( - Rational[512, 5] Pi $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr^4 ( - Complex[0, -2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + - Rational[64, 5] Pi $CellContext`r0 - Teukolsky`PN`\[CapitalOmega]Kerr^4 ( - Complex[0, 8] $CellContext`r0^Rational[1, 2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + Complex[0, - Rational[256, 15]] Pi $CellContext`r0^2 - Teukolsky`PN`\[CapitalOmega]Kerr^5 ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0])) + - Rational[-1, 6400] - Teukolsky`PN`\[CapitalOmega]Kerr^(-4) ( - Rational[-256, 5] Pi $CellContext`r0 - Teukolsky`PN`\[CapitalOmega]Kerr^4 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + $CellContext`r0^Rational[-1, 2] ( - Rational[-512, 5] $CellContext`a Pi $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr^4 + Complex[0, - Rational[512, 3]] Pi $CellContext`r0^3 - Teukolsky`PN`\[CapitalOmega]Kerr^5) ( - Complex[0, -2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + Complex[0, - Rational[256, 15]] Pi $CellContext`r0^2 - Teukolsky`PN`\[CapitalOmega]Kerr^5 ( - Complex[0, 8] $CellContext`r0^Rational[1, 2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + - Rational[64, 5] Pi $CellContext`r0 - Teukolsky`PN`\[CapitalOmega]Kerr^4 ((-4) ( - Complex[0, 1] $CellContext`a + - 2 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr) - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - 2 (Complex[0, 1] $CellContext`a + - 2 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr) - Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + ( - Rational[96, 5] Pi Teukolsky`PN`\[CapitalOmega]Kerr^4 + - Rational[-128, 105] Pi - Teukolsky`PN`\[CapitalOmega]Kerr^4 (42 + - Complex[0, 21] $CellContext`a + - 11 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2)) ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0])))}, -4, 2, 2], + Rational[-3, 2] (Rational[1, 5] Pi)^ + Rational[1, + 2] $CellContext`r^(-1) $CellContext`r0^(-3) ($CellContext`r0^5 + HeavisideTheta[$CellContext`r - $CellContext`r0] + $CellContext`r^5 + HeavisideTheta[-$CellContext`r + $CellContext`r0]), 0, + Complex[0, -1] (Rational[1, 5] Pi)^ + Rational[1, 2] $CellContext`r^(-1) $CellContext`r0^ + Rational[-7, 2] ($CellContext`r0^5 (-2 - + 3 $CellContext`r $CellContext`r0^Rational[1, 2] + Teukolsky`PN`\[CapitalOmega]Kerr + + 2 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr) + HeavisideTheta[$CellContext`r - $CellContext`r0] + $CellContext`r^5 (3 + + 2 $CellContext`r $CellContext`r0^Rational[1, 2] + Teukolsky`PN`\[CapitalOmega]Kerr - 3 $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr) + HeavisideTheta[-$CellContext`r + $CellContext`r0]), 0, + Rational[1, 28] (Rational[1, 5] Pi)^ + Rational[1, + 2] $CellContext`r^(-2) $CellContext`r0^(-4) ($CellContext`r0^5 ((-42) \ +$CellContext`r0 + + 84 $CellContext`r^3 $CellContext`r0 Teukolsky`PN`\[CapitalOmega]Kerr^2 - + 112 $CellContext`r^2 $CellContext`r0^Rational[1, 2] + Teukolsky`PN`\[CapitalOmega]Kerr (-1 + $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr) + $CellContext`r (119 - 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Rational[-1, 5] Pi $CellContext`r^(-1) $CellContext`r0 - HeavisideTheta[$CellContext`r - $CellContext`r0] ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, - 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, - 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + - Rational[-1, 5] Pi $CellContext`r^4 $CellContext`r0^(-4) - HeavisideTheta[-$CellContext`r + $CellContext`r0] ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, - 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, - 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]), 0, - HeavisideTheta[-$CellContext`r + $CellContext`r0] (Complex[0, - Rational[-4, 15]] Pi $CellContext`r^5 $CellContext`r0^(-4) - Teukolsky`PN`\[CapitalOmega]Kerr ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + Complex[0, - Rational[-2, 15]] $CellContext`r^4 - Teukolsky`PN`\[CapitalOmega]Kerr ( - Complex[0, 3] Pi $CellContext`r0^Rational[-7, 2] - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ( - Complex[0, -2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + Complex[0, - Rational[-3, 2]] Pi $CellContext`r0^(-4) - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ( - Complex[0, 8] $CellContext`r0^Rational[1, 2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) - 3 - Pi $CellContext`r0^(-3) ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]))) + - HeavisideTheta[$CellContext`r - $CellContext`r0] (Complex[0, - Rational[2, 5]] Pi $CellContext`r0 - Teukolsky`PN`\[CapitalOmega]Kerr ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + - Rational[-1, 64] $CellContext`r^(-1) - Teukolsky`PN`\[CapitalOmega]Kerr^(-4) ( - Rational[512, 5] Pi $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr^4 ( - Complex[0, -2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + - Rational[64, 5] Pi $CellContext`r0 - Teukolsky`PN`\[CapitalOmega]Kerr^4 ( - Complex[0, 8] $CellContext`r0^Rational[1, 2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + Complex[0, - Rational[256, 15]] Pi $CellContext`r0^2 - Teukolsky`PN`\[CapitalOmega]Kerr^5 ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]))), 0, - HeavisideTheta[-$CellContext`r + $CellContext`r0] ( - Rational[2, 105] - Pi $CellContext`r^3 $CellContext`r0^(-4) (42 + - Complex[0, 21] $CellContext`a + - 11 $CellContext`r^3 Teukolsky`PN`\[CapitalOmega]Kerr^2) ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + - Rational[8, 45] $CellContext`r^5 - Teukolsky`PN`\[CapitalOmega]Kerr^2 ( - Complex[0, 3] Pi $CellContext`r0^Rational[-7, 2] - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ( - Complex[0, -2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + Complex[0, - Rational[-3, 2]] Pi $CellContext`r0^(-4) - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ( - Complex[0, 8] $CellContext`r0^Rational[1, 2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) - 3 - Pi $CellContext`r0^(-3) ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0])) + Complex[0, - Rational[-2, 15]] $CellContext`r^4 - Teukolsky`PN`\[CapitalOmega]Kerr ( - Complex[0, 6] Pi $CellContext`r0^(-4) - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - Complex[0, -3] $CellContext`a Pi $CellContext`r0^(-4) - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ( - Complex[0, -2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) - 3 - Pi $CellContext`r0^(-3) ( - Complex[0, 8] $CellContext`r0^Rational[1, 2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - Complex[0, -4] $CellContext`r0^Rational[1, 2] Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + Complex[0, - Rational[-3, 2]] Pi $CellContext`r0^(-4) - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) ((-4) ( - Complex[0, 1] $CellContext`a + - 2 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr) - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - 2 (Complex[0, 1] $CellContext`a + - 2 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr) - Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]) + (Complex[0, - Rational[-9, 4]] Pi $CellContext`r0^(-5)/ - Teukolsky`PN`\[CapitalOmega]Kerr + - Rational[1, 2] Pi $CellContext`r0^(-5) - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) (Complex[0, -3] + - 4 $CellContext`a + - Complex[0, 6] $CellContext`r0^3 - Teukolsky`PN`\[CapitalOmega]Kerr^2)) ( - 2 $CellContext`r0 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 $CellContext`r0 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]))) + - HeavisideTheta[$CellContext`r - $CellContext`r0] (Complex[0, - Rational[-1, 15]] - Pi $CellContext`r^(-2) $CellContext`r0 (Complex[0, -3] + - 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2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0 Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0])) + - Rational[-1, 64] $CellContext`r^(-1) - Teukolsky`PN`\[CapitalOmega]Kerr^(-4) ( - Rational[-256, 5] Pi $CellContext`r0 Teukolsky`PN`\[CapitalOmega]Kerr^4 - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, - 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + $CellContext`r0^Rational[-1, 2] ( - Rational[-512, 5] $CellContext`a Pi $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr^4 + Complex[0, - Rational[512, 3]] Pi $CellContext`r0^3 - Teukolsky`PN`\[CapitalOmega]Kerr^5) ( - Complex[0, -2] - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + Complex[0, 1] Derivative[1, 0][ - 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Rational[1, 2] Pi, 0]))}, -4, 2, 2], + Rational[-3, 2] (Rational[1, 5] Pi)^ + Rational[1, + 2] $CellContext`r^(-1) $CellContext`r0^(-3) ($CellContext`r0^5 + HeavisideTheta[$CellContext`r - $CellContext`r0] + $CellContext`r^5 + HeavisideTheta[-$CellContext`r + $CellContext`r0]), 0, + Complex[0, -1] (Rational[1, 5] Pi)^ + Rational[1, 2] $CellContext`r^(-1) $CellContext`r0^ + Rational[-7, 2] ($CellContext`r0^5 (-2 - + 3 $CellContext`r $CellContext`r0^Rational[1, 2] + Teukolsky`PN`\[CapitalOmega]Kerr + + 2 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr) + HeavisideTheta[$CellContext`r - $CellContext`r0] + $CellContext`r^5 (3 + + 2 $CellContext`r $CellContext`r0^Rational[1, 2] + Teukolsky`PN`\[CapitalOmega]Kerr - 3 $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr) + HeavisideTheta[-$CellContext`r + $CellContext`r0]), 0, + Rational[1, 28] (Rational[1, 5] Pi)^ + Rational[1, + 2] $CellContext`r^(-2) $CellContext`r0^(-4) ($CellContext`r0^5 ((-42) \ +$CellContext`r0 + + 84 $CellContext`r^3 $CellContext`r0 Teukolsky`PN`\[CapitalOmega]Kerr^2 - + 112 $CellContext`r^2 $CellContext`r0^Rational[1, 2] + Teukolsky`PN`\[CapitalOmega]Kerr (-1 + $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr) + $CellContext`r (119 - + 56 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr + + 44 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2)) + HeavisideTheta[$CellContext`r - $CellContext`r0] + $CellContext`r^5 ( + 168 $CellContext`r0 + + 44 $CellContext`r^3 $CellContext`r0 Teukolsky`PN`\[CapitalOmega]Kerr^2 - + 112 $CellContext`r^2 $CellContext`r0^Rational[1, 2] + Teukolsky`PN`\[CapitalOmega]Kerr (-1 + $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr) + + 7 $CellContext`r (-13 - 8 $CellContext`r0^Rational[3, 2] + Teukolsky`PN`\[CapitalOmega]Kerr + + 12 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2)) + HeavisideTheta[-$CellContext`r + $CellContext`r0])}, -4, 2, 2], Editable->False]], "Output", - CellChangeTimes->{{3.9423101905159187`*^9, 3.942310194542695*^9}}, - CellLabel->"Out[25]=", - CellID->219182377,ExpressionUUID->"b761df1b-df70-4705-9fb3-ff23c4ff8b48"] -}, Closed]], + CellChangeTimes->{{3.942309768469756*^9, 3.9423097739478903`*^9}, + 3.946104655802152*^9, 3.946104800405693*^9, 3.946104858368867*^9, + 3.946110075307023*^9, 3.946110151610098*^9, 3.946113965230694*^9, { + 3.946114058160997*^9, 3.946114061675576*^9}, 3.9461141562634907`*^9}, + CellLabel->"Out[11]=", + CellID->910098748,ExpressionUUID->"2e18fd21-42fb-4417-bc02-27ce7517558a"] +}, Open ]], -Cell["\<\ -\"\[Delta]\" gives the contribution on the worldline. The full solution can \ -be pieced together from \[LineSeparator]mode[\"RadialFunction\"] = \ -HeavisideTheta[r-r0] mode[\"ExtendedHomogeneous\" \[Rule] \"\[ScriptCapitalI]\ -\"]+HeavisideTheta[r0-r] mode[\"ExtendedHomogeneous\" \[Rule] \"\ -\[ScriptCapitalH]\"]+DiracDelta[r-r0] mode[\"\[Delta]\"]\ -\>", "ExampleText", - CellChangeTimes->{{3.942310233174951*^9, 3.9423104174852247`*^9}}, - CellID->1000349281,ExpressionUUID->"16e0e578-12be-4417-a68a-d18ad72e0e7e"], +Cell[TextData[{ + "\"ExtendedHomogeneous\" \[Rule] \"\[ScriptCapitalI]\" gives the extended \ +homogenous solution on the infinity side of the particle, i.e., ", + Cell[BoxData[ + RowBox[{ + SubscriptBox["Z", "Up"], + SubscriptBox["R", "Up"]}]], "InlineFormula", + FormatType->StandardForm,ExpressionUUID-> + "bd82a563-c1b2-46df-b6cf-67a7533a06ec"], + " where ", + Cell[BoxData[ + RowBox[{ + RowBox[{ + SubscriptBox["Z", "Up"], + RowBox[{"HeavisideTheta", "[", + RowBox[{"r", "-", "r0"}], "]"}]}], "=", + SubscriptBox["c", "Up"]}]], "InlineFormula", + FormatType->StandardForm,ExpressionUUID-> + "06fddb09-77ce-41ce-a0a0-0ab5fbaf010c"], + ". Likewise for \"ExtendedHomogeneous\" \[Rule] \"\[ScriptCapitalH]\" and In\ +\[LeftRightArrow]Up. (Note that the output is again a Function and thus has \ +the same weird behaviour as \"RadialFunction\" )" +}], "ExampleText", + CellChangeTimes->{{3.942309863097804*^9, 3.942310041198903*^9}, { + 3.9423101107986603`*^9, 3.942310177163575*^9}}, + CellID->715202156,ExpressionUUID->"09af8ff7-a58d-437b-8e6e-0c719590edf6"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ - RowBox[{"mode", "[", "\"\<\[Delta]\>\"", "]"}], "//", "Simplify"}]], "Input",\ - - CellChangeTimes->{{3.94231043256621*^9, 3.942310437955163*^9}}, - CellLabel->"In[27]:=", - CellID->570692951,ExpressionUUID->"a8a843eb-48f4-457a-8b13-a2ce5d7d1d13"], + RowBox[{ + RowBox[{"mode", "[", + RowBox[{ + "\"\\"", "\[Rule]", "\"\<\[ScriptCapitalI]\>\""}], + "]"}], "[", "r", "]"}], "//", "Simplify"}]], "Input", + CellChangeTimes->{{3.942310046450639*^9, 3.942310057727206*^9}, { + 3.946110157737054*^9, 3.9461101589967012`*^9}}, + CellLabel->"In[12]:=", + CellID->1645563088,ExpressionUUID->"2e15e418-5e71-4c97-ac20-def3b9046005"], Cell[BoxData[ InterpretationBox[ RowBox[{ RowBox[{"-", FractionBox[ - 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a/Documentation/English/ReferencePages/Symbols/TeukolskyRadialPN.nb b/Documentation/English/ReferencePages/Symbols/TeukolskyRadialPN.nb index cf55dfd..10c420a 100644 --- a/Documentation/English/ReferencePages/Symbols/TeukolskyRadialPN.nb +++ b/Documentation/English/ReferencePages/Symbols/TeukolskyRadialPN.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 154621, 3686] -NotebookOptionsPosition[ 142129, 3414] -NotebookOutlinePosition[ 142914, 3440] -CellTagsIndexPosition[ 142833, 3435] +NotebookDataLength[ 232131, 5415] +NotebookOptionsPosition[ 213755, 5027] +NotebookOutlinePosition[ 214540, 5053] +CellTagsIndexPosition[ 214459, 5048] WindowFrame->Normal*) (* Beginning of Notebook Content *) @@ -39,20 +39,18 @@ Teukolsky equation for a given mode, ", terms." }], "Usage", CellChangeTimes->{{3.942146008319547*^9, 3.942146008438014*^9}, { - 3.9422227374162283`*^9, 3.942222782742009*^9}, 3.9422228636619577`*^9}, + 3.9422227374162283`*^9, 3.942222782742009*^9}, 3.942222863661958*^9}, CellID->1226706735,ExpressionUUID->"860eeef6-58c4-4c2e-ba29-267e4b80d079"], Cell[TextData[{ "We compute ", Cell[BoxData[ - SubscriptBox["R", "In"]], "InlineFormula", - FormatType->StandardForm,ExpressionUUID-> + SubscriptBox["R", "In"]], "InlineFormula",ExpressionUUID-> "0ab18874-2e89-4e4c-b1e8-6d614322d6f0"], " according to Eq.(147) of Sasaki, Tagoshi \ https://arxiv.org/abs/gr-qc/0306120 where we divide out a factor of ", Cell[BoxData[ - SubscriptBox["K", "\[Nu]"]], "InlineFormula", - FormatType->StandardForm,ExpressionUUID-> + SubscriptBox["K", "\[Nu]"]], "InlineFormula",ExpressionUUID-> "69332f75-3d13-4104-bcca-63654e495bea"], ", i.e., ", Cell[BoxData[ @@ -69,18 +67,16 @@ https://arxiv.org/abs/gr-qc/0306120 where we divide out a factor of ", SubsuperscriptBox["R", "C", RowBox[{ RowBox[{"-", "\[Nu]"}], "-", "1"}]]}]}]}]], "InlineFormula", - FormatType->StandardForm,ExpressionUUID-> - "22f03afe-7450-4da1-9be5-f80612788075"], + ExpressionUUID->"22f03afe-7450-4da1-9be5-f80612788075"], ". ", Cell[BoxData[ - SubscriptBox["R", "Up"]], "InlineFormula", - FormatType->StandardForm,ExpressionUUID-> + SubscriptBox["R", "Up"]], "InlineFormula",ExpressionUUID-> "442a549d-6804-49df-ade2-e854a7ce585d"], " is computed after Eq.(B.7) of Throwe ,Hughes \ https://dspace.mit.edu/handle/1721.1/61270. " }], "Notes", CellChangeTimes->{{3.94223010417549*^9, 3.942230142224959*^9}, { - 3.942230254892722*^9, 3.9422304701669703`*^9}, {3.942230511057706*^9, + 3.942230254892722*^9, 3.9422304701669707`*^9}, {3.942230511057706*^9, 3.942230554319509*^9}, 3.942230599297515*^9, {3.942230711648521*^9, 3.942230712881777*^9}}, CellID->1221797685,ExpressionUUID->"9593db7e-87cc-4897-928e-46f72ca9bb38"], @@ -168,10 +164,10 @@ Cell[TextData[{ "10b7800f-16f4-4306-9eb8-92692ee5eb1a"], DynamicModuleBox[{$CellContext`nbobj$$ = NotebookObject[ "cb00ca4d-1fa2-41b6-98e6-832d08bd84d9", - "332804b2-39d2-4bcb-8344-d71e6d3ba10d"], $CellContext`cellobj$$ = + "a7641fdd-8d7f-4b25-b8b4-5a71fc7a3299"], $CellContext`cellobj$$ = CellObject[ "44eaa3dc-4fda-45ae-8c12-c5c4bc64c656", - "09411fbb-06f6-4de1-b0e2-ecea144dc49e"]}, + "16089017-ecda-44ab-ba1a-7655ea99f07b"]}, TemplateBox[{ GraphicsBox[{{ Thickness[0.06], @@ -290,6 +286,13 @@ Cell[BoxData[ $Line = 0; Null]], "PrimaryExamplesSection", CellID->336632675,ExpressionUUID->"7ffeb3ad-6511-4c6a-adaf-258336a6fc70"], +Cell["\<\ +Here is a basic example to compute the s=-2 l=2 m=2 mode in Kerr up to 4 \ +terms in the Series (1.5PN).\ +\>", "ExampleText", + CellChangeTimes->{{3.946103658010844*^9, 3.9461036974031773`*^9}}, + CellID->2113760115,ExpressionUUID->"94aae506-9436-4ecf-bea7-2359cc22bd8a"], + Cell[CellGroupData[{ Cell[BoxData[ @@ -302,7 +305,7 @@ Cell[BoxData[ CellChangeTimes->{{3.942146462350701*^9, 3.942146474773769*^9}, { 3.9421465581906*^9, 3.942146558375171*^9}, {3.942146610065486*^9, 3.94214663484762*^9}}, - 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Complex[0, -3] - Teukolsky`PN`Private`r^(-1)/$CellContext`\[Omega]}, -1, 0, 1]], + Complex[0, -3] ( + Teukolsky`PN`Private`r^(-1)/$CellContext`\[Omega])}, -1, 0, 1]], "TermCount" -> 4, "Normalization" -> "Default", "Amplitudes" -> <|"Transmission" -> "Not Computed"|>, "Simplified" -> True|>], @@ -861,9 +864,10 @@ Cell[BoxData[ Selectable->False]}]}], "\[RightAssociation]"}]], "Output", CellChangeTimes->{{3.942146476518921*^9, 3.942146505601961*^9}, 3.94214656684116*^9, {3.942146622616212*^9, 3.942146642585205*^9}, - 3.942388590247716*^9, 3.942388959260372*^9, 3.942388991878456*^9}, - CellLabel->"Out[3]=", - CellID->170885151,ExpressionUUID->"1dd5883d-dd2d-4512-9520-b363380acb86"] + 3.942388590247716*^9, 3.942388959260372*^9, 3.942388991878456*^9, + 3.946097134141876*^9, 3.946097183744846*^9}, + CellLabel->"Out[5]=", + CellID->2135881485,ExpressionUUID->"bef86d06-49d7-4cfe-b465-ea71d64444c8"] }, Open ]], Cell[CellGroupData[{ @@ -874,7 +878,7 @@ Cell[BoxData[ RowBox[{"test", "[", "\"\\"", "]"}], "[", "r", "]"}], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.942146573661372*^9, 3.942146586725117*^9}}, - 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It gives the solution to the \ -Teukolsky equation. It can be shortcut by simply querying for a symbol or \ -number. \ +Alternatively to the number of terms, one can put in the desired PN order as \ +a String\ \>", "ExampleText", - CellChangeTimes->{{3.942389387917851*^9, 3.942389457072596*^9}}, - CellID->1955714696,ExpressionUUID->"f4b0d4be-ddb7-4012-bf8c-9a18ad2a5b96"], + CellChangeTimes->{{3.946103489397705*^9, 3.9461035297604647`*^9}, + 3.94610370519433*^9}, + CellID->246899034,ExpressionUUID->"f5f2c554-c444-4609-8d23-91ebce7c64a7"], Cell[CellGroupData[{ Cell[BoxData[ - RowBox[{ - RowBox[{ - RowBox[{"test", "[", "\"\\"", "]"}], "[", "\"\\"", - "]"}], "[", "r", "]"}]], "Input", - CellChangeTimes->{{3.942389459284617*^9, 3.942389466310264*^9}}, - CellLabel->"In[12]:=", - CellID->2134775037,ExpressionUUID->"db939f69-4715-4698-bfa2-456877832e19"], - -Cell[BoxData[ - InterpretationBox[ + RowBox[{"TeukolskyRadialPN", "[", RowBox[{ - RowBox[{ - FractionBox["4", "5"], " ", - SuperscriptBox["r", "4"], " ", - SuperscriptBox["\[Omega]", "4"], " ", - 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Function[Teukolsky`PN`Private`r, SeriesData[$CellContext`\[Eta], 0, { - Rational[1, 16] (-1 + $CellContext`a^2)^(-2) - Teukolsky`PN`Private`r^4, Complex[0, - Rational[1, 24]] (-1 + $CellContext`a^2)^(-2) - Teukolsky`PN`Private`r^5 $CellContext`\[Omega], - Rational[-1, 672] (-1 + $CellContext`a^2)^(-2) - Teukolsky`PN`Private`r^3 (168 + Complex[0, 84] $CellContext`a + + Rational[5, 64] (-1 + $CellContext`a^2)^(-2) + E^(Complex[0, 1] $CellContext`a ( + 1 + (1 + (1 - $CellContext`a^2)^Rational[1, 2])^(-1) + Log[1 - $CellContext`a^2])) + Teukolsky`PN`Private`r^4 $CellContext`\[Omega]^(-4), Complex[0, + Rational[5, 96]] (-1 + $CellContext`a^2)^(-2) + E^(Complex[0, 1] $CellContext`a ( + 1 + (1 + (1 - $CellContext`a^2)^Rational[1, 2])^(-1) + Log[1 - $CellContext`a^2])) + Teukolsky`PN`Private`r^5 $CellContext`\[Omega]^(-3), + Rational[-5, 2688] (-1 + $CellContext`a^2)^(-2) + E^(Complex[0, 1] $CellContext`a ( + 1 + (1 + (1 - $CellContext`a^2)^Rational[1, 2])^(-1) + Log[1 - $CellContext`a^2])) + Teukolsky`PN`Private`r^3 $CellContext`\[Omega]^(-4) (168 + + Complex[0, 84] $CellContext`a + 11 Teukolsky`PN`Private`r^3 $CellContext`\[Omega]^2), - Rational[1, 2016] (-1 + $CellContext`a^2)^(-2) - Teukolsky`PN`Private`r^4 $CellContext`\[Omega] (28 $CellContext`a + - Complex[0, -126] $CellContext`a^2 (1 - $CellContext`a^2)^ - Rational[-1, 2] + - Complex[0, 9] (Rational[35, 3] + - 14 (1 - $CellContext`a^2)^Rational[-1, 2] - 28 EulerGamma - - Teukolsky`PN`Private`r^3 $CellContext`\[Omega]^2) + - Complex[0, -252] + Rational[-5, + 8064] $CellContext`a^(-1) (-1 + $CellContext`a^2)^(-2) (-1 + + 3 $CellContext`a^2 + + Complex[0, -3] $CellContext`a (1 - $CellContext`a^2)^ + Rational[1, 2])^(-1) + E^(Complex[0, 1] $CellContext`a ( + 1 + (1 + (1 - $CellContext`a^2)^Rational[1, 2])^(-1) + Log[1 - $CellContext`a^2])) + Teukolsky`PN`Private`r^4 $CellContext`\[Omega]^(-3) ( + Complex[0, -777] $CellContext`a + 532 $CellContext`a^2 + + Complex[0, 819] $CellContext`a^3 - 588 $CellContext`a^4 - + 126 (1 - $CellContext`a^2)^Rational[1, 2] + + Complex[0, -168] $CellContext`a (1 - $CellContext`a^2)^ + Rational[1, 2] + + 945 $CellContext`a^2 (1 - $CellContext`a^2)^Rational[1, 2] + + Complex[0, 588] $CellContext`a^3 (1 - $CellContext`a^2)^ + Rational[1, 2] + Complex[0, -252] $CellContext`a EulerGamma + + Complex[0, 756] $CellContext`a^3 EulerGamma + + 756 $CellContext`a^2 (1 - $CellContext`a^2)^Rational[1, 2] + EulerGamma + + Complex[0, -9] $CellContext`a + Teukolsky`PN`Private`r^3 $CellContext`\[Omega]^2 + + Complex[0, 27] $CellContext`a^3 + Teukolsky`PN`Private`r^3 $CellContext`\[Omega]^2 + + 27 $CellContext`a^2 (1 - $CellContext`a^2)^Rational[1, 2] + Teukolsky`PN`Private`r^3 $CellContext`\[Omega]^2 + + Complex[0, -126] $CellContext`a Log[1 - $CellContext`a^2] + + Complex[0, 378] $CellContext`a^3 Log[1 - $CellContext`a^2] + + 378 $CellContext`a^2 (1 - $CellContext`a^2)^Rational[1, 2] + Log[1 - $CellContext`a^2] + + 252 $CellContext`a (Complex[0, -1] + + Complex[0, 3] $CellContext`a^2 + + 3 $CellContext`a (1 - $CellContext`a^2)^Rational[1, 2]) + PolyGamma[ + 0, Complex[0, 2] $CellContext`a (1 - $CellContext`a^2)^ + Rational[-1, 2]] + + 252 $CellContext`a (Complex[0, -1] + + Complex[0, 3] $CellContext`a^2 + + 3 $CellContext`a (1 - $CellContext`a^2)^Rational[1, 2]) PolyGamma[ 0, 3 + Complex[0, 2] $CellContext`a (1 - $CellContext`a^2)^ - Rational[-1, 2]])}, -8, -4, 1]], "BoundaryCondition" -> "In", - "LeadingOrder" -> Function[Teukolsky`PN`Private`r, + Rational[-1, 2]])}, -20, -16, 1]], "BoundaryCondition" -> + "In", "LeadingOrder" -> Function[Teukolsky`PN`Private`r, SeriesData[$CellContext`\[Eta], 0, { - Rational[1, 16] (-1 + $CellContext`a^2)^(-2) - Teukolsky`PN`Private`r^4}, -8, -7, 1]], "TermCount" -> 4, - "Normalization" -> "SasakiTagoshi", + Rational[5, 64] (-1 + $CellContext`a^2)^(-2) + E^(Complex[0, 1] $CellContext`a ( + 1 + (1 + (1 - $CellContext`a^2)^Rational[1, 2])^(-1) + Log[1 - $CellContext`a^2])) + Teukolsky`PN`Private`r^4 $CellContext`\[Omega]^(-4)}, -20, -19, + 1]], "TermCount" -> 4, "Normalization" -> "UnitTransmission", "Amplitudes" -> <|"Transmission" -> "Not Computed"|>, "Simplified" -> True|>], Editable->False, @@ -2665,13 +2552,11 @@ Cell[BoxData[ PaneBox[ ButtonBox[ DynamicBox[ - FEPrivate`FrontEndResource["FEBitmaps", "SummaryBoxOpener"], - ImageSizeCache -> { - 10.299843749999999`, {0., 10.299843749999999`}}], - Appearance -> None, BaseStyle -> {}, - ButtonFunction :> (Typeset`open$$ = True), Evaluator -> - Automatic, Method -> "Preemptive"], - Alignment -> {Center, Center}, ImageSize -> + FEPrivate`FrontEndResource[ + "FEBitmaps", "SummaryBoxOpener"]], + ButtonFunction :> (Typeset`open$$ = True), Appearance -> + None, BaseStyle -> {}, Evaluator -> Automatic, Method -> + "Preemptive"], Alignment -> {Center, Center}, ImageSize -> Dynamic[{ Automatic, 3.5 (CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[ @@ -2750,31 +2635,29 @@ Cell[BoxData[ TagBox[ "\"Boundary Condition: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", - TagBox["\"Up\"", "SummaryItem"]}]}}, AutoDelete -> False, - BaseStyle -> { - ShowStringCharacters -> False, NumberMarks -> False, - PrintPrecision -> 3, ShowSyntaxStyles -> False}, + TagBox["\"Up\"", "SummaryItem"]}]}}, GridBoxAlignment -> { - "Columns" -> {{Left}}, "Rows" -> {{Automatic}}}, - GridBoxItemSize -> { + "Columns" -> {{Left}}, "Rows" -> {{Automatic}}}, AutoDelete -> + False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings -> { - "Columns" -> {{2}}, "Rows" -> {{Automatic}}}]}}, AutoDelete -> - False, BaselinePosition -> {1, 1}, + "Columns" -> {{2}}, "Rows" -> {{Automatic}}}, + BaseStyle -> { + ShowStringCharacters -> False, NumberMarks -> False, + PrintPrecision -> 3, ShowSyntaxStyles -> False}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}, "Rows" -> {{Top}}}, + AutoDelete -> False, GridBoxItemSize -> { - "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], True -> - GridBox[{{ + "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, + BaselinePosition -> {1, 1}], True -> GridBox[{{ PaneBox[ ButtonBox[ DynamicBox[ - FEPrivate`FrontEndResource["FEBitmaps", "SummaryBoxCloser"], - ImageSizeCache -> { - 10.299843749999999`, {0., 10.299843749999999`}}], - Appearance -> None, BaseStyle -> {}, - ButtonFunction :> (Typeset`open$$ = False), Evaluator -> - Automatic, Method -> "Preemptive"], - Alignment -> {Center, Center}, ImageSize -> + FEPrivate`FrontEndResource[ + "FEBitmaps", "SummaryBoxCloser"]], + ButtonFunction :> (Typeset`open$$ = False), Appearance -> + None, BaseStyle -> {}, Evaluator -> Automatic, Method -> + "Preemptive"], Alignment -> {Center, Center}, ImageSize -> Dynamic[{ Automatic, 3.5 (CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[ @@ -2860,23 +2743,25 @@ Cell[BoxData[ TagBox[ InterpretationBox[ RowBox[{ - RowBox[{"-", - FractionBox[ - RowBox[{"3", " ", "\[ImaginaryI]"}], - RowBox[{"\"r\"", " ", "\[Omega]", " ", "\[Eta]"}]]}], "+", + FractionBox["3", + RowBox[{"2", " ", "\"r\"", " ", + SuperscriptBox["\[Omega]", "4"], " ", + SuperscriptBox["\[Eta]", "10"]}]], "+", InterpretationBox[ + FractionBox["1", SuperscriptBox[ - RowBox[{"O", "[", "\[Eta]", "]"}], "0"], - SeriesData[$CellContext`\[Eta], 0, {}, -1, 0, 1], + RowBox[{"O", "[", "\[Eta]", "]"}], "9"]], + SeriesData[$CellContext`\[Eta], 0, {}, -10, -9, 1], Editable -> False]}], SeriesData[$CellContext`\[Eta], 0, { - Complex[0, -3] ("r"^(-1)/$CellContext`\[Omega])}, -1, 0, - 1], Editable -> False], "SummaryItem"]}]}, { + Rational[3, 2] + "r"^(-1) $CellContext`\[Omega]^(-4)}, -10, -9, 1], + Editable -> False], "SummaryItem"]}]}, { RowBox[{ TagBox["\"Normalization: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", - TagBox["\"SasakiTagoshi\"", "SummaryItem"]}]}, { + TagBox["\"UnitTransmission\"", "SummaryItem"]}]}, { RowBox[{ TagBox["\"Simplified: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", @@ -2884,20 +2769,21 @@ Cell[BoxData[ RowBox[{ TagBox["\"Number of terms: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", - TagBox["4", "SummaryItem"]}]}}, AutoDelete -> False, - BaseStyle -> { - ShowStringCharacters -> False, NumberMarks -> False, - PrintPrecision -> 3, ShowSyntaxStyles -> False}, + TagBox["4", "SummaryItem"]}]}}, GridBoxAlignment -> { - "Columns" -> {{Left}}, "Rows" -> {{Automatic}}}, - GridBoxItemSize -> { + "Columns" -> {{Left}}, "Rows" -> {{Automatic}}}, AutoDelete -> + False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings -> { - "Columns" -> {{2}}, "Rows" -> {{Automatic}}}]}}, AutoDelete -> - False, BaselinePosition -> {1, 1}, + "Columns" -> {{2}}, "Rows" -> {{Automatic}}}, + BaseStyle -> { + ShowStringCharacters -> False, NumberMarks -> False, + PrintPrecision -> 3, ShowSyntaxStyles -> False}]}}, GridBoxAlignment -> {"Columns" -> {{Left}}, "Rows" -> {{Top}}}, + AutoDelete -> False, GridBoxItemSize -> { - "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}]}, + "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, + BaselinePosition -> {1, 1}]}, Dynamic[Typeset`open$$], ImageSize -> Automatic]}, "SummaryPanel"], DynamicModuleValues:>{}], "]"}], @@ -2907,33 +2793,32 @@ Cell[BoxData[ "a" -> $CellContext`a, "PN" -> {$CellContext`\[Eta], 4}, "RadialFunction" -> Function[Teukolsky`PN`Private`r, SeriesData[$CellContext`\[Eta], 0, { - Complex[0, -3] ( - Teukolsky`PN`Private`r^(-1)/$CellContext`\[Omega]), -3, - Rational[1, 2] - Teukolsky`PN`Private`r^(-2) $CellContext`\[Omega]^(-1) ( - 8 $CellContext`a + - Complex[0, 3] (-2 + - Teukolsky`PN`Private`r^3 $CellContext`\[Omega]^2)), - Rational[1, 6] - Teukolsky`PN`Private`r^(-1) (Complex[0, -4] $CellContext`a + - 12 (1 - $CellContext`a^2)^Rational[1, 2] - 36 EulerGamma + - Complex[0, 36] Pi + - 3 Teukolsky`PN`Private`r^3 $CellContext`\[Omega]^2)}, -1, 3, 1]], - "BoundaryCondition" -> "Up", "LeadingOrder" -> - Function[Teukolsky`PN`Private`r, + Rational[3, 2] + Teukolsky`PN`Private`r^(-1) $CellContext`\[Omega]^(-4), + Complex[0, + Rational[-3, 2]] $CellContext`\[Omega]^(-3), Rational[1, 4] + Teukolsky`PN`Private`r^(-2) $CellContext`\[Omega]^(-4) (6 + + Complex[0, 8] $CellContext`a - 3 + Teukolsky`PN`Private`r^3 $CellContext`\[Omega]^2), Complex[0, + Rational[1, 4]] + Teukolsky`PN`Private`r^(-1) $CellContext`\[Omega]^(-3) (6 - 12 + EulerGamma + Complex[0, 6] Pi + + Teukolsky`PN`Private`r^3 $CellContext`\[Omega]^2 - 12 Log[2] - 12 + Log[2 $CellContext`\[Omega]])}, -10, -6, 1]], "BoundaryCondition" -> + "Up", "LeadingOrder" -> Function[Teukolsky`PN`Private`r, SeriesData[$CellContext`\[Eta], 0, { - Complex[0, -3] ( - Teukolsky`PN`Private`r^(-1)/$CellContext`\[Omega])}, -1, 0, 1]], - "TermCount" -> 4, "Normalization" -> "SasakiTagoshi", + Rational[3, 2] + Teukolsky`PN`Private`r^(-1) $CellContext`\[Omega]^(-4)}, -10, -9, + 1]], "TermCount" -> 4, "Normalization" -> "UnitTransmission", "Amplitudes" -> <|"Transmission" -> "Not Computed"|>, "Simplified" -> True|>], Editable->False, SelectWithContents->True, Selectable->False]}]}], 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CellID->2043357575,ExpressionUUID->"2644f8aa-0d99-4516-ad6e-bd9380d2ed2a"], + +Cell[BoxData[ + InterpretationBox[Cell[ + "Interactive Examples", "ExampleSection",ExpressionUUID-> + "00661863-bd35-4171-b591-abb48d165af1"], + $Line = 0; Null]], "ExampleSection", + CellID->99524524,ExpressionUUID->"63e63013-e191-42fa-b8cd-64d7240ecfa9"], + +Cell[CellGroupData[{ + +Cell[BoxData[ + InterpretationBox[Cell[ + "Neat Examples", "ExampleSection",ExpressionUUID-> + "365d2492-1823-4b69-a81c-88a6f028df9d"], + $Line = 0; Null]], "ExampleSection", + CellID->1283718040,ExpressionUUID->"f6a5eb39-7442-454c-92c7-d98d7fd15347"], + +Cell["\<\ +To get some idea for the timings we can Table over increasing PN orders. Here \ +is an example where we computed some scalar (s=0) modes from 1PN (1 term) to \ +12PN (25 terms). The timings are given in seconds and we can observe an \ +approximate doubling per PN order. \ +\>", "ExampleText", + CellChangeTimes->{{3.946098215637084*^9, 3.946098229891073*^9}, { + 3.9460983557315197`*^9, 3.9460984598559313`*^9}, {3.946098586832157*^9, + 3.946098624801135*^9}}, + CellID->1046221386,ExpressionUUID->"912086c4-4ee1-4183-9892-f33c68077427"], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"Table", "[", "\[IndentingNewLine]", + RowBox[{ + RowBox[{"EchoTiming", "[", "\[IndentingNewLine]", + RowBox[{ + RowBox[{ + RowBox[{"tist", "[", "i", "]"}], "=", + RowBox[{"TeukolskyRadialPN", "[", + RowBox[{"0", ",", "1", ",", "0", ",", "a", ",", "\[Omega]", ",", + RowBox[{"{", + RowBox[{"\[Eta]", ",", "i"}], "}"}]}], "]"}]}], ",", "i"}], "]"}], + "\[IndentingNewLine]", ",", + RowBox[{"{", + RowBox[{"i", ",", "1", ",", "25", ",", "2"}], "}"}]}], "]"}]], "Input", + CellChangeTimes->{{3.945694220862994*^9, 3.9456943062972803`*^9}, { + 3.945694341957625*^9, 3.945694367499048*^9}, {3.945694410849701*^9, + 3.945694453081644*^9}, {3.945694838819032*^9, 3.945694864676091*^9}}, + CellLabel->"In[3]:=", + CellID->310155003,ExpressionUUID->"c23576aa-c276-4c71-9dc8-a55e676d959d"], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{ + TagBox["1", + "EchoLabel"], " ", "4.84072`"}]], "EchoTiming", + CellChangeTimes->{3.945694879416453*^9}, + CellID->1479690548,ExpressionUUID->"5cd32106-918c-40a8-8f0d-4d6bf775772f"], + +Cell[BoxData[ + RowBox[{ + TagBox["3", + "EchoLabel"], " ", "9.482586`"}]], "EchoTiming", + CellChangeTimes->{3.9456948889048433`*^9}, + CellID->957304461,ExpressionUUID->"73dab4e3-cd69-4980-999c-08a0d93c0b9b"], + +Cell[BoxData[ + RowBox[{ + TagBox["5", + "EchoLabel"], " ", "17.482955`"}]], "EchoTiming", + CellChangeTimes->{3.945694906393314*^9}, + CellID->1435544872,ExpressionUUID->"751ef2d5-0dbd-4dc1-9465-2d40cfa0a03f"], + +Cell[BoxData[ + RowBox[{ + TagBox["7", + "EchoLabel"], " ", "34.109019`"}]], "EchoTiming", + CellChangeTimes->{3.9456949405080214`*^9}, + CellID->759566879,ExpressionUUID->"ebbd2894-75de-4d05-9c1f-2f7b9152b504"], + +Cell[BoxData[ + RowBox[{ + TagBox["9", + "EchoLabel"], " ", "54.775601`"}]], "EchoTiming", + CellChangeTimes->{3.945694995288897*^9}, + CellID->1814378353,ExpressionUUID->"52731855-cfb2-4cf3-b654-fecba3fc7562"], + +Cell[BoxData[ + RowBox[{ + TagBox["11", + "EchoLabel"], " ", "86.560144`"}]], "EchoTiming", + CellChangeTimes->{3.945695081854488*^9}, + CellID->301038460,ExpressionUUID->"378e7b2b-e92d-4063-84fc-c7c4eb0b1681"], + +Cell[BoxData[ + RowBox[{ + TagBox["13", + "EchoLabel"], " ", "135.628242`"}]], "EchoTiming", + CellChangeTimes->{3.94569521748816*^9}, + CellID->38739405,ExpressionUUID->"70618e8e-a4e5-4ebf-9a08-910978a2bb8f"], + +Cell[BoxData[ + RowBox[{ + TagBox["15", + "EchoLabel"], " ", "209.171788`"}]], "EchoTiming", + CellChangeTimes->{3.945695426665341*^9}, + CellID->154343254,ExpressionUUID->"fd485d91-8604-478a-9f49-44dceb06b40c"], + +Cell[BoxData[ + RowBox[{ + TagBox["17", + "EchoLabel"], " ", "399.93244`"}]], "EchoTiming", + CellChangeTimes->{3.945695826603092*^9}, + CellID->324905533,ExpressionUUID->"d1fdf454-cdb5-4d2e-9abd-ea00016a7f16"], + +Cell[BoxData[ + RowBox[{ + TagBox["19", + "EchoLabel"], " ", "796.557597`"}]], "EchoTiming", + CellChangeTimes->{3.945696623165621*^9}, + CellID->325877729,ExpressionUUID->"aa539e64-ee37-42c8-85ff-21d826245390"], + +Cell[BoxData[ + RowBox[{ + TagBox["21", + "EchoLabel"], " ", "1881.850263`"}]], "EchoTiming", + CellChangeTimes->{3.945698505021377*^9}, + CellID->803369154,ExpressionUUID->"9ffcb926-c2b2-4692-8f2b-5972a471b1ad"], + +Cell[BoxData[ + RowBox[{ + TagBox["23", + "EchoLabel"], " ", "5236.106216`"}]], "EchoTiming", + CellChangeTimes->{3.945703741132449*^9}, + CellID->1011748486,ExpressionUUID->"a26f4bfd-0746-43a1-aff7-4cdfb3076423"] +}, Open ]], + +Cell[BoxData[ + TemplateBox[{ + "Simplify", "time", + "\"Time spent on a transformation exceeded \ +\\!