三角関数・双曲線関数

% カスタムコマンド定義
\newcommand{\sech}{\operatorname{sech}}
%\newcommand{\arsinh}{\operatorname{arsinh}}
\newcommand{\arcsin}{\sin^{-1}}
\newcommand{\arccos}{\cos^{-1}}
\newcommand{\arctan}{\tan^{-1}}
\newcommand{\arsinh}{\sinh^{-1}}
\newcommand{\arcosh}{\cosh^{-1}}
\newcommand{\artanh}{\tanh^{-1}}
\newcommand{\intz}[1]{\int #1 \, \mathrm{d}z}

\begin{align}
\sin z &= \frac{e^{iz} - e^{-iz}}{2i} \\
\cos z &= \frac{e^{iz} + e^{-iz}}{2} \\
\tan z &= \frac{1}{i} \left( 1 - \frac{2}{e^{2iz} + 1} \right) \\
\sinh z &= \frac{e^{z} - e^{-z}}{2} \\
\cosh z &= \frac{e^{z} + e^{-z}}{2} \\
\tanh z &= 1 - \frac{2}{e^{2z} + 1} \\
\end{align}

回転

\begin{align}
\sin iz &= i \sinh z \\
\cos iz &= \cosh z \\
\tan iz &= i \tanh z \\
\sinh iz &= i \sin z \\
\cosh iz &= \cos z \\
\tanh iz &= i \tan z \\
\end{align}

\begin{align}
\sin(-z) &= -\sin z \\
\cos(-z) &= \cos z \\
\tan(-z) &= -\tan z \\
\sinh(-z) &= -\sinh z \\
\cosh(-z) &= \cosh z \\
\tanh(-z) &= -\tanh z \\
\end{align}

90度シフト

\begin{align}
\sin\left( z + \frac{\pi}{2} \right) &= \cos z \\
\cos\left( z + \frac{\pi}{2} \right) &= -\sin z \\
\tan\left( z + \frac{\pi}{2} \right) &= -\cot z = -\frac{1}{\tan z} \\
\sinh\left( z + \frac{i\pi}{2} \right) &= i \cosh z \\
\cosh\left( z + \frac{i\pi}{2} \right) &= i \sinh z \\
\tanh\left( z + \frac{i\pi}{2} \right) &= i \coth z = \frac{i}{\tanh z}\\
\end{align}

180度シフト

\begin{align}
\sin\left( z + \pi \right) &= -\sin z \\
\cos\left( z + \pi \right) &= -\cos z \\
\tan\left( z + \pi \right) &= \tan z \\
\sinh\left( z + i\pi \right) &= -\sinh z \\
\cosh\left( z + i\pi \right) &= -\cosh z \\
\tanh\left( z + i\pi \right) &= \tanh z \\
\end{align}

加法定理

\begin{align}
\sin\left( z + w \right) &= \sin z \cos w + \cos z \sin w \\
\cos\left( z + w \right) &= \cos z \cos w - \sin z \sin w \\
\tan\left( z + w \right) &= \frac{\tan z + \tan w}{1 - \tan z \tan w} \\
\sinh\left( z + w \right) &= \sinh z \cosh w + \cosh z \sinh w \\
\cosh\left( z + w \right) &= \cosh z \cosh w + \sinh z \sinh w \\
\tanh\left( z + w \right) &= \frac{\tanh z + \tanh w}{1 + \tanh z \tanh w} \\
\end{align}

微分

\begin{align}
(\sin z)' &= \cos z \\
(\cos z)' &= -\sin z \\
(\tan z)' &= \sec^2 z = \frac{1}{\cos^2 z}\\
(\sinh z)' &= \cosh z \\
(\cosh z)' &= \sinh z \\
(\tanh z)' &= \sech^2 z = \frac{1}{\cosh^2 z} \\
\end{align}

積分

$C$ は積分定数。

\begin{align}
\intz{\sin z} &= -\cos z + C \\
\intz{\cos z} &= \sin z + C \\
\intz{\tan z} &= -\log(\cos z) + C \\
\intz{\sinh z} &= \cosh z + C \\
\intz{\cosh z} &= \sinh z + C \\
\intz{\tanh z} &= \log(\cosh z) + C \\
\end{align}

逆三角関数・逆双曲線関数

以下、主値を定めずに多価関数のまま扱います。

\begin{align}
\arcsin z &= -i \log\left( iz \pm \sqrt{1 - z^2} \right) \\
\arccos z &= -i \log\left( z \pm i \sqrt{1 - z^2} \right) \\
\arctan z &= \frac{1}{2i} \log{\frac{1+iz}{1-iz}} \\
\arsinh z &= \log\left( z \pm \sqrt{z^2 + 1} \right) \\
\arcosh z &= \log\left( z \pm \sqrt{z^2 - 1} \right) = i \arccos z \\
\artanh z &= \frac{1}{2} \log{\frac{1+z}{1-z}} \\
\end{align}

回転

\begin{align}
\arcsin iz &= i \arsinh z \\
\arctan iz &= i \artanh z \\
\arsinh iz &= i \arcsin z \\
\artanh iz &= i \arctan z \\
\end{align}

\begin{align}
\arcsin(-z) &= -\arcsin z \\
\arccos(-z) &= \pi - \arccos z \\
\arctan(-z) &= -\arctan z \\
\arsinh(-z) &= -\arsinh z \\
\arcosh(-z) &= i\pi - \arcosh z \\
\artanh(-z) &= -\artanh z \\
\end{align}

対称性

\begin{align}
\arcsin z &= \frac{\pi}{2} - \arccos z \\
\arccos z &= \frac{\pi}{2} - \arcsin z \\
\end{align}

微分

\begin{align}
(\arcsin z)' &= \pm \frac{1}{\sqrt{1 - z^2}} \\
(\arccos z)' &= \mp \frac{1}{\sqrt{1 - z^2}} \\
(\arctan z)' &= \frac{1}{z^2 + 1} \\
(\arsinh z)' &= \pm \frac{1}{\sqrt{z^2 + 1}} \\
(\arcosh z)' &= \pm \frac{1}{\sqrt{z^2 - 1}} \\
(\artanh z)' &= \frac{1}{1 - z^2} \\
\end{align}

積分

$C$ は積分定数。

\begin{align}
\intz{\arcsin z} &= z \arcsin z \pm \sqrt{1 - z^2} + C \\
\intz{\arccos z} &= z \arccos z \mp \sqrt{1 - z^2} + C \\
\intz{\arctan z} &= z \arctan z - \frac{1}{2} \log(z^2 + 1) + C \\
\intz{\arsinh z} &= z \arsinh z \mp \sqrt{z^2 + 1} + C \\
\intz{\arcosh z} &= z \arcosh z \mp \sqrt{z^2 - 1} + C \\
\intz{\artanh z} &= z \artanh z + \frac{1}{2} \log(1 - z^2) + C \\
\end{align}