complex exponential function

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

• $\exp z:=e^x(\cos y+i\sin y)$

• $\exp z:=\underset{n\to \mathrm{\infty }}{lim}{\left(1+{\displaystyle \frac{z}{n}}\right)}^n$

• $\exp z:={\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\displaystyle \frac{z^n}{n!}}$

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 $ℝ$.

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:

$$|e^z|=e^x,\arg e^z=y+2n\pi \mathit{\hspace{1em}}(n=0,\pm 1,\pm 2,\mathrm{\dots }),$$

$$\text{Re}(e^z)=e^x\cos y,\text{Im}(e^z)=e^x\sin y$$

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