\\(\\*RowBox[{\\\"300.`\\\"}]\\) seconds, and the transformation was \ +aborted. Increasing the value of the TimeConstraint option may improve the \ +result of simplification.\"", 2, 3, 1, 27765836861268227613, "Local"}, + "MessageTemplate"]], "Message", "MSG", + CellChangeTimes->{3.945694859662318*^9, 3.945744211216871*^9}, + CellLabel->"During evaluation of In[3]:=", + CellID->24170283,ExpressionUUID->"62467a15-bf87-4eda-8dd7-1ae6b59471ee"], + +Cell[BoxData["$Aborted"], "Output", + CellChangeTimes->{ + 3.945694286645772*^9, 3.945694316768975*^9, 3.9456943853938413`*^9, { + 3.945694428333228*^9, 3.945694451233023*^9}, {3.945694835729683*^9, + 3.9456948596863756`*^9}, 3.945744674882318*^9}, + CellLabel->"Out[3]=", + CellID->843434883,ExpressionUUID->"c5c0805c-7abf-4c37-99c1-063154e468c2"] +}, Open ]], + +Cell["\<\ +As we can see we only got to 11PN (23 terms) because we ran into Simplify \ +timeouts for 12PN. This can be mended by resetting the TimeConstraint Option \ +for Simplify. Here the time scaling seems to increase significantly\ +\>", "ExampleText", + CellChangeTimes->{{3.9460984680468597`*^9, 3.946098558871151*^9}, { + 3.946099061524925*^9, 3.946099078185382*^9}}, + CellID->279466906,ExpressionUUID->"778f4c57-9bcb-4098-ad19-9a19ef29621e"], + +Cell[CellGroupData[{ + +Cell[BoxData[{ + RowBox[{"SetOptions", "[", + RowBox[{"Simplify", ",", + RowBox[{"TimeConstraint", "->", "1000"}]}], "]"}], "\[IndentingNewLine]", + RowBox[{"EchoTiming", "[", + RowBox[{ + RowBox[{ + RowBox[{"tist", "[", "25", "]"}], "=", + RowBox[{"TeukolskyRadialPN", "[", + RowBox[{"0", ",", "1", ",", "0", ",", "a", ",", "\[Omega]", ",", + RowBox[{"{", + RowBox[{"\[Eta]", ",", "25"}], "}"}]}], "]"}]}], ",", "25"}], + "]"}], "\[IndentingNewLine]", + RowBox[{"SetOptions", "[", + RowBox[{"Simplify", ",", + RowBox[{"TimeConstraint", "->", "300"}]}], "]"}]}], "Input", + CellChangeTimes->{{3.94584121255661*^9, 3.94584122975668*^9}, { + 3.945841282359701*^9, 3.945841290745599*^9}, 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What do you even mean by a complex Ke TeukolskyRadialFunctionPN::parama="a=`1`. Numeric values for the Kerr parameter a have to be within [0,1]. It can however be left arbitrary."; TeukolskyRadialFunctionPN::param\[Omega]="\[Omega]=`1`. Complex frequencies are not yet supported"; TeukolskyRadialFunctionPN::paramorder="order=`1`. The given number of terms has to be an Integer"; +TeukolskyRadialFunctionPN::PNInput="Input String does not contain \"PN\". Assume PN orders are desired (calulate `1` terms in the Series)."; (* ::Subsection:: *) @@ -1063,7 +1064,7 @@ MST=Append[MST,Table[a[i]->aMST[i]+If[i==0,0,O[\[Epsilon]]^(ExpOrder+1)],{i,-Exp ] -(* ::Subsection:: *) +(* ::Subsection::Closed:: *) (*Definitions, replacements and auxiliary functions*) @@ -1156,6 +1157,19 @@ aux//PNScalings[#,arguments,\[Eta],opt]&//SeriesTerms[#,{\[Eta],0,termOrder}]& ] +Scalings[list_List,\[Eta]_Symbol][expr_]:=Module[{aux,repls,arguments}, +arguments=list//Partition[#,2]&; +repls=(#[[1]]->#[[1]]\[Eta]^#[[2]]&)/@arguments; +expr/.\[Eta]->1/.repls +] +Scalings[arguments_List,\[Eta]_Symbol][series_SeriesData]:=Module[{aux,termOrder}, +termOrder=series//SeriesLength; +aux=series//RemovePN[#,\[Eta]]&; +aux//Scalings[arguments,\[Eta]]//SeriesTerms[#,{\[Eta],0,termOrder}]& +] +Scalings[arguments_List,\[Eta]_Symbol][list_List]:=Scalings[arguments,\[Eta]][#]&/@list; + + IgnoreExpansionParameter[series_SeriesData,symbol_:1]:=Module[{aux,param,newList}, param=series[[1]]; newList=series[[3]]/.param->symbol; @@ -1163,16 +1177,16 @@ ReplacePart[series,3->newList] ] -Zero[expr_,var_Symbol]:=Module[{aux,repls}, +Zero[var_Symbol][expr_]:=Module[{aux,repls}, repls=var->0; expr/.repls] -Zero[expr_,vars_List]:=Module[{aux,repls}, +Zero[vars_List][expr_]:=Module[{aux,repls}, repls=vars->0//Thread; expr/.repls] -One[expr_,var_Symbol]:=Module[{aux,repls}, +One[var_Symbol][expr_]:=Module[{aux,repls}, repls=var->1; expr/.repls] -One[expr_,vars_List]:=Module[{aux,repls}, +One[vars_List][expr_]:=Module[{aux,repls}, repls=vars->1//Thread; expr/.repls] @@ -1205,7 +1219,7 @@ ExpandSpheroidals[expr_Times,{\[Eta]_,n_}]:=ExpandSpheroidals[#,{\[Eta],n}]&/@ex ExpandSpheroidals[expr_,{\[Eta]_,n_}]:=expr; -(* ::Subsubsection:: *) +(* ::Subsubsection::Closed:: *) (*Tools for Series*) @@ -1228,6 +1242,8 @@ Attributes[SeriesLenght]={Listable}; SeriesCollect[series_SeriesData,var__,func_:Identity]:=Collect[#,var,func]&/@series; +SeriesCollect[list_List,var__,func_:Identity]:=SeriesCollect[#,var,func]&/@list; +SeriesCollect[expr_,var__,func_:Identity]:=Collect[#,var,func]&@expr; SeriesTake[series_SeriesData,order_Integer:1]:=Block[{aux}, @@ -1315,11 +1331,19 @@ ret=\!\( ret ] +ExpandDiracDelta[expr_List,x_]:=ExpandDiracDelta[#,x]&/@expr; + ExpandDiracDelta[expr_,x_]/;!FreeQ[expr,DiracDelta[a_]/;!FreeQ[a,x]]:=Module[{aux,ret}, aux=expr//Collect[#,{DiracDelta[__],Derivative[__][DiracDelta][__]}]&; ExpandDiracDelta[#,x]&/@aux ] +ExpandDiracDelta[expr_SeriesData,x_]/;!FreeQ[expr,DiracDelta[a_]/;!FreeQ[a,x]]:=Module[{aux,ret}, +aux=expr//SeriesCollect[#,{DiracDelta[__],Derivative[__][DiracDelta][__]}]&; +ExpandDiracDelta[#,x]&/@aux +] + + ExpandDiracDelta[expr_Plus,x_]:=(ExpandDiracDelta[#,x]&/@expr); ExpandDiracDelta[expr_,x_]:=expr; @@ -1688,7 +1712,7 @@ Derivative[n_][\[Theta]][arg_]:=Derivative[n-1][\[Delta]][arg]; \[Delta]''[\[Eta]^-2 a_]:=\[Eta]^2 \[Delta]''[a]; -(* ::Subsection::Closed:: *) +(* ::Subsection:: *) (*Amplitudes*) @@ -1736,7 +1760,7 @@ aux/.MSTCoefficientsInternal[\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]+3]// (*These are the amplitudes Subscript[B, trans] and Subscript[B, inc] from Sasaki Tagoshi Eq.(167-169) TODO: Add Subscript[B, ref]*) -Options[BAmplitude]={"Normalization"->"SFPN"} +Options[BAmplitude]={"Normalization"->"Default"} BAmplitude["Inc",OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=Module[{aux,\[CurlyEpsilon],\[Kappa],\[CurlyEpsilon]p,\[Tau],\[ScriptCapitalK]1,\[ScriptCapitalK]2,A}, @@ -1747,11 +1771,11 @@ BAmplitude["Inc",OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\ aux=\[Omega]^-1 (\[ScriptCapitalK]1 -I E^(-I \[Pi] \[Nu]MST) Sin[\[Pi](\[Nu]MST-\[ScriptS]+I \[CurlyEpsilon])]/Sin[\[Pi](\[Nu]MST+\[ScriptS]-I \[CurlyEpsilon])] \[ScriptCapitalK]2) A E^(-I(\[CurlyEpsilon] Log[\[CurlyEpsilon]]-(1-\[Kappa])/2 \[CurlyEpsilon]))//PNScalingsInternal; aux=aux/.MSTCoefficientsInternal[\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]+3]//IgnoreExpansionParameter; \[ScriptCapitalK]1=Switch[OptionValue["Normalization"], - "SFPN",1, + "Default",1, "SasakiTagoshi",\[ScriptCapitalK]Amplitude["\[Nu]"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]//ExpandPolyGamma//ExpandGamma//SeriesCollect[#,PolyGamma[__,__]]& ]; \[ScriptCapitalK]2=Switch[OptionValue["Normalization"], - "SFPN",\[ScriptCapitalK]Amplitude["Ratio"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]//ExpandPolyGamma//ExpandGamma//SeriesCollect[#,PolyGamma[__,__]]&, + "Default",\[ScriptCapitalK]Amplitude["Ratio"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]//ExpandPolyGamma//ExpandGamma//SeriesCollect[#,PolyGamma[__,__]]&, "SasakiTagoshi",\[ScriptCapitalK]Amplitude["-\[Nu]-1"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]//ExpandPolyGamma//ExpandGamma//SeriesCollect[#,PolyGamma[__,__]]& ]; A=AAmplitude["+"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]//SeriesCollect[#,Log[__]]&//IgnoreExpansionParameter; @@ -1759,12 +1783,6 @@ aux//SeriesTake[#,order\[Eta]]&//IgnoreExpansionParameter ] -BAmplitude["Inc","Normalization"->"Toolkit"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=BAmplitude["Inc"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]/BAmplitude["Trans"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]; - - -BAmplitude["Trans","Normalization"->"Toolkit"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=1 (1+O[\[Eta]] \[Eta]^(order\[Eta]-1)); - - BAmplitude["Trans",OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=Module[{aux,\[CurlyEpsilon],\[Kappa],\[CurlyEpsilon]p,\[Tau],\[ScriptCapitalK],nMin,nMax}, \[CurlyEpsilon]=2 \[Omega]; \[Kappa]=Sqrt[1-a^2]; @@ -1773,7 +1791,7 @@ BAmplitude["Trans",OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,orde nMax=order\[Eta]/3//Ceiling; nMin=-(order\[Eta]/3+2)//Floor; \[ScriptCapitalK]=Switch[OptionValue["Normalization"], - "SFPN",\[ScriptCapitalK]Amplitude["\[Nu]"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]//ExpandPolyGamma//ExpandGamma//SeriesCollect[#,PolyGamma[__,__]]&, + "Default",\[ScriptCapitalK]Amplitude["\[Nu]"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]//ExpandPolyGamma//ExpandGamma//SeriesCollect[#,PolyGamma[__,__]]&, "SasakiTagoshi",1 ]; aux=((\[CurlyEpsilon] \[Kappa])/\[Omega])^(2\[ScriptS]) E^(I \[Kappa] \[CurlyEpsilon]p(1+(2 Log[\[Kappa]])/(1+\[Kappa]))) \!\( @@ -1784,7 +1802,25 @@ aux//IgnoreExpansionParameter ] -(* ::Subsubsection::Closed:: *) +BAmplitude["Ref",OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=Module[{aux,\[CurlyEpsilon],\[Kappa],\[CurlyEpsilon]p,\[Tau],\[ScriptCapitalK]1,\[ScriptCapitalK]2,A}, +\[CurlyEpsilon]=2 \[Omega]; +\[Kappa]=Sqrt[1-a^2]; +\[CurlyEpsilon]p=(\[CurlyEpsilon]+\[Tau])/2; +\[Tau]=(-a \[ScriptM]+\[CurlyEpsilon])/\[Kappa]; +aux=(\[Omega]^(-1-2 \[ScriptS]) E^(-I \[CurlyEpsilon](Log[\[CurlyEpsilon]]-(1-\[Kappa])/2)) (\[ScriptCapitalK]1+I E^(I \[Pi] \[Nu]MST) \[ScriptCapitalK]2) A)/\[Omega]//PNScalingsInternal; +aux=IgnoreExpansionParameter[aux/. MSTCoefficientsInternal[\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]+3]]; +\[ScriptCapitalK]1=Switch[OptionValue["Normalization"],"Default",1,"SasakiTagoshi",(SeriesCollect[#1,PolyGamma[__,__]]&)[ExpandGamma[ExpandPolyGamma[\[ScriptCapitalK]Amplitude["\[Nu]"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]]]]]; +\[ScriptCapitalK]2=Switch[OptionValue["Normalization"],"Default",(SeriesCollect[#1,PolyGamma[__,__]]&)[ExpandGamma[ExpandPolyGamma[\[ScriptCapitalK]Amplitude["Ratio"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]]]],"SasakiTagoshi",(SeriesCollect[#1,PolyGamma[__,__]]&)[ExpandGamma[ExpandPolyGamma[\[ScriptCapitalK]Amplitude["-\[Nu]-1"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]]]]]; +A=IgnoreExpansionParameter[(SeriesCollect[#1,Log[__]]&)[AAmplitude["-"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]]]; +IgnoreExpansionParameter[(SeriesTake[#1,order\[Eta]]&)[aux]]] + + +BAmplitude["Inc","Normalization"->"UnitTransmission"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=BAmplitude["Inc"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]/BAmplitude["Trans"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]; +BAmplitude["Ref","Normalization"->"UnitTransmission"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=BAmplitude["Ref"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]/BAmplitude["Trans"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]; +BAmplitude["Trans","Normalization"->"UnitTransmission"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=1 (1+O[\[Eta]] \[Eta]^(order\[Eta]-1)); + + +(* ::Subsubsection:: *) (*C Amplitude*) @@ -1792,10 +1828,10 @@ aux//IgnoreExpansionParameter (*These is the amplitudes Subscript[C, trans] from Sasaki Tagoshi Eq.(170) TODO: Add Subscript[C, ref] and Subscript[C, inc]*) -Options[CAmplitude]={"Normalization"->"SFPN"} +Options[CAmplitude]={"Normalization"->"Default"} -CAmplitude["Trans","Normalization"->"Toolkit"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=1 (1+O[\[Eta]] \[Eta]^(order\[Eta]-1)); +CAmplitude["Trans","Normalization"->"UnitTransmission"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=1 (1+O[\[Eta]] \[Eta]^(order\[Eta]-1)); CAmplitude["Trans",OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=Module[{aux,\[CurlyEpsilon],\[Kappa],\[CurlyEpsilon]p,\[Tau],\[ScriptCapitalK],A}, @@ -1810,6 +1846,16 @@ aux//SeriesTake[#,order\[Eta]]&//IgnoreExpansionParameter ] +CAmplitude["Inc",OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=Module[{aux,\[CurlyEpsilon],\[Kappa],\[CurlyEpsilon]p,\[Tau],\[ScriptCapitalK],A}, +"Not Computed" +] + + +CAmplitude["Ref",OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=Module[{aux,\[CurlyEpsilon],\[Kappa],\[CurlyEpsilon]p,\[Tau],\[ScriptCapitalK],A}, +"Not Computed" +] + + (* ::Subsubsection::Closed:: *) (*\[ScriptCapitalK] Amplitude*) @@ -1925,11 +1971,11 @@ ret//SeriesTake[#,order\[Eta]]& ] -(* ::Subsubsection::Closed:: *) +(* ::Subsubsection:: *) (*Interface*) -Options[TeukolskyAmplitudePN]={"Normalization"->"SFPN"} +Options[TeukolskyAmplitudePN]={"Normalization"->"Default"} TeukolskyAmplitudePN["A+",opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]Var_,{\[Eta]Var_,order\[Eta]_}] :=AAmplitude["+"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]/.{\[Omega]->\[Omega]Var,\[Eta]->\[Eta]Var}; @@ -1938,9 +1984,12 @@ TeukolskyAmplitudePN["A-",opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[Script TeukolskyAmplitudePN["Binc",opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]Var_,{\[Eta]Var_,order\[Eta]_}] :=BAmplitude["Inc",opt][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]/.{\[Omega]->\[Omega]Var,\[Eta]->\[Eta]Var}; TeukolskyAmplitudePN["Btrans",opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]Var_,{\[Eta]Var_,order\[Eta]_}] :=BAmplitude["Trans",opt][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]/.{\[Omega]->\[Omega]Var,\[Eta]->\[Eta]Var}; +TeukolskyAmplitudePN["Bref",opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]Var_,{\[Eta]Var_,order\[Eta]_}] :=BAmplitude["Ref",opt][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]/.{\[Omega]->\[Omega]Var,\[Eta]->\[Eta]Var}; TeukolskyAmplitudePN["Ctrans",opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]Var_,{\[Eta]Var_,order\[Eta]_}] :=CAmplitude["Trans",opt][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]/.{\[Omega]->\[Omega]Var,\[Eta]->\[Eta]Var}; +TeukolskyAmplitudePN["Cinc",opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]Var_,{\[Eta]Var_,order\[Eta]_}] :=CAmplitude["Inc",opt][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]/.{\[Omega]->\[Omega]Var,\[Eta]->\[Eta]Var}; +TeukolskyAmplitudePN["Cref",opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]Var_,{\[Eta]Var_,order\[Eta]_}] :=CAmplitude["Ref",opt][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]/.{\[Omega]->\[Omega]Var,\[Eta]->\[Eta]Var}; TeukolskyAmplitudePN["\[ScriptCapitalK]\[Nu]",opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]Var_,{\[Eta]Var_,order\[Eta]_}] :=\[ScriptCapitalK]Amplitude["\[Nu]"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]/.{\[Omega]->\[Omega]Var,\[Eta]->\[Eta]Var}; @@ -2235,7 +2284,7 @@ InGap[0,0]=0; InGap[\[ScriptL]_,0]:=4\[ScriptL]+2; -Options[RPN]={"Normalization"->"SFPN"} +Options[RPN]={"Normalization"->"Default"} RPN["In",OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=Module[{aux,ret,gap,\[ScriptCapitalK],secondTerm,secondR,normalization}, @@ -2247,7 +2296,7 @@ aux=RPN["C\[Nu]"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]+secondTerm; normalization=Switch[OptionValue["Normalization"], "Default",1, "SasakiTagoshi",\[ScriptCapitalK]Amplitude["\[Nu]"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]], - "UnitTransmission",\[ScriptCapitalK]Amplitude["\[Nu]"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]/BAmplitude["Trans"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]] + "UnitTransmission",1/BAmplitude["Trans"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]] ]; ret=normalization aux; ret//IgnoreExpansionParameter @@ -2299,7 +2348,7 @@ c["Up"][z_]:=2^\[Nu] E^(-\[Pi] \[Epsilon]) E^(-I \[Pi](\[Nu]+1+s)) E^(I z) (z-\[ element["Up"][n_,j_,z_]:=2^n (I z)^n Pochhammer[1+s-I \[Epsilon]+\[Nu],n]/Pochhammer[1-s+I \[Epsilon]+\[Nu],n] ((2^(-1+j-2 n-2 \[Nu]) (-I z)^(-1+j-2 n-2 \[Nu]) Gamma[1+2 n+2 \[Nu]] Pochhammer[-n+s-I \[Epsilon]-\[Nu],j])/(j! Gamma[1+n+s-I \[Epsilon]+\[Nu]] Pochhammer[-2 n-2 \[Nu],j])+(2^j (-I z)^j Gamma[-1-2 n-2 \[Nu]] Pochhammer[1+n+s-I \[Epsilon]+\[Nu],j])/(j! Gamma[-n+s-I \[Epsilon]-\[Nu]] Pochhammer[2+2 n+2 \[Nu],j]))*) -(* ::Subsubsection::Closed:: *) +(* ::Subsubsection:: *) (*Constructions (depreciated)*) @@ -2593,7 +2642,7 @@ icons = <| |>; -(* ::Subsubsection::Closed:: *) +(* ::Subsubsection:: *) (*TeukolskyRadialPN*) @@ -2601,7 +2650,7 @@ Options[RadialAssociation]={"Normalization"->"Default", "Amplitudes"->False, "Si Options[TeukolskyRadialPN]={"Normalization"->"Default", "Amplitudes"->False, "Simplify"->True} -RadialAssociation[sol_String,opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]_,{varPN_,order_}]/;MemberQ[PossibleSols,sol]:=Module[{aux,ret,R,BC,lead,termCount,normalization,amplitudes,trans}, +RadialAssociation[sol_String,opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]_,{varPN_,order_}]/;MemberQ[PossibleSols,sol]:=Module[{aux,ret,R,BC,lead,termCount,normalization,amplitudes,trans,inc,ref}, CheckInput[sol,\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,order}]; R=RPNF[sol,"Normalization"->OptionValue["Normalization"],"Simplify"->OptionValue["Simplify"]][\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,order}]; BC=sol; @@ -2609,12 +2658,34 @@ lead=RPNF[sol,"Normalization"->OptionValue["Normalization"],"Simplify"->OptionVa termCount=R[r]//SeriesLength; normalization=OptionValue["Normalization"]; trans=If[OptionValue["Amplitudes"],TeukolskyAmplitudePN[Switch[sol,"In","Btrans","Up","Ctrans"],"Normalization"->OptionValue["Normalization"]][\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,order}],"Not Computed"]; -amplitudes=<|"Transmission"->trans|>; -ret=<|"s"->\[ScriptS],"l"->\[ScriptL],"m"->\[ScriptM],"a"->a,"PN"->{varPN,order},"RadialFunction"->R,"BoundaryCondition"->BC,"LeadingOrder"->lead,"TermCount"->termCount,"Normalization"->normalization,"Amplitudes"->amplitudes,"Simplified"->OptionValue["Simplify"]|>; +inc=If[OptionValue["Amplitudes"],TeukolskyAmplitudePN[Switch[sol,"In","Binc","Up","Cinc"],"Normalization"->OptionValue["Normalization"]][\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,order}],"Not Computed"]; +ref=If[OptionValue["Amplitudes"],TeukolskyAmplitudePN[Switch[sol,"In","Bref","Up","Cref"],"Normalization"->OptionValue["Normalization"]][\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,order}],"Not Computed"]; +If[OptionValue["Simplify"],{trans,inc,ref}={trans,inc,ref}//Simplify]; +amplitudes=<|"Incidence"->inc,"Transmission"->trans,"Reflection"->ref|>; +ret=<|"s"->\[ScriptS],"l"->\[ScriptL],"m"->\[ScriptM],"a"->a,"PN"->{varPN,order},"RadialFunction"->R,"BoundaryCondition"->BC,"LeadingOrder"->lead,"TermCount"->termCount,"Normalization"->normalization,"Amplitudes"->amplitudes,"Simplify"->OptionValue["Simplify"]|>; ret ] +PNStringToOrder[pn_String]:=Module[{aux,ret,check1,check2}, +check1=StringContainsQ[pn,"PN"]; +aux=pn//StringReplace[#,"PN"->""]&; +aux=aux//ToExpression; +aux=2aux+1; +ret=aux//IntegerPart; +check2=ret-aux===0`; +If[!check1,Message[TeukolskyRadialFunctionPN::PNInput,ret]]; +If[!check2,ret=aux]; +ret +] + + +TeukolskyRadialPN[\[ScriptS]_, \[ScriptL]_, \[ScriptM]_, a_, \[Omega]_,{varPN_,order_String},opt:OptionsPattern[]]:=Module[{aux}, +aux=order//PNStringToOrder; +TeukolskyRadialPN[\[ScriptS], \[ScriptL], \[ScriptM], a, \[Omega],{varPN,aux},opt] +] + + TeukolskyRadialPN[\[ScriptS]_, \[ScriptL]_, \[ScriptM]_, a_, \[Omega]_,{varPN_,order_},opt:OptionsPattern[]]:=Module[{aux,Rin,Rup,assocUp,assocIn,retIn,retUp}, assocIn=RadialAssociation["In",opt][\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,order}]; retIn=TeukolskyRadialFunctionPN[\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,order},assocIn]; @@ -2624,7 +2695,7 @@ retUp=TeukolskyRadialFunctionPN[\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{var ] -(* ::Subsubsection::Closed:: *) +(* ::Subsubsection:: *) (*TeukolskyRadialFunctionPN*) @@ -2646,7 +2717,7 @@ TeukolskyRadialFunctionPN /: BoxForm`SummaryItem[{"Boundary Condition: ", assoc["BoundaryCondition"]}]}; extended = {BoxForm`SummaryItem[{"Leading order: ",assoc["LeadingOrder"]["r"]}], BoxForm`SummaryItem[{"Normalization: ",assoc["Normalization"]}], - BoxForm`SummaryItem[{"Simplified: ",assoc["Simplified"]}], + BoxForm`SummaryItem[{"Simplify: ",assoc["Simplify"]}], BoxForm`SummaryItem[{"Number of terms: ",assoc["TermCount"]}]}; BoxForm`ArrangeSummaryBox[ @@ -2670,19 +2741,21 @@ ret (*TeukolskyPointParticleModePN*) -Options[RadialSourcedAssociation]={"Normalization"->"Default"} +Options[RadialSourcedAssociation]={"Normalization"->"Default","Simplify"->True} -RadialSourcedAssociation["CO",opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,r0_,{varPN_,order_}]:=Assuming[{varPN>0,r0>0,r>0,1>a>=0,\[ScriptA]>=0},Module[{aux,ret,Rin,dRin,ddRin,Rup,dRup,ddRup,wronskian,source,sourceCoeffs,cUp,cIn,deltaCoeff,innerF,outerF,radialF}, -CheckInput["Up",\[ScriptS],\[ScriptL],\[ScriptM],a,\[ScriptM]/Sqrt[r0^3],{varPN,order}]; -aux=TeukolskyRadialPN[\[ScriptS],\[ScriptL],\[ScriptM],a,If[\[ScriptM]!=0,\[ScriptM],Style["0",Red]]\[CapitalOmega]Kerr,{varPN,order},"Normalization"->OptionValue["Normalization"]]; +RadialSourcedAssociation["CO",opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,aVar_,r0Var_,{varPN_,order_}]:=Assuming[{varPN>0,r0>0,r>0,1>a>=0,\[ScriptA]>=0},Module[{aux,ret,Rin,dRin,ddRin,Rup,dRup,ddRup,wronskian,source,sourceCoeffs,cUp,cIn,deltaCoeff,innerF,outerF,inner,outer,radialF,ampAssoc}, +CheckInput["Up",\[ScriptS],\[ScriptL],\[ScriptM],aVar,\[ScriptM]/Sqrt[r0Var^3],{varPN,order}]; +aux=TeukolskyRadialPN[\[ScriptS],\[ScriptL],\[ScriptM],aVar,If[\[ScriptM]!=0,\[ScriptM],Style["0",Red]]\[CapitalOmega]Kerr,{varPN,order},"Normalization"->OptionValue["Normalization"]]; Rin=aux["In"]["RadialFunction"]; Rup=aux["Up"]["RadialFunction"]; dRup=Rup'; dRin=Rin'; ddRup=dRup'; ddRin=dRin'; -wronskian=(Simplify[#1,Assumptions->r>2]&)[Kerr\[CapitalDelta][a,r/varPN^2]^(\[ScriptS]+1) varPN^2 (Rin[r] dRup[r]-dRin[r] Rup[r])]; +(*The replacements in the wronskian are a quick fix for non vanishing r dependence in case a has a numerical value*) +wronskian=(Simplify[#1,Assumptions->r>2]&)[Kerr\[CapitalDelta][aVar,r/varPN^2]^(\[ScriptS]+1) varPN^2 (Rin[r] dRup[r]-dRin[r] Rup[r])]; +wronskian=wronskian/.Log[__ r]->0/.r^a_/;a<0:>r^-a/.r->0; source=TeukolskySourceCircularOrbit[\[ScriptS],\[ScriptL],\[ScriptM],a,{#,r0},"Form"->"InvariantWronskian"]&; sourceCoeffs=source[r]//Coefficient[#,{DiracDelta[r-r0],Derivative[1][DiracDelta][r-r0],Derivative[2][DiracDelta][r-r0]}]&; sourceCoeffs=Collect[#,{SpinWeightedSpheroidalHarmonicS[__][__],Derivative[__][SpinWeightedSpheroidalHarmonicS[__]][__]},Simplify]&/@sourceCoeffs; @@ -2691,10 +2764,17 @@ cIn=1/wronskian Total[sourceCoeffs {Rup[r0],-varPN^2dRup[r0],varPN^4 ddRup[r0]}] cUp=1/wronskian Total[sourceCoeffs {Rin[r0],-varPN^2dRin[r0],varPN^4 ddRin[r0]}]//SeriesTake[#,order]&; deltaCoeff=Coefficient[source[r],Derivative[2][DiracDelta][r-r0]]/Kerr\[CapitalDelta][a,r0]; deltaCoeff=Assuming[{varPN>0},If[deltaCoeff===0,0,deltaCoeff//PNScalings[#,{{r0,-2},{\[CapitalOmega]Kerr,3}},varPN,"IgnoreHarmonics"->True]&//SeriesTerms[#,{varPN,0,order}]&]]; -innerF=cIn Rin[#]&; -outerF=cUp Rup[#]&; -radialF=innerF[#] HeavisideTheta[r0-#] + outerF[#] HeavisideTheta[#-r0]+deltaCoeff DiracDelta[#-r0]&; -ret=<|"s"->\[ScriptS],"l"->\[ScriptL],"m"->\[ScriptM],"a"->a,"r0"->r0,"PN"->{varPN,order},"RadialFunction"->radialF,("ExtendedHomogeneous"->"\[ScriptCapitalI]")->innerF,("ExtendedHomogeneous"->"\[ScriptCapitalH]")->outerF,"\[Delta]"->deltaCoeff,"cUp"->cUp,"cIn"->cIn,"Wronskian"->wronskian,"Source"->source,"In"->aux["In"],"Up"->aux["Up"]|>; +{cIn,cUp,deltaCoeff,source}={cIn,cUp,deltaCoeff,source}/.r0->r0Var/.a->aVar; +If[OptionValue["Simplify"],{cIn,cUp,deltaCoeff,source}={cIn,cUp,deltaCoeff,source}//SeriesCollect[#,{SpinWeightedSpheroidalHarmonicS[__],Derivative[__][SpinWeightedSpheroidalHarmonicS][__]},(Simplify[#,{aVar>=0,r0Var>0,varPN>0}]&)]&]; +inner=cIn Rin[r]; +outer=cUp Rup[r]; +If[OptionValue["Simplify"],{inner,outer}={inner,outer}//SeriesCollect[#,{SpinWeightedSpheroidalHarmonicS[__],Derivative[__][SpinWeightedSpheroidalHarmonicS][__]},(Simplify[#,{aVar>=0,r0Var>0,varPN>0}]&)]&]; +innerF=inner/.r->#&; +outerF=outer/.r->#&; + +ampAssoc=<|"\[ScriptCapitalI]"->cUp,"\[ScriptCapitalH]"->cIn|>; +radialF=innerF[#] HeavisideTheta[r0Var-#] + outerF[#] HeavisideTheta[#-r0Var]+deltaCoeff DiracDelta[#-r0Var]&; +ret=<|"s"->\[ScriptS],"l"->\[ScriptL],"m"->\[ScriptM],"a"->aVar,"r0"->r0Var,"PN"->{varPN,order},"RadialFunction"->radialF,("ExtendedHomogeneous"->"\[ScriptCapitalI]")->innerF,("ExtendedHomogeneous"->"\[ScriptCapitalH]")->outerF,"\[Delta]"->deltaCoeff,"Amplitudes"->ampAssoc,"Wronskian"->wronskian,"Source"->source,"In"->aux["In"],"Up"->aux["Up"],"Simplify"->OptionValue["Simplify"],"Normalization"->OptionValue["Normalization"]|>; ret ] ] @@ -2710,7 +2790,7 @@ ret (*RadialSourcedAssociation["CO"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,r0Var_,{varPN_,order_}]/;NumericQ[r0Var]:=RadialSourcedAssociation["CO"][\[ScriptS],\[ScriptL],\[ScriptM],a,r0,{varPN,order}]/.r0->r0Var;*)*) -Options[TeukolskyPointParticleModePN]={"Normalization"->"Default"} +Options[TeukolskyPointParticleModePN]={"Normalization"->"Default","Simplify"->True} TeukolskyModePN /: @@ -2727,7 +2807,8 @@ TeukolskyModePN /: BoxForm`SummaryItem[{"PN order: ", N[(order-1)/2]"PN"}] }], BoxForm`SummaryItem[{"Orbit: ", "Circular Equatorial"}]}; - extended = {}; + extended = {BoxForm`SummaryItem[{"Simplify: ",assoc["Simplify"]}], + BoxForm`SummaryItem[{"Homogeneous Normalization: ",assoc["Normalization"]}]}; BoxForm`ArrangeSummaryBox[ TeukolskyModePN, @@ -2749,7 +2830,13 @@ ret ] -(* ::Subsubsection::Closed:: *) +TeukolskyPointParticleModePN[\[ScriptS]_, \[ScriptL]_, \[ScriptM]_,orbit_KerrGeodesics`KerrGeoOrbit`KerrGeoOrbitFunction,{varPN_,order_String},opt:OptionsPattern[]]:=Module[{aux}, +aux=order//PNStringToOrder; +TeukolskyPointParticleModePN[\[ScriptS], \[ScriptL], \[ScriptM],orbit,{varPN,aux},opt] +] + + +(* ::Subsubsection:: *) (*Accessing functions and keys*) diff --git a/Kernel/Tools.wl b/Kernel/Tools.wl index 97bb8ec..4cd549f 100644 --- a/Kernel/Tools.wl +++ b/Kernel/Tools.wl @@ -57,7 +57,7 @@ StyleBox[\"\[ScriptL]\",\nFontSlant->\"Italic\"]\), \!\(\* StyleBox[\"\[ScriptM]\",\nFontSlant->\"Italic\"]\)} mode up to \!\(\*SuperscriptBox[\(\[Epsilon]\), \(order\[Epsilon]\)]\). Where the relation to \[Eta] is given by \[Epsilon]=2 \[Omega] \!\(\*SuperscriptBox[\(\[Eta]\), \(3\)]\)."*) -(* ::Subsection::Closed:: *) +(* ::Subsection:: *) (*General Tools for Series*) @@ -70,11 +70,12 @@ SeriesTerms::usage="SeriesTerms[series, {x, x0, n}] works exactly like Series, w IgnoreExpansionParameter::usage="IgnoreExpansionParameter[series,x] sets all occurences of the expansion parameter in the series coefficients to x. If no value is entered x defaults to 1." -(* ::Subsection::Closed:: *) +(* ::Subsection:: *) (*Tools for PN Scalings*) -PNScalings::usage="PNScalings[expr,params,var] applies the given powercounting scalings to the expression. E.g. PNScalings[\[Omega] r,{{\[Omega],3},{r,-2},\[Eta]]" +Scalings::usage="Scalings[params,var][expr] applies the given powercounting scalings to the expression. E.g. Scalings[{{\[Omega],3,r,-2},\[Eta]][\[Omega] r]" +PNScalings::usage="Same as Scalings but with different input. Just here to not break my older code but you should use Scalings instead" RemovePN::usage="PNScalings[expr,var] takes the Normal[] and sets var to 1" Zero::usage="Zero[expr,vars] sets all vars in expr to 0" One::usage="One[expr,vars] sets all vars in expr to 1" @@ -149,7 +150,7 @@ TeukolskySourceCircularOrbit::usage="TeukolskySource[\[ScriptS],\[ScriptL],\[Scr TeukolskyEquation::usage="TeukolskyEquation[\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{\[Eta],order},R[r]] gives the Teukolsky equation with for a given {\[ScriptS],\[ScriptL],\[ScriptM]} mode with included \[Eta] scalings. The {\[Eta],order} argument can be left out for a general expression." -(* ::Section::Closed:: *) +(* ::Section:: *) (*Private*) @@ -176,11 +177,12 @@ SeriesTerms=Teukolsky`PN`Private`SeriesTerms IgnoreExpansionParameter=Teukolsky`PN`Private`IgnoreExpansionParameter -(* ::Subsection::Closed:: *) +(* ::Subsection:: *) (*Tools for PN Scalings*) PNScalings=Teukolsky`PN`Private`PNScalings +Scalings=Teukolsky`PN`Private`Scalings RemovePN=Teukolsky`PN`Private`RemovePN Zero=Teukolsky`PN`Private`Zero One=Teukolsky`PN`Private`One @@ -257,7 +259,10 @@ TeukolskyEquation=Teukolsky`PN`Private`TeukolskyEquation SetAttributes[{\[Nu]MST, aMST,MSTCoefficients}, {Protected, ReadProtected}]; -SetAttributes[{SeriesTake, SeriesMinOrder,SeriesMaxOrder,SeriesLength,SeriesCollect,SeriesTerms,IgnoreExpansionParameter}, {Protected, ReadProtected}]; +SetAttributes[{SeriesTake, SeriesMinOrder,SeriesMaxOrder,SeriesLength,SeriesTerms,IgnoreExpansionParameter}, {Protected, ReadProtected}]; + + +SetAttributes[{SeriesCollect}, {Protected, ReadProtected,Listable}]; SetAttributes[{PNScalings, RemovePN,Zero,One}, {Protected, ReadProtected}]; From f9533e8421820168c36d793ddc86dd6aee5ef712 Mon Sep 17 00:00:00 2001 From: jakobneef Date: Thu, 23 Jan 2025 10:17:15 +0000 Subject: [PATCH 04/13] added Cref and Cinc. (still slow and unchecked) --- Kernel/PN.wl | 74 ++++++++++++++++++++++++++++++++++++++-------------- 1 file changed, 54 insertions(+), 20 deletions(-) diff --git a/Kernel/PN.wl b/Kernel/PN.wl index c71b00a..1c81025 100644 --- a/Kernel/PN.wl +++ b/Kernel/PN.wl @@ -145,10 +145,10 @@ ClearAttributes[{TeukolskyRadialPN, TeukolskyRadialFunctionPN,TeukolskyPointPart (* ::Input:: *) -(*AAmplitude::usage="AAmplitude[\"+\"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]] gives the factor \!