From c38a2cc349e38ce6b136ca729286b2a80e148301 Mon Sep 17 00:00:00 2001 From: Mathieu Boudreau Date: Mon, 7 Oct 2024 14:26:00 -0300 Subject: [PATCH] Try removing \ warning in latex --- .../03-IR_DataFitting.md | 2 +- .../A1-Appendix A.md | 38 +++++++++---------- 2 files changed, 20 insertions(+), 20 deletions(-) diff --git a/2 T1 Mapping/2-1 Inversion Recovery/03-IR_DataFitting.md b/2 T1 Mapping/2-1 Inversion Recovery/03-IR_DataFitting.md index 4a34556..94b6695 100644 --- a/2 T1 Mapping/2-1 Inversion Recovery/03-IR_DataFitting.md +++ b/2 T1 Mapping/2-1 Inversion Recovery/03-IR_DataFitting.md @@ -21,7 +21,7 @@ Early implementations of [inversion recovery](wiki:Inversion_recovery) fitting a ```{math} :label: irEq4 :enumerator:2.4 -\begin{equation}\label{eq:1.4} +\begin{equation} S(TI) = a + be^{- \frac{TI}{T_1}} \end{equation} ``` diff --git a/4 B1 Mapping/01-Double Angle technique/A1-Appendix A.md b/4 B1 Mapping/01-Double Angle technique/A1-Appendix A.md index 993841d..448b866 100644 --- a/4 B1 Mapping/01-Double Angle technique/A1-Appendix A.md +++ b/4 B1 Mapping/01-Double Angle technique/A1-Appendix A.md @@ -24,28 +24,28 @@ This content of this section is still a work-in-progress and has not been proofr :enumerator: \begin{equation} \begin{split} -\text{e}^{i2\alpha} &= \text{e}^{i\left( \alpha+\alpha \right)} \\ -&= \text{e}^{i\alpha+i\alpha} \\ -&= \text{e}^{i\alpha}\text{e}^{i\alpha} \\ -&= \left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right)\left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right) \\ -&= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+i^{2}\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\\ -&= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+\left( -1 \right)\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\\ -&= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right)\\ -&= \text{cos}^{2}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\\ -&= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right) +\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\\ -&= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right)+\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right)\\ -\text{e}^{i2\alpha} &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\\ -\text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right)\\ +\text{e}^{i2\alpha} &= \text{e}^{i\left( \alpha+\alpha \right)} +&= \text{e}^{i\alpha+i\alpha} +&= \text{e}^{i\alpha}\text{e}^{i\alpha} +&= \left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right)\left( \text{cos}\left( \alpha \right)+i\text{sin}\left( \alpha \right) \right) +&= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+i^{2}\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right) +&= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)+\left( -1 \right)\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right) +&= \text{cos}\left( \alpha \right)\text{cos}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}\left( \alpha \right)\text{sin}\left( \alpha \right) +&= \text{cos}^{2}\left( \alpha \right)+i\text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right)+i\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right) +&= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{cos}\left( \alpha \right)\text{sin}\left( \alpha \right) +\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right) +&= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( \text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right)+\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right) \right) +\text{e}^{i2\alpha} &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right) +\text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) &= \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right) \end{split} \\ \\ -\text{For }z \in \mathbb{C} \text{ and }q \in \mathbb{C}\text{,}\\ -\text{if }z=q \\ -\text{then }\text{Re}\left( z \right)=\text{Re}\left( q \right) \\ -\text{ and }\text{Im}\left( z \right)=\text{Im}\left( q \right) \\ +\text{For }z \in \mathbb{C} \text{ and }q \in \mathbb{C}\text{,} +\text{if }z=q +\text{then }\text{Re}\left( z \right)=\text{Re}\left( q \right) +\text{ and }\text{Im}\left( z \right)=\text{Im}\left( q \right) \text{thus,} \\ \\ \begin{split} -\text{Im}\left( \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) \right) &= \text{Im}\left( \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right) \right)\\ -\text{sin}\left( 2\alpha \right) &= 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \\ -\end{split}\\ +\text{Im}\left( \text{cos}\left( 2\alpha \right)+i\text{sin}\left( 2\alpha \right) \right) &= \text{Im}\left( \left( \text{cos}^{2}\left( \alpha \right)-\text{sin}^{2}\left( \alpha \right)\right)+i\left( 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) \right) \right) +\text{sin}\left( 2\alpha \right) &= 2\text{sin}\left( \alpha \right) \text{cos}\left( \alpha \right) +\end{split} Q.E.D. \end{equation} ```