From 72a7b4a197206716553bf4d2f196a429ed8853a9 Mon Sep 17 00:00:00 2001 From: Mathieu Boudreau Date: Mon, 7 Oct 2024 14:31:47 -0300 Subject: [PATCH] Revert --- .../A1-Appendix A.md | 38 +++++++++---------- 1 file changed, 19 insertions(+), 19 deletions(-) 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 448b866..993841d 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} ```