複素解析 (Complex Analysis)

オイラーの公式 (Euler's formula)

e^{ix} = \cos x + i \sin x

\begin{aligned} \cos x & = \Re \left( e^{ix} \right) & = \frac{e^{ix} + e^{-ix}}{2} \\ \sin x & = \Im \left( e^{ix} \right) & = \frac{e^{ix} - e^{-ix}}{2i} \\ \end{aligned}

\begin{aligned} e^{ix} & = \cos x + i \sin x \\ e^{-ix} & = \cos x - i \sin x \\ \end{aligned}

\begin{aligned} \cos (ix) & = \frac{e^{-x} + e^{x}}{2} & = \cosh x \\ \sin (ix) & = \frac{e^{-x} - e^{x}}{2i} & = i \sinh x \\ \end{aligned}