\(\*SubscriptBox[\(A\), \(+\)]\) from Sasaki Tagoshi Eq.(157). Likewise for [\"-\"]"*) +(*(*AAmplitude::usage="AAmplitude[\"+\"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]] gives the factor \!\(\*SubscriptBox[\(A\), \(+\)]\) from Sasaki Tagoshi Eq.(157). Likewise for [\"-\"]"*) (*BAmplitude::usage="BAmplitude[\"trans\"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]] gives the factor \!\(\*SuperscriptBox[\(B\), \(trans\)]\) from Sasaki Tagoshi Eq.(167). Likewise for [\"inc\"]"*) (*CAmplitude::usage="CAmplitude[\"trans\"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]] gives the factor \!\(\*SuperscriptBox[\(C\), \(trans\)]\) from Sasaki Tagoshi Eq.(170)."*) -(*\[ScriptCapitalK]Amplitude::usage="\[ScriptCapitalK]Amplitude[\"\[Nu]\"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]] gives the \!\(\*SuperscriptBox[\(\[ScriptCapitalK]\), \(\[Nu]\)]\) factor for \!\(\*SubscriptBox[\(R\), \(In\)]\). Likewise for \[ScriptCapitalK]Amplitude[\"-\[Nu]-1\"]. \[ScriptCapitalK]Amplitude[\"Ratio\"] gives the tidal response function \!\(\*FractionBox[SuperscriptBox[\(\[ScriptCapitalK]\), \(\(-\[Nu]\) - 1\)], SuperscriptBox[\(\[ScriptCapitalK]\), \(\[Nu]\)]]\)."*) +(*\[ScriptCapitalK]Amplitude::usage="\[ScriptCapitalK]Amplitude[\"\[Nu]\"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]] gives the \!\(\*SuperscriptBox[\(\[ScriptCapitalK]\), \(\[Nu]\)]\) factor for \!\(\*SubscriptBox[\(R\), \(In\)]\). Likewise for \[ScriptCapitalK]Amplitude[\"-\[Nu]-1\"]. \[ScriptCapitalK]Amplitude[\"Ratio\"] gives the tidal response function \!\(\*FractionBox[SuperscriptBox[\(\[ScriptCapitalK]\), \(\(-\[Nu]\) - 1\)], SuperscriptBox[\(\[ScriptCapitalK]\), \(\[Nu]\)]]\)."*)*) (* ::Input:: *) @@ -171,7 +171,7 @@ ClearAttributes[{TeukolskyRadialPN, TeukolskyRadialFunctionPN,TeukolskyPointPart (*(*InvariantWronskian::usage="InvariantWronskian[\[ScriptS], \[ScriptL], \[ScriptM], a, \[Omega], {\[Eta], n}] gives the invariant Wronskian."*)*) -(* ::Subsection:: *) +(* ::Subsection::Closed:: *) (*Radial solutions*) @@ -190,7 +190,7 @@ TeukolskyRadialFunctionPN::paramorder="order=`1`. The given number of terms has TeukolskyRadialFunctionPN::PNInput="Input String does not contain \"PN\". Assume PN orders are desired (calulate `1` terms in the Series)."; -(* ::Subsection:: *) +(* ::Subsection::Closed:: *) (*Sourced things*) @@ -1752,7 +1752,7 @@ aux/.MSTCoefficientsInternal[\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]+3]// ] -(* ::Subsubsection::Closed:: *) +(* ::Subsubsection:: *) (*B Amplitudes*) @@ -1831,9 +1831,6 @@ BAmplitude["Trans","Normalization"->"UnitTransmission"][\[ScriptS]_,\[ScriptL]_, Options[CAmplitude]={"Normalization"->"Default"} -CAmplitude["Trans","Normalization"->"UnitTransmission"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=1 (1+O[\[Eta]] \[Eta]^(order\[Eta]-1)); - - CAmplitude["Trans",OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=Module[{aux,\[CurlyEpsilon],\[Kappa],\[CurlyEpsilon]p,\[Tau],\[ScriptCapitalK],A}, \[CurlyEpsilon]=2 \[Omega]; \[Kappa]=Sqrt[1-a^2]; @@ -1846,17 +1843,54 @@ aux//SeriesTake[#,order\[Eta]]&//IgnoreExpansionParameter ] -CAmplitude["Inc",OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=Module[{aux,\[CurlyEpsilon],\[Kappa],\[CurlyEpsilon]p,\[Tau],\[ScriptCapitalK],A}, -"Not Computed" +CAmplitude["Inc",OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=Module[{aux,\[CurlyEpsilon],\[Kappa],\[CurlyEpsilon]p,\[Tau],\[ScriptCapitalK]1,\[ScriptCapitalK]2,coeff,D1,D2,nMin,nMax,repls,repls\[Nu],jumpCount}, +\[CurlyEpsilon]=2 \[Omega]; +\[Kappa]=Sqrt[1-a^2]; +\[CurlyEpsilon]p=(\[CurlyEpsilon]+\[Tau])/2; +\[Tau]=(-a \[ScriptM]+\[CurlyEpsilon])/\[Kappa]; +repls=MSTCoefficientsInternal[\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]+3]; +jumpCount=1; +repls\[Nu]=<|\[Nu]MST->(repls[\[Nu]MST]//SeriesTake[#,order\[Eta]+4+3jumpCount]&)|>; +nMax=order\[Eta]/3//Ceiling; +nMin=-(order\[Eta]/3+2)//Floor; +D1=E^(-I \[Kappa](\[CurlyEpsilon]+\[Tau])(1/2+Log[\[Kappa]]/(1+\[Kappa]))) (Sin[\[Pi](\[Nu]MST+I \[CurlyEpsilon])]Sin[\[Pi](\[Nu]MST+I \[Tau])]Gamma[1-\[ScriptS]-I(\[CurlyEpsilon]+\[Tau])])/(Sin[2 \[Pi] \[Nu]MST]Sin[I \[Pi](\[CurlyEpsilon]+\[Tau])]Gamma[1+\[ScriptS]+I(\[CurlyEpsilon]+\[Tau])]); +D2=D1/.\[Nu]MST->-\[Nu]MST-1; +coeff=E^-(\[Pi] \[CurlyEpsilon]+I\[NonBreakingSpace]\[Pi] \[ScriptS])/Sin[2 \[Pi] \[Nu]MST]; +aux=(PNScalingsInternal[coeff ((E^(-I \[Pi] \[Nu]MST) Sin[\[Pi](\[Nu]MST-\[ScriptS]-I \[CurlyEpsilon])])/\[ScriptCapitalK]1 D1-(I Sin[\[Pi](\[Nu]MST+\[ScriptS]-I \[CurlyEpsilon])])/\[ScriptCapitalK]2 D2)]/.repls\[Nu])(( \!\( +\*UnderoverscriptBox[\(\[Sum]\), \(n = nMin\), \(nMax\)]\(aMST[n]\)\))/.repls)//IgnoreExpansionParameter//ExpandGamma//ExpandPolyGamma; +\[ScriptCapitalK]1=\[ScriptCapitalK]Amplitude["\[Nu]"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]//ExpandPolyGamma//ExpandGamma//SeriesCollect[#,PolyGamma[__,__]]&; +\[ScriptCapitalK]2=\[ScriptCapitalK]Amplitude["-\[Nu]-1"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]//ExpandPolyGamma//ExpandGamma//SeriesCollect[#,PolyGamma[__,__]]&; +aux//SeriesTake[#,order\[Eta]]&//IgnoreExpansionParameter ] -CAmplitude["Ref",OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=Module[{aux,\[CurlyEpsilon],\[Kappa],\[CurlyEpsilon]p,\[Tau],\[ScriptCapitalK],A}, -"Not Computed" +CAmplitude["Ref",OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=Module[{aux,\[CurlyEpsilon],\[Kappa],\[CurlyEpsilon]p,\[Tau],\[ScriptCapitalK]1,\[ScriptCapitalK]2,coeff,D1,D2,nMin,nMax,repls,repls\[Nu],jumpCount}, +\[CurlyEpsilon]=2 \[Omega]; +\[Kappa]=Sqrt[1-a^2]; +\[CurlyEpsilon]p=(\[CurlyEpsilon]+\[Tau])/2; +\[Tau]=(-a \[ScriptM]+\[CurlyEpsilon])/\[Kappa]; +repls=MSTCoefficientsInternal[\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]+3]; +jumpCount=1; +repls\[Nu]=<|\[Nu]MST->(repls[\[Nu]MST]//SeriesTake[#,order\[Eta]+4+3jumpCount]&)|>; +nMax=order\[Eta]/3//Ceiling; +nMin=-(order\[Eta]/3+2)//Floor; +D1=-E^(I \[Kappa](\[CurlyEpsilon]+\[Tau])(1/2+Log[\[Kappa]]/(1+\[Kappa]))) (2\[Kappa])^(2\[ScriptS]) (Sin[\[Pi](\[Nu]MST-I \[CurlyEpsilon])]Sin[\[Pi](\[Nu]MST-I \[Tau])])/(Sin[2 \[Pi] \[Nu]MST]Sin[I \[Pi](\[CurlyEpsilon]+\[Tau])]); +D2=D1/.\[Nu]MST->-\[Nu]MST-1; +coeff=E^-(\[Pi] \[CurlyEpsilon]+I\[NonBreakingSpace]\[Pi] \[ScriptS])/Sin[2 \[Pi] \[Nu]MST]; +aux=(PNScalingsInternal[coeff ((E^(-I \[Pi] \[Nu]MST) Sin[\[Pi](\[Nu]MST-\[ScriptS]+I \[CurlyEpsilon])])/\[ScriptCapitalK]1 D1-(I Sin[\[Pi](\[Nu]MST+\[ScriptS]-I \[CurlyEpsilon])])/\[ScriptCapitalK]2 D2)]/.repls\[Nu])(( \!\( +\*UnderoverscriptBox[\(\[Sum]\), \(n = nMin\), \(nMax\)]\(aMST[n]\)\))/.repls)//IgnoreExpansionParameter//ExpandGamma//ExpandPolyGamma; +\[ScriptCapitalK]1=\[ScriptCapitalK]Amplitude["\[Nu]"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]//ExpandPolyGamma//ExpandGamma//SeriesCollect[#,PolyGamma[__,__]]&; +\[ScriptCapitalK]2=\[ScriptCapitalK]Amplitude["-\[Nu]-1"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]//ExpandPolyGamma//ExpandGamma//SeriesCollect[#,PolyGamma[__,__]]&; +aux//SeriesTake[#,order\[Eta]]&//IgnoreExpansionParameter ] -(* ::Subsubsection::Closed:: *) +CAmplitude["Inc","Normalization"->"UnitTransmission"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=CAmplitude["Inc"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]/CAmplitude["Trans"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]; +CAmplitude["Ref","Normalization"->"UnitTransmission"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=CAmplitude["Ref"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]/CAmplitude["Trans"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]; +CAmplitude["Trans","Normalization"->"UnitTransmission"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=1 (1+O[\[Eta]] \[Eta]^(order\[Eta]-1)); + + +(* ::Subsubsection:: *) (*\[ScriptCapitalK] Amplitude*) @@ -1890,7 +1924,7 @@ sumUpPH=\!\( \*UnderoverscriptBox[\(\[Sum]\), \(n = \[ScriptR]\), \(nMax\)]\(\((\( \*FractionBox[\( \*SuperscriptBox[\((\(-1\))\), \(n\)]\ PH[1 + \[ScriptS] + I\ \[CurlyEpsilon] + \[Nu]MST, n]\ PH[1 + \[ScriptR] + 2\ \[Nu]MST, n]\ PH[1 + \[Nu]MST + I\ \[Tau], n]\), \(\(\((n - \[ScriptR])\)!\)\ PH[1 - \[ScriptS] - I\ \[CurlyEpsilon] + \[Nu]MST, n]\ PH[1 + \[Nu]MST - I\ \[Tau], n]\)] /. replsPN\) /. repls\[Nu])\) \((aMST[n]\ /. repls)\)\)\)//IgnoreExpansionParameter; -sumUpPHCoeff= Gamma[1+\[ScriptR]+2 \[Nu]MST] /(Gamma[1-\[ScriptS]-I \[CurlyEpsilon]+\[Nu]MST] Gamma[1+\[Nu]MST-I \[Tau]]) If[\[ScriptR]==0,1,Gamma[1+\[ScriptS]+I \[CurlyEpsilon]+\[Nu]MST] Gamma[1+\[Nu]MST+I \[Tau]]]//IgnoreExpansionParameter; +sumUpPHCoeff= Gamma[1+\[ScriptR]+2 \[Nu]MST] /(Gamma[1-\[ScriptS]-I \[CurlyEpsilon]+\[Nu]MST] Gamma[1+\[Nu]MST-I \[Tau]]) If[\[ScriptR]==0,1,Gamma[1+\[ScriptS]+I \[CurlyEpsilon]+\[Nu]MST] Gamma[1+\[Nu]MST+I \[Tau]]]; coeff=(coeff If[OptionValue["PochhammerForm"],sumUpPHCoeff,1])/.replsPN/.repls//IgnoreExpansionParameter; sumDown=\!\( @@ -1903,7 +1937,7 @@ ret//SeriesTake[#,order\[Eta]]& ] -\[ScriptCapitalK]Amplitude["C-\[Nu]-1",OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_,OptionsPattern[]]:=Module[{\[ScriptR]=0,ret,\[CapitalGamma]=Gamma,PH=Pochhammer,coeff,sumUp,sumUpPHCoeff,sumUpPH,sumDown,\[CurlyEpsilon]p,\[Tau],\[CurlyEpsilon],\[Kappa]=Sqrt[1-a^2],nMax,nMin,jump,jumpCount,repls,repls\[Nu]}, +\[ScriptCapitalK]Amplitude["-\[Nu]-1",OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_,OptionsPattern[]]:=Module[{\[ScriptR]=0,ret,\[CapitalGamma]=Gamma,PH=Pochhammer,coeff,sumUp,sumUpPHCoeff,sumUpPH,sumDown,\[CurlyEpsilon]p,\[Tau],\[CurlyEpsilon],\[Kappa]=Sqrt[1-a^2],nMax,nMin,jump,jumpCount,repls,repls\[Nu]}, \[CurlyEpsilon]=2 \[Omega]; \[CurlyEpsilon]p=(\[CurlyEpsilon]+\[Tau])/2; \[Tau]=(-a \[ScriptM]+\[CurlyEpsilon])/\[Kappa]; @@ -1924,14 +1958,14 @@ sumUpPH=\!\( \*UnderoverscriptBox[\(\[Sum]\), \(n = nMin\), \(-\[ScriptR]\)]\(\((\( \*FractionBox[\( \*SuperscriptBox[\((\(-1\))\), \(-n\)]\ PH[\(-1\) + \[ScriptR] - 2\ \[Nu]MST, \(-n\)]\ PH[\[ScriptS] + I\ \[CurlyEpsilon] - \[Nu]MST, \(-n\)]\ PH[\(-\[Nu]MST\) + I\ \[Tau], \(-n\)]\), \(\(\((\(-n\) - \[ScriptR])\)!\)\ PH[\(-\[ScriptS]\) - I\ \[CurlyEpsilon] - \[Nu]MST, \(-n\)]\ PH[\(-\[Nu]MST\) - I\ \[Tau], \(-n\)]\)] /. replsPN\) /. repls\[Nu])\) \((aMST[n]\ /. repls)\)\)\)//IgnoreExpansionParameter; -sumUpPHCoeff=Gamma[-1+\[ScriptR]-2 \[Nu]MST] /(Gamma[-\[ScriptS]-I \[CurlyEpsilon]-\[Nu]MST] Gamma[-\[Nu]MST-I \[Tau]]) If[\[ScriptR]==0,1,Gamma[\[ScriptS]+I \[CurlyEpsilon]-\[Nu]MST] Gamma[-\[Nu]MST+I \[Tau]]]//IgnoreExpansionParameter; -coeff=(coeff If[OptionValue["PochhammerForm"],sumUpPHCoeff,1])/.replsPN/.repls; +sumUpPHCoeff=Gamma[-1+\[ScriptR]-2 \[Nu]MST] /(Gamma[-\[ScriptS]-I \[CurlyEpsilon]-\[Nu]MST] Gamma[-\[Nu]MST-I \[Tau]]) If[\[ScriptR]==0,1,Gamma[\[ScriptS]+I \[CurlyEpsilon]-\[Nu]MST] Gamma[-\[Nu]MST+I \[Tau]]]; +coeff=(coeff If[OptionValue["PochhammerForm"],sumUpPHCoeff,1])/.replsPN/.repls//IgnoreExpansionParameter; sumDown=\!\( \*UnderoverscriptBox[\(\[Sum]\), \(n = \(-\[ScriptR]\)\), \(nMax\)]\(\((\( \*FractionBox[\( \*SuperscriptBox[\((\(-1\))\), \(-n\)]\ PH[\[ScriptS] - I\ \[CurlyEpsilon] - \[Nu]MST, \(-n\)]\), \(\(\((n + \[ScriptR])\)!\)\ PH[\[ScriptR] - 2\ \[Nu]MST, \(-n\)]\ PH[\(-\[ScriptS]\) + I\ \[CurlyEpsilon] - \[Nu]MST, \(-n\)]\)] /. replsPN\) /. repls\[Nu])\) \((aMST[n] /. repls)\)\)\)//IgnoreExpansionParameter; -ret=coeff/sumDown If[OptionValue["PochhammerForm"],sumUpPHCoeff sumUpPH,sumUp]//IgnoreExpansionParameter; +ret=coeff/sumDown If[OptionValue["PochhammerForm"],sumUpPH,sumUp]//IgnoreExpansionParameter; ret//SeriesTake[#,order\[Eta]]& ] @@ -1993,7 +2027,7 @@ TeukolskyAmplitudePN["Cref",opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[Scri TeukolskyAmplitudePN["\[ScriptCapitalK]\[Nu]",opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]Var_,{\[Eta]Var_,order\[Eta]_}] :=\[ScriptCapitalK]Amplitude["\[Nu]"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]/.{\[Omega]->\[Omega]Var,\[Eta]->\[Eta]Var}; -TeukolskyAmplitudePN["\[ScriptCapitalK]-\[Nu]-1",opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]Var_,{\[Eta]Var_,order\[Eta]_}] :=\[ScriptCapitalK]Amplitude["C-\[Nu]-1"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]/.{\[Omega]->\[Omega]Var,\[Eta]->\[Eta]Var}; +TeukolskyAmplitudePN["\[ScriptCapitalK]-\[Nu]-1",opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]Var_,{\[Eta]Var_,order\[Eta]_}] :=\[ScriptCapitalK]Amplitude["-\[Nu]-1"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]/.{\[Omega]->\[Omega]Var,\[Eta]->\[Eta]Var}; TeukolskyAmplitudePN["\[ScriptCapitalK]",opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]Var_,{\[Eta]Var_,order\[Eta]_}] :=\[ScriptCapitalK]Amplitude["Ratio"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]]/.{\[Omega]->\[Omega]Var,\[Eta]->\[Eta]Var}; @@ -2270,7 +2304,7 @@ ret (*,{status,n,j}]]*)*) -(* ::Subsubsection::Closed:: *) +(* ::Subsubsection:: *) (*Constructing Subscript[R, In]*) From 5fdd26662af63cb363180f87947508889c8913b5 Mon Sep 17 00:00:00 2001 From: jakobneef Date: Tue, 28 Jan 2025 11:30:45 +0000 Subject: [PATCH 05/13] added CoefficientList, fixed TKEq, Changed OmegaKerr to KerrGeoFrequencies --- .../Symbols/TeukolskyPointParticleModePN.nb | 2522 +++++++++++++---- .../Symbols/TeukolskyRadialPN.nb | 318 ++- Kernel/PN.wl | 68 +- 3 files changed, 2171 insertions(+), 737 deletions(-) diff --git a/Documentation/English/ReferencePages/Symbols/TeukolskyPointParticleModePN.nb b/Documentation/English/ReferencePages/Symbols/TeukolskyPointParticleModePN.nb index ef5a17a..0d0b211 100644 --- a/Documentation/English/ReferencePages/Symbols/TeukolskyPointParticleModePN.nb +++ b/Documentation/English/ReferencePages/Symbols/TeukolskyPointParticleModePN.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 163844, 3940] -NotebookOptionsPosition[ 150332, 3654] -NotebookOutlinePosition[ 151116, 3680] -CellTagsIndexPosition[ 151035, 3675] +NotebookDataLength[ 227392, 5362] +NotebookOptionsPosition[ 213566, 5070] +NotebookOutlinePosition[ 214351, 5096] +CellTagsIndexPosition[ 214270, 5091] WindowFrame->Normal*) (* Beginning of Notebook Content *) @@ -137,10 +137,10 @@ Cell[TextData[{ "83d1757e-96d2-44c4-8211-de196c886561"], DynamicModuleBox[{$CellContext`nbobj$$ = NotebookObject[ "4cdc53be-4474-4e7d-8aa9-8c4b6af9a759", - "f8b797ce-49f5-4b23-9cd8-fb5daf2e0891"], $CellContext`cellobj$$ = + "ad63e4bb-636e-4942-b5b8-e2e163d40a61"], $CellContext`cellobj$$ = CellObject[ "bfafe1ae-4382-474b-97c7-755897deb3ff", - "d085d55a-1770-4305-89b3-1c7e08d2a004"]}, + "7fefafaa-83da-447f-ad9b-121d20996490"]}, TemplateBox[{ GraphicsBox[{{ Thickness[0.06], @@ -283,7 +283,7 @@ Cell[BoxData[{ CellChangeTimes->{{3.942221956582928*^9, 3.94222199381381*^9}, { 3.942222147391078*^9, 3.942222148633972*^9}, 3.946104722586904*^9, 3.946104814370336*^9}, - CellLabel->"In[1]:=", + CellLabel->"In[34]:=", CellID->103538757,ExpressionUUID->"3ed8b034-7711-4c9d-b4d6-76f204a012e2"], Cell[BoxData[ @@ -295,6 +295,17 @@ Cell[BoxData[ TemplateBox[{ PaneSelectorBox[{False -> GridBox[{{ + PaneBox[ + ButtonBox[ + DynamicBox[ + FEPrivate`FrontEndResource["FEBitmaps", "SummaryBoxOpener"]], + ButtonFunction :> (Typeset`open$$ = True), Appearance -> None, + BaseStyle -> {}, Evaluator -> Automatic, Method -> + "Preemptive"], Alignment -> {Center, Center}, ImageSize -> + Dynamic[{ + Automatic, + 3.5 (CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[ + Magnification])}]], GraphicsBox[{ GrayLevel[0], DiskBox[{0, 0}, {1, 1}, {3.041592653589793, 6.383185307179586}], @@ -333,8 +344,10 @@ Cell[BoxData[ TagBox["\"\[Omega]: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox[ - RowBox[{"2", " ", "\[CapitalOmega]Kerr"}], - "SummaryItem"]}], "\" \"", + FractionBox["2", + RowBox[{"a", "+", + SuperscriptBox["r0", + RowBox[{"3", "/", "2"}]]}]], "SummaryItem"]}], "\" \"", RowBox[{ TagBox[ "\"\\!\\(\\*SubscriptBox[\\(r\\), \\(0\\)]\\): \"", @@ -367,6 +380,17 @@ Cell[BoxData[ GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, BaselinePosition -> {1, 1}], True -> GridBox[{{ + PaneBox[ + ButtonBox[ + DynamicBox[ + FEPrivate`FrontEndResource["FEBitmaps", "SummaryBoxCloser"]], + ButtonFunction :> (Typeset`open$$ = False), Appearance -> None, + BaseStyle -> {}, Evaluator -> Automatic, Method -> + "Preemptive"], Alignment -> {Center, Center}, ImageSize -> + Dynamic[{ + Automatic, + 3.5 (CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[ + Magnification])}]], GraphicsBox[{ GrayLevel[0], DiskBox[{0, 0}, {1, 1}, {3.041592653589793, 6.383185307179586}], @@ -405,8 +429,10 @@ Cell[BoxData[ TagBox["\"\[Omega]: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox[ - RowBox[{"2", " ", "\[CapitalOmega]Kerr"}], - "SummaryItem"]}], "\" \"", + FractionBox["2", + RowBox[{"a", "+", + SuperscriptBox["r0", + RowBox[{"3", "/", "2"}]]}]], "SummaryItem"]}], "\" \"", RowBox[{ TagBox[ "\"\\!\\(\\*SubscriptBox[\\(r\\), \\(0\\)]\\): \"", @@ -425,7 +451,16 @@ Cell[BoxData[ RowBox[{ TagBox["\"Orbit: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", - TagBox["\"Circular Equatorial\"", "SummaryItem"]}]}}, + TagBox["\"Circular Equatorial\"", "SummaryItem"]}]}, { + RowBox[{ + TagBox["\"Simplify: \"", "SummaryItemAnnotation"], + "\[InvisibleSpace]", + TagBox["True", "SummaryItem"]}]}, { + RowBox[{ + TagBox[ + "\"Homogeneous Normalization: \"", "SummaryItemAnnotation"], + "\[InvisibleSpace]", + TagBox["\"Default\"", "SummaryItem"]}]}}, GridBoxAlignment -> { "Columns" -> {{Left}}, "Rows" -> {{Automatic}}}, AutoDelete -> False, GridBoxItemSize -> { @@ -447,193 +482,476 @@ Cell[BoxData[ "s" -> -2, "l" -> 2, "m" -> 2, "a" -> $CellContext`a, "r0" -> $CellContext`r0, "PN" -> {$CellContext`\[Eta], 3}, "RadialFunction" -> ( - Teukolsky`PN`Private`innerF$4169[#] HeavisideTheta[$CellContext`r0 - #] + - Teukolsky`PN`Private`outerF$4169[#] HeavisideTheta[# - $CellContext`r0] + - Teukolsky`PN`Private`deltaCoeff$4169 - DiracDelta[# - $CellContext`r0]& ), ("ExtendedHomogeneous" -> - "\[ScriptCapitalI]") -> (Teukolsky`PN`Private`cIn$4169 - Teukolsky`PN`Private`Rin$4169[#]& ), ("ExtendedHomogeneous" -> - "\[ScriptCapitalH]") -> (Teukolsky`PN`Private`cUp$4169 - Teukolsky`PN`Private`Rup$4169[#]& ), "\[Delta]" -> + Teukolsky`PN`Private`innerF$239381[#] + HeavisideTheta[$CellContext`r0 - #] + + Teukolsky`PN`Private`outerF$239381[#] + HeavisideTheta[# - $CellContext`r0] + + Teukolsky`PN`Private`deltaCoeff$239381 + DiracDelta[# - $CellContext`r0]& ), "SCoefficients" -> (ReplaceAll[ + Teukolsky`PN`Private`SeriesToSCoeffs[ + Teukolsky`PN`Private`radialF$239381[Teukolsky`PN`Private`r]], + Teukolsky`PN`Private`r -> #]& ), ("ExtendedHomogeneous" -> + "\[ScriptCapitalI]") -> (ReplaceAll[ + Teukolsky`PN`Private`inner$239381, Teukolsky`PN`Private`r -> #]& ), ( + "ExtendedHomogeneous" -> "\[ScriptCapitalH]") -> (ReplaceAll[ + Teukolsky`PN`Private`outer$239381, Teukolsky`PN`Private`r -> #]& ), + "\[Delta]" -> SeriesData[$CellContext`\[Eta], 0, {-Pi $CellContext`r0^(-3) SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, - 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`a,\ + $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ Rational[1, 2] Pi, 0], 0, 2 $CellContext`a Pi $CellContext`r0^Rational[-7, 2] SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, - 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`a,\ + $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ Rational[1, 2] Pi, 0], 0, Rational[-1, 2] (7 + 2 $CellContext`a^2) Pi $CellContext`r0^(-4) SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, - 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`a,\ + $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ Rational[1, 2] Pi, 0]}, 12, 17, 2], "Amplitudes" -> <| "\[ScriptCapitalI]" -> SeriesData[$CellContext`\[Eta], 0, {Complex[0, - Rational[-2, 15]] Pi $CellContext`r0^2 - Teukolsky`PN`\[CapitalOmega]Kerr ( + Rational[-2, 15]] + Pi $CellContext`r0^2 ( 2 SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ +2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ Rational[1, 2] Pi, 0] - 4 Derivative[1, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ +2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ Rational[1, 2] Pi, 0] + Derivative[2, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]), 0, Rational[8, 45] - Pi $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr ( - 2 (-3 + $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr) +2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0]) + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`a,\ + $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"], 0, + Rational[8, 45] Pi $CellContext`r0^Rational[3, 2] + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`a,\ + $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"] ((-6) SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + (3 - 4 $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr) Derivative[1, 0][ + 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ + Rational[1, 2] Pi, 0] + 3 Derivative[1, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr Derivative[2, 0][ - +2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0] + $CellContext`r0^Rational[3, 2] ( + 2 SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[\ +-2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ + Rational[1, 2] Pi, 0] - 4 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[\ +-2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0] + Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[\ +-2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0]) + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`\ +a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]), 0, + Rational[1, 315] Pi $CellContext`r0 + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`a,\ + $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"] ( + 21 (2 (Complex[0, 9] + 8 $CellContext`a) SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]), 0, Complex[0, - Rational[1, 315]] Pi $CellContext`r0 - Teukolsky`PN`\[CapitalOmega]Kerr ((378 + - Complex[0, -336] $CellContext`a - - 336 $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr + - 88 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2) - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - 4 (-105 + Complex[0, -21] $CellContext`a + - 42 $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr - 44 $CellContext`r0^3 - Teukolsky`PN`\[CapitalOmega]Kerr^2) Derivative[1, 0][ +2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ + Rational[1, 2] Pi, 0] + + 4 (Complex[0, -5] + $CellContext`a) Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`\ +SpinWeightedSpheroidalHarmonicS[-2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0] + (Complex[0, 5] - 4 $CellContext`a) + Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`\ +SpinWeightedSpheroidalHarmonicS[-2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0]) + + Complex[0, -168] $CellContext`r0^Rational[3, 2] ( + 2 SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[\ +-2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ + Rational[1, 2] Pi, 0] - Derivative[1, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + (105 + Complex[0, 84] $CellContext`a + - 44 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2) - Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0])}, -2, 4, 2], "\[ScriptCapitalH]" -> +2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0]) + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`\ +a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"] + + Complex[0, 44] $CellContext`r0^3 ( + 2 SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[\ +-2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ + Rational[1, 2] Pi, 0] - 4 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[\ +-2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0] + Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[\ +-2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0]) + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^2)}, -2, + 4, 2], "\[ScriptCapitalH]" -> SeriesData[$CellContext`\[Eta], 0, { - Rational[-1, 64] Pi $CellContext`r0^(-3) - Teukolsky`PN`\[CapitalOmega]Kerr^(-4) ( + Rational[-1, 64] + Pi $CellContext`r0^(-3) ( 2 SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ +2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ Rational[1, 2] Pi, 0] - 4 Derivative[1, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ +2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ Rational[1, 2] Pi, 0] + Derivative[2, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]), 0, Complex[0, +2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0]) + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`\ +a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^(-4), 0, + Complex[0, Rational[1, 32]] Pi $CellContext`r0^Rational[-7, 2] - Teukolsky`PN`\[CapitalOmega]Kerr^(-4) ( - 2 (-3 + $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr) - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + (3 - 4 $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr) Derivative[1, 0][ - + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`\ +a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^(-4) ((-6) SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr Derivative[2, 0][ +2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ + Rational[1, 2] Pi, 0] + 3 Derivative[1, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]), 0, Rational[1, 384] - Pi $CellContext`r0^(-4) - Teukolsky`PN`\[CapitalOmega]Kerr^(-4) ((-6 + - Complex[0, 32] $CellContext`a - 48 $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr + - 24 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2) +2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0] + $CellContext`r0^Rational[3, 2] ( + 2 SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[\ +-2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ + Rational[1, 2] Pi, 0] - 4 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[\ +-2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0] + Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[\ +-2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0]) + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`\ +a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]), 0, + Rational[1, 384] Pi $CellContext`r0^(-4) + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`\ +a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^(-4) ((-6 + + Complex[0, 32] $CellContext`a) SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ + Rational[1, 2] Pi, 0] + 60 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ Rational[1, 2] Pi, 0] + - 4 (15 + Complex[0, 2] $CellContext`a + - 6 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr - - 12 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2) - Derivative[1, 0][ + Complex[0, 8] $CellContext`a Derivative[1, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + (-15 + Complex[0, -8] $CellContext`a + - 12 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2) - Derivative[2, 0][ +2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0] - 15 Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0] + + Complex[0, -8] $CellContext`a Derivative[2, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0])}, -12, -6, 2]|>, "Wronskian" -> +2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0] + + 24 $CellContext`r0^ + Rational[3, 2] ((-2) + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ + Rational[1, 2] Pi, 0] + Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[\ +-2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0]) + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`\ +a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"] + + 12 $CellContext`r0^3 ( + 2 SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[\ +-2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ + Rational[1, 2] Pi, 0] - 4 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[\ +-2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0] + Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[\ +-2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0]) + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^2)}, \ +-12, -6, 2]|>, "Wronskian" -> SeriesData[$CellContext`\[Eta], 0, { - Complex[0, 96] Teukolsky`PN`\[CapitalOmega]Kerr^3}, 9, 12, 1], - "Source" -> ( + Complex[0, 96] + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`a, \ +$CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^3}, 9, 12, + 1], "Source" -> ( Teukolsky`PN`Private`TeukolskySourceCircularOrbit[-2, 2, 2, $CellContext`a, {#, $CellContext`r0}, "Form" -> "InvariantWronskian"]& ), "In" -> Teukolsky`PN`TeukolskyRadialFunctionPN[-2, 2, 2, $CellContext`a, 2 - Teukolsky`PN`\[CapitalOmega]Kerr, {$CellContext`\[Eta], 3}, <| + Inactive[KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"], \ +{$CellContext`\[Eta], 3}, <| "s" -> -2, "l" -> 2, "m" -> 2, "a" -> $CellContext`a, "PN" -> {$CellContext`\[Eta], 3}, "RadialFunction" -> Function[Teukolsky`PN`Private`r, SeriesData[$CellContext`\[Eta], 0, { Rational[64, 5] Teukolsky`PN`Private`r^4 - Teukolsky`PN`\[CapitalOmega]Kerr^4, Complex[0, + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^4, + Complex[0, Rational[256, 15]] Teukolsky`PN`Private`r^5 - Teukolsky`PN`\[CapitalOmega]Kerr^5, Rational[-128, 105] - Teukolsky`PN`Private`r^3 - Teukolsky`PN`\[CapitalOmega]Kerr^4 (42 + + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^5, + Rational[-128, 105] Teukolsky`PN`Private`r^3 + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^4 (42 + Complex[0, 21] $CellContext`a + - 11 Teukolsky`PN`Private`r^3 Teukolsky`PN`\[CapitalOmega]Kerr^2)}, + 11 Teukolsky`PN`Private`r^3 + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^2)}, 4, 7, 1]], "BoundaryCondition" -> "In", "LeadingOrder" -> Function[Teukolsky`PN`Private`r, SeriesData[$CellContext`\[Eta], 0, { Rational[64, 5] Teukolsky`PN`Private`r^4 - Teukolsky`PN`\[CapitalOmega]Kerr^4}, 4, 5, 1]], "TermCount" -> 3, - "Normalization" -> "Default", - "Amplitudes" -> <|"Transmission" -> "Not Computed"|>, "Simplified" -> - True|>], "Up" -> + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^4}, 4, + 5, 1]], "TermCount" -> 3, "Normalization" -> "Default", + "Amplitudes" -> <| + "Incidence" -> "Not Computed", "Transmission" -> "Not Computed", + "Reflection" -> "Not Computed"|>, "Simplify" -> True|>], "Up" -> Teukolsky`PN`TeukolskyRadialFunctionPN[-2, 2, 2, $CellContext`a, 2 - Teukolsky`PN`\[CapitalOmega]Kerr, {$CellContext`\[Eta], 3}, <| + Inactive[KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"], \ +{$CellContext`\[Eta], 3}, <| "s" -> -2, "l" -> 2, "m" -> 2, "a" -> $CellContext`a, "PN" -> {$CellContext`\[Eta], 3}, "RadialFunction" -> Function[Teukolsky`PN`Private`r, SeriesData[$CellContext`\[Eta], 0, {Complex[0, - Rational[-3, 2]] Teukolsky`PN`Private`r^(-1)/ - Teukolsky`PN`\[CapitalOmega]Kerr, -3, Rational[1, 2] - Teukolsky`PN`Private`r^(-2) - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) (Complex[0, -3] + - 4 $CellContext`a + + Rational[-3, 2]] Teukolsky`PN`Private`r^(-1)/Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`\ +a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"], -3, + Rational[1, 2] Teukolsky`PN`Private`r^(-2) + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^(-1) ( + Complex[0, -3] + 4 $CellContext`a + Complex[0, 6] Teukolsky`PN`Private`r^3 - Teukolsky`PN`\[CapitalOmega]Kerr^2)}, -1, 2, 1]], - "BoundaryCondition" -> "Up", "LeadingOrder" -> + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^2)}, \ +-1, 2, 1]], "BoundaryCondition" -> "Up", "LeadingOrder" -> Function[Teukolsky`PN`Private`r, SeriesData[$CellContext`\[Eta], 0, {Complex[0, - Rational[-3, 2]] Teukolsky`PN`Private`r^(-1)/ - Teukolsky`PN`\[CapitalOmega]Kerr}, -1, 0, 1]], "TermCount" -> 3, - "Normalization" -> "Default", - "Amplitudes" -> <|"Transmission" -> "Not Computed"|>, "Simplified" -> - True|>], "Simplified" -> True|>], + Rational[-3, 2]] Teukolsky`PN`Private`r^(-1)/Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`\ +a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]}, -1, 0, + 1]], "TermCount" -> 3, "Normalization" -> "Default", + "Amplitudes" -> <| + "Incidence" -> "Not Computed", "Transmission" -> "Not Computed", + "Reflection" -> "Not Computed"|>, "Simplify" -> True|>], "Simplify" -> + True, "Normalization" -> "Default"|>], Editable->False, SelectWithContents->True, Selectable->False]], "Output", CellChangeTimes->{{3.94222200433076*^9, 3.942222016566121*^9}, 3.942222158382213*^9, 3.946104634610979*^9, 3.9461047796987343`*^9, 3.9461048377087297`*^9, 3.94611005563437*^9, 3.946113940645184*^9, - 3.9461141370507*^9}, - CellLabel->"Out[2]=", - CellID->331545821,ExpressionUUID->"7ad3ffb0-7f8c-4ad8-96c9-086bd974b68c"] + 3.9461141370507*^9, 3.946986818632866*^9}, + CellLabel->"Out[35]=", + CellID->2116508496,ExpressionUUID->"e750d263-2bf2-4b6e-a0b7-366441a950c6"] }, Open ]], Cell["The main use is now to use it as a Function[] ", "ExampleText", @@ -644,416 +962,1483 @@ Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ - RowBox[{ - RowBox[{"mode", "[", "r", "]"}], "/.", - RowBox[{"a", "->", "0"}]}], "//", "Simplify"}]], "Input", - 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Teukolsky`PN`\[CapitalOmega]Kerr (-1 + $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr) + - 7 $CellContext`r (-13 - 8 $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr + - 12 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2)) - HeavisideTheta[-$CellContext`r + $CellContext`r0])}, -4, 2, 2], + HeavisideTheta[-$CellContext`r + $CellContext`r0]) ( + 2 SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, + 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`a, \ +$CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ + Rational[1, 2] Pi, 0] - 4 Derivative[1, 0][ + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, + 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`a, \ +$CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0] + Derivative[2, 0][ + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, + 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`a, \ +$CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0]), 0, Complex[0, + Rational[-2, 15]] $CellContext`r^(-1) ( + Pi $CellContext`r^5 $CellContext`r0^Rational[-7, 2] + HeavisideTheta[-$CellContext`r + $CellContext`r0] ( + 18 SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`\ +a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ + Rational[1, 2] Pi, 0] - 9 Derivative[1, 0][ + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`\ +a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0] + (2 $CellContext`r - + 3 $CellContext`r0) $CellContext`r0^Rational[1, 2] ( + 2 SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ + Rational[1, 2] Pi, 0] - 4 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0] + Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0]) + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`a, \ +$CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]) - + HeavisideTheta[$CellContext`r - $CellContext`r0] ( + 6 Pi $CellContext`r0^Rational[3, 2] ( + 2 SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`\ +a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ + Rational[1, 2] Pi, 0] - Derivative[1, 0][ + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`\ +a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0]) + + Pi (3 $CellContext`r - 2 $CellContext`r0) $CellContext`r0^2 ( + 2 SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`\ +a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ + Rational[1, 2] Pi, 0] - 4 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`\ +a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0] + Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`\ +a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0]) + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`a, \ +$CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"])), 0, + Rational[1, 210] + Pi $CellContext`r^(-2) $CellContext`r0^(-4) ($CellContext`r0^5 + HeavisideTheta[$CellContext`r - $CellContext`r0] ( + Complex[0, -14] (Complex[0, -3] + 4 $CellContext`a) $CellContext`r0 ( + 2 SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ + Rational[1, 2] Pi, 0] - 4 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0] + Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0]) + + 21 $CellContext`r ( + 2 (9 + Complex[0, -8] $CellContext`a) + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ + Rational[1, 2] Pi, 0] + + Complex[0, -4] (Complex[0, -5] + $CellContext`a) Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0] + (5 + Complex[0, 4] $CellContext`a) + Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0]) + + 168 $CellContext`r ( + 2 $CellContext`r - $CellContext`r0) $CellContext`r0^Rational[1, 2] ( + 2 SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ + Rational[1, 2] Pi, 0] - Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`\ +a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0]) + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`a, \ +$CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"] + + 4 $CellContext`r $CellContext`r0 (21 $CellContext`r^2 - + 28 $CellContext`r $CellContext`r0 + 11 $CellContext`r0^2) ( + 2 SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ + Rational[1, 2] Pi, 0] - 4 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0] + Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0]) + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`a,\ + $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^2) + \ +$CellContext`r^5 + HeavisideTheta[-$CellContext`r + $CellContext`r0] ( + 84 (2 + Complex[0, 1] $CellContext`a) $CellContext`r0 ( + 2 SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ + Rational[1, 2] Pi, 0] - 4 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0] + Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0]) + + 7 $CellContext`r ((-6 + Complex[0, 32] $CellContext`a) + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ + Rational[1, 2] Pi, 0] + (60 + Complex[0, 8] $CellContext`a) + Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0] + (-15 + Complex[0, -8] $CellContext`a) + Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ +2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0]) + + 168 $CellContext`r ( + 2 $CellContext`r - $CellContext`r0) $CellContext`r0^Rational[1, 2] ( + 2 SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ + Rational[1, 2] Pi, 0] - Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`\ +a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0]) + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`a, \ +$CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"] + + 4 $CellContext`r $CellContext`r0 (11 $CellContext`r^2 - + 28 $CellContext`r $CellContext`r0 + 21 $CellContext`r0^2) ( + 2 SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, + 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ + Rational[1, 2] Pi, 0] - 4 Derivative[1, 0][ + + 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https://dspace.mit.edu/handle/1721.1/61270. 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MST=Append[MST,Table[a[i]->aMST[i]+If[i==0,0,O[\[Epsilon]]^(ExpOrder+1)],{i,-Exp ] -(* ::Subsection::Closed:: *) +(* ::Subsection:: *) (*Definitions, replacements and auxiliary functions*) @@ -1352,7 +1352,7 @@ ExpandDiracDelta[expr_,x_]:=expr; (*Point particle source*) -(* ::Subsubsection::Closed:: *) +(* ::Subsubsection:: *) (*Interface*) @@ -1372,7 +1372,7 @@ Options[TeukolskySource]={"Form"->"Default"}; TeukolskySource[-2,\[ScriptL]_,\[ScriptM]_,a_,{r_,r0_},OptionsPattern[]] :=Assuming[{r0>0,r>0,1>a>=0}, - Module[{\[ScriptS]=-2,aux,auxFactor, \[ScriptCapitalE], \[ScriptCapitalL], \[Theta]0, \[CapitalDelta],\[CapitalDelta]1d,\[CapitalDelta]2d, Kt, \[CapitalUpsilon]t, \[Omega],\[CapitalOmega], SH,S0, dS0, d2S0, L1, L2, L2S, L2p, L1Sp, L1L2S, rcomp, invFactor0,invFactor1,invFactor2,\[Theta]comp, \[Rho], \[Rho]bar, \[CapitalSigma], Ann0, Anmbar0, Anmbar1, Ambarmbar0, Ambarmbar1, Ambarmbar2, Cnnp1p1, Cnmbarp1p1, Cmbarmbarp1p1,ret}, + Module[{\[ScriptS]=-2,aux,auxFactor,\[ScriptCapitalE], \[ScriptCapitalL], \[Theta]0, \[CapitalDelta],\[CapitalDelta]1d,\[CapitalDelta]2d, Kt, \[CapitalUpsilon]t, \[Omega],\[CapitalOmega], SH,S0, dS0, d2S0, L1, L2, L2S, L2p, L1Sp, L1L2S, rcomp, invFactor0,invFactor1,invFactor2,\[Theta]comp, \[Rho], \[Rho]bar, \[CapitalSigma], Ann0, Anmbar0, Anmbar1, Ambarmbar0, Ambarmbar1, Ambarmbar2, Cnnp1p1, Cnmbarp1p1, Cmbarmbarp1p1,ret}, \[ScriptCapitalE]=(a+(-2+r0) Sqrt[r0])/Sqrt[2 a r0^(3/2)+(-3+r0) r0^2]; \[ScriptCapitalL]=(a^2-2 a Sqrt[r0]+r0^2)/(Sqrt[2 a+(-3+r0) Sqrt[r0]] r0^(3/4)); @@ -1569,7 +1569,6 @@ TeukolskySource[2,\[ScriptL]_,\[ScriptM]_,a_,{r_,r0_},OptionsPattern[]] :=Assumi \[ScriptCapitalL]=(a^2-2 a Sqrt[r0]+r0^2)/(Sqrt[2 a+(-3+r0) Sqrt[r0]] r0^(3/4)); \[CapitalUpsilon]t = (r0^(5/4) (a+r0^(3/2)))/Sqrt[2 a+(-3+r0) Sqrt[r0]]; \[Theta]0 = \[Pi]/2; - \[Omega]=\[ScriptM] \[CapitalOmega]Kerr; (*\[CapitalOmega]=1/Sqrt[r0^3];*) @@ -1615,7 +1614,7 @@ ret ]] -(* ::Subsection::Closed:: *) +(* ::Subsection:: *) (*Teukolsky Equation*) @@ -1637,12 +1636,16 @@ aux/.\[Gamma]->\[ScriptA] \[Omega]/.replsPN//Series[#,{\[Eta],0,order\[Eta]}]& ] -TeukolskyEquation[\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_] := Collect[equation[\[ScriptS], \[ScriptL], \[ScriptM], \[Omega], a, 1, r]/.R->(R[# \[Eta]^2]&)/.replsPN,Derivative[__][R][__],Simplify]; -TeukolskyEquation[\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_] := Collect[equation[\[ScriptS], \[ScriptL], \[ScriptM], \[Omega], a, 1, r,order\[Eta]]/.R->(R[# \[Eta]^2]&)/.replsPN/.eigenValue->\[Lambda],{R[__],Derivative[__][R][__]},Simplify]; +(*TeukolskyEquation[\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_] := Collect[equation[\[ScriptS], \[ScriptL], \[ScriptM], \[Omega], a, 1, r]/.R->(R[# \[Eta]^2]&)/.replsPN,Derivative[__][R][__],Simplify]; +TeukolskyEquation[\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_] := Collect[equation[\[ScriptS], \[ScriptL], \[ScriptM], \[Omega], a, 1, r,order\[Eta]]/.R->(R[# \[Eta]^2]&)/.replsPN/.eigenValue->\[Lambda],{R[__],Derivative[__][R][__]},Simplify];*) + + +Options[TeukolskyEquation]={"ScaleR"->False} -TeukolskyEquation[\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]Var_,{\[Eta]Var_,order_},RVar_[rvar_]]:=Module[{aux}, -aux=SeriesCollect[equation[\[ScriptS], \[ScriptL], \[ScriptM], \[Omega], a, 1, r,order]/.R->(R[# \[Eta]^2]&)/.replsPN/.eigenValue->\[Lambda],{R[__],Derivative[__][R][__]},Simplify]; +TeukolskyEquation[\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]Var_,{\[Eta]Var_,order_},RVar_[rvar_],OptionsPattern[]]:=Module[{aux,replsR}, +replsR=If[OptionValue["ScaleR"],R->(R[# \[Eta]^2]&),{R[r_]:>R[r \[Eta]^2],Derivative[n_][R][r_]:>Derivative[n][R][\[Eta]^2 r]}]; +aux=SeriesCollect[equation[\[ScriptS], \[ScriptL], \[ScriptM], \[Omega], a, 1, r,order]/.replsR/.replsPN/.eigenValue->\[Lambda],{R[__],Derivative[__][R][__]},Simplify]; aux/.{\[Eta]->\[Eta]Var,\[Omega]->\[Omega]Var,R->RVar,r->rvar}] @@ -1712,7 +1715,7 @@ Derivative[n_][\[Theta]][arg_]:=Derivative[n-1][\[Delta]][arg]; \[Delta]''[\[Eta]^-2 a_]:=\[Eta]^2 \[Delta]''[a]; -(* ::Subsection:: *) +(* ::Subsection::Closed:: *) (*Amplitudes*) @@ -1752,7 +1755,7 @@ aux/.MSTCoefficientsInternal[\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]+3]// ] -(* ::Subsubsection:: *) +(* ::Subsubsection::Closed:: *) (*B Amplitudes*) @@ -1820,7 +1823,7 @@ BAmplitude["Ref","Normalization"->"UnitTransmission"][\[ScriptS]_,\[ScriptL]_,\[ BAmplitude["Trans","Normalization"->"UnitTransmission"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=1 (1+O[\[Eta]] \[Eta]^(order\[Eta]-1)); -(* ::Subsubsection:: *) +(* ::Subsubsection::Closed:: *) (*C Amplitude*) @@ -1890,7 +1893,7 @@ CAmplitude["Ref","Normalization"->"UnitTransmission"][\[ScriptS]_,\[ScriptL]_,\[ CAmplitude["Trans","Normalization"->"UnitTransmission"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,order\[Eta]_]:=1 (1+O[\[Eta]] \[Eta]^(order\[Eta]-1)); -(* ::Subsubsection:: *) +(* ::Subsubsection::Closed:: *) (*\[ScriptCapitalK] Amplitude*) @@ -2005,7 +2008,7 @@ ret//SeriesTake[#,order\[Eta]]& ] -(* ::Subsubsection:: *) +(* ::Subsubsection::Closed:: *) (*Interface*) @@ -2646,7 +2649,7 @@ If[!MatchQ[order,_Integer],Message[TeukolskyRadialFunctionPN::paramorder,order]; ] -(* ::Subsection:: *) +(* ::Subsection::Closed:: *) (*TeukolskyRadialFunctionPN*) @@ -2676,7 +2679,7 @@ icons = <| |>; -(* ::Subsubsection:: *) +(* ::Subsubsection::Closed:: *) (*TeukolskyRadialPN*) @@ -2691,9 +2694,9 @@ BC=sol; lead=RPNF[sol,"Normalization"->OptionValue["Normalization"],"Simplify"->OptionValue["Simplify"]][\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,1}]; termCount=R[r]//SeriesLength; normalization=OptionValue["Normalization"]; -trans=If[OptionValue["Amplitudes"],TeukolskyAmplitudePN[Switch[sol,"In","Btrans","Up","Ctrans"],"Normalization"->OptionValue["Normalization"]][\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,order}],"Not Computed"]; -inc=If[OptionValue["Amplitudes"],TeukolskyAmplitudePN[Switch[sol,"In","Binc","Up","Cinc"],"Normalization"->OptionValue["Normalization"]][\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,order}],"Not Computed"]; -ref=If[OptionValue["Amplitudes"],TeukolskyAmplitudePN[Switch[sol,"In","Bref","Up","Cref"],"Normalization"->OptionValue["Normalization"]][\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,order}],"Not Computed"]; +trans=If[OptionValue["Amplitudes"],TeukolskyAmplitudePN[Switch[sol,"In","Btrans","Up","Ctrans"],"Normalization"->OptionValue["Normalization"]][\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,order}],Missing["NotComputed"]]; +inc=If[OptionValue["Amplitudes"],TeukolskyAmplitudePN[Switch[sol,"In","Binc","Up","Cinc"],"Normalization"->OptionValue["Normalization"]][\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,order}],Missing["NotComputed"]]; +ref=If[OptionValue["Amplitudes"],TeukolskyAmplitudePN[Switch[sol,"In","Bref","Up","Cref"],"Normalization"->OptionValue["Normalization"]][\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,order}],Missing["NotComputed"]]; If[OptionValue["Simplify"],{trans,inc,ref}={trans,inc,ref}//Simplify]; amplitudes=<|"Incidence"->inc,"Transmission"->trans,"Reflection"->ref|>; ret=<|"s"->\[ScriptS],"l"->\[ScriptL],"m"->\[ScriptM],"a"->a,"PN"->{varPN,order},"RadialFunction"->R,"BoundaryCondition"->BC,"LeadingOrder"->lead,"TermCount"->termCount,"Normalization"->normalization,"Amplitudes"->amplitudes,"Simplify"->OptionValue["Simplify"]|>; @@ -2729,7 +2732,7 @@ retUp=TeukolskyRadialFunctionPN[\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{var ] -(* ::Subsubsection:: *) +(* ::Subsubsection::Closed:: *) (*TeukolskyRadialFunctionPN*) @@ -2771,16 +2774,26 @@ ret ] -(* ::Subsubsection:: *) +(* ::Subsubsection::Closed:: *) (*TeukolskyPointParticleModePN*) Options[RadialSourcedAssociation]={"Normalization"->"Default","Simplify"->True} -RadialSourcedAssociation["CO",opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,aVar_,r0Var_,{varPN_,order_}]:=Assuming[{varPN>0,r0>0,r>0,1>a>=0,\[ScriptA]>=0},Module[{aux,ret,Rin,dRin,ddRin,Rup,dRup,ddRup,wronskian,source,sourceCoeffs,cUp,cIn,deltaCoeff,innerF,outerF,inner,outer,radialF,ampAssoc}, +SeriesToSCoeffs[expr_SeriesData]:=Module[{aux}, +aux=expr[[3]]; +aux=aux/.Derivative[n_,0][SpinWeightedSpheroidalHarmonicS[a___]][b___]:>(SS)^n; +aux=aux/.SpinWeightedSpheroidalHarmonicS[a___][b___]:>(1)//Simplify; +aux=aux//CoefficientList[#,SS]&//Simplify; +aux=aux//.{}->Nothing; +aux +] + + +RadialSourcedAssociation["CO",opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,aVar_,r0Var_,{varPN_,order_}]:=Assuming[{varPN>0,r0>0,r>0,1>a>=0,\[ScriptA]>=0},Module[{aux,ret,Scoeffs,Rin,dRin,ddRin,Rup,dRup,ddRup,wronskian,source,sourceCoeffs,cUp,cIn,deltaCoeff,innerF,outerF,inner,outer,radialF,ampAssoc}, CheckInput["Up",\[ScriptS],\[ScriptL],\[ScriptM],aVar,\[ScriptM]/Sqrt[r0Var^3],{varPN,order}]; -aux=TeukolskyRadialPN[\[ScriptS],\[ScriptL],\[ScriptM],aVar,If[\[ScriptM]!=0,\[ScriptM],Style["0",Red]]\[CapitalOmega]Kerr,{varPN,order},"Normalization"->OptionValue["Normalization"]]; +aux=TeukolskyRadialPN[\[ScriptS],\[ScriptL],\[ScriptM],aVar,If[\[ScriptM]!=0,\[ScriptM],Style["0",Red]]Inactive[KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][aVar,r0Var,0,1]["\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"],{varPN,order},"Normalization"->OptionValue["Normalization"]]; Rin=aux["In"]["RadialFunction"]; Rup=aux["Up"]["RadialFunction"]; dRup=Rup'; @@ -2798,7 +2811,7 @@ cIn=1/wronskian Total[sourceCoeffs {Rup[r0],-varPN^2dRup[r0],varPN^4 ddRup[r0]}] cUp=1/wronskian Total[sourceCoeffs {Rin[r0],-varPN^2dRin[r0],varPN^4 ddRin[r0]}]//SeriesTake[#,order]&; deltaCoeff=Coefficient[source[r],Derivative[2][DiracDelta][r-r0]]/Kerr\[CapitalDelta][a,r0]; deltaCoeff=Assuming[{varPN>0},If[deltaCoeff===0,0,deltaCoeff//PNScalings[#,{{r0,-2},{\[CapitalOmega]Kerr,3}},varPN,"IgnoreHarmonics"->True]&//SeriesTerms[#,{varPN,0,order}]&]]; -{cIn,cUp,deltaCoeff,source}={cIn,cUp,deltaCoeff,source}/.r0->r0Var/.a->aVar; +{cIn,cUp,deltaCoeff,source}={cIn,cUp,deltaCoeff,source}/.r0->r0Var/.a->aVar/.\[CapitalOmega]Kerr->Inactive[KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][aVar,r0Var,0,1]["\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]; If[OptionValue["Simplify"],{cIn,cUp,deltaCoeff,source}={cIn,cUp,deltaCoeff,source}//SeriesCollect[#,{SpinWeightedSpheroidalHarmonicS[__],Derivative[__][SpinWeightedSpheroidalHarmonicS][__]},(Simplify[#,{aVar>=0,r0Var>0,varPN>0}]&)]&]; inner=cIn Rin[r]; outer=cUp Rup[r]; @@ -2808,7 +2821,8 @@ outerF=outer/.r->#&; ampAssoc=<|"\[ScriptCapitalI]"->cUp,"\[ScriptCapitalH]"->cIn|>; radialF=innerF[#] HeavisideTheta[r0Var-#] + outerF[#] HeavisideTheta[#-r0Var]+deltaCoeff DiracDelta[#-r0Var]&; -ret=<|"s"->\[ScriptS],"l"->\[ScriptL],"m"->\[ScriptM],"a"->aVar,"r0"->r0Var,"PN"->{varPN,order},"RadialFunction"->radialF,("ExtendedHomogeneous"->"\[ScriptCapitalI]")->innerF,("ExtendedHomogeneous"->"\[ScriptCapitalH]")->outerF,"\[Delta]"->deltaCoeff,"Amplitudes"->ampAssoc,"Wronskian"->wronskian,"Source"->source,"In"->aux["In"],"Up"->aux["Up"],"Simplify"->OptionValue["Simplify"],"Normalization"->OptionValue["Normalization"]|>; +Scoeffs=SeriesToSCoeffs[radialF[r]]/.r->#&; +ret=<|"s"->\[ScriptS],"l"->\[ScriptL],"m"->\[ScriptM],"a"->aVar,"r0"->r0Var,"PN"->{varPN,order},"RadialFunction"->radialF,"CoefficientList"->Scoeffs,("ExtendedHomogeneous"->"\[ScriptCapitalI]")->innerF,("ExtendedHomogeneous"->"\[ScriptCapitalH]")->outerF,"\[Delta]"->deltaCoeff,"Amplitudes"->ampAssoc,"Wronskian"->wronskian,"Source"->source,"In"->aux["In"],"Up"->aux["Up"],"Simplify"->OptionValue["Simplify"],"Normalization"->OptionValue["Normalization"]|>; ret ] ] @@ -2835,7 +2849,7 @@ TeukolskyModePN /: BoxForm`SummaryItem[{"l: ", l}], " ", BoxForm`SummaryItem[{"m: ", m}], " ", BoxForm`SummaryItem[{"a: ", a}], " ", - BoxForm`SummaryItem[{"\[Omega]: ", m \[CapitalOmega]Kerr}]," ", + BoxForm`SummaryItem[{"\[Omega]: ", m KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies[a,r0,0,1]["\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]}]," ", BoxForm`SummaryItem[{"\!\(\*SubscriptBox[\(r\), \(0\)]\): ", r0}]," ", BoxForm`SummaryItem[{"PN parameter: ", varPN}]," ", BoxForm`SummaryItem[{"PN order: ", N[(order-1)/2]"PN"}] @@ -2870,7 +2884,7 @@ TeukolskyPointParticleModePN[\[ScriptS], \[ScriptL], \[ScriptM],orbit,{varPN,aux ] -(* ::Subsubsection:: *) +(* ::Subsubsection::Closed:: *) (*Accessing functions and keys*) From f987e3e48bbe20b4e7c8ab5eff5d5cdc38b6fe3d Mon Sep 17 00:00:00 2001 From: jakobneef Date: Tue, 28 Jan 2025 11:47:43 +0000 Subject: [PATCH 06/13] Cleaned up public section --- Kernel/PN.wl | 247 ++------------------------------------------------- 1 file changed, 6 insertions(+), 241 deletions(-) diff --git a/Kernel/PN.wl b/Kernel/PN.wl index 04d1eb5..e0dbcc9 100644 --- a/Kernel/PN.wl +++ b/Kernel/PN.wl @@ -26,157 +26,12 @@ ClearAttributes[{TeukolskyRadialPN, TeukolskyRadialFunctionPN,TeukolskyPointPart (*Public *) -(* ::Subsection::Closed:: *) -(*MST Coefficients*) - - -(* ::Input:: *) -(*(*\[Nu]MST::usage="\[Nu]MST is representative of the \[Nu] coefficient in the MST solutions"*) -(*aMST::usage="aMST[\!\(\**) -(*StyleBox[\"n\",\nFontSlant->\"Italic\"]\)] is the \!\(\*SuperscriptBox[*) -(*StyleBox[\"n\",\nFontSlant->\"Italic\"], \(th\)]\) MST coefficient";*) -(*MSTCoefficients::usage="MSTCoefficients[\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]] gives the PN expanded MST coefficients aMST[n] for a given {\!\(\**) -(*StyleBox[\"\[ScriptS]\",\nFontSlant->\"Italic\"]\),\!\(\**) -(*StyleBox[\"\[ScriptL]\",\nFontSlant->\"Italic\"]\),\!\(\**) -(*StyleBox[\"\[ScriptM]\",\nFontSlant->\"Italic\"]\)} mode up to \!\(\*SuperscriptBox[\(\[Eta]\), \(\!\(\**) -(*StyleBox[\"order\[Eta]\",\nFontSlant->\"Italic\"]\)\)]\)."*) -(*KerrMSTSeries::usage="KerrMSTSeries[\!\(\**) -(*StyleBox[\"\[ScriptS]\",\nFontSlant->\"Italic\"]\),\!\(\**) -(*StyleBox[\"\[ScriptL]\",\nFontSlant->\"Italic\"]\),\!\(\**) -(*StyleBox[\"\[ScriptM]\",\nFontSlant->\"Italic\"]\),\!\(\**) -(*StyleBox[\"order\[Epsilon]\",\nFontSlant->\"Italic\"]\)] gives the PN expanded MST coefficients a[n] for a given {\!\(\**) -(*StyleBox[\"\[ScriptS]\",\nFontSlant->\"Italic\"]\), \!\(\**) -(*StyleBox[\"\[ScriptL]\",\nFontSlant->\"Italic\"]\), \!\(\**) -(*StyleBox[\"\[ScriptM]\",\nFontSlant->\"Italic\"]\)} mode up to \!\(\*SuperscriptBox[\(\[Epsilon]\), \(order\[Epsilon]\)]\). Where the relation to \[Eta] is given by \[Epsilon]=2 \[Omega] \!\(\*SuperscriptBox[\(\[Eta]\), \(3\)]\)."*)*) - - -(* ::Subsection::Closed:: *) -(*Spacetime replacements*) - - -(* ::Input:: *) -(*(*Kerr\[CapitalDelta]::usage="Kerr\[CapitalDelta][a,r] gives the Kerr \[CapitalDelta] \!\(\*SuperscriptBox[\(r\), \(2\)]\)-2r+\!\(\*SuperscriptBox[\(a\), \(2\)]\)"*)*) - - -(* ::Input:: *) -(*(*(*replsKerr::usage="a list of replacements for a Kerr spacetime."*) -(*replsSchwarzschild::usage="a list of replacements for Schwarzschild spacetime."*)*) -(*(*Schwarzschild::usage="Schwarzschild[\!\(\**) -(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)] will set a to 0 in \!\(\**) -(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)."*) -(**)*)*) - - -(* ::Subsection::Closed:: *) -(*Tools for Series*) - - -(* ::Subsubsection::Closed:: *) -(*General Tools for Series*) - - -(* ::Input:: *) -(*(*SeriesTake::usage="SeriesTake[series, n] takes the first n terms of series"*) -(*SeriesMinOrder::usage="SeriesMinOrder[series] gives the leading order of series"*) -(*SeriesMaxOrder::usage="SeriesMaxOrder[series] gives the first surpressed order of series"*) -(*SeriesLength::usage="SeriesLenght[series] gives the number of terms in series"*) -(*SeriesCollect::usage="SeriesCollect[expr, var, func] works like Collect but applied to each order individually. Crucially, unlike Collect it keeps the SeriesData structure."*) -(*SeriesTerms::usage="SeriesTerms[series, {x, \!\(\*SubscriptBox[\(x\), \(0\)]\), n}] works exactly like Series, with the difference that n gives the desired number of terms instead of a maximum order"*) -(*IgnoreExpansionParameter::usage="IgnoreExpansionParameter[series,x] sets all occurences of the expansion parameter in the series coefficients to x. If no value is entered x defaults to 1." *)*) - - -(* ::Subsubsection::Closed:: *) -(*Tools for PN Scalings*) - - -(* ::Input:: *) -(*(*PNScalings::usage="PNScalings[expr,params,var] applies the given powercounting scalings to the expression. E.g. PNScalings[\[Omega] r,{{\[Omega],3},{r,-2},\[Eta]]"*) -(*RemovePN::usage="PNScalings[expr,var] takes the Normal[] and sets var to 1"*) -(*Zero::usage="Zero[expr,vars] sets all vars in expr to 0"*) -(*One::usage="One[expr,vars] sets all vars in expr to 1"*)*) - - -(* ::Input:: *) -(*(*ExpandSpheroidals::usage="ExpandSpheroidal[expr,{param,order}] returns a all SpinWeightedSpheroidalHarmonicS in expr have been Series expanded around param->0 to order."*)*) - - -(* ::Input:: *) -(*(*CollectDerivatives::usage="CollectDerivatives[expr,f] works exactly like Collect[] but also collects for derivatives of f."*)*) - - -(* ::Subsection::Closed:: *) -(*Tools for Logs, Gammas, and PolyGammas*) - - -(* ::Input:: *) -(*(*ExpandLog::usage="ExpandLog[\!\(\**) -(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)] replaces all Logs in \!\(\**) -(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)\!\(\**) -(*StyleBox[\" \",\nFontSlant->\"Italic\"]\)with a PowerExpanded version"*) -(*ExpandGamma::usage="ExpandGamma[\!\(\**) -(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)] factors out all Integer facors out of the Gammas in \!\(\**) -(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\). E.g. Gamma[x+1]->x Gamma[x]"*) -(*ExpandPolyGamma::usage="ExpandPolyGamma[\!\(\**) -(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)] factors out all Integer facors out of the PolyGammas in \!\(\**) -(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\). E.g. PolyGamma[x+1]->\!\(\*FractionBox[\(1\), \(x\)]\) PolyGamma[x]"*) -(*PochhammerToGamma::usage="PochhammerToGamma[\!\(\**) -(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)] replaces all Pochhammer in \!\(\**) -(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\) with the respecive Gamma."*) -(*GammaToPochhammer::usage="PochhammerToGamma[\!\(\**) -(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\),\!\(\**) -(*StyleBox[\"n\",\nFontSlant->\"Italic\"]\)] replaces all Gamma in \!\(\**) -(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)\!\(\**) -(*StyleBox[\" \",\nFontSlant->\"Italic\"]\)that contain \!\(\**) -(*StyleBox[\"n\",\nFontSlant->\"Italic\"]\) with the respective Pochhammer[__,\!\(\**) -(*StyleBox[\"n\",\nFontSlant->\"Italic\"]\)]"*) -(**)*) - - -(* ::Subsection::Closed:: *) -(*Tools for DiracDelta*) - - -(* ::Input:: *) -(*(*ExpandDiracDelta::usage="ExpandDiracDelta[expr,r] applies identities for Dirac deltas and it's derivatives to expr."*)*) - - -(* ::Subsection::Closed:: *) -(*Amplitudes*) - - -(* ::Input:: *) -(*(*AAmplitude::usage="AAmplitude[\"+\"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]] gives the factor \!\(\*SubscriptBox[\(A\), \(+\)]\) from Sasaki Tagoshi Eq.(157). Likewise for [\"-\"]"*) -(*BAmplitude::usage="BAmplitude[\"trans\"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]] gives the factor \!\(\*SuperscriptBox[\(B\), \(trans\)]\) from Sasaki Tagoshi Eq.(167). Likewise for [\"inc\"]"*) -(*CAmplitude::usage="CAmplitude[\"trans\"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]] gives the factor \!\(\*SuperscriptBox[\(C\), \(trans\)]\) from Sasaki Tagoshi Eq.(170)."*) -(*\[ScriptCapitalK]Amplitude::usage="\[ScriptCapitalK]Amplitude[\"\[Nu]\"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]] gives the \!\(\*SuperscriptBox[\(\[ScriptCapitalK]\), \(\[Nu]\)]\) factor for \!\(\*SubscriptBox[\(R\), \(In\)]\). Likewise for \[ScriptCapitalK]Amplitude[\"-\[Nu]-1\"]. \[ScriptCapitalK]Amplitude[\"Ratio\"] gives the tidal response function \!\(\*FractionBox[SuperscriptBox[\(\[ScriptCapitalK]\), \(\(-\[Nu]\) - 1\)], SuperscriptBox[\(\[ScriptCapitalK]\), \(\[Nu]\)]]\)."*)*) - - -(* ::Input:: *) -(*(*TeukolskyAmplitudePN::usage="TeukolskyAmplitudePN[\"sol\"][\[ScriptS], \[ScriptL], \[ScriptM], a, \[Omega], {\[Eta], n}] gives the desired PN expanded amplitude. Options for sol are as follows: *) -(*\"A+\": Sasaki Tagoshi Eq.(157), *) -(*\"A-\": ST Eq.(158), *) -(*\"Btrans\": ST Eq.(167), *) -(*\"Binc\": ST Eq.(168) divided by \!\(\*SubscriptBox[\(\[ScriptCapitalK]\), \(\[Nu]\)]\), *) -(*\"Ctrans\": Eq.(170) ST, *) -(*\"\[ScriptCapitalK]\": , *) -(*\"\[ScriptCapitalK]\[Nu]\": , *) -(*\"\[ScriptCapitalK]-\[Nu]-1\": "*)*) - - -(* ::Subsection::Closed:: *) -(*Wronskian*) - - -(* ::Input:: *) -(*(*InvariantWronskian::usage="InvariantWronskian[\[ScriptS], \[ScriptL], \[ScriptM], a, \[Omega], {\[Eta], n}] gives the invariant Wronskian."*)*) - - -(* ::Subsection::Closed:: *) -(*Radial solutions*) +(* ::Subsection:: *) +(*Homogeneous solutions*) TeukolskyRadialPN::usage="TeukolskyRadialPN[\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{\[Eta],n}] gives the In and Up solution to the radial Teukolsky equation. {\[ScriptS],\[ScriptL],\[ScriptM]} specify the mode, a is the Kerr spin parameter, \[Omega] is the frequency, \[Eta] is the PN expansion parameter and n the number of terms." -TeukolskyRadialFunctionPN::usage="TeukolskyRadialFunctionPN[\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{\[Eta],n},\"sol\"] gives the specified solution to the radial Teukolsky equation. {\[ScriptS],\[ScriptL],\[ScriptM]} specify the mode, a is the Kerr spin parameter, \[Omega] is the frequency, \[Eta] is the PN expansion parameter and n the number of terms. Possible options for sol are In, Up, C\[Nu], and C-\[Nu]-1" +TeukolskyRadialFunctionPN::usage="is an object representing a PN expanded homogeneous solution to the radial Teukolsky equation." TeukolskyRadialFunctionPN::optx="`1` is not a valid boundary condition. Possible options are In, Up, C\[Nu], and C-\[Nu]-1"; @@ -190,108 +45,18 @@ TeukolskyRadialFunctionPN::paramorder="order=`1`. The given number of terms has TeukolskyRadialFunctionPN::PNInput="Input String does not contain \"PN\". Assume PN orders are desired (calulate `1` terms in the Series)."; -(* ::Subsection::Closed:: *) +(* ::Subsection:: *) (*Sourced things*) -(* ::Input:: *) -(*(*TeukolskySourceCircularOrbit::usage="TeukolskySource[\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{r,r\:2080}] gives an analytical expression for the Teukolsky point particle source for a given {\[ScriptS],\[ScriptL],\[ScriptM]} mode. "*)*) - - TeukolskyPointParticleModePN::usage="TeukolskyPointParticleModePN[s, l, m, orbit, {\[Eta], n}] produces a TeukolskyModePN representing a PN expanded analyitcal solution to the radial Teukolsky equation with a point particle source. s, l, and m specify the mode and need to be Integers. orbit needs to be a KerrGeoOrbitFunction (computed with KerrGeoOrbit[]). {\[Eta], n} specify the PN information. \[Eta] needs to be a symbol, while n is an integer specifying the amount of terms (including the Newtonian order), i.e., n=2 PNorder+1. " -TeukolskyModePN::usage="aa" +TeukolskyModePN::usage="is an object which represents a PN expanded Teukolsky mode." TeukolskyPointParticleModePN::orbit="As of now TeukolskyPointParticleModePN only supports circular equatorial orbits, i.e., e=0 and x=1."; -TeukolskyPointParticleModePN::particle="TeukolskyPointParticleModePN cannot be evaluated at the particle."; - - -\[CapitalOmega]Kerr::usage="This is here as a quick fix. It is the particles orbital frequency, i.e., \!