双曲函数

h-hyperbolic functions

利用指数函数和三角函数定义：
$c h = \cosh s h = \sinh t h = \tanh$

\begin{aligned} \sinh x = \frac{e^{x} - e^{- x}}{2} & , \cosh x = \frac{e^{x} + e^{- x}}{2} \\ \tanh x = \frac{e^{x} - e^{- x}}{e^{x} + e^{- x}} & , \coth x = \frac{e^{x} + e^{- x}}{e^{x} - e^{- x}} \\ s e c h x = \frac{2}{e^{x} + e^{- x}} & , c s c h x = \frac{2}{e^{x} - e^{- x}} \end{aligned}

\begin{array}{r} (\sinh x)^{'} = \cosh x \\ (\cosh x)^{'} = \sinh x \end{array}

$\cosh y i = \cos y \sinh y i = i \sin y$
$\tanh y i = i \tan y$

以 $2 π i$ 为周期

\begin{aligned} A r c s i n h z & = L n (z + \sqrt{z^{2} + 1}) \\ A r c o s h z & = L n (z + \sqrt{z^{2} - 1}) \\ A r t a n h z & = \frac{1}{2} L n \frac{1 + z}{1 - z} \end{aligned}