## 2019年11月3日日曜日

### 数学 - 解析学 - “ε-δ”その他 - 複素数 - 極形式 - 極形式を通常の形に変換

1. $\begin{array}{l}{e}^{3i\pi }\\ =\mathrm{cos}\left(3\pi \right)+i\mathrm{sin}\left(3\pi \right)\\ =-1\end{array}$

2. $\begin{array}{l}{e}^{\frac{2i\pi }{3}}\\ =\mathrm{cos}\left(\frac{2}{3}\pi \right)+i\mathrm{sin}\left(\frac{2}{3}\pi \right)\\ =-\frac{1}{2}+\frac{\sqrt{3}}{2}i\end{array}$

3. $\begin{array}{l}3{e}^{\frac{i\pi }{4}}\\ =3\left(\mathrm{cos}\frac{\pi }{4}+i\mathrm{sin}\frac{\pi }{4}\right)\\ =3\left(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}i\right)\\ =\frac{3}{\sqrt{2}}+\frac{3}{\sqrt{3}}i\end{array}$

4. $\begin{array}{l}\pi {e}^{\frac{-i\pi }{3}}\\ =\pi \left(\mathrm{cos}\left(-\frac{\pi }{3}\right)+i\mathrm{sin}\left(-\frac{\pi }{3}\right)\right)\\ =\pi \left(\frac{1}{2}-\frac{\sqrt{3}}{2}i\right)\\ =\frac{\pi }{2}-\frac{\sqrt{3}}{2}\pi i\end{array}$

5. $\begin{array}{l}{e}^{\frac{2\pi }{6}i}\\ =\mathrm{cos}\frac{\pi }{3}+i\mathrm{sin}\frac{\pi }{3}\\ =\frac{1}{2}+\frac{\sqrt{3}}{2}i\end{array}$

6. $\begin{array}{l}{e}^{-\frac{i\pi }{2}}\\ =\mathrm{cos}\left(-\frac{\pi }{2}\right)+i\mathrm{sin}\left(-\frac{\pi }{2}\right)\\ =-i\end{array}$

7. $\begin{array}{l}{e}^{-i\pi }\\ =\mathrm{cos}\left(-\pi \right)+i\mathrm{sin}\left(-\pi \right)\\ =-1\end{array}$

8. $\begin{array}{l}{e}^{-\frac{5i\pi }{4}}\\ =\mathrm{cos}\left(-\frac{5}{4}\pi \right)+i\mathrm{sin}\left(-\frac{5}{4}\pi \right)\\ =-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}i\end{array}$