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complex exponential function
The complex exponential function $\exp:\,\mathbb{C}\to\mathbb{C}$ may be defined in many equivalent ways: Let $z=x\!+\!iy$ where $x,\,y\in\mathbb{R}$.

$\displaystyle\exp{z}\;:=\;e^{x}(\cos{y}+i\sin{y})$

$\displaystyle\exp{z}\;:=\;\lim_{{n\to\infty}}\left(1+\frac{z}{n}\right)^{n}$

$\displaystyle\exp{z}\;:=\;\sum_{{n=0}}^{\infty}\frac{z^{n}}{n!}$
The complex exponential function is usually denoted in power form:
$e^{z}\;:=\;\exp{z},$ 
where $e$ is the Napier’s constant. It also coincincides with the real exponential function when $z$ is real (choose $y=0$). It has all the properties of power, e.g. $e^{{z}}=\frac{1}{e^{z}}$; these are consequences of the addition formula
$e^{{z_{1}+z_{2}}}\;=\;e^{{z_{1}}}e^{{z_{2}}}$ 
of the complex exponential function.
The function gets all complex values except 0 and is periodic having the prime period (the period with least nonzero modulus) $2\pi i$. The $\exp$ is holomorphic, its derivative
$\frac{d}{dz}e^{z}\;=\;e^{z},$ 
which is obtained from the series form via termwise differentiation, is similar as in $\mathbb{R}$.
So we have a fourth way to define

$\exp{z}\;:=\;w(z)$
with $w$ the solution of the differential equation $\displaystyle\frac{dw}{dz}=w$ under the initial condition $w(0)=1$.
Some formulae:
$e^{z}\;=\;e^{x},\quad\arg{e^{z}}\;=\;y+2n\pi\quad(n=0,\,\pm 1,\,\pm 2,\,% \ldots),$ 
$\mbox{Re}(e^{z})\;=\;e^{x}\cos{y},\quad\mbox{Im}(e^{z})\;=\;e^{x}\sin{y}$ 
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