쌍곡함수

정의

$z \in \mathbb{C}$에 대해서,

$$ \begin{align*} \sinh z &:= \frac{e^{z}-e^{-z}}{2} \\ \cosh z &:= \frac{e^{z}+e^{-z}}{2} \\ \tanh z &:= \frac{\sinh z}{\cosh z} \end{align*} $$

$$ \begin{align*} \mathrm{csch}x&=\frac{1}{\sinh x} \\ \mathrm{sech} x&=\frac{1}{\cosh x} \\ \coth x &=\frac{1}{\tanh x} \end{align*} $$

삼각함수와의 관계

$$ \begin{align*} \sinh (iz) &= i\sin z \\ \sin (iz) &= i\sinh z \\ \cosh (iz) &= \cos z \\ \cos (iz) &= \cosh z \end{align*} $$

미분

$$ \begin{align*} (\sinh x)^{\prime} &= \cosh x \\ (\cosh x )^{\prime} &= \sinh x \\ (\tanh x)^{\prime} &= \frac{1}{\cosh^{2} x}=\mathrm{sech}^{2}x \end{align*} $$

항등식

$$ \begin{align*} \sinh(-x) &= -\sinh x \\ \cosh(-x) &= \cosh x \\ \tanh(-x)&=- \tanh x \\ \cosh x + \sinh x &=e^{x} \\ \cosh x - \sinh x &= e^{-x} \\ \cosh^{2}x -\sinh^{2}x &=1 \end{align*} $$

$$ \begin{align*} \sinh (x\pm y) &=\sinh x \cosh y \pm \sinh y \cosh x \\ \cosh (x \pm y) &= \cosh x \cosh y \pm \sinh x \sinh y \\ \tanh{x \pm y}&=\frac{\tanh x \pm \tanh y}{1 \pm \tanh x \tanh y} \end{align*} $$

배각공식

$$ \begin{align*} \sinh (2x) &= 2\sinh x \cosh x \\ \cosh (2x) &= \cosh^{2} x + \sinh^{2} x \\ &=2\cosh ^{2 } x -1 = 2\sinh ^{2} x +1 \\ \tanh (2x) &= \frac{2\tanh x}{1+\tanh^{2}x} \end{align*} $$

반각 공식

$$ \begin{align*} \sinh^{2} \frac{x}{2} &=\frac{\cosh x -1 }{2} \\ \cosh^{2} \frac{x}{2} &=\frac{\cosh x +1 }{2} \\ \tanh ^{2} \frac{x}{2} &= \frac{\cosh x -1}{\cosh x +1} \end{align*} $$

$$ \begin{align*} \sinh \frac{x}{2}&=\frac{\sinh x}{\sqrt{2(\cosh x +1)}} \\ \cosh \frac{x}{2}&=\frac{\sinh x}{\mathrm{sgn}(x)\sqrt{2(\cosh x -1)}} \end{align*} $$

라플라스 변환

$$ \begin{align*} \mathcal{L} \left\{ \sinh (at) \right\} &= \dfrac{a}{s^2-a^2} ,&s>|a| \\ \mathcal{L} \left\{ \cosh (at) \right\} &= \dfrac{s}{s^2-a^2}, &s>|a| \end{align*} $$

합차 공식, 곱셈공식

$$ \begin{align*} \sinh x +\sinh y &=2\sinh \frac{x+y}{2} \cosh \frac{x-y}{2} \\ \sinh x -\sinh y &=2\sinh \frac{x-y}{2} \cosh \frac{x+y}{2} \\ \cosh x + \cosh y &= 2 \cosh \frac{x+y}{2} \cosh \frac{x-y}{2} \\ \cosh x -\cosh y &= 2 \sinh \frac{x+y}{2}\sinh \frac{x-y}{2} \end{align*} $$

$$ \begin{align*} \sinh x \sinh y &= \frac{\cosh (x+y)-\cosh (x-y)}{2} \\ \sinh x \cosh y &= \frac{\sinh (x+y)+\sinh (x-y)}{2} \\ \cosh x \sinh y &= \frac{\sinh (x+y)-\sinh (x-y)}{2} \\ \cosh x \cosh y &= \frac{\cosh (x+y)+\cosh (x-y)}{2} \end{align*} $$