\(\*SqrtBox[FractionBox[\(1\), SuperscriptBox[SubscriptBox[\(r\), \(0\)], \(3\)]]]\) in Schwarzschild." - - -(* ::Subsection::Closed:: *) -(*Teukolsky Equation*) - - -(* ::Input:: *) -(*(*TeukolskyEquation::usage="TeukolskyEquation[\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{\[Eta],order},R[r]] gives the Teukolsky equation with for a given {\[ScriptS],\[ScriptL],\[ScriptM]} mode with included \[Eta] scalings. The {\[Eta],order} argument can be left out for a general expression."*)*) - - -(* ::Subsection::Closed:: *) -(*Developer options*) - - -(* ::Input:: *) -(*(*integrateDelta::usage="aala"*) -(*\[Delta]*) -(*\[Theta]*) -(**) -(*PNScalings*)*) - - -(* ::Input:: *) -(*(*z::usage="iuaeouia"*) -(*\[CapitalDelta]::usage="\[CapitalDelta][\!\(\**) -(*StyleBox[\"a\",\nFontSlant->\"Italic\"]\),\!\(\**) -(*StyleBox[\"M\",\nFontSlant->\"Italic\"]\),\!\(\**) -(*StyleBox[\"r\",\nFontSlant->\"Italic\"]\)] is the usual length scale of the Kerr spacetime"*)*) - - -(* ::Subsubsection::Closed:: *) -(*Post Newtonian Scalings*) - - -(* ::Input:: *) -(*(*replsPN="Replacements for PN scalings"*) -(*PNScalingsInternal::usage="PNScalingsInternal[\!\(\**) -(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)] reapplies the PN scalings to \!\(\**) -(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\). Should not be used on SeriesData objects. "*) -(*redo\[Eta]Repls::usage="iuaeuiae"*) -(*RemovePNInternal::usage="RemovePNInternal[expr] removes all PN scalings in the expression, i.e., taking the Normal and setting \[Eta]->1"*) -(*polyToSeries*) -(*IgnoreLog\[Eta]::usage="IgnoreLog\[Eta][\!\(\**) -(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)] sets all \[Eta] factors within \!\(\**) -(*StyleBox[\"expr\",\nFontSlant->\"Italic\"]\) to 1. This can be helpful when using SeriesCollect"*)*) - - -(* ::Subsubsection::Closed:: *) -(*Radial functions*) - - -(* ::Input:: *) -(*(*RPN::usage="RPN[\"In\"][\!\(\**) -(*StyleBox[\"\[ScriptS]\",\nFontSlant->\"Italic\"]\),\!\(\**) -(*StyleBox[\"\[ScriptL]\",\nFontSlant->\"Italic\"]\),\!\(\**) -(*StyleBox[\"\[ScriptM]\",\nFontSlant->\"Italic\"]\),\!\(\**) -(*StyleBox[\"a\",\nFontSlant->\"Italic\"]\),\!\(\**) -(*StyleBox[\"order\[Eta]\",\nFontSlant->\"Italic\"]\)] Gives the In solution to the radial Teukolsky equation. Analogously for \"Up\", \"C\", or \"secondTerm\". The inhomogeneous solution for a circular orbit can be obtained with \"CO\""*) -(*RPNF::usage="RPNF[\!\(\**) -(*StyleBox[\"\[ScriptS]\",\nFontSlant->\"Italic\"]\),\!\(\**) -(*StyleBox[\"\[ScriptL]\",\nFontSlant->\"Italic\"]\),\!\(\**) -(*StyleBox[\"\[ScriptM]\",\nFontSlant->\"Italic\"]\),\!\(\**) -(*StyleBox[\"a\",\nFontSlant->\"Italic\"]\),\!\(\**) -(*StyleBox[\"\[Omega]\",\nFontSlant->\"Italic\"]\),{\!\(\**) -(*StyleBox[\"\[Eta]\",\nFontSlant->\"Italic\"]\),order\[Eta]}] gives a function"*) -(*integrateDelta::usage="Function that integrates Dirac delta distributions. Used when getting the inhomogeneous solutions."*) -(*Normalization::usage="Normalization[\"In\"][\[ScriptS],\[ScriptL],\[ScriptM],a,order\[Eta]] gives a PN expanded normaliztion coefficient. If you divide RPN by it it will match the result of the toolkit"*)*) - - -(* ::Input:: *) -(*(*CCoefficient::usage="CCoefficient[\"In\"][\!\(\**) -(*StyleBox[\"\[ScriptS]\",\nFontSlant->\"Italic\"]\),\!\(\**) -(*StyleBox[\"\[ScriptL]\",\nFontSlant->\"Italic\"]\),\!\(\**) -(*StyleBox[\"\[ScriptM]\",\nFontSlant->\"Italic\"]\),\!\(\**) -(*StyleBox[\"a\",\nFontSlant->\"Italic\"]\),\!\(\**) -(*StyleBox[\"order\[Eta]\",\nFontSlant->\"Italic\"]\)] gives the \!\(\*SubscriptBox[\(c\), \(\(in\)\(\\\ \)\)]\)coefficient for the sourced solution R=\!\(\*SubscriptBox[\(c\), \(in\)]\) \!\(\*SubscriptBox[\(R\), \(in\)]\) + \!\(\*SubscriptBox[\(c\), \(up\)]\) \!\(\*SubscriptBox[\(R\), \(up\)]\). Likewise for [\"Up\"]"*) -(*InvariantWronskian::usage="InvariantWronskian[\!\(\**) -(*StyleBox[\"\[ScriptS]\",\nFontSlant->\"Italic\"]\),\!\(\**) -(*StyleBox[\"\[ScriptL]\",\nFontSlant->\"Italic\"]\),\!\(\**) -(*StyleBox[\"\[ScriptM]\",\nFontSlant->\"Italic\"]\),\!\(\**) -(*StyleBox[\"a\",\nFontSlant->\"Italic\"]\),\!\(\**) -(*StyleBox[\"order\[Eta]\",\nFontSlant->\"Italic\"]\)] gives the invariant Wronskian. C.f. Sasaki Tagoshi Eq.(23)"*) -(**)*) +TeukolskyPointParticleModePN::particle="TeukolskyPointParticleModePN cannot be evaluated directly at the particle. Try the Keys \"ExtendedHomogeneous\"\[Rule]\"\[ScriptCapitalI]\",\"ExtendedHomogeneous\"\[Rule]\"\[ScriptCapitalH]\" and \"\[Delta]\" "; (* ::Section:: *) From 2646f9c6582e744350bdba49faf6356d7634ae67 Mon Sep 17 00:00:00 2001 From: jakobneef Date: Tue, 28 Jan 2025 14:48:45 +0000 Subject: [PATCH 07/13] fixed Keys --- Kernel/PN.wl | 96 +++++++++++++++++++++++++++++++++++----------------- 1 file changed, 65 insertions(+), 31 deletions(-) diff --git a/Kernel/PN.wl b/Kernel/PN.wl index e0dbcc9..26d8c45 100644 --- a/Kernel/PN.wl +++ b/Kernel/PN.wl @@ -2414,8 +2414,8 @@ If[!MatchQ[order,_Integer],Message[TeukolskyRadialFunctionPN::paramorder,order]; ] -(* ::Subsection::Closed:: *) -(*TeukolskyRadialFunctionPN*) +(* ::Subsection:: *) +(*TeukolskyRadialPN*) (* ::Subsubsection::Closed:: *) @@ -2444,19 +2444,20 @@ icons = <| |>; -(* ::Subsubsection::Closed:: *) -(*TeukolskyRadialPN*) +(* ::Subsubsection:: *) +(*Getting internal association*) Options[RadialAssociation]={"Normalization"->"Default", "Amplitudes"->False, "Simplify"->True} Options[TeukolskyRadialPN]={"Normalization"->"Default", "Amplitudes"->False, "Simplify"->True} -RadialAssociation[sol_String,opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]_,{varPN_,order_}]/;MemberQ[PossibleSols,sol]:=Module[{aux,ret,R,BC,lead,termCount,normalization,amplitudes,trans,inc,ref}, +RadialAssociation[sol_String,opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,\[Omega]_,{varPN_,order_}]/;MemberQ[PossibleSols,sol]:=Module[{aux,ret,R,BC,lead,minOrder,termCount,normalization,amplitudes,trans,inc,ref}, CheckInput[sol,\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,order}]; R=RPNF[sol,"Normalization"->OptionValue["Normalization"],"Simplify"->OptionValue["Simplify"]][\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,order}]; BC=sol; lead=RPNF[sol,"Normalization"->OptionValue["Normalization"],"Simplify"->OptionValue["Simplify"]][\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,1}]; +minOrder=lead[r]//SeriesMinOrder; termCount=R[r]//SeriesLength; normalization=OptionValue["Normalization"]; trans=If[OptionValue["Amplitudes"],TeukolskyAmplitudePN[Switch[sol,"In","Btrans","Up","Ctrans"],"Normalization"->OptionValue["Normalization"]][\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,order}],Missing["NotComputed"]]; @@ -2464,11 +2465,15 @@ inc=If[OptionValue["Amplitudes"],TeukolskyAmplitudePN[Switch[sol,"In","Binc","Up ref=If[OptionValue["Amplitudes"],TeukolskyAmplitudePN[Switch[sol,"In","Bref","Up","Cref"],"Normalization"->OptionValue["Normalization"]][\[ScriptS],\[ScriptL],\[ScriptM],a,\[Omega],{varPN,order}],Missing["NotComputed"]]; If[OptionValue["Simplify"],{trans,inc,ref}={trans,inc,ref}//Simplify]; amplitudes=<|"Incidence"->inc,"Transmission"->trans,"Reflection"->ref|>; -ret=<|"s"->\[ScriptS],"l"->\[ScriptL],"m"->\[ScriptM],"a"->a,"PN"->{varPN,order},"RadialFunction"->R,"BoundaryCondition"->BC,"LeadingOrder"->lead,"TermCount"->termCount,"Normalization"->normalization,"Amplitudes"->amplitudes,"Simplify"->OptionValue["Simplify"]|>; +ret=<|"s"->\[ScriptS],"l"->\[ScriptL],"m"->\[ScriptM],"a"->a,"PN"->{varPN,order},"RadialFunction"->R,"BoundaryCondition"->BC,"SeriesMinOrder"->minOrder,"LeadingOrder"->lead,"TermCount"->termCount,"Normalization"->normalization,"Amplitudes"->amplitudes,"Simplify"->OptionValue["Simplify"],"AmplitudesBool"->OptionValue["Amplitudes"]|>; ret ] +(* ::Subsubsection::Closed:: *) +(*TeukolskyRadialPN*) + + PNStringToOrder[pn_String]:=Module[{aux,ret,check1,check2}, check1=StringContainsQ[pn,"PN"]; aux=pn//StringReplace[#,"PN"->""]&; @@ -2518,9 +2523,11 @@ TeukolskyRadialFunctionPN /: }], BoxForm`SummaryItem[{"Boundary Condition: ", assoc["BoundaryCondition"]}]}; extended = {BoxForm`SummaryItem[{"Leading order: ",assoc["LeadingOrder"]["r"]}], + BoxForm`SummaryItem[{"Min order: ",assoc["SeriesMinOrder"]}], + BoxForm`SummaryItem[{"Number of terms: ",assoc["TermCount"]}], BoxForm`SummaryItem[{"Normalization: ",assoc["Normalization"]}], BoxForm`SummaryItem[{"Simplify: ",assoc["Simplify"]}], - BoxForm`SummaryItem[{"Number of terms: ",assoc["TermCount"]}]}; + BoxForm`SummaryItem[{"Amplitudes: ",assoc["AmplitudesBool"]}]}; BoxForm`ArrangeSummaryBox[ TeukolskyRadialFunctionPN, @@ -2540,9 +2547,31 @@ ret (* ::Subsubsection::Closed:: *) +(*Accessing functions and keys*) + + +TeukolskyRadialFunctionPN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_][y_String]/;!MemberQ[{"RadialFunction","AmplitudesBool","TermCount"}, y]:= + assoc[y]; + + +TeukolskyRadialFunctionPN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_][r_Symbol] :=assoc["RadialFunction"][r] +TeukolskyRadialFunctionPN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_][r_/;NumericQ[r]] :=assoc["RadialFunction"][r] + + +Derivative[n_Integer][trf_TeukolskyRadialFunctionPN][r_Symbol]:=trf[[6,1]]^(2 n) Derivative[n][trf["RadialFunction"]][r] + + +Keys[trfpn_TeukolskyRadialFunctionPN] ^:= DeleteElements[Join[Keys[trfpn[[-1]]], {}], {"RadialFunction","AmplitudesBool","TermCount"}]; + + +(* ::Subsection:: *) (*TeukolskyPointParticleModePN*) +(* ::Subsubsection:: *) +(*Getting internal association*) + + Options[RadialSourcedAssociation]={"Normalization"->"Default","Simplify"->True} @@ -2556,11 +2585,11 @@ aux ] -RadialSourcedAssociation["CO",opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,aVar_,r0Var_,{varPN_,order_}]:=Assuming[{varPN>0,r0>0,r>0,1>a>=0,\[ScriptA]>=0},Module[{aux,ret,Scoeffs,Rin,dRin,ddRin,Rup,dRup,ddRup,wronskian,source,sourceCoeffs,cUp,cIn,deltaCoeff,innerF,outerF,inner,outer,radialF,ampAssoc}, +RadialSourcedAssociation["CO",opt:OptionsPattern[]][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,aVar_,r0Var_,{varPN_,order_}]:=Assuming[{varPN>0,r0>0,r>0,1>a>=0,\[ScriptA]>=0},Module[{aux,ret,Scoeffs,Rin,dRin,ddRin,Rup,dRup,ddRup,wronskian,source,sourceF,sourceCoeffs,minOrder,cUp,cIn,deltaCoeff,innerF,outerF,inner,outer,radialF,ampAssoc}, CheckInput["Up",\[ScriptS],\[ScriptL],\[ScriptM],aVar,\[ScriptM]/Sqrt[r0Var^3],{varPN,order}]; aux=TeukolskyRadialPN[\[ScriptS],\[ScriptL],\[ScriptM],aVar,If[\[ScriptM]!=0,\[ScriptM],Style["0",Red]]Inactive[KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][aVar,r0Var,0,1]["\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"],{varPN,order},"Normalization"->OptionValue["Normalization"]]; -Rin=aux["In"]["RadialFunction"]; -Rup=aux["Up"]["RadialFunction"]; +Rin=aux["In"][[-1]]["RadialFunction"]; +Rup=aux["Up"][[-1]]["RadialFunction"]; dRup=Rup'; dRin=Rin'; ddRup=dRup'; @@ -2583,11 +2612,12 @@ outer=cUp Rup[r]; If[OptionValue["Simplify"],{inner,outer}={inner,outer}//SeriesCollect[#,{SpinWeightedSpheroidalHarmonicS[__],Derivative[__][SpinWeightedSpheroidalHarmonicS][__]},(Simplify[#,{aVar>=0,r0Var>0,varPN>0}]&)]&]; innerF=inner/.r->#&; outerF=outer/.r->#&; - ampAssoc=<|"\[ScriptCapitalI]"->cUp,"\[ScriptCapitalH]"->cIn|>; radialF=innerF[#] HeavisideTheta[r0Var-#] + outerF[#] HeavisideTheta[#-r0Var]+deltaCoeff DiracDelta[#-r0Var]&; +minOrder=radialF[r]//SeriesMinOrder; +sourceF=source[r]/.\[CapitalOmega]Kerr->Inactive[KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][aVar,r0Var,0,1]["\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]/.r->#&; Scoeffs=SeriesToSCoeffs[radialF[r]]/.r->#&; -ret=<|"s"->\[ScriptS],"l"->\[ScriptL],"m"->\[ScriptM],"a"->aVar,"r0"->r0Var,"PN"->{varPN,order},"RadialFunction"->radialF,"CoefficientList"->Scoeffs,("ExtendedHomogeneous"->"\[ScriptCapitalI]")->innerF,("ExtendedHomogeneous"->"\[ScriptCapitalH]")->outerF,"\[Delta]"->deltaCoeff,"Amplitudes"->ampAssoc,"Wronskian"->wronskian,"Source"->source,"In"->aux["In"],"Up"->aux["Up"],"Simplify"->OptionValue["Simplify"],"Normalization"->OptionValue["Normalization"]|>; +ret=<|"s"->\[ScriptS],"l"->\[ScriptL],"m"->\[ScriptM],"a"->aVar,"r0"->r0Var,"PN"->{varPN,order},"RadialFunction"->radialF,"CoefficientList"->Scoeffs,("ExtendedHomogeneous"->"\[ScriptCapitalI]")->innerF,("ExtendedHomogeneous"->"\[ScriptCapitalH]")->outerF,"\[Delta]"->deltaCoeff,"Amplitudes"->ampAssoc,"Wronskian"->wronskian,"Source"->sourceF,"SeriesMinOrder"->minOrder,"In"->aux["In"],"Up"->aux["Up"],"Simplify"->OptionValue["Simplify"],"Normalization"->OptionValue["Normalization"]|>; ret ] ] @@ -2603,7 +2633,8 @@ ret (*RadialSourcedAssociation["CO"][\[ScriptS]_,\[ScriptL]_,\[ScriptM]_,a_,r0Var_,{varPN_,order_}]/;NumericQ[r0Var]:=RadialSourcedAssociation["CO"][\[ScriptS],\[ScriptL],\[ScriptM],a,r0,{varPN,order}]/.r0->r0Var;*)*) -Options[TeukolskyPointParticleModePN]={"Normalization"->"Default","Simplify"->True} +(* ::Subsubsection:: *) +(*TeukolskyModePN*) TeukolskyModePN /: @@ -2620,7 +2651,9 @@ TeukolskyModePN /: BoxForm`SummaryItem[{"PN order: ", N[(order-1)/2]"PN"}] }], BoxForm`SummaryItem[{"Orbit: ", "Circular Equatorial"}]}; - extended = {BoxForm`SummaryItem[{"Simplify: ",assoc["Simplify"]}], + extended = { + BoxForm`SummaryItem[{"Min order: ",assoc["SeriesMinOrder"]}], + BoxForm`SummaryItem[{"Simplify: ",assoc["Simplify"]}], BoxForm`SummaryItem[{"Homogeneous Normalization: ",assoc["Normalization"]}]}; BoxForm`ArrangeSummaryBox[ @@ -2633,6 +2666,13 @@ TeukolskyModePN /: ]; +(* ::Subsubsection::Closed:: *) +(*TeukolskyPointParticleModePN*) + + +Options[TeukolskyPointParticleModePN]={"Normalization"->"Default","Simplify"->True} + + TeukolskyPointParticleModePN[\[ScriptS]_, \[ScriptL]_, \[ScriptM]_,orbit_KerrGeodesics`KerrGeoOrbit`KerrGeoOrbitFunction,{varPN_,order_},opt:OptionsPattern[]]:=Module[{aux,assoc,ret,a,r0Var,eccentricity,inclination}, {a,r0Var,eccentricity,inclination}=orbit[#]&/@{"a","p","e","Inclination"}; If[!(eccentricity===0),Message[TeukolskyPointParticleModePN::orbit];Abort[];]; @@ -2649,25 +2689,11 @@ TeukolskyPointParticleModePN[\[ScriptS], \[ScriptL], \[ScriptM],orbit,{varPN,aux ] -(* ::Subsubsection::Closed:: *) +(* ::Subsubsection:: *) (*Accessing functions and keys*) -TeukolskyRadialFunctionPN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_][y_String]:= - assoc[y]; - - -TeukolskyRadialFunctionPN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_][r_Symbol] :=assoc["RadialFunction"][r] -TeukolskyRadialFunctionPN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_][r_/;NumericQ[r]] :=assoc["RadialFunction"][r] - - -Derivative[n_Integer][trf_TeukolskyRadialFunctionPN][r_Symbol]:=trf[[6,1]]^(2 n) Derivative[n][trf["RadialFunction"]][r] - - -Keys[trfpn_TeukolskyRadialFunctionPN]^:=trfpn[[-1]]//Keys - - -TeukolskyModePN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_][y:(_String|("ExtendedHomogeneous"->"\[ScriptCapitalH]")|("ExtendedHomogeneous"->"\[ScriptCapitalI]"))]:= +TeukolskyModePN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_][y_String]/;!MemberQ[{"RadialFunction","ExtendedHomogeneous"->"\[ScriptCapitalH]","ExtendedHomogeneous"->"\[ScriptCapitalH]","Source"}, y]:= assoc[y]; @@ -2676,7 +2702,15 @@ TeukolskyModePN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_][r_Symbol] :=assoc["R TeukolskyModePN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_][r_/;NumericQ[r]] :=assoc["RadialFunction"][r] -Keys[trfpn_TeukolskyModePN]^:=trfpn[[-1]]//Keys +TeukolskyModePN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_]["ExtendedHomogeneous"->"\[ScriptCapitalH]"][r_Symbol] :=assoc["ExtendedHomogeneous"->"\[ScriptCapitalH]"][r] +TeukolskyModePN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_]["ExtendedHomogeneous"->"\[ScriptCapitalH]"][r_/;NumericQ[r]] :=assoc["ExtendedHomogeneous"->"\[ScriptCapitalH]"][r] +TeukolskyModePN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_]["ExtendedHomogeneous"->"\[ScriptCapitalI]"][r_Symbol] :=assoc["ExtendedHomogeneous"->"\[ScriptCapitalI]"][r] +TeukolskyModePN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_]["ExtendedHomogeneous"->"\[ScriptCapitalI]"][r_/;NumericQ[r]] :=assoc["ExtendedHomogeneous"->"\[ScriptCapitalI]"][r] +TeukolskyModePN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_]["Source"][r_Symbol] :=assoc["Source"][r] +TeukolskyModePN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_]["Source"][r_/;NumericQ[r]] :=assoc["Source"][r] + + +Keys[trfpn_TeukolskyModePN]^:= DeleteElements[Join[Keys[trfpn[[-1]]], {}], {"RadialFunction"}]; Derivative[n_Integer][tppm_TeukolskyModePN][r_Symbol]:=tppm[[6,1]]^(2 n) Derivative[n][tppm["RadialFunction"]][r] From dee8e9772f7efcae34db79a3047ada03055cb87d Mon Sep 17 00:00:00 2001 From: jakobneef Date: Tue, 28 Jan 2025 16:07:12 +0000 Subject: [PATCH 08/13] fixed CoefficientList --- Kernel/PN.wl | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/Kernel/PN.wl b/Kernel/PN.wl index 26d8c45..84a84b8 100644 --- a/Kernel/PN.wl +++ b/Kernel/PN.wl @@ -2575,12 +2575,10 @@ Keys[trfpn_TeukolskyRadialFunctionPN] ^:= DeleteElements[Join[Keys[trfpn[[-1]]], Options[RadialSourcedAssociation]={"Normalization"->"Default","Simplify"->True} -SeriesToSCoeffs[expr_SeriesData]:=Module[{aux}, -aux=expr[[3]]; -aux=aux/.Derivative[n_,0][SpinWeightedSpheroidalHarmonicS[a___]][b___]:>(SS)^n; -aux=aux/.SpinWeightedSpheroidalHarmonicS[a___][b___]:>(1)//Simplify; -aux=aux//CoefficientList[#,SS]&//Simplify; -aux=aux//.{}->Nothing; +SeriesToSCoeffs[series_SeriesData]:=Module[{aux}, +aux=series/.SpinWeightedSpheroidalHarmonicS[___]->SS;aux=Table[aux//Coefficient[#,Derivative[i,0][SS][\[Pi]/2,0]]&,{i,0,2}]; +aux=aux//Series[#,{series[[1]],series[[2]],SeriesMaxOrder[series]-1}]&; +aux=Association[{C["S"],C["S'"],C["S''"]}->aux//Thread]; aux ] @@ -2617,7 +2615,7 @@ radialF=innerF[#] HeavisideTheta[r0Var-#] + outerF[#] HeavisideTheta[#-r0Var]+de minOrder=radialF[r]//SeriesMinOrder; sourceF=source[r]/.\[CapitalOmega]Kerr->Inactive[KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][aVar,r0Var,0,1]["\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]/.r->#&; Scoeffs=SeriesToSCoeffs[radialF[r]]/.r->#&; -ret=<|"s"->\[ScriptS],"l"->\[ScriptL],"m"->\[ScriptM],"a"->aVar,"r0"->r0Var,"PN"->{varPN,order},"RadialFunction"->radialF,"CoefficientList"->Scoeffs,("ExtendedHomogeneous"->"\[ScriptCapitalI]")->innerF,("ExtendedHomogeneous"->"\[ScriptCapitalH]")->outerF,"\[Delta]"->deltaCoeff,"Amplitudes"->ampAssoc,"Wronskian"->wronskian,"Source"->sourceF,"SeriesMinOrder"->minOrder,"In"->aux["In"],"Up"->aux["Up"],"Simplify"->OptionValue["Simplify"],"Normalization"->OptionValue["Normalization"]|>; +ret=<|"s"->\[ScriptS],"l"->\[ScriptL],"m"->\[ScriptM],"a"->aVar,"r0"->r0Var,"PN"->{varPN,order},"RadialFunction"->radialF,"CoefficientList"->Scoeffs,("ExtendedHomogeneous"->"\[ScriptCapitalI]")->innerF,("ExtendedHomogeneous"->"\[ScriptCapitalH]")->outerF,"\[Delta]"->deltaCoeff,"Amplitudes"->ampAssoc,"Wronskian"->wronskian,"Source"->sourceF,"SeriesMinOrder"->minOrder,"RadialFunctions"->aux,"Simplify"->OptionValue["Simplify"],"Normalization"->OptionValue["Normalization"]|>; ret ] ] @@ -2693,7 +2691,7 @@ TeukolskyPointParticleModePN[\[ScriptS], \[ScriptL], \[ScriptM],orbit,{varPN,aux (*Accessing functions and keys*) -TeukolskyModePN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_][y_String]/;!MemberQ[{"RadialFunction","ExtendedHomogeneous"->"\[ScriptCapitalH]","ExtendedHomogeneous"->"\[ScriptCapitalH]","Source"}, y]:= +TeukolskyModePN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_][y_String]/;!MemberQ[{"RadialFunction","ExtendedHomogeneous"->"\[ScriptCapitalH]","ExtendedHomogeneous"->"\[ScriptCapitalH]","Source","CoefficientList"}, y]:= assoc[y]; @@ -2708,6 +2706,8 @@ TeukolskyModePN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_]["ExtendedHomogeneous TeukolskyModePN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_]["ExtendedHomogeneous"->"\[ScriptCapitalI]"][r_/;NumericQ[r]] :=assoc["ExtendedHomogeneous"->"\[ScriptCapitalI]"][r] TeukolskyModePN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_]["Source"][r_Symbol] :=assoc["Source"][r] TeukolskyModePN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_]["Source"][r_/;NumericQ[r]] :=assoc["Source"][r] +TeukolskyModePN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_]["CoefficientList"][r_Symbol] :=assoc["CoefficientList"][r] +TeukolskyModePN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_]["CoefficientList"][r_/;NumericQ[r]] :=assoc["CoefficientList"][r] Keys[trfpn_TeukolskyModePN]^:= DeleteElements[Join[Keys[trfpn[[-1]]], {}], {"RadialFunction"}]; From dd3d6a61280d97cc6ba73d705a3e0ddb7249a93d Mon Sep 17 00:00:00 2001 From: Barry Wardell Date: Tue, 28 Jan 2025 17:25:11 +0000 Subject: [PATCH 09/13] Fix problem with computing the derivative of a PN radial function --- Kernel/PN.wl | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Kernel/PN.wl b/Kernel/PN.wl index 84a84b8..c288bca 100644 --- a/Kernel/PN.wl +++ b/Kernel/PN.wl @@ -2558,7 +2558,7 @@ TeukolskyRadialFunctionPN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_][r_Symbol] TeukolskyRadialFunctionPN[s_, l_, m_, a_, r0_,{varPN_,order_},assoc_][r_/;NumericQ[r]] :=assoc["RadialFunction"][r] -Derivative[n_Integer][trf_TeukolskyRadialFunctionPN][r_Symbol]:=trf[[6,1]]^(2 n) Derivative[n][trf["RadialFunction"]][r] +Derivative[n_Integer][trf_TeukolskyRadialFunctionPN][r_Symbol]:=trf[[6,1]]^(2 n) Derivative[n][trf[[-1]]["RadialFunction"]][r] Keys[trfpn_TeukolskyRadialFunctionPN] ^:= DeleteElements[Join[Keys[trfpn[[-1]]], {}], {"RadialFunction","AmplitudesBool","TermCount"}]; From 498edb5cd4cc1a2f4b4e25f9cec54cf3210bc494 Mon Sep 17 00:00:00 2001 From: Barry Wardell Date: Tue, 28 Jan 2025 18:17:56 +0000 Subject: [PATCH 10/13] Tweak PN documentation --- .../Symbols/TeukolskyPointParticleModePN.nb | 3289 +++++++++-------- .../Symbols/TeukolskyRadialPN.nb | 1688 ++++----- 2 files changed, 2466 insertions(+), 2511 deletions(-) diff --git a/Documentation/English/ReferencePages/Symbols/TeukolskyPointParticleModePN.nb b/Documentation/English/ReferencePages/Symbols/TeukolskyPointParticleModePN.nb index 0d0b211..4cc75dc 100644 --- a/Documentation/English/ReferencePages/Symbols/TeukolskyPointParticleModePN.nb +++ b/Documentation/English/ReferencePages/Symbols/TeukolskyPointParticleModePN.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 227392, 5362] -NotebookOptionsPosition[ 213566, 5070] -NotebookOutlinePosition[ 214351, 5096] -CellTagsIndexPosition[ 214270, 5091] +NotebookDataLength[ 232472, 5421] +NotebookOptionsPosition[ 219695, 5151] +NotebookOutlinePosition[ 220474, 5177] +CellTagsIndexPosition[ 220393, 5172] WindowFrame->Normal*) (* Beginning of Notebook Content *) @@ -41,68 +41,93 @@ particle source." 3.942307934869256*^9, 3.942307958753386*^9}}, CellID->502739332,ExpressionUUID->"7fc170f0-fb0a-4785-bb5a-972c73accea2"], -Cell["\<\ -This Function computes a TeukolskyModePN[] for a given mode and orbit. As of \ -now it only supports circular equatorial orbits in Kerr. \ -\>", "Notes", +Cell[TextData[{ + "This Function produces a ", + Cell[BoxData[ + ButtonBox["TeukolskyModePN", + BaseStyle->"Link", + ButtonData->"paclet:Teukolsky/ref/TeukolskyModePN"]], "InlineFormula", + ExpressionUUID->"915101e2-6f6d-445f-8319-5596b59ac61e"], + " for a given mode and orbit. It currently only supports circular equatorial \ +orbits in Kerr. " +}], "Notes", CellChangeTimes->{{3.942308248878138*^9, 3.9423082558379707`*^9}, { - 3.942308287433778*^9, 3.942308330953247*^9}}, + 3.942308287433778*^9, 3.942308330953247*^9}, {3.9470745135848227`*^9, + 3.9470745312051783`*^9}, 3.9470750464485617`*^9}, CellID->308730800,ExpressionUUID->"2006ab6f-1bbf-439f-918e-919e9054ec76"], Cell[TextData[{ - "The main output is accessed by the Key \"RadialFuncition\" which gives ", + "The output represents a function of the form ", Cell[BoxData[ RowBox[{ - SubscriptBox["R", "\[ScriptL]\[ScriptM]\[Omega]"], "=", " ", + RowBox[{ + SubscriptBox["\[Psi]", "\[ScriptL]\[ScriptM]\[Omega]"], + RowBox[{"(", "r", ")"}]}], "=", " ", RowBox[{ RowBox[{ - SubscriptBox["c", "in"], " ", - SubscriptBox["R", "In"]}], "+", + SubsuperscriptBox["c", "\[ScriptL]\[ScriptM]\[Omega]", "In"], " ", + SubsuperscriptBox["R", "\[ScriptL]\[ScriptM]\[Omega]", "In"], + RowBox[{"(", "r", ")"}]}], "+", RowBox[{ - SubscriptBox["c", "up"], " ", - SubscriptBox["R", "In"]}]}]}]], "InlineFormula", - FormatType->StandardForm,ExpressionUUID-> + SubsuperscriptBox["c", "\[ScriptL]\[ScriptM]\[Omega]", "Up"], " ", + SubsuperscriptBox["R", "\[ScriptL]\[ScriptM]\[Omega]", "Up"], + RowBox[{"(", "r", ")"}]}]}]}]], "InlineFormula",ExpressionUUID-> "5f6cb3b1-e152-49f4-8768-a1cf725872b8"], " where ", Cell[BoxData[ RowBox[{ - SubscriptBox["c", "In"], "=", + SubsuperscriptBox["c", "\[ScriptL]\[ScriptM]\[Omega]", "In"], "=", RowBox[{ SubsuperscriptBox["\[Integral]", "r", "\[Infinity]"], RowBox[{ FractionBox[ RowBox[{ - SubscriptBox["R", "Up"], "S"}], "Wronskian"], - RowBox[{"\[DifferentialD]", "r"}]}]}]}]], "InlineFormula", - FormatType->StandardForm,ExpressionUUID-> + SubsuperscriptBox["R", "\[ScriptL]\[ScriptM]\[Omega]", "Up"], "S"}], + RowBox[{"W", "[", + RowBox[{ + SubsuperscriptBox["R", "\[ScriptL]\[ScriptM]\[Omega]", "In"], ",", + SubsuperscriptBox["R", "\[ScriptL]\[ScriptM]\[Omega]", "Up"]}], + "]"}]], + RowBox[{"\[DifferentialD]", "r"}]}]}]}]], "InlineFormula",ExpressionUUID-> "42fe0dd7-b2af-4d80-9b6d-7f1648eb4d21"], " and ", Cell[BoxData[ RowBox[{ - SubscriptBox["c", "Up"], "=", + SuperscriptBox["c", "Up"], "=", RowBox[{ - SubsuperscriptBox["\[Integral]", "0", "r"], + SubsuperscriptBox["\[Integral]", + SubscriptBox["r", "\[ScriptCapitalH]"], "r"], RowBox[{ FractionBox[ RowBox[{ - SubscriptBox["R", "In"], "S"}], "Wronskian"], - RowBox[{"\[DifferentialD]", "r"}]}]}]}]], "InlineFormula", - FormatType->StandardForm,ExpressionUUID-> + SubsuperscriptBox["R", "\[ScriptL]\[ScriptM]\[Omega]", "In"], "S"}], + RowBox[{"W", "[", + RowBox[{ + SubsuperscriptBox["R", "\[ScriptL]\[ScriptM]\[Omega]", "In"], ",", + SubsuperscriptBox["R", "\[ScriptL]\[ScriptM]\[Omega]", "Up"]}], + "]"}]], + RowBox[{"\[DifferentialD]", "r"}]}]}]}]], "InlineFormula",ExpressionUUID-> "8e15cf07-4e97-4cca-9829-df51585d360f"], ". In this case S is the Teukolsky point particle source and ", Cell[BoxData[ - SubscriptBox["R", "In"]], "InlineFormula", - FormatType->StandardForm,ExpressionUUID-> - "1e090fb3-e977-43eb-b64a-d598dd2af087"], + SubsuperscriptBox["R", "\[ScriptL]\[ScriptM]\[Omega]", "In"]], + "InlineFormula",ExpressionUUID->"1e090fb3-e977-43eb-b64a-d598dd2af087"], " and ", Cell[BoxData[ - SubscriptBox["R", "Up"]], "InlineFormula", - FormatType->StandardForm,ExpressionUUID-> - "a2b2281f-07c0-4745-9cd8-7b3959dd9bd0"], - " are solutions to the homogeneous Teukolsky. The homogeneous solutions are \ -generated by TeukolskyRadial[]" + SubscriptBox["R", + SubsuperscriptBox["R", "\[ScriptL]\[ScriptM]\[Omega]", "Up"]]], + "InlineFormula",ExpressionUUID->"a2b2281f-07c0-4745-9cd8-7b3959dd9bd0"], + " are solutions to the homogeneous Teukolsky equation generated by ", + Cell[BoxData[ + ButtonBox["TeukolskyRadial", + BaseStyle->"Link", + ButtonData->"paclet:Teukolsky/ref/TeukolskyRadial"]], "InlineFormula", + ExpressionUUID->"f46a0564-b652-4614-8bc7-7d3eae541b3a"], + "." }], "Notes", - CellChangeTimes->{{3.942308363282289*^9, 3.942308656941688*^9}}, + CellChangeTimes->{{3.942308363282289*^9, 3.942308656941688*^9}, { + 3.947074572261559*^9, 3.9470747330533657`*^9}, {3.9470749262168303`*^9, + 3.947075002645296*^9}, 3.947075050411436*^9}, CellID->509672919,ExpressionUUID->"9f61c655-7649-49e3-9121-c64f2678cbef"] }, Open ]], @@ -125,11 +150,30 @@ Cell[TextData[{ Cell[TextData[{ Cell[BoxData[ - TagBox[ - FrameBox["\<\"XXXX\"\>"], - "FunctionPlaceholder"]], "InlineSeeAlsoFunction", + ButtonBox["TeukolskyRadialPN", + BaseStyle->"Link", + ButtonData->"paclet:Teukolsky/ref/TeukolskyRadialPN"]], + "InlineSeeAlsoFunction", TaggingRules->{"PageType" -> "Function"},ExpressionUUID-> "b5198d57-55ab-44a9-86aa-ff44349e4640"], + StyleBox[" \[FilledVerySmallSquare] ", "InlineSeparator"], + Cell[BoxData[ + ButtonBox["TeukolskyPointParticleMode", + BaseStyle->"Link", + ButtonData->"paclet:Teukolsky/ref/TeukolskyPointParticleMode"]], + "InlineSeeAlsoFunction", + TaggingRules->{"PageType" -> "Function"}, + CellTags->"e62fbaaf-f773-4b82-b2c3-05cc8f9064ba",ExpressionUUID-> + "65f53e37-ca6b-4fea-97fb-8d91bf296dd2"], + StyleBox[" \[FilledVerySmallSquare] ", "InlineSeparator"], + Cell[BoxData[ + ButtonBox["TeukolskyRadial", + BaseStyle->"Link", + ButtonData->"paclet:Teukolsky/ref/TeukolskyRadial"]], + "InlineSeeAlsoFunction", + TaggingRules->{"PageType" -> "Function"}, + CellTags->"8c158b8e-603a-4953-8558-714b6eb7cb29",ExpressionUUID-> + "8dfeff03-ba3b-4757-a095-cd965f43beb3"], Cell[BoxData[ RowBox[{ Cell[TextData[StyleBox[ @@ -137,10 +181,10 @@ Cell[TextData[{ "83d1757e-96d2-44c4-8211-de196c886561"], DynamicModuleBox[{$CellContext`nbobj$$ = NotebookObject[ "4cdc53be-4474-4e7d-8aa9-8c4b6af9a759", - "ad63e4bb-636e-4942-b5b8-e2e163d40a61"], $CellContext`cellobj$$ = + "001cf066-157e-4934-bc86-d3341b673070"], $CellContext`cellobj$$ = CellObject[ "bfafe1ae-4382-474b-97c7-755897deb3ff", - "7fefafaa-83da-447f-ad9b-121d20996490"]}, + "f22649e9-2a18-44f0-9208-11837ba17fca"]}, TemplateBox[{ GraphicsBox[{{ Thickness[0.06], @@ -165,6 +209,7 @@ Cell[TextData[{ "InlineListingAddButton",ExpressionUUID-> "bfafe1ae-4382-474b-97c7-755897deb3ff"] }], "SeeAlso", + CellChangeTimes->{{3.947075060402141*^9, 3.947075087591421*^9}}, CellID->1678894427,ExpressionUUID->"c30537f5-4ea3-4221-8aac-aecd10168b9a"] }, Open ]], @@ -193,7 +238,10 @@ Cell[CellGroupData[{ Cell["Related Guides", "MoreAboutSection", CellID->1298664429,ExpressionUUID->"9c457c6a-6e77-4d9a-ba18-47c43a72c2d8"], -Cell["XXXX", "MoreAbout", +Cell[TextData[ButtonBox["Teukolsky", + BaseStyle->"Link", + ButtonData->"paclet:Teukolsky/guide/Teukolsky"]], "MoreAbout", + CellChangeTimes->{{3.947075091838523*^9, 3.947075096251676*^9}}, CellID->335865233,ExpressionUUID->"10ea345a-3b78-4901-9e3c-a03ecbb4e72d"] }, Open ]], @@ -237,11 +285,11 @@ Needs[\[Ellipsis]].", "MoreInfoText"], BaseStyle -> "IFrameBox"]], CellID->1635936226,ExpressionUUID->"10055f99-e284-42ce-8373-920c44836dbc"], Cell[BoxData[ - RowBox[{"Needs", "[", "\"\\"", - "]"}]], "ExampleInitialization", + RowBox[{"Needs", "[", "\"\\"", "]"}]], "ExampleInitialization", CellChangeTimes->{{3.942221953532112*^9, 3.9422219536727867`*^9}, { - 3.942222002531625*^9, 3.942222003209047*^9}, {3.942308886556801*^9, - 3.942308888306569*^9}}, + 3.942222002531625*^9, 3.942222003209047*^9}, {3.942308886556801*^9, + 3.942308888306569*^9}, 3.9470751042457733`*^9, {3.947075142055503*^9, + 3.9470751428229094`*^9}}, CellID->1211450235,ExpressionUUID->"3aa18ed7-d030-4d18-a84d-4f27b3801a37"] }, Open ]], @@ -272,8 +320,7 @@ Cell[BoxData[{ RowBox[{ RowBox[{"orbit", "=", RowBox[{"KerrGeoOrbit", "[", - RowBox[{"a", ",", "r0", ",", "0", ",", "1"}], "]"}]}], - ";"}], "\[IndentingNewLine]", + RowBox[{"a", ",", "r0", ",", "0", ",", "1"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"mode", "=", RowBox[{"TeukolskyPointParticleModePN", "[", RowBox[{ @@ -283,7 +330,7 @@ Cell[BoxData[{ CellChangeTimes->{{3.942221956582928*^9, 3.94222199381381*^9}, { 3.942222147391078*^9, 3.942222148633972*^9}, 3.946104722586904*^9, 3.946104814370336*^9}, - CellLabel->"In[34]:=", + CellLabel->"In[32]:=", CellID->103538757,ExpressionUUID->"3ed8b034-7711-4c9d-b4d6-76f204a012e2"], Cell[BoxData[ @@ -452,6 +499,11 @@ Cell[BoxData[ TagBox["\"Orbit: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox["\"Circular Equatorial\"", "SummaryItem"]}]}, { + RowBox[{ + TagBox["\"Min order: \"", "SummaryItemAnnotation"], + "\[InvisibleSpace]", + TagBox[ + RowBox[{"-", "2"}], "SummaryItem"]}]}, { RowBox[{ TagBox["\"Simplify: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", @@ -482,19 +534,19 @@ Cell[BoxData[ "s" -> -2, "l" -> 2, "m" -> 2, "a" -> $CellContext`a, "r0" -> $CellContext`r0, "PN" -> {$CellContext`\[Eta], 3}, "RadialFunction" -> ( - Teukolsky`PN`Private`innerF$239381[#] + Teukolsky`PN`Private`innerF$267608[#] HeavisideTheta[$CellContext`r0 - #] + - Teukolsky`PN`Private`outerF$239381[#] + Teukolsky`PN`Private`outerF$267608[#] HeavisideTheta[# - $CellContext`r0] + - Teukolsky`PN`Private`deltaCoeff$239381 - DiracDelta[# - $CellContext`r0]& ), "SCoefficients" -> (ReplaceAll[ + Teukolsky`PN`Private`deltaCoeff$267608 + DiracDelta[# - $CellContext`r0]& ), "CoefficientList" -> (ReplaceAll[ Teukolsky`PN`Private`SeriesToSCoeffs[ - Teukolsky`PN`Private`radialF$239381[Teukolsky`PN`Private`r]], + Teukolsky`PN`Private`radialF$267608[Teukolsky`PN`Private`r]], Teukolsky`PN`Private`r -> #]& ), ("ExtendedHomogeneous" -> "\[ScriptCapitalI]") -> (ReplaceAll[ - Teukolsky`PN`Private`inner$239381, Teukolsky`PN`Private`r -> #]& ), ( + Teukolsky`PN`Private`inner$267608, Teukolsky`PN`Private`r -> #]& ), ( "ExtendedHomogeneous" -> "\[ScriptCapitalH]") -> (ReplaceAll[ - Teukolsky`PN`Private`outer$239381, Teukolsky`PN`Private`r -> #]& ), + Teukolsky`PN`Private`outer$267608, Teukolsky`PN`Private`r -> #]& ), "\[Delta]" -> SeriesData[$CellContext`\[Eta], 0, {-Pi $CellContext`r0^(-3) @@ -860,88 +912,101 @@ CellContext`a, $CellContext`r0, 0, 1][ KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`a, \ $CellContext`r0, 0, 1][ "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^3}, 9, 12, - 1], "Source" -> ( - Teukolsky`PN`Private`TeukolskySourceCircularOrbit[-2, 2, - 2, $CellContext`a, {#, $CellContext`r0}, "Form" -> - "InvariantWronskian"]& ), "In" -> - Teukolsky`PN`TeukolskyRadialFunctionPN[-2, 2, 2, $CellContext`a, 2 - Inactive[KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ -CellContext`a, $CellContext`r0, 0, 1][ - "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"], \ + 1], "Source" -> (ReplaceAll[ + ReplaceAll[ + Teukolsky`PN`Private`source$267608[Teukolsky`PN`Private`r], + Teukolsky`PN`Private`\[CapitalOmega]Kerr -> + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`a, \ +$CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]], + Teukolsky`PN`Private`r -> #]& ), "SeriesMinOrder" -> -2, + "RadialFunctions" -> <| + "In" -> Teukolsky`PN`TeukolskyRadialFunctionPN[-2, 2, 2, $CellContext`a, + 2 Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`a, \ +$CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"], \ {$CellContext`\[Eta], 3}, <| - "s" -> -2, "l" -> 2, "m" -> 2, "a" -> $CellContext`a, - "PN" -> {$CellContext`\[Eta], 3}, "RadialFunction" -> - Function[Teukolsky`PN`Private`r, - SeriesData[$CellContext`\[Eta], 0, { - Rational[64, 5] Teukolsky`PN`Private`r^4 - Inactive[ - KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ -CellContext`a, $CellContext`r0, 0, 1][ - "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^4, - Complex[0, - Rational[256, 15]] Teukolsky`PN`Private`r^5 - Inactive[ - KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ + "s" -> -2, "l" -> 2, "m" -> 2, "a" -> $CellContext`a, + "PN" -> {$CellContext`\[Eta], 3}, "RadialFunction" -> + Function[Teukolsky`PN`Private`r, + + SeriesData[$CellContext`\[Eta], 0, { + Rational[64, 5] Teukolsky`PN`Private`r^4 + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ CellContext`a, $CellContext`r0, 0, 1][ - "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^5, - Rational[-128, 105] Teukolsky`PN`Private`r^3 - Inactive[ - KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^4, + Complex[0, + Rational[256, 15]] Teukolsky`PN`Private`r^5 + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ CellContext`a, $CellContext`r0, 0, 1][ - "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^4 (42 + - Complex[0, 21] $CellContext`a + - 11 Teukolsky`PN`Private`r^3 + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^5, + Rational[-128, 105] Teukolsky`PN`Private`r^3 Inactive[ KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ CellContext`a, $CellContext`r0, 0, 1][ - "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^2)}, - 4, 7, 1]], "BoundaryCondition" -> "In", "LeadingOrder" -> - Function[Teukolsky`PN`Private`r, - SeriesData[$CellContext`\[Eta], 0, { - Rational[64, 5] Teukolsky`PN`Private`r^4 - Inactive[ - KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^4 (42 + + Complex[0, 21] $CellContext`a + + 11 Teukolsky`PN`Private`r^3 + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ CellContext`a, $CellContext`r0, 0, 1][ - "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^4}, 4, - 5, 1]], "TermCount" -> 3, "Normalization" -> "Default", - "Amplitudes" -> <| - "Incidence" -> "Not Computed", "Transmission" -> "Not Computed", - "Reflection" -> "Not Computed"|>, "Simplify" -> True|>], "Up" -> - Teukolsky`PN`TeukolskyRadialFunctionPN[-2, 2, 2, $CellContext`a, 2 - Inactive[KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^2)}, + 4, 7, 1]], "BoundaryCondition" -> "In", "SeriesMinOrder" -> 4, + "LeadingOrder" -> Function[Teukolsky`PN`Private`r, + + SeriesData[$CellContext`\[Eta], 0, { + Rational[64, 5] Teukolsky`PN`Private`r^4 + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ CellContext`a, $CellContext`r0, 0, 1][ - "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"], \ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^4}, 4, + 5, 1]], "TermCount" -> 3, "Normalization" -> "Default", + "Amplitudes" -> <| + "Incidence" -> Missing["NotComputed"], "Transmission" -> + Missing["NotComputed"], "Reflection" -> Missing["NotComputed"]|>, + "Simplify" -> True, "AmplitudesBool" -> False|>], "Up" -> + Teukolsky`PN`TeukolskyRadialFunctionPN[-2, 2, 2, $CellContext`a, 2 + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`a, \ +$CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"], \ {$CellContext`\[Eta], 3}, <| - "s" -> -2, "l" -> 2, "m" -> 2, "a" -> $CellContext`a, - "PN" -> {$CellContext`\[Eta], 3}, "RadialFunction" -> - Function[Teukolsky`PN`Private`r, - SeriesData[$CellContext`\[Eta], 0, {Complex[0, - Rational[-3, 2]] Teukolsky`PN`Private`r^(-1)/Inactive[ - KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`\ -a, $CellContext`r0, 0, 1][ - "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"], -3, - Rational[1, 2] Teukolsky`PN`Private`r^(-2) - Inactive[ + "s" -> -2, "l" -> 2, "m" -> 2, "a" -> $CellContext`a, + "PN" -> {$CellContext`\[Eta], 3}, "RadialFunction" -> + Function[Teukolsky`PN`Private`r, + SeriesData[$CellContext`\[Eta], 0, {Complex[0, + Rational[-3, 2]] Teukolsky`PN`Private`r^(-1)/Inactive[ KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ CellContext`a, $CellContext`r0, 0, 1][ - "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^(-1) ( - Complex[0, -3] + 4 $CellContext`a + - Complex[0, 6] Teukolsky`PN`Private`r^3 + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"], -3, + Rational[1, 2] Teukolsky`PN`Private`r^(-2) Inactive[ KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ CellContext`a, $CellContext`r0, 0, 1][ - "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^2)}, \ --1, 2, 1]], "BoundaryCondition" -> "Up", "LeadingOrder" -> - Function[Teukolsky`PN`Private`r, - SeriesData[$CellContext`\[Eta], 0, {Complex[0, - Rational[-3, 2]] Teukolsky`PN`Private`r^(-1)/Inactive[ - KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`\ -a, $CellContext`r0, 0, 1][ - "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]}, -1, 0, - 1]], "TermCount" -> 3, "Normalization" -> "Default", - "Amplitudes" -> <| - "Incidence" -> "Not Computed", "Transmission" -> "Not Computed", - "Reflection" -> "Not Computed"|>, "Simplify" -> True|>], "Simplify" -> + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^(-1) ( + Complex[0, -3] + 4 $CellContext`a + + Complex[0, 6] Teukolsky`PN`Private`r^3 + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \ +\(\[Phi]\)]\)"]^2)}, -1, 2, 1]], "BoundaryCondition" -> "Up", + "SeriesMinOrder" -> -1, "LeadingOrder" -> + Function[Teukolsky`PN`Private`r, + SeriesData[$CellContext`\[Eta], 0, {Complex[0, + Rational[-3, 2]] Teukolsky`PN`Private`r^(-1)/Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]}, -1, 0, + 1]], "TermCount" -> 3, "Normalization" -> "Default", + "Amplitudes" -> <| + "Incidence" -> Missing["NotComputed"], "Transmission" -> + Missing["NotComputed"], "Reflection" -> Missing["NotComputed"]|>, + "Simplify" -> True, "AmplitudesBool" -> False|>]|>, "Simplify" -> True, "Normalization" -> "Default"|>], Editable->False, SelectWithContents->True, @@ -949,13 +1014,14 @@ a, $CellContext`r0, 0, 1][ CellChangeTimes->{{3.94222200433076*^9, 3.942222016566121*^9}, 3.942222158382213*^9, 3.946104634610979*^9, 3.9461047796987343`*^9, 3.9461048377087297`*^9, 3.94611005563437*^9, 3.946113940645184*^9, - 3.9461141370507*^9, 3.946986818632866*^9}, - CellLabel->"Out[35]=", - CellID->2116508496,ExpressionUUID->"e750d263-2bf2-4b6e-a0b7-366441a950c6"] + 3.9461141370507*^9, 3.946986818632866*^9, 3.947075161584251*^9}, + CellLabel->"Out[33]=", + CellID->920257162,ExpressionUUID->"4034adf7-4baf-434c-9b53-a4b30bb1767e"] }, Open ]], -Cell["The main use is now to use it as a Function[] ", "ExampleText", - CellChangeTimes->{{3.9422225202895*^9, 3.9422225372103558`*^9}}, +Cell["The main use is now to use it as a Function of radius:", "ExampleText", + CellChangeTimes->{{3.9422225202895*^9, 3.9422225372103558`*^9}, { + 3.947075122819428*^9, 3.947075128137085*^9}}, CellID->334635788,ExpressionUUID->"864cd375-bb7b-437b-8f73-52b511703b31"], Cell[CellGroupData[{ @@ -965,7 +1031,7 @@ Cell[BoxData[ RowBox[{"mode", "[", "r", "]"}], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.9422221535452967`*^9, 3.942222176310635*^9}, 3.94698681874024*^9}, - CellLabel->"In[36]:=", + CellLabel->"In[34]:=", CellID->1460972587,ExpressionUUID->"3432ad75-e4ce-45cb-83cb-35cba5c8b09c"], Cell[BoxData[ @@ -1501,8 +1567,7 @@ Cell[BoxData[ Derivative], MultilineFunction->None], "[", RowBox[{ - FractionBox["\[Pi]", "2"], ",", "0"}], "]"}]}]}], ")"}]}], - "+", + FractionBox["\[Pi]", "2"], ",", "0"}], "]"}]}]}], ")"}]}], "+", RowBox[{"168", " ", "r", " ", RowBox[{"(", RowBox[{ @@ -1806,8 +1871,7 @@ Cell[BoxData[ Derivative], MultilineFunction->None], "[", RowBox[{ - FractionBox["\[Pi]", "2"], ",", "0"}], "]"}]}]}], ")"}]}], - "+", + FractionBox["\[Pi]", "2"], ",", "0"}], "]"}]}]}], ")"}]}], "+", RowBox[{"168", " ", "r", " ", RowBox[{"(", RowBox[{ @@ -2285,23 +2349,26 @@ CellContext`a, $CellContext`r0, 0, 1][ CellChangeTimes->{{3.942222158467584*^9, 3.9422221766919403`*^9}, 3.946104634790491*^9, 3.94610477978925*^9, 3.946104837797861*^9, 3.946110055728364*^9, 3.9461139408392878`*^9, {3.9461141199141*^9, - 3.946114137291688*^9}, 3.946986819083281*^9}, - CellLabel->"Out[36]=", - CellID->714681410,ExpressionUUID->"23d024ed-1c76-4b13-92a8-a5f7fd6a4541"] + 3.946114137291688*^9}, 3.946986819083281*^9, 3.94707516744296*^9}, + CellLabel->"Out[34]=", + CellID->1351772274,ExpressionUUID->"35d76024-bb76-4b14-a38c-a1519a884dab"] }, Open ]], Cell[TextData[{ - "Note that by default the KerrGeoFrequencies are deactivated, i.e., ", + "Note that by default the KerrGeoFrequencies is deactivated, i.e., ", StyleBox["not ", FontSlant->"Italic"], - "PN expanded", - ". They can be activated using Activate[]. Note that this does not include \ -further PN scalings, so in Kerr the power counting will be off. The \ -SpinWeightedSpheroidalHarmonicS are also not PN expanded. " + "PN expanded. They can be activated using ", + ButtonBox["Activate", + BaseStyle->"Link", + ButtonData->"paclet:ref/Activate"], + ". Note that this does not include further PN scalings, so in Kerr the power \ +counting will be off. The SpinWeightedSpheroidalHarmonicS are also not PN \ +expanded. " }], "ExampleText", CellChangeTimes->{{3.946986880227598*^9, 3.946986900027214*^9}, { 3.946986931636042*^9, 3.946987049714376*^9}, {3.9469872105942507`*^9, - 3.946987225008074*^9}}, + 3.946987225008074*^9}, {3.947075212134715*^9, 3.9470752266871557`*^9}}, CellID->1026498030,ExpressionUUID->"ba97f58c-9fcf-423a-9864-2d8127361d67"], Cell[CellGroupData[{ @@ -2309,13 +2376,14 @@ Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ - RowBox[{"test", "[", "r", "]"}], "//", + RowBox[{"mode", "[", "r", "]"}], "//", RowBox[{ RowBox[{"SeriesCoefficient", "[", RowBox[{"#", ",", RowBox[{"-", "2"}]}], "]"}], "&"}]}], "//", "Activate"}]], "Input", - CellChangeTimes->{{3.94698705201591*^9, 3.94698708644901*^9}}, - CellLabel->"In[40]:=", + CellChangeTimes->{{3.94698705201591*^9, 3.94698708644901*^9}, { + 3.9470753050099087`*^9, 3.947075305567692*^9}}, + CellLabel->"In[37]:=", CellID->986513007,ExpressionUUID->"a32ed7c7-df48-4bd4-bae4-555faa161b8b"], Cell[BoxData[ @@ -2436,9 +2504,10 @@ Cell[BoxData[ RowBox[{ FractionBox["\[Pi]", "2"], ",", "0"}], "]"}]}], ")"}]}]}]}]], "Output",\ - CellChangeTimes->{{3.946987059901534*^9, 3.9469870868901*^9}}, - CellLabel->"Out[40]=", - CellID->1231142507,ExpressionUUID->"8ba86513-44fb-4445-91a0-27517f23bbe1"] + CellChangeTimes->{{3.946987059901534*^9, 3.9469870868901*^9}, + 3.947075306064691*^9}, + CellLabel->"Out[37]=", + CellID->1855279845,ExpressionUUID->"41e3b175-5ffc-4df5-8881-59f7a8e271d0"] }, Open ]], Cell["\<\ @@ -2456,7 +2525,7 @@ Cell[BoxData[ RowBox[{"-", "2"}], ",", "2", ",", "2", ",", "orbit", ",", RowBox[{"{", RowBox[{"\[Eta]", ",", "\"\<1PN\>\""}], "}"}]}], "]"}]], "Input", - CellLabel->"In[5]:=", + CellLabel->"In[35]:=", CellID->1255405975,ExpressionUUID->"47676817-e187-4bfa-8a88-2ab8a30da9cc"], Cell[BoxData[ @@ -2468,6 +2537,17 @@ Cell[BoxData[ TemplateBox[{ PaneSelectorBox[{False -> GridBox[{{ + PaneBox[ + ButtonBox[ + DynamicBox[ + FEPrivate`FrontEndResource["FEBitmaps", "SummaryBoxOpener"]], + ButtonFunction :> (Typeset`open$$ = True), Appearance -> None, + BaseStyle -> {}, Evaluator -> Automatic, Method -> + "Preemptive"], Alignment -> {Center, Center}, ImageSize -> + Dynamic[{ + Automatic, + 3.5 (CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[ + Magnification])}]], GraphicsBox[{ GrayLevel[0], DiskBox[{0, 0}, {1, 1}, {3.041592653589793, 6.383185307179586}], @@ -2506,8 +2586,10 @@ Cell[BoxData[ TagBox["\"\[Omega]: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox[ - RowBox[{"2", " ", "\[CapitalOmega]Kerr"}], - "SummaryItem"]}], "\" \"", + FractionBox["2", + RowBox[{"a", "+", + SuperscriptBox["r0", + RowBox[{"3", "/", "2"}]]}]], "SummaryItem"]}], "\" \"", RowBox[{ TagBox[ "\"\\!\\(\\*SubscriptBox[\\(r\\), \\(0\\)]\\): \"", @@ -2540,6 +2622,17 @@ Cell[BoxData[ GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, BaselinePosition -> {1, 1}], True -> GridBox[{{ + PaneBox[ + ButtonBox[ + DynamicBox[ + FEPrivate`FrontEndResource["FEBitmaps", "SummaryBoxCloser"]], + ButtonFunction :> (Typeset`open$$ = False), Appearance -> None, + BaseStyle -> {}, Evaluator -> Automatic, Method -> + "Preemptive"], Alignment -> {Center, Center}, ImageSize -> + Dynamic[{ + Automatic, + 3.5 (CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[ + Magnification])}]], GraphicsBox[{ GrayLevel[0], DiskBox[{0, 0}, {1, 1}, {3.041592653589793, 6.383185307179586}], @@ -2578,8 +2671,10 @@ Cell[BoxData[ TagBox["\"\[Omega]: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox[ - RowBox[{"2", " ", "\[CapitalOmega]Kerr"}], - "SummaryItem"]}], "\" \"", + FractionBox["2", + RowBox[{"a", "+", + SuperscriptBox["r0", + RowBox[{"3", "/", "2"}]]}]], "SummaryItem"]}], "\" \"", RowBox[{ TagBox[ "\"\\!\\(\\*SubscriptBox[\\(r\\), \\(0\\)]\\): \"", @@ -2598,7 +2693,21 @@ Cell[BoxData[ RowBox[{ TagBox["\"Orbit: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", - TagBox["\"Circular Equatorial\"", "SummaryItem"]}]}}, + TagBox["\"Circular Equatorial\"", "SummaryItem"]}]}, { + RowBox[{ + TagBox["\"Min order: \"", "SummaryItemAnnotation"], + "\[InvisibleSpace]", + TagBox[ + RowBox[{"-", "2"}], "SummaryItem"]}]}, { + RowBox[{ + TagBox["\"Simplify: \"", "SummaryItemAnnotation"], + "\[InvisibleSpace]", + TagBox["True", "SummaryItem"]}]}, { + RowBox[{ + TagBox[ + "\"Homogeneous Normalization: \"", "SummaryItemAnnotation"], + "\[InvisibleSpace]", + TagBox["\"Default\"", "SummaryItem"]}]}}, GridBoxAlignment -> { "Columns" -> {{Left}}, "Rows" -> {{Automatic}}}, AutoDelete -> False, GridBoxItemSize -> { @@ -2620,1076 +2729,723 @@ Cell[BoxData[ "s" -> -2, "l" -> 2, "m" -> 2, "a" -> $CellContext`a, "r0" -> $CellContext`r0, "PN" -> {$CellContext`\[Eta], 3}, "RadialFunction" -> ( - Teukolsky`PN`Private`innerF$54832[#] HeavisideTheta[$CellContext`r0 - #] + - Teukolsky`PN`Private`outerF$54832[#] HeavisideTheta[# - $CellContext`r0] + - Teukolsky`PN`Private`deltaCoeff$54832 - DiracDelta[# - $CellContext`r0]& ), ("ExtendedHomogeneous" -> - "\[ScriptCapitalI]") -> (Teukolsky`PN`Private`cIn$54832 - Teukolsky`PN`Private`Rin$54832[#]& ), ("ExtendedHomogeneous" -> - "\[ScriptCapitalH]") -> (Teukolsky`PN`Private`cUp$54832 - Teukolsky`PN`Private`Rup$54832[#]& ), "\[Delta]" -> + Teukolsky`PN`Private`innerF$321011[#] + HeavisideTheta[$CellContext`r0 - #] + + Teukolsky`PN`Private`outerF$321011[#] + HeavisideTheta[# - $CellContext`r0] + + Teukolsky`PN`Private`deltaCoeff$321011 + DiracDelta[# - $CellContext`r0]& ), "CoefficientList" -> (ReplaceAll[ + Teukolsky`PN`Private`SeriesToSCoeffs[ + Teukolsky`PN`Private`radialF$321011[Teukolsky`PN`Private`r]], + Teukolsky`PN`Private`r -> #]& ), ("ExtendedHomogeneous" -> + "\[ScriptCapitalI]") -> (ReplaceAll[ + Teukolsky`PN`Private`inner$321011, Teukolsky`PN`Private`r -> #]& ), ( + "ExtendedHomogeneous" -> "\[ScriptCapitalH]") -> (ReplaceAll[ + Teukolsky`PN`Private`outer$321011, Teukolsky`PN`Private`r -> #]& ), + "\[Delta]" -> SeriesData[$CellContext`\[Eta], 0, {-Pi $CellContext`r0^(-3) SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, - 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`a,\ + $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ Rational[1, 2] Pi, 0], 0, 2 $CellContext`a Pi $CellContext`r0^Rational[-7, 2] SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, - 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`a,\ + $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ Rational[1, 2] Pi, 0], 0, Rational[-1, 2] (7 + 2 $CellContext`a^2) Pi $CellContext`r0^(-4) SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, 2, - 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ + 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`a,\ + $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ Rational[1, 2] Pi, 0]}, 12, 17, 2], "Amplitudes" -> <| "\[ScriptCapitalI]" -> SeriesData[$CellContext`\[Eta], 0, {Complex[0, - Rational[-2, 15]] Pi $CellContext`r0^2 - Teukolsky`PN`\[CapitalOmega]Kerr ( - 2 SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] - 4 Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]), 0, Rational[8, 45] - Pi $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr ( - 2 (-3 + $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr) - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + (3 - 4 $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr) Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]), 0, Complex[0, - Rational[1, 315]] Pi $CellContext`r0 - Teukolsky`PN`\[CapitalOmega]Kerr ((378 + - Complex[0, -336] $CellContext`a - - 336 $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr + - 88 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2) - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - 4 (-105 + Complex[0, -21] $CellContext`a + - 42 $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr - 44 $CellContext`r0^3 - Teukolsky`PN`\[CapitalOmega]Kerr^2) Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + (105 + Complex[0, 84] $CellContext`a + - 44 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2) - Derivative[2, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0])}, -2, 4, 2], "\[ScriptCapitalH]" -> - SeriesData[$CellContext`\[Eta], 0, { - Rational[-1, 64] Pi $CellContext`r0^(-3) - Teukolsky`PN`\[CapitalOmega]Kerr^(-4) ( + Rational[-2, 15]] + Pi $CellContext`r0^2 ( 2 SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ +2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ Rational[1, 2] Pi, 0] - 4 Derivative[1, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ +2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ Rational[1, 2] Pi, 0] + Derivative[2, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]), 0, Complex[0, - Rational[1, 32]] Pi $CellContext`r0^Rational[-7, 2] - Teukolsky`PN`\[CapitalOmega]Kerr^(-4) ( - 2 (-3 + $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr) +2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0]) + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`a,\ + $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"], 0, + Rational[8, 45] Pi $CellContext`r0^Rational[3, 2] + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`a,\ + $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"] ((-6) SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + (3 - 4 $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr) Derivative[1, 0][ - - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr Derivative[2, 0][ + 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ + Rational[1, 2] Pi, 0] + 3 Derivative[1, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0]), 0, Rational[1, 384] - Pi $CellContext`r0^(-4) - Teukolsky`PN`\[CapitalOmega]Kerr^(-4) ((-6 + - Complex[0, 32] $CellContext`a - 48 $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr + - 24 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2) - SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-2, - 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr][ - Rational[1, 2] Pi, 0] + - 4 (15 + Complex[0, 2] $CellContext`a + - 6 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr - - 12 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2) - Derivative[1, 0][ - +2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0] + $CellContext`r0^Rational[3, 2] ( + 2 SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[\ +-2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ + Rational[1, 2] Pi, 0] - 4 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[\ +-2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0] + Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[\ +-2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0]) + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`\ +a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]), 0, + Rational[1, 315] Pi $CellContext`r0 + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`a,\ + $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"] ( + 21 (2 (Complex[0, 9] + 8 $CellContext`a) SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0] + (-15 + Complex[0, -8] $CellContext`a + - 12 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2) - Derivative[2, 0][ +2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ + Rational[1, 2] Pi, 0] + + 4 (Complex[0, -5] + $CellContext`a) Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`\ +SpinWeightedSpheroidalHarmonicS[-2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0] + (Complex[0, 5] - 4 $CellContext`a) + Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`\ +SpinWeightedSpheroidalHarmonicS[-2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0]) + + Complex[0, -168] $CellContext`r0^Rational[3, 2] ( + 2 SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[\ +-2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ + Rational[1, 2] Pi, 0] - Derivative[1, 0][ SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[-\ -2, 2, 2, 2 $CellContext`a Teukolsky`PN`\[CapitalOmega]Kerr]][ - Rational[1, 2] Pi, 0])}, -12, -6, 2]|>, "Wronskian" -> - SeriesData[$CellContext`\[Eta], 0, { - Complex[0, 96] Teukolsky`PN`\[CapitalOmega]Kerr^3}, 9, 12, 1], - "Source" -> ( - Teukolsky`PN`Private`TeukolskySourceCircularOrbit[-2, 2, - 2, $CellContext`a, {#, $CellContext`r0}, "Form" -> - "InvariantWronskian"]& ), "In" -> - Teukolsky`PN`TeukolskyRadialFunctionPN[-2, 2, 2, $CellContext`a, 2 - Teukolsky`PN`\[CapitalOmega]Kerr, {$CellContext`\[Eta], 3}, <| - "s" -> -2, "l" -> 2, "m" -> 2, "a" -> $CellContext`a, - "PN" -> {$CellContext`\[Eta], 3}, "RadialFunction" -> - Function[Teukolsky`PN`Private`r, - SeriesData[$CellContext`\[Eta], 0, { - Rational[64, 5] Teukolsky`PN`Private`r^4 - Teukolsky`PN`\[CapitalOmega]Kerr^4, Complex[0, - Rational[256, 15]] Teukolsky`PN`Private`r^5 - Teukolsky`PN`\[CapitalOmega]Kerr^5, Rational[-128, 105] - Teukolsky`PN`Private`r^3 - Teukolsky`PN`\[CapitalOmega]Kerr^4 (42 + - Complex[0, 21] $CellContext`a + - 11 Teukolsky`PN`Private`r^3 Teukolsky`PN`\[CapitalOmega]Kerr^2)}, - 4, 7, 1]], "BoundaryCondition" -> "In", "LeadingOrder" -> - Function[Teukolsky`PN`Private`r, - SeriesData[$CellContext`\[Eta], 0, { - Rational[64, 5] Teukolsky`PN`Private`r^4 - Teukolsky`PN`\[CapitalOmega]Kerr^4}, 4, 5, 1]], "TermCount" -> 3, - "Normalization" -> "Default", - "Amplitudes" -> <|"Transmission" -> "Not Computed"|>, "Simplified" -> - True|>], "Up" -> - Teukolsky`PN`TeukolskyRadialFunctionPN[-2, 2, 2, $CellContext`a, 2 - Teukolsky`PN`\[CapitalOmega]Kerr, {$CellContext`\[Eta], 3}, <| - "s" -> -2, "l" -> 2, "m" -> 2, "a" -> $CellContext`a, - "PN" -> {$CellContext`\[Eta], 3}, "RadialFunction" -> - Function[Teukolsky`PN`Private`r, - SeriesData[$CellContext`\[Eta], 0, {Complex[0, - Rational[-3, 2]] Teukolsky`PN`Private`r^(-1)/ - Teukolsky`PN`\[CapitalOmega]Kerr, -3, Rational[1, 2] - Teukolsky`PN`Private`r^(-2) - Teukolsky`PN`\[CapitalOmega]Kerr^(-1) (Complex[0, -3] + - 4 $CellContext`a + - Complex[0, 6] Teukolsky`PN`Private`r^3 - Teukolsky`PN`\[CapitalOmega]Kerr^2)}, -1, 2, 1]], - "BoundaryCondition" -> "Up", "LeadingOrder" -> - Function[Teukolsky`PN`Private`r, - SeriesData[$CellContext`\[Eta], 0, {Complex[0, - Rational[-3, 2]] Teukolsky`PN`Private`r^(-1)/ - Teukolsky`PN`\[CapitalOmega]Kerr}, -1, 0, 1]], "TermCount" -> 3, - "Normalization" -> "Default", - "Amplitudes" -> <|"Transmission" -> "Not Computed"|>, "Simplified" -> - True|>], "Simplified" -> True|>], - Editable->False, - SelectWithContents->True, - Selectable->False]], "Output", - CellChangeTimes->{3.946103993320241*^9, 3.946104644969741*^9, - 3.946104789864366*^9, 3.946104847705056*^9, 3.9461100653902884`*^9, - 3.946113953055612*^9, 3.94611414630647*^9}, - CellLabel->"Out[5]=", - CellID->1219125808,ExpressionUUID->"e1e31aa2-9c2f-4197-ae2b-d4722af5d437"] -}, Open ]], - -Cell["However we can extract a lot more from it", "ExampleText", - CellChangeTimes->{{3.942222569937722*^9, 3.9422225814378433`*^9}, { - 3.946987172131329*^9, 3.946987178016773*^9}}, - CellID->1440359689,ExpressionUUID->"a0164e99-140b-4343-89f0-cb2466a073e7"], - -Cell[CellGroupData[{ - -Cell[BoxData[ - RowBox[{"mode", "//", "Keys"}]], "Input", - CellChangeTimes->{{3.942222496725218*^9, 3.942222499178196*^9}}, - CellLabel->"In[4]:=", - CellID->600713610,ExpressionUUID->"cd2451b2-6582-4b09-b355-3a34583caef9"], - -Cell[BoxData[ - RowBox[{"{", - RowBox[{"\<\"s\"\>", ",", "\<\"l\"\>", ",", "\<\"m\"\>", ",", "\<\"a\"\>", - ",", "\<\"r0\"\>", ",", "\<\"PN\"\>", ",", "\<\"RadialFunction\"\>", ",", - RowBox[{"\<\"ExtendedHomogeneous\"\>", - "\[Rule]", "\<\"\[ScriptCapitalI]\"\>"}], ",", - RowBox[{"\<\"ExtendedHomogeneous\"\>", - "\[Rule]", "\<\"\[ScriptCapitalH]\"\>"}], ",", "\<\"\[Delta]\"\>", - ",", "\<\"Amplitudes\"\>", ",", "\<\"Wronskian\"\>", ",", "\<\"Source\"\>", - ",", "\<\"In\"\>", ",", "\<\"Up\"\>", ",", "\<\"Simplified\"\>"}], - "}"}]], "Output", - CellChangeTimes->{3.942222499430403*^9, 3.9423091918116503`*^9, - 3.946104634822709*^9, 3.946104779809112*^9, 3.946104837827648*^9, - 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"ac0dec70-1428-454e-9f36-4bf89982db6b"], - $Line = 0; Null]], "ExampleSection", - CellID->1512661770,ExpressionUUID->"57c3a8c5-4f1c-4c78-9e9e-2b52d024a06f"], - -Cell["\<\ -Here we want to display the possible Keys that can be used to query a \ -TeukolskyModePN (we set a=0 here to improve the readability of the output)\ -\>", "ExampleText", - CellChangeTimes->{{3.942308774836884*^9, 3.942308779609502*^9}, { - 3.942308822367017*^9, 3.942308846368334*^9}, {3.946104822516281*^9, - 3.946104822735633*^9}}, - CellID->214174106,ExpressionUUID->"01d5f544-7788-4a34-bfce-b0fe3f48d890"], - -Cell[BoxData[{ - RowBox[{ - RowBox[{"orbit", "=", - RowBox[{"KerrGeoOrbit", "[", - RowBox[{"0", ",", "r0", ",", "0", ",", "1"}], "]"}]}], - ";"}], "\[IndentingNewLine]", - RowBox[{ - RowBox[{"mode", "=", - RowBox[{"TeukolskyPointParticleModePN", "[", - RowBox[{ - RowBox[{"-", "2"}], ",", "2", ",", "2", ",", "orbit", ",", - RowBox[{"{", - RowBox[{"\[Eta]", ",", "3"}], "}"}]}], "]"}]}], ";"}]}], "Input", - CellChangeTimes->{3.94230876201133*^9, 3.946104810927847*^9}, - CellLabel->"In[1]:=", - CellID->51974984,ExpressionUUID->"07e74c9d-c70c-49c1-8a3e-17290b6b2191"], - -Cell[CellGroupData[{ - -Cell[BoxData[ - RowBox[{"mode", "//", "Keys"}]], "Input", - CellLabel->"In[3]:=", - CellID->921934977,ExpressionUUID->"0f0a7412-ec66-4cb7-8f17-1ccf4d894668"], - -Cell[BoxData[ - RowBox[{"{", - RowBox[{"\<\"s\"\>", ",", "\<\"l\"\>", ",", "\<\"m\"\>", ",", "\<\"a\"\>", - ",", "\<\"r0\"\>", ",", "\<\"PN\"\>", ",", "\<\"RadialFunction\"\>", ",", - RowBox[{"\<\"ExtendedHomogeneous\"\>", - "\[Rule]", "\<\"\[ScriptCapitalI]\"\>"}], ",", - RowBox[{"\<\"ExtendedHomogeneous\"\>", - "\[Rule]", "\<\"\[ScriptCapitalH]\"\>"}], ",", "\<\"\[Delta]\"\>", - ",", "\<\"Amplitudes\"\>", ",", "\<\"Wronskian\"\>", ",", "\<\"Source\"\>", - ",", "\<\"In\"\>", ",", "\<\"Up\"\>", ",", "\<\"Simplified\"\>"}], - "}"}]], "Output", - CellChangeTimes->{{3.9423088493082743`*^9, 3.942308873691544*^9}, - 3.942308960743535*^9, 3.946104655186612*^9, 3.946104799859556*^9, - 3.946104858134309*^9, 3.946110075141685*^9, 3.946113964908204*^9, - 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Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[\ +-2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0]) + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`\ +a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"] + + 12 $CellContext`r0^3 ( + 2 SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[\ +-2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]][ + Rational[1, 2] Pi, 0] - 4 Derivative[1, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[\ +-2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0] + Derivative[2, 0][ + + SpinWeightedSpheroidalHarmonics`SpinWeightedSpheroidalHarmonicS[\ +-2, 2, 2, 2 $CellContext`a + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]]][ + Rational[1, 2] Pi, 0]) + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^2)}, \ +-12, -6, 2]|>, "Wronskian" -> + SeriesData[$CellContext`\[Eta], 0, { + Complex[0, 96] + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`a, \ +$CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^3}, 9, 12, + 1], "Source" -> (ReplaceAll[ + ReplaceAll[ + Teukolsky`PN`Private`source$321011[Teukolsky`PN`Private`r], + Teukolsky`PN`Private`\[CapitalOmega]Kerr -> + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`a, \ +$CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]], + Teukolsky`PN`Private`r -> #]& ), "SeriesMinOrder" -> -2, + "RadialFunctions" -> <| + "In" -> Teukolsky`PN`TeukolskyRadialFunctionPN[-2, 2, 2, $CellContext`a, + 2 Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$CellContext`a, \ +$CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"], \ +{$CellContext`\[Eta], 3}, <| + "s" -> -2, "l" -> 2, "m" -> 2, "a" -> $CellContext`a, + "PN" -> {$CellContext`\[Eta], 3}, "RadialFunction" -> + Function[Teukolsky`PN`Private`r, + + SeriesData[$CellContext`\[Eta], 0, { + Rational[64, 5] Teukolsky`PN`Private`r^4 + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][$\ +CellContext`a, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^4, + Complex[0, + 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CellLabel->"Out[41]=", + CellID->604703027,ExpressionUUID->"5bcead93-b8ff-4662-8e58-5f957e0bb71f"] }, Open ]], Cell["\<\ -Note that the blank output of \"RadialFunction\" looks weird but is \ -operational. This is due to the fact that it is a Function[] object \ -internally that will not be evaluated unless given an explicit variable.