cosh(i x), sinh(i x) (純虚数に対するcosh, sinh)

$\cosh \left(ix\right)$

$$\begin{array}{rcl}\cosh \left(x\right)& =& \frac{e^x+e^{-x}}{2}\\ \cosh \left(ix\right)& =& \frac{e^{ix}+e^{-ix}}{2}\\ & =& \frac{\{\cos \left(x\right)+i\sin \left(x\right)\}+\{\cos (-x)+i\sin (-x)\}}{2}\\ & & \dots e^{ix}=\cos \left(x\right)+i\sin \left(x\right)\\ & =& \frac{\{\cos \left(x\right)+i\sin \left(x\right)\}+\{\cos \left(x\right)-i\sin \left(x\right)\}}{2}\\ & & \dots \cos (-x)=\cos \left(x\right),\sin (-x)=-\sin (-x)\\ & =& \frac{\cos \left(x\right)+i\sin \left(x\right)+\cos \left(x\right)-i\sin \left(x\right)}{2}\\ & =& \frac{\cos \left(x\right)+\cos \left(x\right)}{2}\\ & =& \frac{2\cos \left(x\right)}{2}\\ & =& \cos \left(x\right)\end{array}$$

$\sinh \left(ix\right)$

$$\begin{array}{rcl}\sinh \left(x\right)& =& \frac{e^x-e^{-x}}{2}\\ \sinh \left(ix\right)& =& \frac{e^{ix}-e^{-ix}}{2}\\ & =& \frac{\{\cos \left(x\right)+i\sin \left(x\right)\}-\{\cos (-x)+i\sin (-x)\}}{2}\\ & & \dots e^{ix}=\cos \left(x\right)+i\sin \left(x\right)\\ & =& \frac{\{\cos \left(x\right)+i\sin \left(x\right)\}-\{\cos \left(x\right)-i\sin \left(x\right)\}}{2}\\ & & \dots \cos (-x)=\cos \left(x\right),\sin (-x)=-\sin (-x)\\ & =& \frac{\cos \left(x\right)+i\sin \left(x\right)-\cos \left(x\right)+i\sin \left(x\right)}{2}\\ & =& \frac{i\sin \left(x\right)+i\sin \left(x\right)}{2}\\ & =& \frac{2i\sin \left(x\right)}{2}\\ & =& i\sin \left(x\right)\end{array}$$

$cos(ix),sin(ix)$ (純虚数に対する$\cos ,\sin$)

$cos(ix),sin(ix)$ (純虚数に対する$\cos ,\sin$)