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Try removing \ warning in latex
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@mathieuboudreau
mathieuboudreau committed Oct 7, 2024
1 parent 7752211 commit c38a2cc
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2 changes: 1 addition & 1 deletion 2 T1 Mapping/2-1 Inversion Recovery/03-IR_DataFitting.md
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Expand Up @@ -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}
```
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38 changes: 19 additions & 19 deletions 4 B1 Mapping/01-Double Angle technique/A1-Appendix A.md
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Expand Up @@ -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}
```
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