Exponential function

Definition 2..1 $w = e^{z} = e^{x}(\cos{y} + i\sin{y})$ is called an exponential function and expressed as $w = e^{z}$ or $w = \exp{z}$ .

Theorem 2..1 (1) $e^{z_{1} + z_{2}} = e^{z_{1}}e^{z_{2}}, (e^{z})^{n} = e^{nz} (n is integer)$
(2) $e^{z}$ has a period of $2\pi i$ .

Proof (1) Let $z_{1} = x_{1} + i y_{1}, z_{2} = x_{2} + i y_{2}$ . Then

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

(2) When $e^{z + c} = e^{z}$ holds, the function $e^{z}$ is said to have a period $c$

. Then we let $e^{z + c} = e^{z}$ . Since $e^{z}e^{c} = e^{z}$ , we have $e^{c} = 1$ . Let $c = a + bi$

. Then $e^{c} = e^{a + bi} = e^{a}(\cos{b} + i\sin{b}) = 1$ implies $e^{a}\cos{b} = 1, e^{a}\sin{b} = 0$ . Thus, $a = 0, \cos{b} = 1, \sin{b} = 0$ . $b$