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:

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

of the complex exponential function.

The function gets all complex values except 0 and is periodic having the prime period (the period with least non-zero modulus) $2\pi i$.  The $\exp$ is holomorphic, its derivative

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: