指数関数

定義 2..1 $w = e^{z} = e^{x}(\cos{y} + i\sin{y})$ を指数関数といい， $w = e^{z}$ あるいは $w = \exp{z}$ と表わす．

定理 2..1 (1) $e^{z_{1} + z_{2}} = e^{z_{1}}e^{z_{2}}, \ (e^{z})^{n} = e^{nz} \ (nは整数)$
(2) $e^{z}$ は周期 $2\pi i$ を持つ．

証明 (1) $z_{1} = x_{1} + i y_{1}, z_{2} = x_{2} + i y_{2}$ とすると，

$\displaystyle e^{z_{1} + z_{2}}$	$\displaystyle =$	$\displaystyle e^{x_{1} + x_{2} + i(y_{1} + y_{2})}$
	$\displaystyle =$	$\displaystyle e^{x_{1} + x_{2}}(\cos{(y_{1} + y_{2})} + i\sin(y_{1} + y_{2}))$
	$\displaystyle =$	$\displaystyle e^{x_{1} + x_{2}}[\cos{y_{1}}\cos{y_{2}} - \sin{y_{1}}\sin{y_{2}} + i(\sin{y_{1}}\cos{y_{2}} + \cos{y_{1}}\sin{y_{2}})]$
	$\displaystyle =$	$\displaystyle e^{x_{1} + x_{2}}[(\cos{y_{1}} + i\sin{y_{1}})(\cos{y_{2}}+i\sin{y_{2}})]$
	$\displaystyle =$	$\displaystyle e^{x_{1}}(\cos{y_{1}} + i\sin{y_{1}})e^{x_{2}}(\cos{y_{2}}+i\sin{y_{2}})$
	$\displaystyle =$	$\displaystyle e^{z_{1}}e^{z_{2}}$