# 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 (http://planetmath.org/PeriodicityOfExponentialFunction) having the (the with least non-zero 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:

 $|e^{z}|\;=\;e^{x},\quad\arg{e^{z}}\;=\;y+2n\pi\quad(n=0,\,\pm 1,\,\pm 2,\,% \ldots),$
 $\mbox{Re}(e^{z})\;=\;e^{x}\cos{y},\quad\mbox{Im}(e^{z})\;=\;e^{x}\sin{y}$
 Title complex exponential function Canonical name ComplexExponentialFunction Date of creation 2013-03-22 14:43:08 Last modified on 2013-03-22 14:43:08 Owner pahio (2872) Last modified by pahio (2872) Numerical id 22 Author pahio (2872) Entry type Definition Classification msc 30D20 Classification msc 32A05 Related topic ExponentialFunctionDefinedAsLimitOfPowers Related topic ExponentialFunction Related topic ComplexSineAndCosine Related topic ProofOfEquivalenceOfFormulasForExp Related topic DerivativeOfExponentialFunction Related topic ConvergenceOfRiemannZetaSeries Defines exponential function Defines prime period