\ +\"s\", \"l\", and \"m\" give the mode values. \"a\" and \"r0\" give the Kerr \ +spin parameter and the particles radius respectively. \"PN\" gives the \ +expansion parameter and the number of terms. \ \>", "ExampleText", - CellChangeTimes->{{3.9423096373480053`*^9, 3.9423097501991253`*^9}}, - CellID->1605161360,ExpressionUUID->"9e0f866e-55a7-49f8-aaae-dce309c4b4ad"], + CellChangeTimes->{{3.9423091876035557`*^9, 3.942309300576494*^9}, { + 3.94230935001857*^9, 3.942309413828618*^9}}, + CellID->1057982257,ExpressionUUID->"0721f743-8a90-4335-9c9c-cc3d54b19b1b"], Cell[CellGroupData[{ Cell[BoxData[ - RowBox[{"rafu", "=", - RowBox[{"mode", "[", "\"\\"", "]"}]}]], "Input", - CellChangeTimes->{{3.942309752143576*^9, 3.942309762792602*^9}}, - CellLabel->"In[10]:=", - CellID->566631531,ExpressionUUID->"2ae0a031-bb99-4836-b23b-42ba66156178"], + RowBox[{ + RowBox[{ + RowBox[{"mode", "[", "#", "]"}], "&"}], "/@", + RowBox[{"{", + RowBox[{"\"\\"", ",", "\"\\"", ",", "\"\\""}], + "}"}]}]], "Input", + CellChangeTimes->{{3.942309303978402*^9, 3.942309329641718*^9}}, + CellLabel->"In[42]:=", + CellID->2068679207,ExpressionUUID->"0a02f4f3-c7c5-48bb-b672-7483b592c56a"], + +Cell[BoxData[ + RowBox[{"{", + RowBox[{ + RowBox[{"-", "2"}], ",", "2", ",", "2"}], "}"}]], "Output", + CellChangeTimes->{{3.9423093042864017`*^9, 3.942309330314762*^9}, + 3.946104655222256*^9, 3.946104799886143*^9, 3.946104858165707*^9, + 3.946110075159411*^9, 3.946113964941649*^9, 3.946114156030138*^9, + 3.947075363529108*^9}, + CellLabel->"Out[42]=", + CellID->1178210196,ExpressionUUID->"5b75dc66-df93-4c7a-a340-1ad53bfb1263"] +}, Open ]], + +Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ + RowBox[{ + RowBox[{"mode", "[", "#", "]"}], "&"}], "/@", + RowBox[{"{", + RowBox[{"\"\\"", ",", "\"\\""}], "}"}]}]], "Input", + CellChangeTimes->{{3.942309338141388*^9, 3.942309341596808*^9}}, + CellLabel->"In[43]:=", + CellID->351510573,ExpressionUUID->"38dc0fbd-b77d-493e-be7c-8f2c8d90a7b0"], + +Cell[BoxData[ + RowBox[{"{", + RowBox[{"0", ",", "r0"}], "}"}]], "Output", + CellChangeTimes->{3.942309341926709*^9, 3.946104655255844*^9, + 3.9461047999154058`*^9, 3.946104858195342*^9, 3.946110075176362*^9, + 3.9461139649737587`*^9, 3.9461141560613413`*^9, 3.9470753644117107`*^9}, + CellLabel->"Out[43]=", + CellID->459068503,ExpressionUUID->"e147b481-4a45-4b85-8af6-01ee5711be2e"] +}, Open ]], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"mode", "[", "\"\\"", "]"}]], "Input", + CellChangeTimes->{{3.942309385749609*^9, 3.9423093935755873`*^9}}, + CellLabel->"In[44]:=", + CellID->960309762,ExpressionUUID->"6295dc6d-2bce-496c-9935-8fcb719c91aa"], + +Cell[BoxData[ + RowBox[{"{", + RowBox[{"\[Eta]", ",", "3"}], "}"}]], "Output", + CellChangeTimes->{3.942309394027725*^9, 3.946104655286149*^9, + 3.946104799939814*^9, 3.946104858224215*^9, 3.946110075193413*^9, + 3.946113965001844*^9, 3.9461141560905323`*^9, 3.947075365097116*^9}, + CellLabel->"Out[44]=", + CellID->271120064,ExpressionUUID->"fb43f8d9-7005-47ae-bbbe-ada7ea293ca9"] +}, Open ]], + +Cell[TextData[{ + "Querying with an arbitrary symbol or numeric number causes produces the \ +radial function ", + Cell[BoxData[ RowBox[{ RowBox[{ - RowBox[{"Teukolsky`PN`Private`innerF$103743", "[", "#1", "]"}], " ", - RowBox[{"HeavisideTheta", "[", - RowBox[{"r0", "-", "#1"}], "]"}]}], "+", + SubscriptBox["\[Psi]", "\[ScriptL]\[ScriptM]\[Omega]"], + RowBox[{"(", "r", ")"}]}], "=", " ", RowBox[{ - RowBox[{"Teukolsky`PN`Private`outerF$103743", "[", "#1", "]"}], " ", - RowBox[{"HeavisideTheta", "[", - RowBox[{"#1", "-", "r0"}], "]"}]}], "+", - RowBox[{"Teukolsky`PN`Private`deltaCoeff$103743", " ", - RowBox[{"DiracDelta", "[", - RowBox[{"#1", "-", "r0"}], "]"}]}]}], "&"}]], "Output", - CellChangeTimes->{{3.942309756979131*^9, 3.942309763973527*^9}, - 3.946104655761898*^9, 3.946104800369116*^9, 3.9461048583450136`*^9, - 3.9461100752869477`*^9, 3.946113965205282*^9, 3.94611415623333*^9}, - CellLabel->"Out[10]=", - CellID->2134043283,ExpressionUUID->"6727bb2b-8dd0-41e7-821c-c9ec901527cf"] -}, Open ]], + RowBox[{ + SubsuperscriptBox["c", "\[ScriptL]\[ScriptM]\[Omega]", "In"], " ", + SubsuperscriptBox["R", "\[ScriptL]\[ScriptM]\[Omega]", "In"], + RowBox[{"(", "r", ")"}]}], "+", + RowBox[{ + SubsuperscriptBox["c", "\[ScriptL]\[ScriptM]\[Omega]", "Up"], " ", + SubsuperscriptBox["R", "\[ScriptL]\[ScriptM]\[Omega]", "Up"], + RowBox[{"(", "r", ")"}]}]}]}]], "InlineFormula",ExpressionUUID-> + "4df3ff7b-c94d-4e0d-90dc-f618ab8255f6"], + "." +}], "ExampleText", + CellChangeTimes->{{3.942309425140519*^9, 3.942309473152443*^9}, { + 3.947075382405175*^9, 3.947075423815789*^9}}, + CellID->617540728,ExpressionUUID->"d73dc925-8c83-4803-93b0-fb958364a7d1"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ - 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RowBox[{"7", "/", "2"}]], " ", "\[Eta]"}]], + RowBox[{"7", "/", "2"}]], " ", + SqrtBox[ + SuperscriptBox["r0", "5"]], " ", "\[Eta]"}]], RowBox[{"\[ImaginaryI]", " ", SqrtBox[ FractionBox["\[Pi]", "5"]], " ", RowBox[{"(", RowBox[{ RowBox[{ - SuperscriptBox["r0", "5"], " ", RowBox[{"(", RowBox[{ - RowBox[{"-", "2"}], "-", RowBox[{"3", " ", "r", " ", - SqrtBox["r0"], " ", "\[CapitalOmega]Kerr"}], "+", + SuperscriptBox["r0", + RowBox[{"13", "/", "2"}]]}], "-", RowBox[{"2", " ", SuperscriptBox["r0", - RowBox[{"3", "/", "2"}]], " ", "\[CapitalOmega]Kerr"}]}], ")"}], - " ", + RowBox[{"15", "/", "2"}]]}], "+", + RowBox[{"2", " ", + SuperscriptBox[ + RowBox[{"(", + SuperscriptBox["r0", "5"], ")"}], + RowBox[{"3", "/", "2"}]]}]}], ")"}], " ", RowBox[{"HeavisideTheta", "[", RowBox[{"r", "-", "r0"}], "]"}]}], "+", RowBox[{ SuperscriptBox["r", "5"], " ", RowBox[{"(", - RowBox[{"3", "+", - RowBox[{"2", " ", "r", " ", - SqrtBox["r0"], " ", "\[CapitalOmega]Kerr"}], "-", + RowBox[{ + RowBox[{ + RowBox[{"-", "2"}], " ", "r", " ", + SuperscriptBox["r0", + RowBox[{"3", "/", "2"}]]}], "+", RowBox[{"3", " ", SuperscriptBox["r0", - RowBox[{"3", "/", "2"}]], " ", "\[CapitalOmega]Kerr"}]}], ")"}], - " ", + RowBox[{"5", "/", "2"}]]}], "-", + RowBox[{"3", " ", + SqrtBox[ + SuperscriptBox["r0", "5"]]}]}], ")"}], " ", RowBox[{"HeavisideTheta", "[", RowBox[{ RowBox[{"-", "r"}], "+", "r0"}], "]"}]}]}], ")"}]}]}], "+", @@ -3767,62 +3532,60 @@ Cell[BoxData[ SuperscriptBox["r0", "5"], " ", RowBox[{"(", RowBox[{ - RowBox[{ - RowBox[{"-", "42"}], " ", "r0"}], "+", - RowBox[{"84", " ", - SuperscriptBox["r", "3"], " ", "r0", " ", - SuperscriptBox["\[CapitalOmega]Kerr", "2"]}], "-", - RowBox[{"112", " ", - SuperscriptBox["r", "2"], " ", - SqrtBox["r0"], " ", "\[CapitalOmega]Kerr", " ", + RowBox[{"7", " ", RowBox[{"(", RowBox[{ - RowBox[{"-", "1"}], "+", + RowBox[{"17", " ", "r"}], "-", + RowBox[{"6", " ", "r0"}]}], ")"}]}], "-", + FractionBox[ + RowBox[{"56", " ", "r", " ", + SuperscriptBox["r0", + RowBox[{"3", "/", "2"}]], " ", + RowBox[{"(", RowBox[{ - SuperscriptBox["r0", - RowBox[{"3", "/", "2"}]], " ", "\[CapitalOmega]Kerr"}]}], - 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$CellContext`r0] + $CellContext`r^5 HeavisideTheta[-$CellContext`r + $CellContext`r0]), 0, - Complex[0, -1] (Rational[1, 5] Pi)^ + Complex[0, 1] (Rational[1, 5] Pi)^ Rational[1, 2] $CellContext`r^(-1) $CellContext`r0^ - Rational[-7, 2] ($CellContext`r0^5 (-2 - - 3 $CellContext`r $CellContext`r0^Rational[1, 2] - Teukolsky`PN`\[CapitalOmega]Kerr + - 2 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr) - HeavisideTheta[$CellContext`r - $CellContext`r0] + $CellContext`r^5 (3 + - 2 $CellContext`r $CellContext`r0^Rational[1, 2] - Teukolsky`PN`\[CapitalOmega]Kerr - 3 $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr) + Rational[-7, 2] ($CellContext`r0^5)^ + Rational[-1, 2] ((3 $CellContext`r $CellContext`r0^Rational[13, 2] - + 2 $CellContext`r0^Rational[15, 2] + + 2 ($CellContext`r0^5)^Rational[3, 2]) + HeavisideTheta[$CellContext`r - $CellContext`r0] + $CellContext`r^5 \ +((-2) $CellContext`r $CellContext`r0^Rational[3, 2] + + 3 $CellContext`r0^Rational[5, 2] - 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Rational[1, 2] $CellContext`r^(-1) $CellContext`r0^Rational[3, 2] (2 + - 3 $CellContext`r $CellContext`r0^Rational[1, 2] - Teukolsky`PN`\[CapitalOmega]Kerr - 2 $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr), 0, + Rational[1, 2] $CellContext`r^(-1) $CellContext`r0^Rational[3, 2] ( + 2 + (3 $CellContext`r - 2 $CellContext`r0) $CellContext`r0^ + Rational[3, 2] ($CellContext`r0^5)^Rational[-1, 2]), 0, Rational[1, 28] (Rational[1, 5] Pi)^ - Rational[1, - 2] $CellContext`r^(-2) $CellContext`r0 ((-42) $CellContext`r0 + - 84 $CellContext`r^3 $CellContext`r0 Teukolsky`PN`\[CapitalOmega]Kerr^2 - - 112 $CellContext`r^2 $CellContext`r0^Rational[1, 2] - Teukolsky`PN`\[CapitalOmega]Kerr (-1 + $CellContext`r0^Rational[3, 2] - Teukolsky`PN`\[CapitalOmega]Kerr) + $CellContext`r (119 - - 56 $CellContext`r0^Rational[3, 2] Teukolsky`PN`\[CapitalOmega]Kerr + - 44 $CellContext`r0^3 Teukolsky`PN`\[CapitalOmega]Kerr^2))}, -4, 2, 2], + Rational[1, 2] $CellContext`r^(-2) $CellContext`r0 ( + 7 (17 $CellContext`r - 6 $CellContext`r0) - + 56 $CellContext`r $CellContext`r0^Rational[3, 2] ($CellContext`r0^5)^ + Rational[-1, 2] ((-2) $CellContext`r + $CellContext`r0) + + 4 $CellContext`r $CellContext`r0^(-2) (21 $CellContext`r^2 - + 28 $CellContext`r $CellContext`r0 + 11 $CellContext`r0^2))}, -4, 2, 2], + Editable->False]], "Output", CellChangeTimes->{{3.942310190515919*^9, 3.942310194542695*^9}, 3.946104655890276*^9, 3.94610480049261*^9, 3.946104858420904*^9, 3.946110075344924*^9, 3.946110156182564*^9, 3.946113965340487*^9, - 3.9461141563778133`*^9}, - CellLabel->"Out[13]=", - CellID->1954445087,ExpressionUUID->"4dbbac21-e97a-49dd-a166-439c106e6dcd"] + 3.9461141563778133`*^9, 3.947075541328116*^9}, + CellLabel->"Out[51]=", + CellID->2111808784,ExpressionUUID->"711d1fe5-1dc2-45cd-84dc-d6520c0aa2e2"] }, Open ]], Cell["\<\ \"\[Delta]\" gives the contribution on the worldline. The full solution can \ -be pieced together from \[LineSeparator]mode[\"RadialFunction\"] = \ -HeavisideTheta[r-r0] mode[\"ExtendedHomogeneous\" \[Rule] \"\[ScriptCapitalI]\ -\"]+HeavisideTheta[r0-r] mode[\"ExtendedHomogeneous\" \[Rule] \"\ -\[ScriptCapitalH]\"]+DiracDelta[r-r0] mode[\"\[Delta]\"]\ +be pieced together from \[LineSeparator]mode[r] = HeavisideTheta[r-r0] \ +mode[\"ExtendedHomogeneous\" \[Rule] \ +\"\[ScriptCapitalI]\"][r]+HeavisideTheta[r0-r] mode[\"ExtendedHomogeneous\" \ +\[Rule] \"\[ScriptCapitalH]\"][r]+DiracDelta[r-r0] mode[\"\[Delta]\"]\ \>", "ExampleText", - CellChangeTimes->{{3.942310233174951*^9, 3.9423104174852247`*^9}}, + CellChangeTimes->{{3.942310233174951*^9, 3.9423104174852247`*^9}, { + 3.947075556004992*^9, 3.9470755791568727`*^9}}, CellID->1000349281,ExpressionUUID->"16e0e578-12be-4417-a68a-d18ad72e0e7e"], Cell[CellGroupData[{ @@ -4137,7 +4080,7 @@ Cell[BoxData[ RowBox[{"mode", "[", "\"\<\[Delta]\>\"", "]"}], "//", "Simplify"}]], "Input",\ CellChangeTimes->{{3.94231043256621*^9, 3.942310437955163*^9}}, - CellLabel->"In[14]:=", + CellLabel->"In[52]:=", CellID->570692951,ExpressionUUID->"a8a843eb-48f4-457a-8b13-a2ce5d7d1d13"], Cell[BoxData[ @@ -4170,9 +4113,10 @@ Cell[BoxData[ Editable->False]], "Output", CellChangeTimes->{{3.942310434641239*^9, 3.942310438209908*^9}, 3.946104655930561*^9, 3.946104800527301*^9, 3.946104858445613*^9, - 3.946110075362335*^9, 3.946113965367964*^9, 3.9461141564096727`*^9}, - CellLabel->"Out[14]=", - CellID->474113964,ExpressionUUID->"d66c1bd4-b8fe-4d49-a11c-63dac67180f1"] + 3.946110075362335*^9, 3.946113965367964*^9, 3.9461141564096727`*^9, + 3.9470755660423326`*^9}, + CellLabel->"Out[52]=", + CellID->402144827,ExpressionUUID->"6d7b51ee-e04a-4642-a55c-0f08cc2218fe"] }, Open ]], Cell[TextData[{ @@ -4201,48 +4145,60 @@ Cell[TextData[{ Cell[CellGroupData[{ Cell[BoxData[ - RowBox[{"mode", "[", "\"\\"", "]"}]], "Input", - CellChangeTimes->{{3.942310590572832*^9, 3.9423105969431887`*^9}}, - CellLabel->"In[15]:=", + RowBox[{ + RowBox[{"mode", "[", "\"\\"", "]"}], "//", + "Activate"}]], "Input", + CellChangeTimes->{{3.942310590572832*^9, 3.9423105969431887`*^9}, { + 3.947075587818475*^9, 3.9470755892007637`*^9}}, + CellLabel->"In[54]:=", CellID->1179931945,ExpressionUUID->"0c11346b-e974-47a1-af84-ed0adde5f6cc"], Cell[BoxData[ InterpretationBox[ RowBox[{ - RowBox[{"96", " ", "\[ImaginaryI]", " ", - SuperscriptBox["\[CapitalOmega]Kerr", "3"], " ", - SuperscriptBox["\[Eta]", "9"]}], "+", + FractionBox[ + RowBox[{"96", " ", "\[ImaginaryI]", " ", + SuperscriptBox["r0", "3"], " ", + SuperscriptBox["\[Eta]", "9"]}], + SuperscriptBox[ + RowBox[{"(", + SuperscriptBox["r0", "5"], ")"}], + RowBox[{"3", "/", "2"}]]], "+", InterpretationBox[ SuperscriptBox[ RowBox[{"O", "[", "\[Eta]", "]"}], "12"], SeriesData[$CellContext`\[Eta], 0, {}, 9, 12, 1], Editable->False]}], SeriesData[$CellContext`\[Eta], 0, { - Complex[0, 96] Teukolsky`PN`\[CapitalOmega]Kerr^3}, 9, 12, 1], + Complex[0, 96] $CellContext`r0^3 ($CellContext`r0^5)^Rational[-3, 2]}, 9, + 12, 1], Editable->False]], "Output", CellChangeTimes->{3.94231059720576*^9, 3.946104655957692*^9, 3.946104800553637*^9, 3.946104858465288*^9, 3.946110075381483*^9, - 3.946113965390458*^9, 3.946114156437705*^9}, - CellLabel->"Out[15]=", - CellID->559435070,ExpressionUUID->"6451be53-f5fe-4c52-aba8-a94f9d59c839"] + 3.946113965390458*^9, 3.946114156437705*^9, 3.947075589795309*^9}, + CellLabel->"Out[54]=", + CellID->222719337,ExpressionUUID->"58012f39-eebd-4fec-a80e-74bc26aceb9e"] }, Open ]], Cell["\<\ \"Source\" gives the source term on the right hand side of \[ScriptCapitalO] \ -R = S. It is again a Function and thus plagued by the same aesthetic issue as \ -\"RadialFunction\".\ +R = S.\ \>", "ExampleText", - CellChangeTimes->{{3.942310735055651*^9, 3.942310839484604*^9}}, + CellChangeTimes->{{3.942310735055651*^9, 3.942310839484604*^9}, + 3.947075603423586*^9}, CellID->310614131,ExpressionUUID->"d9409b08-30fc-4e76-ae5d-3c21244c3a0c"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ - RowBox[{"mode", "[", "\"\\"", "]"}], "[", "r", "]"}]], "Input", + RowBox[{ + RowBox[{"mode", "[", "\"\\"", "]"}], "[", "r", "]"}], "//", + "Activate"}]], "Input", CellChangeTimes->{{3.94231078153257*^9, 3.942310808500852*^9}, { - 3.946110163140851*^9, 3.94611016688568*^9}}, - CellLabel->"In[16]:=", + 3.946110163140851*^9, 3.94611016688568*^9}, {3.947075606989831*^9, + 3.947075609053946*^9}}, + CellLabel->"In[56]:=", CellID->1327405499,ExpressionUUID->"edb10b0f-5bf5-4a17-af4b-698874fb4b8e"], Cell[BoxData[ @@ -4255,36 +4211,41 @@ Cell[BoxData[ RowBox[{"(", RowBox[{ RowBox[{"16", " ", "\[ImaginaryI]"}], "+", - RowBox[{"12", " ", + RowBox[{"13", " ", SqrtBox["r0"]}], "-", - RowBox[{"16", " ", "\[ImaginaryI]", " ", "r0"}], "+", + RowBox[{"16", " ", "\[ImaginaryI]", " ", "r0"}], "-", + FractionBox[ + RowBox[{"4", " ", + SuperscriptBox["r0", "4"]}], + SqrtBox[ + SuperscriptBox["r0", "5"]]], "+", RowBox[{ SuperscriptBox["r0", RowBox[{"5", "/", "2"}]], " ", RowBox[{"(", RowBox[{"3", "-", - RowBox[{"\[ImaginaryI]", " ", "\[CapitalOmega]Kerr"}]}], ")"}]}], - "-", - RowBox[{"4", " ", - SuperscriptBox["r0", "3"], " ", "\[CapitalOmega]Kerr"}], "+", - RowBox[{ - SuperscriptBox["r0", - RowBox[{"7", "/", "2"}]], " ", - SuperscriptBox["\[CapitalOmega]Kerr", "2"]}], "+", + FractionBox[ + RowBox[{"\[ImaginaryI]", " ", "r0"}], + SqrtBox[ + SuperscriptBox["r0", "5"]]]}], ")"}]}], "+", RowBox[{"\[ImaginaryI]", " ", SuperscriptBox["r0", RowBox[{"3", "/", "2"}]], " ", RowBox[{"(", RowBox[{ - RowBox[{"12", " ", "\[ImaginaryI]"}], "+", "\[CapitalOmega]Kerr"}], - ")"}]}], "+", + RowBox[{"12", " ", "\[ImaginaryI]"}], "+", + FractionBox["r0", + SqrtBox[ + SuperscriptBox["r0", "5"]]]}], ")"}]}], "+", RowBox[{ SuperscriptBox["r0", "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"4", " ", "\[ImaginaryI]"}], "+", - RowBox[{"8", " ", "\[CapitalOmega]Kerr"}]}], ")"}]}]}], ")"}], " ", - + FractionBox[ + RowBox[{"8", " ", "r0"}], + SqrtBox[ + SuperscriptBox["r0", "5"]]]}], ")"}]}]}], ")"}], " ", RowBox[{"DiracDelta", "[", RowBox[{"r", "-", "r0"}], "]"}]}], ")"}], "/", RowBox[{"(", @@ -4308,9 +4269,11 @@ Cell[BoxData[ RowBox[{"4", " ", "\[ImaginaryI]", " ", SuperscriptBox["r0", RowBox[{"3", "/", "2"}]]}], "-", - RowBox[{"2", " ", "\[ImaginaryI]", " ", - SuperscriptBox["r0", "2"], " ", "\[CapitalOmega]Kerr"}]}], ")"}], " ", - + FractionBox[ + RowBox[{"2", " ", "\[ImaginaryI]", " ", + SuperscriptBox["r0", "3"]}], + SqrtBox[ + SuperscriptBox["r0", "5"]]]}], ")"}], " ", RowBox[{ SuperscriptBox["DiracDelta", "\[Prime]", MultilineFunction->None], "[", @@ -4341,22 +4304,29 @@ Cell[BoxData[ CellChangeTimes->{{3.942310783957175*^9, 3.942310809904798*^9}, 3.946104657140086*^9, 3.946104801694309*^9, 3.946104859570797*^9, 3.946110075564424*^9, {3.946110164831791*^9, 3.946110167124357*^9}, - 3.946113965582231*^9, 3.9461140751777983`*^9, 3.946114156625847*^9}, - CellLabel->"Out[16]=", - CellID->1408011242,ExpressionUUID->"74acd0b4-f7e6-4f07-b3be-4f252408d715"] + 3.946113965582231*^9, 3.9461140751777983`*^9, 3.946114156625847*^9, { + 3.94707560528516*^9, 3.947075609465626*^9}}, + CellLabel->"Out[56]=", + CellID->2040704292,ExpressionUUID->"3f916d9f-d567-4ffb-b29d-954822b57ee1"] }, Open ]], -Cell["\"In\" and \"Up\" return the homogeneous solutions used. ", \ -"ExampleText", - CellChangeTimes->{{3.942310854482686*^9, 3.942310884392668*^9}}, +Cell["\<\ +\"RadialFunctions\" returns an Association with the \"In\" and \"Up\" \ +homogeneous solutions\ +\>", "ExampleText", + CellChangeTimes->{{3.942310854482686*^9, 3.942310884392668*^9}, { + 3.947075618089176*^9, 3.947075636367235*^9}}, CellID->785083211,ExpressionUUID->"994c7a77-533c-4141-8cd2-fbf8d168c342"], Cell[CellGroupData[{ Cell[BoxData[ - RowBox[{"mode", "[", "\"\\"", "]"}]], "Input", - CellChangeTimes->{{3.942310885786623*^9, 3.942310887571189*^9}}, - CellLabel->"In[17]:=", + RowBox[{ + RowBox[{"mode", "[", "\"\\"", "]"}], "[", "\"\\"", + "]"}]], "Input", + CellChangeTimes->{{3.942310885786623*^9, 3.942310887571189*^9}, { + 3.947075640009712*^9, 3.947075647631733*^9}}, + CellLabel->"In[57]:=", CellID->729034064,ExpressionUUID->"19643dba-5b72-4cec-a646-44b216827324"], Cell[BoxData[ @@ -4439,8 +4409,17 @@ Cell[BoxData[ TagBox["\"\[Omega]: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox[ - RowBox[{"2", " ", "\[CapitalOmega]Kerr"}], - "SummaryItem"]}], "\" \"", + RowBox[{"2", " ", + RowBox[{ + RowBox[{ + + TemplateBox[{"KerrGeoFrequencies"}, "InactiveHead", + BaseStyle -> "Inactive", Tooltip -> + "Inactive[KerrGeoFrequencies]", SyntaxForm -> "Symbol"], + "[", + RowBox[{"0", ",", "r0", ",", "0", ",", "1"}], "]"}], "[", + "\"\\!\\(\\*SubscriptBox[\\(\[CapitalOmega]\\), \\(\[Phi]\ +\\)]\\)\"", "]"}]}], "SummaryItem"]}], "\" \"", RowBox[{ TagBox["\"PN parameter: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", @@ -4539,8 +4518,17 @@ Cell[BoxData[ TagBox["\"\[Omega]: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox[ - RowBox[{"2", " ", "\[CapitalOmega]Kerr"}], - "SummaryItem"]}], "\" \"", + RowBox[{"2", " ", + RowBox[{ + RowBox[{ + + TemplateBox[{"KerrGeoFrequencies"}, "InactiveHead", + BaseStyle -> "Inactive", Tooltip -> + "Inactive[KerrGeoFrequencies]", SyntaxForm -> "Symbol"], + "[", + RowBox[{"0", ",", "r0", ",", "0", ",", "1"}], "]"}], "[", + "\"\\!\\(\\*SubscriptBox[\\(\[CapitalOmega]\\), \\(\[Phi]\ +\\)]\\)\"", "]"}]}], "SummaryItem"]}], "\" \"", RowBox[{ TagBox["\"PN parameter: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", @@ -4564,7 +4552,17 @@ Cell[BoxData[ RowBox[{ FractionBox["64", "5"], " ", SuperscriptBox["\"r\"", "4"], " ", - SuperscriptBox["\[CapitalOmega]Kerr", "4"], " ", + SuperscriptBox[ + RowBox[{ + RowBox[{ + + TemplateBox[{"KerrGeoFrequencies"}, "InactiveHead", + BaseStyle -> "Inactive", Tooltip -> + "Inactive[KerrGeoFrequencies]", SyntaxForm -> "Symbol"], + "[", + RowBox[{"0", ",", "r0", ",", "0", ",", "1"}], "]"}], "[", + "\"\\!\\(\\*SubscriptBox[\\(\[CapitalOmega]\\), \\(\[Phi]\ +\\)]\\)\"", "]"}], "4"], " ", SuperscriptBox["\[Eta]", "4"]}], "+", InterpretationBox[ SuperscriptBox[ @@ -4573,20 +4571,32 @@ Cell[BoxData[ False]}], SeriesData[$CellContext`\[Eta], 0, { - Rational[64, 5] "r"^4 Teukolsky`PN`\[CapitalOmega]Kerr^4}, - 4, 5, 1], Editable -> False], "SummaryItem"]}]}, { + Rational[64, 5] "r"^4 + Inactive[ + KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][ + 0, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \ +\(\[Phi]\)]\)"]^4}, 4, 5, 1], Editable -> False], "SummaryItem"]}]}, { + RowBox[{ + TagBox["\"Min order: \"", "SummaryItemAnnotation"], + "\[InvisibleSpace]", + TagBox["4", "SummaryItem"]}]}, { + RowBox[{ + TagBox["\"Number of terms: \"", "SummaryItemAnnotation"], + "\[InvisibleSpace]", + TagBox["3", "SummaryItem"]}]}, { RowBox[{ TagBox["\"Normalization: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox["\"Default\"", "SummaryItem"]}]}, { RowBox[{ - TagBox["\"Simplified: \"", "SummaryItemAnnotation"], + TagBox["\"Simplify: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox["True", "SummaryItem"]}]}, { RowBox[{ - TagBox["\"Number of terms: \"", "SummaryItemAnnotation"], + TagBox["\"Amplitudes: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", - TagBox["3", "SummaryItem"]}]}}, + TagBox["False", "SummaryItem"]}]}}, GridBoxAlignment -> { "Columns" -> {{Left}}, "Rows" -> {{Automatic}}}, AutoDelete -> False, GridBoxItemSize -> { @@ -4604,41 +4614,61 @@ Cell[BoxData[ "SummaryPanel"], DynamicModuleValues:>{}], "]"}], Teukolsky`PN`TeukolskyRadialFunctionPN[-2, 2, 2, 0, 2 - Teukolsky`PN`\[CapitalOmega]Kerr, {$CellContext`\[Eta], 3}, <| + Inactive[KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][ + 0, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"], {$CellContext`\ +\[Eta], 3}, <| "s" -> -2, "l" -> 2, "m" -> 2, "a" -> 0, "PN" -> {$CellContext`\[Eta], 3}, "RadialFunction" -> Function[Teukolsky`PN`Private`r, SeriesData[$CellContext`\[Eta], 0, { Rational[64, 5] Teukolsky`PN`Private`r^4 - Teukolsky`PN`\[CapitalOmega]Kerr^4, Complex[0, + Inactive[KerrGeodesics`OrbitalFrequencies`KerrGeoFrequencies][ + 0, $CellContext`r0, 0, 1][ + "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(\[Phi]\)]\)"]^4, + Complex[0, Rational[256, 15]] Teukolsky`PN`Private`r^5 - 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211723, 4941] +NotebookOutlinePosition[ 212502, 4967] +CellTagsIndexPosition[ 212421, 4962] WindowFrame->Normal*) (* Beginning of Notebook Content *) @@ -26,7 +26,7 @@ Cell["TeukolskyRadialPN", "ObjectName", Cell[TextData[{ Cell[" ", "ModInfo",ExpressionUUID->"dde4c2fa-3238-4980-ae69-60aeac927121"], "TeukolskyRadialPN[ \[ScriptS], \[ScriptL], \[ScriptM], a, \[Omega], \ -{\[Eta], n}]\[LineSeparator]gives the In and Up solution to the radial \ +{\[Eta], n}]\[LineSeparator]gives the In and Up solutiona to the radial \ Teukolsky equation for a given mode, ", Cell[BoxData[ StyleBox["a", "TI"]], "InlineFormula",ExpressionUUID-> @@ -39,16 +39,21 @@ Teukolsky equation for a given mode, ", terms." }], "Usage", CellChangeTimes->{{3.942146008319547*^9, 3.942146008438014*^9}, { - 3.9422227374162283`*^9, 3.942222782742009*^9}, 3.942222863661958*^9}, + 3.9422227374162283`*^9, 3.942222782742009*^9}, 3.942222863661958*^9, { + 3.947075682920389*^9, 3.947075683823448*^9}}, CellID->1226706735,ExpressionUUID->"860eeef6-58c4-4c2e-ba29-267e4b80d079"], Cell[TextData[{ "We compute ", Cell[BoxData[ - SubscriptBox["R", "In"]], "InlineFormula",ExpressionUUID-> - "0ab18874-2e89-4e4c-b1e8-6d614322d6f0"], - " according to Eq.(147) of Sasaki, Tagoshi \ -https://arxiv.org/abs/gr-qc/0306120 where we divide out a factor of ", + SubsuperscriptBox["R", "\[ScriptL]\[ScriptM]\[Omega]", "In"]], + "InlineFormula",ExpressionUUID->"0ab18874-2e89-4e4c-b1e8-6d614322d6f0"], + " according to Eq. (147) of [", + ButtonBox["Sasaki, Tagoshi, Living Rev. Relativity 6", + BaseStyle->"Hyperlink", + ButtonData->{ + URL["http://www.livingreviews.org/lrr-2003-6"], None}], + "], modified to divide out a factor of ", Cell[BoxData[ SubscriptBox["K", "\[Nu]"]], "InlineFormula",ExpressionUUID-> "69332f75-3d13-4104-bcca-63654e495bea"], @@ -72,13 +77,18 @@ https://arxiv.org/abs/gr-qc/0306120 where we divide out a factor of ", Cell[BoxData[ SubscriptBox["R", "Up"]], "InlineFormula",ExpressionUUID-> "442a549d-6804-49df-ade2-e854a7ce585d"], - " is computed after Eq.(B.7) of Throwe ,Hughes \ -https://dspace.mit.edu/handle/1721.1/61270. " + " is computed using Eq. (B.7) of ", + ButtonBox["Will Throwe's MSc thesis", + BaseStyle->"Hyperlink", + ButtonData->{ + URL["https://dspace.mit.edu/handle/1721.1/61270%5D"], None}], + "." }], "Notes", CellChangeTimes->{{3.94223010417549*^9, 3.942230142224959*^9}, { 3.942230254892722*^9, 3.942230470166971*^9}, {3.942230511057706*^9, 3.942230554319509*^9}, 3.942230599297515*^9, {3.942230711648521*^9, - 3.942230712881777*^9}}, + 3.942230712881777*^9}, {3.947074453476533*^9, 3.9470744547454433`*^9}, { + 3.9470756951399527`*^9, 3.947075875401649*^9}}, CellID->1221797685,ExpressionUUID->"9593db7e-87cc-4897-928e-46f72ca9bb38"], Cell["\<\ @@ -95,20 +105,26 @@ Cell["The following options can be given: ", "Notes", Cell[BoxData[GridBox[{ {Cell[" ", "ModInfo",ExpressionUUID-> - "42680162-874f-4d15-a339-87a795cf9f8b"], "\"\\"", - "\"\\"", Cell[TextData[{ - "Determines the normalization for the In and Up solution. The \ -\"Default\" option is geared towards efficiency. \"SasakiTagoshi\" multiplies \ -", + "42680162-874f-4d15-a339-87a795cf9f8b"], "\"\\"", \ +"\"\\"", Cell[TextData[{ + "The normalization for the In and Up solution. The \"Default\" option is \ +geared towards efficiency. \"SasakiTagoshi\" multiplies ", Cell[BoxData[ - SubscriptBox["R", "In"]], "InlineFormula",ExpressionUUID-> - "e25fcfca-6b9f-425c-a0bb-840202144b28"], - " with a factor of ", + SubsuperscriptBox["R", "\[ScriptL]\[ScriptM]\[Omega]", "In"]], + "InlineFormula",ExpressionUUID->"e25fcfca-6b9f-425c-a0bb-840202144b28"], + + " by a factor of ", Cell[BoxData[ SubscriptBox["K", "\[Nu]"]], "InlineFormula",ExpressionUUID-> "0a678336-6d24-43d9-9c73-dd6508f677d1"], ". \"UnitTransmission\" makes it such that the transmission coefficients \ -are 1. This is also the normalization used in TeukolskyRadial[] (no PN). " +are 1. This is also the normalization used in ", + Cell[BoxData[ + ButtonBox["TeukolskyRadial", + BaseStyle->"Link", + ButtonData->"paclet:Teukolsky/ref/TeukolskyRadial"]], "InlineFormula", + ExpressionUUID->"b874f762-de47-44ae-8aec-a69ec3f61430"], + " (no PN). " }], "TableText",ExpressionUUID->"e02324e4-e763-4bba-9ac3-711147d5ef88"]}, {Cell[" ", "ModInfo",ExpressionUUID-> "2f974a1d-1d9e-4eff-98a8-fe47c90934a9"], "\"\\"", "True", @@ -119,7 +135,7 @@ time, but increase the ByteCount by about a factor of 10.\ {Cell[" ", "ModInfo",ExpressionUUID-> "d720dc96-d1e6-47ba-bef9-4f8e06257b7b"], "\"\\"", "False", Cell["\<\ -Determines weather or not to compute the transmission Amplitudes.\ +Determines weather or not to compute the asymptotic Amplitudes.\ \>", "TableText",ExpressionUUID->"df19df64-5e08-4573-bd66-f53298808fff"]} }]], "3ColumnTableMod", CellChangeTimes->{{3.942226969137558*^9, 3.9422269693066587`*^9}, { @@ -127,7 +143,8 @@ Determines weather or not to compute the transmission Amplitudes.\ 3.942227185895194*^9, 3.94222734696238*^9}, {3.942229499518452*^9, 3.94222955524142*^9}, {3.942230656164405*^9, 3.94223066232033*^9}, { 3.942230809279039*^9, 3.942230938905239*^9}, {3.942230975767381*^9, - 3.942230990215602*^9}, {3.9469865615409327`*^9, 3.9469865656218557`*^9}}, + 3.942230990215602*^9}, {3.9469865615409327`*^9, 3.9469865656218557`*^9}, { + 3.947075893567389*^9, 3.9470759420856953`*^9}}, CellID->276668925,ExpressionUUID->"7163bf51-103e-4897-892a-1cda9c934003"] }, Open ]], @@ -162,10 +179,10 @@ Cell[TextData[{ "10b7800f-16f4-4306-9eb8-92692ee5eb1a"], DynamicModuleBox[{$CellContext`nbobj$$ = NotebookObject[ "cb00ca4d-1fa2-41b6-98e6-832d08bd84d9", - "62e8b7ce-9e2c-4792-9c8b-914f4c1a3211"], $CellContext`cellobj$$ = + "0a16f9ef-84ae-43e9-b1f1-9687ff99968c"], $CellContext`cellobj$$ = CellObject[ "44eaa3dc-4fda-45ae-8c12-c5c4bc64c656", - "c30d01d0-8568-4278-bfe0-9d530ff9a57f"]}, + "89b6caa5-4569-4607-a6fd-1e42219d3f1e"]}, TemplateBox[{ GraphicsBox[{{ Thickness[0.06], @@ -262,9 +279,9 @@ Needs[\[Ellipsis]].", "MoreInfoText"], BaseStyle -> "IFrameBox"]], CellID->699830455,ExpressionUUID->"7cb0a5f9-86b2-4507-a404-9a55fe2941d0"], Cell[BoxData[ - RowBox[{"Needs", "[", "\"\\"", - "]"}]], "ExampleInitialization", - CellChangeTimes->{{3.942146478955165*^9, 3.942146497771613*^9}}, + RowBox[{"Needs", "[", "\"\\"", "]"}]], "ExampleInitialization", + CellChangeTimes->{{3.942146478955165*^9, 3.942146497771613*^9}, { + 3.947075938339942*^9, 3.947075938577417*^9}}, CellID->1582090681,ExpressionUUID->"8dccde0a-dc8e-4e53-828c-822e855954c5"] }, Open ]], @@ -285,10 +302,11 @@ Cell[BoxData[ CellID->336632675,ExpressionUUID->"7ffeb3ad-6511-4c6a-adaf-258336a6fc70"], Cell["\<\ -Here is a basic example to compute the s=-2 l=2 m=2 mode in Kerr up to 4 \ -terms in the Series (1.5PN).\ +As a basic example we compute the s=-2 l=2 m=2 mode in Kerr up to 4 terms in \ +the Series (1.5PN).