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쌍곡함수 📂함수

쌍곡함수

정의

z∈Cz \in \mathbb{C}에 대해서,

sinh⁡z:=ez−e−z2cosh⁡z:=ez+e−z2tanh⁡z:=sinh⁡zcosh⁡z \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*}

cschx=1sinh⁡xsechx=1cosh⁡xcoth⁡x=1tanh⁡x \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

sinh⁡(iz)=isin⁡zsin⁡(iz)=isinh⁡zcosh⁡(iz)=cos⁡zcos⁡(iz)=cosh⁡z \begin{align*} \sinh (iz) &= i\sin z \\ \sin (iz) &= i\sinh z \\ \cosh (iz) &= \cos z \\ \cos (iz) &= \cosh z \end{align*}

미분

(sinh⁡x)′=cosh⁡x(cosh⁡x)′=sinh⁡x(tanh⁡x)′=1cosh⁡2x=sech2x \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

sinh⁡(−x)=−sinh⁡xcosh⁡(−x)=cosh⁡xtanh⁡(−x)=−tanh⁡xcosh⁡x+sinh⁡x=excosh⁡x−sinh⁡x=e−xcosh⁡2x−sinh⁡2x=1 \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*}

덧셈정리

sinh⁡(x±y)=sinh⁡xcosh⁡y±sinh⁡ycosh⁡xcosh⁡(x±y)=cosh⁡xcosh⁡y±sinh⁡xsinh⁡ytanh⁡x±y=tanh⁡x±tanh⁡y1±tanh⁡xtanh⁡y \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

sinh⁡(2x)=2sinh⁡xcosh⁡xcosh⁡(2x)=cosh⁡2x+sinh⁡2x=2cosh⁡2x−1=2sinh⁡2x+1tanh⁡(2x)=2tanh⁡x1+tanh⁡2x \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

sinh⁡2x2=cosh⁡x−12cosh⁡2x2=cosh⁡x+12tanh⁡2x2=cosh⁡x−1cosh⁡x+1 \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*}

sinh⁡x2=sinh⁡x2(cosh⁡x+1)cosh⁡x2=sinh⁡xsgn(x)2(cosh⁡x−1) \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

L{sinh⁡(at)}=as2−a2,s>∣a∣L{cosh⁡(at)}=ss2−a2,s>∣a∣ \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

sinh⁡x+sinh⁡y=2sinh⁡x+y2cosh⁡x−y2sinh⁡x−sinh⁡y=2sinh⁡x−y2cosh⁡x+y2cosh⁡x+cosh⁡y=2cosh⁡x+y2cosh⁡x−y2cosh⁡x−cosh⁡y=2sinh⁡x+y2sinh⁡x−y2 \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*}

sinh⁡xsinh⁡y=cosh⁡(x+y)−cosh⁡(x−y)2sinh⁡xcosh⁡y=sinh⁡(x+y)+sinh⁡(x−y)2cosh⁡xsinh⁡y=sinh⁡(x+y)−sinh⁡(x−y)2cosh⁡xcosh⁡y=cosh⁡(x+y)+cosh⁡(x−y)2 \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*}