\ \>", "ExampleText", - CellChangeTimes->{{3.946103658010844*^9, 3.9461036974031773`*^9}}, + CellChangeTimes->{{3.946103658010844*^9, 3.9461036974031773`*^9}, { + 3.947075950842733*^9, 3.94707595333359*^9}}, CellID->2113760115,ExpressionUUID->"94aae506-9436-4ecf-bea7-2359cc22bd8a"], Cell[CellGroupData[{ @@ -303,7 +321,7 @@ Cell[BoxData[ CellChangeTimes->{{3.942146462350701*^9, 3.942146474773769*^9}, { 3.9421465581906*^9, 3.942146558375171*^9}, {3.942146610065486*^9, 3.94214663484762*^9}}, - CellLabel->"In[5]:=", + CellLabel->"In[59]:=", CellID->1569970270,ExpressionUUID->"83e713ae-7909-403e-94c5-eac3d0f8f20c"], Cell[BoxData[ @@ -527,18 +545,26 @@ Cell[BoxData[ SeriesData[$CellContext`\[Eta], 0, { Rational[4, 5] "r"^4 $CellContext`\[Omega]^4}, 4, 5, 1], Editable -> False], "SummaryItem"]}]}, { + RowBox[{ + TagBox["\"Min order: \"", "SummaryItemAnnotation"], + "\[InvisibleSpace]", + TagBox["4", "SummaryItem"]}]}, { + RowBox[{ + TagBox["\"Number of terms: \"", "SummaryItemAnnotation"], + "\[InvisibleSpace]", + TagBox["4", "SummaryItem"]}]}, { RowBox[{ TagBox["\"Normalization: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox["\"Default\"", "SummaryItem"]}]}, { RowBox[{ - TagBox["\"Simplified: \"", "SummaryItemAnnotation"], + TagBox["\"Simplify: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox["True", "SummaryItem"]}]}, { RowBox[{ - TagBox["\"Number of terms: \"", "SummaryItemAnnotation"], + TagBox["\"Amplitudes: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", - TagBox["4", "SummaryItem"]}]}}, + TagBox["False", "SummaryItem"]}]}}, GridBoxAlignment -> { "Columns" -> {{Left}}, "Rows" -> {{Automatic}}}, AutoDelete -> False, GridBoxItemSize -> { @@ -576,13 +602,15 @@ Cell[BoxData[ Complex[0, 1] (35 + 28 (1 - $CellContext`a^2)^Rational[1, 2] - 84 EulerGamma - 3 Teukolsky`PN`Private`r^3 $CellContext`\[Omega]^2))}, 4, 8, 1]], - "BoundaryCondition" -> "In", "LeadingOrder" -> + "BoundaryCondition" -> "In", "SeriesMinOrder" -> 4, "LeadingOrder" -> Function[Teukolsky`PN`Private`r, SeriesData[$CellContext`\[Eta], 0, { Rational[4, 5] Teukolsky`PN`Private`r^4 $CellContext`\[Omega]^4}, 4, 5, 1]], "TermCount" -> 4, "Normalization" -> "Default", - "Amplitudes" -> <|"Transmission" -> "Not Computed"|>, "Simplified" -> - True|>], + "Amplitudes" -> <| + "Incidence" -> Missing["NotComputed"], "Transmission" -> + Missing["NotComputed"], "Reflection" -> Missing["NotComputed"]|>, + "Simplify" -> True, "AmplitudesBool" -> False|>], Editable->False, SelectWithContents->True, Selectable->False]}], ",", @@ -800,20 +828,29 @@ Cell[BoxData[ Editable -> False]}], SeriesData[$CellContext`\[Eta], 0, { - Complex[0, -3] ("r"^(-1)/$CellContext`\[Omega])}, -1, 0, - 1], Editable -> False], "SummaryItem"]}]}, { + Complex[0, -3] "r"^(-1)/$CellContext`\[Omega]}, -1, 0, 1], + Editable -> False], "SummaryItem"]}]}, { + RowBox[{ + TagBox["\"Min order: \"", "SummaryItemAnnotation"], + "\[InvisibleSpace]", + TagBox[ + RowBox[{"-", "1"}], "SummaryItem"]}]}, { + RowBox[{ + TagBox["\"Number of terms: \"", "SummaryItemAnnotation"], + "\[InvisibleSpace]", + TagBox["4", "SummaryItem"]}]}, { RowBox[{ TagBox["\"Normalization: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox["\"Default\"", "SummaryItem"]}]}, { RowBox[{ - TagBox["\"Simplified: \"", "SummaryItemAnnotation"], + TagBox["\"Simplify: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox["True", "SummaryItem"]}]}, { RowBox[{ - TagBox["\"Number of terms: \"", "SummaryItemAnnotation"], + TagBox["\"Amplitudes: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", - TagBox["4", "SummaryItem"]}]}}, + TagBox["False", "SummaryItem"]}]}}, GridBoxAlignment -> { "Columns" -> {{Left}}, "Rows" -> {{Automatic}}}, AutoDelete -> False, GridBoxItemSize -> { @@ -837,8 +874,8 @@ Cell[BoxData[ "PN" -> {$CellContext`\[Eta], 4}, "RadialFunction" -> Function[Teukolsky`PN`Private`r, SeriesData[$CellContext`\[Eta], 0, { - Complex[0, -3] ( - Teukolsky`PN`Private`r^(-1)/$CellContext`\[Omega]), -3, + Complex[0, -3] + Teukolsky`PN`Private`r^(-1)/$CellContext`\[Omega], -3, Rational[1, 2] Teukolsky`PN`Private`r^(-2) $CellContext`\[Omega]^(-1) ( 8 $CellContext`a + @@ -849,25 +886,31 @@ Cell[BoxData[ 12 (1 - $CellContext`a^2)^Rational[1, 2] - 36 EulerGamma + Complex[0, 36] Pi + 3 Teukolsky`PN`Private`r^3 $CellContext`\[Omega]^2)}, -1, 3, 1]], - "BoundaryCondition" -> "Up", "LeadingOrder" -> + "BoundaryCondition" -> "Up", "SeriesMinOrder" -> -1, "LeadingOrder" -> Function[Teukolsky`PN`Private`r, SeriesData[$CellContext`\[Eta], 0, { - Complex[0, -3] ( - Teukolsky`PN`Private`r^(-1)/$CellContext`\[Omega])}, -1, 0, 1]], + Complex[0, -3] + Teukolsky`PN`Private`r^(-1)/$CellContext`\[Omega]}, -1, 0, 1]], "TermCount" -> 4, "Normalization" -> "Default", - "Amplitudes" -> <|"Transmission" -> "Not Computed"|>, "Simplified" -> - True|>], + "Amplitudes" -> <| + "Incidence" -> Missing["NotComputed"], "Transmission" -> + Missing["NotComputed"], "Reflection" -> Missing["NotComputed"]|>, + "Simplify" -> True, "AmplitudesBool" -> False|>], Editable->False, SelectWithContents->True, Selectable->False]}]}], "\[RightAssociation]"}]], "Output", CellChangeTimes->{{3.942146476518921*^9, 3.942146505601961*^9}, 3.94214656684116*^9, {3.942146622616212*^9, 3.942146642585205*^9}, 3.942388590247716*^9, 3.942388959260372*^9, 3.942388991878456*^9, - 3.946097134141876*^9, 3.946097183744846*^9}, - CellLabel->"Out[5]=", - CellID->2135881485,ExpressionUUID->"bef86d06-49d7-4cfe-b465-ea71d64444c8"] + 3.946097134141876*^9, 3.946097183744846*^9, 3.9470759808782463`*^9}, + CellLabel->"Out[59]=", + CellID->1752191333,ExpressionUUID->"c324e158-7f80-4e15-9197-cd26141ecb3d"] }, Open ]], +Cell["Evaluate the radial functions symbolically:", "ExampleText", + CellChangeTimes->{{3.9470759808366737`*^9, 3.9470759888491096`*^9}}, + CellID->1500657367,ExpressionUUID->"2cb2a6f2-66fb-4d9f-b035-35d140e328d9"], + Cell[CellGroupData[{ Cell[BoxData[ @@ -876,7 +919,7 @@ Cell[BoxData[ RowBox[{"test", "[", "\"\\"", "]"}], "[", "r", "]"}], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.942146573661372*^9, 3.942146586725117*^9}}, - CellLabel->"In[6]:=", + CellLabel->"In[60]:=", CellID->1260920295,ExpressionUUID->"855a404e-e30c-4792-9d64-473ea95e84f1"], Cell[BoxData[ @@ -942,9 +985,10 @@ Cell[BoxData[ Editable->False]], "Output", CellChangeTimes->{{3.942146578715905*^9, 3.942146587032299*^9}, { 3.942146622694558*^9, 3.942146642658656*^9}, 3.942388590323811*^9, - 3.9423889593325644`*^9, 3.9423889920466423`*^9, 3.946097183886037*^9}, - CellLabel->"Out[6]=", - CellID->1543820165,ExpressionUUID->"0f9a9e2a-d478-496e-b8d1-8f7d2fa42035"] + 3.9423889593325644`*^9, 3.9423889920466423`*^9, 3.946097183886037*^9, + 3.947075990311345*^9}, + CellLabel->"Out[60]=", + CellID->966587139,ExpressionUUID->"8e5e52c2-cd12-4215-9bbc-98da12442912"] }, Open ]], Cell[CellGroupData[{ @@ -955,7 +999,7 @@ Cell[BoxData[ RowBox[{"test", "[", "\"\\"", "]"}], "[", "r", "]"}], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.94214664372406*^9, 3.942146648449054*^9}}, - CellLabel->"In[7]:=", + CellLabel->"In[61]:=", CellID->1545593839,ExpressionUUID->"31ea1d30-efc1-4882-a0cf-f09829d65edb"], Cell[BoxData[ @@ -1013,17 +1057,18 @@ Cell[BoxData[ 1], Editable->False]], "Output", CellChangeTimes->{3.942146648761879*^9, 3.942388590358964*^9, - 3.9423889593682127`*^9, 3.942388992086396*^9, 3.9460971839179068`*^9}, - CellLabel->"Out[7]=", - CellID->661595076,ExpressionUUID->"3713996e-2a5d-48d7-bcf5-8480f21b55c6"] + 3.9423889593682127`*^9, 3.942388992086396*^9, 3.9460971839179068`*^9, + 3.947075992138709*^9}, + CellLabel->"Out[61]=", + CellID->1638878797,ExpressionUUID->"e433d181-7c16-4560-bc1a-d029c5b4c2b7"] }, Open ]], Cell["\<\ -Alternatively to the number of terms, one can put in the desired PN order as \ +Alternatively to the number of terms, one can specify the desired PN order as \ a String\ \>", "ExampleText", CellChangeTimes->{{3.946103489397705*^9, 3.9461035297604647`*^9}, - 3.94610370519433*^9}, + 3.94610370519433*^9, {3.947076177173936*^9, 3.947076178418899*^9}}, CellID->246899034,ExpressionUUID->"f5f2c554-c444-4609-8d23-91ebce7c64a7"], Cell[CellGroupData[{ @@ -1035,7 +1080,7 @@ Cell[BoxData[ RowBox[{"{", RowBox[{"\[Eta]", ",", "\"\<1.5PN\>\""}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.946103535588821*^9, 3.946103544660578*^9}}, - CellLabel->"In[10]:=", + CellLabel->"In[62]:=", CellID->131943848,ExpressionUUID->"a7bd2a52-7b2b-4476-9178-593dbdd60c7b"], Cell[BoxData[ @@ -1259,18 +1304,26 @@ Cell[BoxData[ SeriesData[$CellContext`\[Eta], 0, { Rational[4, 5] "r"^4 $CellContext`\[Omega]^4}, 4, 5, 1], Editable -> False], "SummaryItem"]}]}, { + RowBox[{ + TagBox["\"Min order: \"", "SummaryItemAnnotation"], + "\[InvisibleSpace]", + TagBox["4", "SummaryItem"]}]}, { + RowBox[{ + TagBox["\"Number of terms: \"", "SummaryItemAnnotation"], + "\[InvisibleSpace]", + TagBox["4", "SummaryItem"]}]}, { RowBox[{ TagBox["\"Normalization: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox["\"Default\"", "SummaryItem"]}]}, { RowBox[{ - TagBox["\"Simplified: \"", "SummaryItemAnnotation"], + TagBox["\"Simplify: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox["True", "SummaryItem"]}]}, { RowBox[{ - TagBox["\"Number of terms: \"", "SummaryItemAnnotation"], + TagBox["\"Amplitudes: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", - TagBox["4", "SummaryItem"]}]}}, + TagBox["False", "SummaryItem"]}]}}, GridBoxAlignment -> { "Columns" -> {{Left}}, "Rows" -> {{Automatic}}}, AutoDelete -> False, GridBoxItemSize -> { @@ -1308,13 +1361,15 @@ Cell[BoxData[ Complex[0, 1] (35 + 28 (1 - $CellContext`a^2)^Rational[1, 2] - 84 EulerGamma - 3 Teukolsky`PN`Private`r^3 $CellContext`\[Omega]^2))}, 4, 8, 1]], - "BoundaryCondition" -> "In", "LeadingOrder" -> + "BoundaryCondition" -> "In", "SeriesMinOrder" -> 4, "LeadingOrder" -> Function[Teukolsky`PN`Private`r, SeriesData[$CellContext`\[Eta], 0, { Rational[4, 5] Teukolsky`PN`Private`r^4 $CellContext`\[Omega]^4}, 4, 5, 1]], "TermCount" -> 4, "Normalization" -> "Default", - "Amplitudes" -> <|"Transmission" -> "Not Computed"|>, "Simplified" -> - True|>], + "Amplitudes" -> <| + "Incidence" -> Missing["NotComputed"], "Transmission" -> + Missing["NotComputed"], "Reflection" -> Missing["NotComputed"]|>, + "Simplify" -> True, "AmplitudesBool" -> False|>], Editable->False, SelectWithContents->True, Selectable->False]}], ",", @@ -1532,20 +1587,29 @@ Cell[BoxData[ Editable -> False]}], SeriesData[$CellContext`\[Eta], 0, { - Complex[0, -3] ("r"^(-1)/$CellContext`\[Omega])}, -1, 0, - 1], Editable -> False], "SummaryItem"]}]}, { + Complex[0, -3] "r"^(-1)/$CellContext`\[Omega]}, -1, 0, 1], + Editable -> False], "SummaryItem"]}]}, { + RowBox[{ + TagBox["\"Min order: \"", "SummaryItemAnnotation"], + "\[InvisibleSpace]", + TagBox[ + RowBox[{"-", "1"}], "SummaryItem"]}]}, { + RowBox[{ + TagBox["\"Number of terms: \"", "SummaryItemAnnotation"], + "\[InvisibleSpace]", + TagBox["4", "SummaryItem"]}]}, { RowBox[{ TagBox["\"Normalization: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox["\"Default\"", "SummaryItem"]}]}, { RowBox[{ - TagBox["\"Simplified: \"", "SummaryItemAnnotation"], + TagBox["\"Simplify: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox["True", "SummaryItem"]}]}, { RowBox[{ - TagBox["\"Number of terms: \"", "SummaryItemAnnotation"], + TagBox["\"Amplitudes: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", - TagBox["4", "SummaryItem"]}]}}, + TagBox["False", "SummaryItem"]}]}}, GridBoxAlignment -> { "Columns" -> {{Left}}, "Rows" -> {{Automatic}}}, AutoDelete -> False, GridBoxItemSize -> { @@ -1569,8 +1633,8 @@ Cell[BoxData[ "PN" -> {$CellContext`\[Eta], 4}, "RadialFunction" -> Function[Teukolsky`PN`Private`r, SeriesData[$CellContext`\[Eta], 0, { - Complex[0, -3] ( - Teukolsky`PN`Private`r^(-1)/$CellContext`\[Omega]), -3, + Complex[0, -3] + Teukolsky`PN`Private`r^(-1)/$CellContext`\[Omega], -3, Rational[1, 2] Teukolsky`PN`Private`r^(-2) $CellContext`\[Omega]^(-1) ( 8 $CellContext`a + @@ -1581,20 +1645,23 @@ Cell[BoxData[ 12 (1 - $CellContext`a^2)^Rational[1, 2] - 36 EulerGamma + Complex[0, 36] Pi + 3 Teukolsky`PN`Private`r^3 $CellContext`\[Omega]^2)}, -1, 3, 1]], - "BoundaryCondition" -> "Up", "LeadingOrder" -> + "BoundaryCondition" -> "Up", "SeriesMinOrder" -> -1, "LeadingOrder" -> Function[Teukolsky`PN`Private`r, SeriesData[$CellContext`\[Eta], 0, { - Complex[0, -3] ( - Teukolsky`PN`Private`r^(-1)/$CellContext`\[Omega])}, -1, 0, 1]], + Complex[0, -3] + Teukolsky`PN`Private`r^(-1)/$CellContext`\[Omega]}, -1, 0, 1]], "TermCount" -> 4, "Normalization" -> "Default", - "Amplitudes" -> <|"Transmission" -> "Not Computed"|>, "Simplified" -> - True|>], + "Amplitudes" -> <| + "Incidence" -> Missing["NotComputed"], "Transmission" -> + Missing["NotComputed"], "Reflection" -> Missing["NotComputed"]|>, + "Simplify" -> True, "AmplitudesBool" -> False|>], Editable->False, SelectWithContents->True, Selectable->False]}]}], "\[RightAssociation]"}]], "Output", - CellChangeTimes->{3.946103555122784*^9, 3.946103640924468*^9}, - CellLabel->"Out[10]=", - CellID->1532006408,ExpressionUUID->"2b9ac16a-1bb3-48b6-ac7d-b9f1a76a673d"] + CellChangeTimes->{3.946103555122784*^9, 3.946103640924468*^9, + 3.94707619497829*^9}, + CellLabel->"Out[62]=", + CellID->1241075495,ExpressionUUID->"0b607248-1d70-4393-b08f-7e78f7924ed3"] }, Open ]] }, Open ]], @@ -1625,10 +1692,11 @@ Cell[BoxData[ CellID->1458198461,ExpressionUUID->"20f5d4a8-a69f-434b-888d-946f8ff93cdd"], Cell["\<\ -Here we want to go through the possible Keys we can query a \ -TeukolskyRadialFunctionPN for.\ +Obtain a list of the possible Keys we can query a TeukolskyRadialFunctionPN \ +for.\ \>", "ExampleText", - CellChangeTimes->{{3.942389277599101*^9, 3.942389296377552*^9}}, + CellChangeTimes->{{3.942389277599101*^9, 3.942389296377552*^9}, { + 3.947076193552033*^9, 3.9470761960188026`*^9}}, CellID->1484792943,ExpressionUUID->"80ac1cd4-5cd4-4e34-92e6-f8f7bc6b6b2f"], Cell[CellGroupData[{ @@ -1640,27 +1708,26 @@ Cell[BoxData[{ RowBox[{ RowBox[{"-", "2"}], ",", "2", ",", "2", ",", "a", ",", "\[Omega]", ",", RowBox[{"{", - RowBox[{"\[Eta]", ",", "4"}], "}"}]}], "]"}]}], - ";"}], "\[IndentingNewLine]", + RowBox[{"\[Eta]", ",", "4"}], "}"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"test", "[", "\"\\"", "]"}], "//", "Keys"}]}], "Input", CellChangeTimes->{{3.9423885701442223`*^9, 3.942388597647747*^9}, { 3.946097122992432*^9, 3.9460971300140343`*^9}}, - CellLabel->"In[1]:=", + CellLabel->"In[63]:=", CellID->946679123,ExpressionUUID->"304c9bef-d103-433f-8a7b-5124afe67499"], Cell[BoxData[ RowBox[{"{", RowBox[{"\<\"s\"\>", ",", "\<\"l\"\>", ",", "\<\"m\"\>", ",", "\<\"a\"\>", - ",", "\<\"PN\"\>", ",", "\<\"RadialFunction\"\>", - ",", "\<\"BoundaryCondition\"\>", ",", "\<\"LeadingOrder\"\>", - ",", "\<\"TermCount\"\>", ",", "\<\"Normalization\"\>", - ",", "\<\"Amplitudes\"\>", ",", "\<\"Simplified\"\>"}], "}"}]], "Output", + ",", "\<\"PN\"\>", ",", "\<\"BoundaryCondition\"\>", + ",", "\<\"SeriesMinOrder\"\>", ",", "\<\"LeadingOrder\"\>", + ",", "\<\"Normalization\"\>", ",", "\<\"Amplitudes\"\>", + ",", "\<\"Simplify\"\>"}], "}"}]], "Output", CellChangeTimes->{{3.9423885751187773`*^9, 3.942388597923132*^9}, 3.942388959401473*^9, 3.942388992122604*^9, {3.946097126650717*^9, - 3.946097130876627*^9}, 3.946097193148*^9}, - CellLabel->"Out[2]=", - CellID->170984950,ExpressionUUID->"c3e812e7-828b-4437-91c8-0371eebf62cd"] + 3.946097130876627*^9}, 3.946097193148*^9, 3.947076215735023*^9}, + CellLabel->"Out[64]=", + CellID->269282304,ExpressionUUID->"9e885709-83c1-48d5-a9bd-9d6a39e6b7c6"] }, Open ]], Cell["\<\ @@ -1677,11 +1744,10 @@ Cell[BoxData[ RowBox[{ RowBox[{"test", "[", "\"\\"", "]"}], "[", "#", "]"}], "&"}], "/@", RowBox[{"{", - RowBox[{ - "\"\\"", ",", "\"\\"", ",", "\"\\"", ",", "\"\\""}], - "}"}]}]], "Input", + RowBox[{"\"\\"", ",", "\"\\"", ",", "\"\\"", + ",", "\"\\""}], "}"}]}]], "Input", CellChangeTimes->{{3.942389213243572*^9, 3.942389264423088*^9}}, - CellLabel->"In[3]:=", + CellLabel->"In[65]:=", CellID->918794442,ExpressionUUID->"76d91407-d515-441d-9e2f-c2a442dadbe8"], Cell[BoxData[ @@ -1689,9 +1755,9 @@ Cell[BoxData[ RowBox[{ RowBox[{"-", "2"}], ",", "2", ",", "2", ",", "a"}], "}"}]], "Output", CellChangeTimes->{{3.9423892239843597`*^9, 3.942389241907608*^9}, - 3.9460971931996937`*^9}, - CellLabel->"Out[3]=", - CellID->705759984,ExpressionUUID->"329975c0-c2d3-4f76-9c6f-094a376a87e3"] + 3.9460971931996937`*^9, 3.947076221012776*^9}, + CellLabel->"Out[65]=", + CellID->161793318,ExpressionUUID->"3893a159-09ec-4298-abe5-f92766f2fa5d"] }, Open ]], Cell["\"PN\" gives the PN expansion parameter and number of terms", \ @@ -1706,111 +1772,35 @@ Cell[BoxData[ RowBox[{"test", "[", "\"\\"", "]"}], "[", "\"\\"", "]"}]], "Input",\ CellChangeTimes->{{3.942389248351698*^9, 3.942389272594774*^9}}, - CellLabel->"In[4]:=", + CellLabel->"In[66]:=", CellID->1458245567,ExpressionUUID->"cd2bcf42-9a74-4acb-8c4b-c8322b27f0a4"], Cell[BoxData[ RowBox[{"{", RowBox[{"\[Eta]", ",", "4"}], "}"}]], "Output", CellChangeTimes->{{3.942389255116514*^9, 3.942389272831227*^9}, - 3.9460971932274637`*^9}, - CellLabel->"Out[4]=", - CellID->1025792732,ExpressionUUID->"0bd674f6-6834-4558-81c4-6f6fe1f31795"] + 3.9460971932274637`*^9, 3.9470762237276287`*^9}, + CellLabel->"Out[66]=", + CellID->871406021,ExpressionUUID->"18789eff-03a2-4d3d-b984-122e586c1a43"] }, Open ]], Cell["\<\ -\"RadialFunction\" is the most important Key. It gives the solution to the \ -Teukolsky equation. It can be shortcut by simply querying for a symbol or \ +The radial function can be evaluated symbolically by querying for a symbol or \ number. \ \>", "ExampleText", - CellChangeTimes->{{3.942389387917851*^9, 3.942389457072596*^9}}, + CellChangeTimes->{{3.942389387917851*^9, 3.942389457072596*^9}, { + 3.947076247574236*^9, 3.947076255819105*^9}}, CellID->1955714696,ExpressionUUID->"f4b0d4be-ddb7-4012-bf8c-9a18ad2a5b96"], Cell[CellGroupData[{ -Cell[BoxData[ - RowBox[{ - RowBox[{ - RowBox[{"test", "[", "\"\\"", "]"}], "[", "\"\\"", - "]"}], "[", "r", "]"}]], "Input", - CellChangeTimes->{{3.942389459284617*^9, 3.942389466310264*^9}}, - CellLabel->"In[5]:=", - CellID->2134775037,ExpressionUUID->"db939f69-4715-4698-bfa2-456877832e19"], - -Cell[BoxData[ - InterpretationBox[ - RowBox[{ - RowBox[{ - FractionBox["4", "5"], " ", - SuperscriptBox["r", "4"], " ", - SuperscriptBox["\[Omega]", "4"], " ", - SuperscriptBox["\[Eta]", "4"]}], "+", - RowBox[{ - FractionBox["8", "15"], " ", "\[ImaginaryI]", " ", - SuperscriptBox["r", "5"], " ", - SuperscriptBox["\[Omega]", "5"], " ", - SuperscriptBox["\[Eta]", "5"]}], "-", - RowBox[{ - FractionBox["2", "105"], " ", - RowBox[{"(", - RowBox[{ - SuperscriptBox["r", "3"], " ", - SuperscriptBox["\[Omega]", "4"], " ", - RowBox[{"(", - RowBox[{"168", "+", - RowBox[{"84", " ", "\[ImaginaryI]", " ", "a"}], "+", - RowBox[{"11", " ", - SuperscriptBox["r", "3"], " ", - SuperscriptBox["\[Omega]", "2"]}]}], ")"}]}], ")"}], " ", - SuperscriptBox["\[Eta]", "6"]}], "+", - RowBox[{ - FractionBox["2", "105"], " ", - SuperscriptBox["r", "4"], " ", - SuperscriptBox["\[Omega]", "5"], " ", - RowBox[{"(", - RowBox[{ - RowBox[{"56", " ", "a"}], "+", - RowBox[{"\[ImaginaryI]", " ", - RowBox[{"(", - RowBox[{"35", "+", - RowBox[{"28", " ", - SqrtBox[ - RowBox[{"1", "-", - SuperscriptBox["a", "2"]}]]}], "-", - RowBox[{"84", " ", "EulerGamma"}], "-", - RowBox[{"3", " ", - SuperscriptBox["r", "3"], " ", - SuperscriptBox["\[Omega]", "2"]}]}], ")"}]}]}], ")"}], " ", - SuperscriptBox["\[Eta]", "7"]}], "+", - InterpretationBox[ - SuperscriptBox[ - RowBox[{"O", "[", "\[Eta]", "]"}], "8"], - SeriesData[$CellContext`\[Eta], 0, {}, 4, 8, 1], - Editable->False]}], - SeriesData[$CellContext`\[Eta], 0, { - Rational[4, 5] $CellContext`r^4 $CellContext`\[Omega]^4, Complex[0, - Rational[8, 15]] $CellContext`r^5 $CellContext`\[Omega]^5, - Rational[-2, 105] $CellContext`r^3 $CellContext`\[Omega]^4 (168 + - Complex[0, 84] $CellContext`a + - 11 $CellContext`r^3 $CellContext`\[Omega]^2), - Rational[2, 105] $CellContext`r^4 $CellContext`\[Omega]^5 ( - 56 $CellContext`a + - Complex[0, 1] (35 + 28 (1 - $CellContext`a^2)^Rational[1, 2] - 84 - EulerGamma - 3 $CellContext`r^3 $CellContext`\[Omega]^2))}, 4, 8, 1], - Editable->False]], "Output", - CellChangeTimes->{3.942389466596204*^9, 3.9460971932540293`*^9}, - CellLabel->"Out[5]=", - CellID->1500075295,ExpressionUUID->"3d9b3e35-8e4f-4146-aad6-71050192f58a"] -}, Open ]], - -Cell[CellGroupData[{ - Cell[BoxData[ RowBox[{ RowBox[{"test", "[", "\"\\"", "]"}], "[", "r", "]"}]], "Input", - 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CellLabel->"In[7]:=", + CellLabel->"In[69]:=", CellID->939982762,ExpressionUUID->"c7dc13ce-5ac7-4640-919a-5f4b6f11983f"], Cell[BoxData[ @@ -1940,9 +1931,10 @@ Cell[BoxData[ Complex[0, 1] (35 + 28 (1 - $CellContext`a^2)^Rational[1, 2] - 84 EulerGamma - 24 $CellContext`\[Omega]^2))}, 4, 8, 1], Editable->False]], "Output", - CellChangeTimes->{3.942389482706994*^9, 3.946097193306814*^9}, - CellLabel->"Out[7]=", - CellID->2011406770,ExpressionUUID->"56645f2c-e53e-42db-86ec-68b373ada068"] + CellChangeTimes->{3.942389482706994*^9, 3.946097193306814*^9, + 3.947076268004302*^9}, + CellLabel->"Out[69]=", + CellID->2068155108,ExpressionUUID->"bfd3cfda-83dd-4b6d-8843-9d6a6f77ecfb"] }, Open ]], Cell["\<\ @@ -1958,13 +1950,14 @@ Cell[BoxData[ RowBox[{"test", "[", "\"\\"", "]"}], "[", "\"\\"", "]"}]], "Input", CellChangeTimes->{{3.942389711001149*^9, 3.94238971771292*^9}}, - CellLabel->"In[8]:=", + CellLabel->"In[70]:=", CellID->1148358653,ExpressionUUID->"8b72465f-5130-49c7-bfc3-2b65f94a5d2c"], Cell[BoxData["\<\"In\"\>"], "Output", - CellChangeTimes->{3.942389717964938*^9, 3.946097193333819*^9}, - CellLabel->"Out[8]=", - CellID->1125389869,ExpressionUUID->"5e26e7ad-7f75-422b-9e42-a023f9fa27d9"] + CellChangeTimes->{3.942389717964938*^9, 3.946097193333819*^9, + 3.947076272315714*^9}, + CellLabel->"Out[70]=", + CellID->1566898002,ExpressionUUID->"5d971526-3ba9-4413-8046-bbf9c52a28f5"] }, Open ]], Cell["\"LeadingOrder\" returns the leading order as a Function.", \ @@ -1980,7 +1973,7 @@ Cell[BoxData[ RowBox[{"test", "[", "\"\\"", "]"}], "[", "\"\\"", "]"}], "[", "r", "]"}]], "Input", CellChangeTimes->{{3.9460974197978*^9, 3.94609742733181*^9}}, - CellLabel->"In[10]:=", + CellLabel->"In[71]:=", CellID->1012977236,ExpressionUUID->"698bfb9e-cfc9-4f00-9c37-4845c7598a83"], Cell[BoxData[ @@ -1999,31 +1992,10 @@ Cell[BoxData[ SeriesData[$CellContext`\[Eta], 0, { Rational[4, 5] $CellContext`r^4 $CellContext`\[Omega]^4}, 4, 5, 1], Editable->False]], "Output", - CellChangeTimes->{{3.946097423832304*^9, 3.9460974276211567`*^9}}, - CellLabel->"Out[10]=", - CellID->487448549,ExpressionUUID->"f024c0ff-4569-4afa-adbe-b9616c22ad4b"] -}, Open ]], - -Cell["\<\ -\"TermCount\" returns the number of terms in the Series. It should always be \ -equivalent to the input. It crucially is not a simple copy of the input and \ -can an easy check that no orders got lost during the computation.\ -\>", "ExampleText", - CellChangeTimes->{{3.9460974455316772`*^9, 3.94609755360815*^9}, { - 3.94609760050492*^9, 3.946097625542478*^9}}, - CellID->23867073,ExpressionUUID->"7214ae85-c00f-487c-9614-00d82540e54c"], - -Cell[CellGroupData[{ - -Cell["test[\"In\"][\"TermCount\"]", "Input", - CellChangeTimes->{{3.946097555064722*^9, 3.946097567115387*^9}}, - CellLabel->"In[11]:=", - CellID->287138323,ExpressionUUID->"db436153-8f85-4af0-807b-080cd3345d3b"], - -Cell[BoxData["4"], "Output", - CellChangeTimes->{3.94609756743054*^9}, - CellLabel->"Out[11]=", - CellID->649025900,ExpressionUUID->"47e1263f-9a6b-45ec-bbbe-181cdee5b889"] + CellChangeTimes->{{3.946097423832304*^9, 3.9460974276211567`*^9}, + 3.947076277559806*^9}, + CellLabel->"Out[71]=", + CellID->1391491703,ExpressionUUID->"e0b0ee9e-954f-408e-8537-ab05bc9cb041"] }, Open ]], Cell["\"Normalization\" returns the OptionValue of \"Normalization\"", \ @@ -2039,20 +2011,21 @@ Cell[BoxData[ RowBox[{"test", "[", "\"\\"", "]"}], "[", "\"\\"", "]"}]], "Input", CellChangeTimes->{{3.946097806293185*^9, 3.946097812471332*^9}}, - CellLabel->"In[12]:=", + CellLabel->"In[73]:=", CellID->1845776202,ExpressionUUID->"ffdd38dc-acab-42f4-aa61-14fa3c18d13d"], Cell[BoxData["\<\"Default\"\>"], "Output", - CellChangeTimes->{3.946097812733303*^9}, - CellLabel->"Out[12]=", - CellID->40065728,ExpressionUUID->"14780fe6-5da8-4050-9623-4ec6a8e25a00"] + CellChangeTimes->{3.946097812733303*^9, 3.94707637835293*^9}, + CellLabel->"Out[73]=", + CellID->844595420,ExpressionUUID->"f4e4d711-0cc5-4f92-873e-8fcfbce154ff"] }, Open ]], Cell["\<\ \"Amplitudes\" returns the Amplitudes. By default they are not computed to \ -safe time. They can be computed when setting the Option \"Amplitudes\"->True\ +save time. They can be computed when setting the Option \"Amplitudes\"->True\ \>", "ExampleText", - CellChangeTimes->{{3.94609782578507*^9, 3.94609790087504*^9}}, + CellChangeTimes->{{3.94609782578507*^9, 3.94609790087504*^9}, { + 3.9470763854972878`*^9, 3.947076385593438*^9}}, CellID->561683896,ExpressionUUID->"50cb2bbf-719b-4890-9a44-bc18efbe52ea"], Cell[CellGroupData[{ @@ -2063,21 +2036,28 @@ Cell[BoxData[ "]"}]], "Input", CellChangeTimes->{{3.946097840119429*^9, 3.946097844176486*^9}, { 3.946097950292618*^9, 3.946097950594594*^9}}, - CellLabel->"In[18]:=", + CellLabel->"In[74]:=", CellID->1248178274,ExpressionUUID->"bbcc42e7-d588-4777-8c3d-be03c42f256c"], Cell[BoxData[ RowBox[{"\[LeftAssociation]", - RowBox[{"\<\"Transmission\"\>", "\[Rule]", "\<\"Not Computed\"\>"}], + RowBox[{ + RowBox[{"\<\"Incidence\"\>", "\[Rule]", + RowBox[{"Missing", "[", "\<\"NotComputed\"\>", "]"}]}], ",", + RowBox[{"\<\"Transmission\"\>", "\[Rule]", + RowBox[{"Missing", "[", "\<\"NotComputed\"\>", "]"}]}], ",", + RowBox[{"\<\"Reflection\"\>", "\[Rule]", + RowBox[{"Missing", "[", "\<\"NotComputed\"\>", "]"}]}]}], "\[RightAssociation]"}]], "Output", - 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TagBox["\"Number of terms: \"", "SummaryItemAnnotation"], + TagBox["\"Amplitudes: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", - TagBox["4", "SummaryItem"]}]}}, + TagBox["False", "SummaryItem"]}]}}, GridBoxAlignment -> { "Columns" -> {{Left}}, "Rows" -> {{Automatic}}}, AutoDelete -> False, GridBoxItemSize -> { @@ -2787,9 +2764,9 @@ Cell[BoxData[ DynamicModuleValues:>{}], "]"}], Teukolsky`PN`TeukolskyRadialFunctionPN[-2, 2, 2, $CellContext`a, $CellContext`\[Omega], {$CellContext`\[Eta], 4}, <| - "\[ScriptS]" -> -2, "\[ScriptL]" -> 2, "\[ScriptM]" -> 2, - "a" -> $CellContext`a, "PN" -> {$CellContext`\[Eta], 4}, - "RadialFunction" -> Function[Teukolsky`PN`Private`r, + "s" -> -2, "l" -> 2, "m" -> 2, "a" -> $CellContext`a, + "PN" -> {$CellContext`\[Eta], 4}, "RadialFunction" -> + Function[Teukolsky`PN`Private`r, SeriesData[$CellContext`\[Eta], 0, { Rational[3, 2] Teukolsky`PN`Private`r^(-1) $CellContext`\[Omega]^(-4), @@ -2803,20 +2780,23 @@ Cell[BoxData[ EulerGamma + Complex[0, 6] Pi + Teukolsky`PN`Private`r^3 $CellContext`\[Omega]^2 - 12 Log[2] - 12 Log[2 $CellContext`\[Omega]])}, -10, -6, 1]], "BoundaryCondition" -> - "Up", "LeadingOrder" -> Function[Teukolsky`PN`Private`r, + "Up", "SeriesMinOrder" -> -10, "LeadingOrder" -> + Function[Teukolsky`PN`Private`r, SeriesData[$CellContext`\[Eta], 0, { Rational[3, 2] Teukolsky`PN`Private`r^(-1) $CellContext`\[Omega]^(-4)}, -10, -9, 1]], "TermCount" -> 4, "Normalization" -> "UnitTransmission", - "Amplitudes" -> <|"Transmission" -> "Not Computed"|>, "Simplified" -> - True|>], + "Amplitudes" -> <| + "Incidence" -> Missing["NotComputed"], "Transmission" -> + Missing["NotComputed"], "Reflection" -> Missing["NotComputed"]|>, + "Simplify" -> True, "AmplitudesBool" -> False|>], Editable->False, SelectWithContents->True, Selectable->False]}]}], "\[RightAssociation]"}]], "Output", CellChangeTimes->{3.9421467318186274`*^9, 3.942146770079395*^9, - 3.9422320846656523`*^9}, - CellLabel->"Out[8]=", - CellID->317740830,ExpressionUUID->"56f61ead-29d2-4990-8696-d670b0b26d02"] + 3.9422320846656523`*^9, 3.947076482493676*^9}, + CellLabel->"Out[80]=", + CellID->1952047290,ExpressionUUID->"2737002e-3768-45a8-b1c5-cf5a5879e2a3"] }, Open ]], Cell[CellGroupData[{ @@ -2827,14 +2807,14 @@ Cell[BoxData[ RowBox[{"test", "[", "\"\\"", "]"}], "[", "r", "]"}], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.9421467364582863`*^9, 3.942146741679969*^9}}, - 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$CellContext`a^2)^ Rational[-1, 2]])}, -8, -4, 1]], "BoundaryCondition" -> "In", - "LeadingOrder" -> Function[Teukolsky`PN`Private`r, + "SeriesMinOrder" -> -8, "LeadingOrder" -> + Function[Teukolsky`PN`Private`r, SeriesData[$CellContext`\[Eta], 0, { Rational[1, 16] (-1 + $CellContext`a^2)^(-2) Teukolsky`PN`Private`r^4}, -8, -7, 1]], "TermCount" -> 4, "Normalization" -> "SasakiTagoshi", - "Amplitudes" -> <|"Transmission" -> "Not Computed"|>, "Simplified" -> - True|>], + "Amplitudes" -> <| + "Incidence" -> Missing["NotComputed"], "Transmission" -> + Missing["NotComputed"], "Reflection" -> Missing["NotComputed"]|>, + "Simplify" -> True, "AmplitudesBool" -> False|>], Editable->False, SelectWithContents->True, Selectable->False]}], ",", @@ -3436,13 +3361,11 @@ Cell[BoxData[ PaneBox[ ButtonBox[ DynamicBox[ - FEPrivate`FrontEndResource["FEBitmaps", "SummaryBoxOpener"], - ImageSizeCache -> { - 10.299843749999999`, {0., 10.299843749999999`}}], - Appearance -> None, BaseStyle -> {}, - ButtonFunction :> (Typeset`open$$ = True), Evaluator -> - Automatic, Method -> "Preemptive"], - Alignment -> {Center, Center}, ImageSize -> + FEPrivate`FrontEndResource[ + "FEBitmaps", "SummaryBoxOpener"]], + ButtonFunction :> (Typeset`open$$ = True), Appearance -> + None, BaseStyle -> {}, Evaluator -> Automatic, Method -> + "Preemptive"], Alignment -> {Center, Center}, ImageSize -> Dynamic[{ Automatic, 3.5 (CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[ @@ -3521,31 +3444,29 @@ Cell[BoxData[ TagBox[ "\"Boundary Condition: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", - TagBox["\"Up\"", "SummaryItem"]}]}}, AutoDelete -> False, - BaseStyle -> { - ShowStringCharacters -> False, NumberMarks -> False, - PrintPrecision -> 3, ShowSyntaxStyles -> False}, + TagBox["\"Up\"", "SummaryItem"]}]}}, GridBoxAlignment -> { - "Columns" -> {{Left}}, "Rows" -> {{Automatic}}}, - GridBoxItemSize -> { + "Columns" -> {{Left}}, "Rows" -> {{Automatic}}}, AutoDelete -> + False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings -> { - 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This can be mended by resetting the TimeConstraint Option \ -for Simplify. Here the time scaling seems to increase significantly\ +timeouts for 12PN. This can be fixed by setting the TimeConstraint Option for \ +Simplify. 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+Cell[211418, 4931, 136, 2, 70, "Template",ExpressionUUID->"074a6791-46ce-46f7-ab01-5b08fc7160db", CellID->1195948677], -Cell[213556, 5019, 138, 2, 70, "Template",ExpressionUUID->"c5e972c0-7706-446e-a514-90545f995ec6", +Cell[211557, 4935, 138, 2, 70, "Template",ExpressionUUID->"c5e972c0-7706-446e-a514-90545f995ec6", CellID->1220537702] }, Closed]] }, Open ]] From 755da6006a7f6d7f95803af06d4263d2941db207 Mon Sep 17 00:00:00 2001 From: Barry Wardell Date: Tue, 28 Jan 2025 18:22:57 +0000 Subject: [PATCH 11/13] Add links to the PN functions in the guide page of the docs --- Documentation/English/Guides/Teukolsky.nb | 111 ++++++++++++++++------ 1 file changed, 80 insertions(+), 31 deletions(-) diff --git a/Documentation/English/Guides/Teukolsky.nb b/Documentation/English/Guides/Teukolsky.nb index 8565b8a..070f88a 100644 --- a/Documentation/English/Guides/Teukolsky.nb +++ b/Documentation/English/Guides/Teukolsky.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest 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Closed]], Cell[CellGroupData[{ -Cell[7323, 204, 111, 1, 21, "KeywordsSection",ExpressionUUID->"ee0d1ebb-e321-4689-a801-bfe5b2945617", +Cell[9034, 245, 111, 1, 20, "KeywordsSection",ExpressionUUID->"ee0d1ebb-e321-4689-a801-bfe5b2945617", CellID->1447286583], -Cell[7437, 207, 100, 1, 70, "Keywords",ExpressionUUID->"70a8ff1f-e2ad-430e-a9f5-e8760f6b54ba", +Cell[9148, 248, 100, 1, 70, "Keywords",ExpressionUUID->"70a8ff1f-e2ad-430e-a9f5-e8760f6b54ba", CellID->2145814559] }, Closed]] }, Open ]] From 4b0cb7d621f8e9fff503223f199b0a6dfcd08e3c Mon Sep 17 00:00:00 2001 From: Barry Wardell Date: Tue, 28 Jan 2025 18:27:07 +0000 Subject: [PATCH 12/13] Produce a message and return $Failed when invalide parameters are specified --- Kernel/TeukolskyRadial.m | 11 +++++++++++ 1 file changed, 11 insertions(+) diff --git a/Kernel/TeukolskyRadial.m b/Kernel/TeukolskyRadial.m index 0c59e8c..71373f6 100644 --- a/Kernel/TeukolskyRadial.m +++ b/Kernel/TeukolskyRadial.m @@ -44,6 +44,8 @@ TeukolskyRadial::precw = "The precision of `1`=`2` is less than WorkingPrecision (`3`)."; TeukolskyRadial::optx = "Unknown options in `1`"; +TeukolskyRadial::params = "Invalid parameters s=`1`, l=`2`, m=`3`"; +TeukolskyRadial::cmplx = "Only real values of a are allowed, but a=`1` specified."; TeukolskyRadial::dm = "Option `1` is not valid with BoundaryConditions \[RightArrow] `2`."; TeukolskyRadial::sopt = "Option `1` not supported for static (\[Omega]=0) modes."; TeukolskyRadial::hc = "Method HeunC is only supported with Mathematica version 12.1 and later."; @@ -392,6 +394,15 @@ }; +TeukolskyRadial[s_?NumericQ, l_?NumericQ, m_?NumericQ, a_, \[Omega]_, OptionsPattern[]] /; + l < Abs[s] || Abs[m] > l || !AllTrue[{2s, 2l, 2m}, IntegerQ] || !IntegerQ[l-s] || !IntegerQ[m-s] := + (Message[TeukolskyRadial::params, s, l, m]; $Failed); + + +TeukolskyRadial[s_, l_, m_, a_Complex, \[Omega]_, OptionsPattern[]] := + (Message[TeukolskyRadial::cmplx, a]; $Failed); + + (* ::Subsubsection::Closed:: *) (*Static modes*) From cb8c805d44825d0dd8835926a0d2eb6dc183267d Mon Sep 17 00:00:00 2001 From: Barry Wardell Date: Tue, 28 Jan 2025 18:27:33 +0000 Subject: [PATCH 13/13] Add NHoldAll attribute for TeukolskyRadialFunction This ensures that N is never applied to the parameters in its association. --- Kernel/TeukolskyRadial.m | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Kernel/TeukolskyRadial.m b/Kernel/TeukolskyRadial.m index 71373f6..763da49 100644 --- a/Kernel/TeukolskyRadial.m +++ b/Kernel/TeukolskyRadial.m @@ -625,7 +625,7 @@ (*Numerical evaluation*) -SetAttributes[TeukolskyRadialFunction, {NumericFunction}]; +SetAttributes[TeukolskyRadialFunction, {NHoldAll}]; outsideDomainQ[r_, rmin_, rmax_] := Min[r]rmax;