complex exponential function

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

  • expz:=ex(cosy+isiny)

  • expz:=limn(1+zn)n

  • expz:=n=0znn!

The complex exponential function is usually denoted in power form:


where e is the Napier’s constant.  It also coincincides with the real exponential functionDlmfDlmfMathworldPlanetmathPlanetmath when z is real (choose  y=0).  It has all the properties of power, e.g.  e-z=1ez;  these are consequences of the addition formulaPlanetmathPlanetmath


of the complex exponential function.

The functionMathworldPlanetmath gets all complex values except 0 and is periodic ( having the (the with least non-zero modulus) 2πi.  The exp is holomorphic, its derivative


which is obtained from the series form via termwise differentiation, is similar as in .

So we have a fourth way to define

  • expz:=w(z)

with w the solution of the differential equationMathworldPlanetmathdwdz=w  under the initial conditionMathworldPlanetmathw(0)=1.

Some formulae:

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 periodPlanetmathPlanetmathPlanetmath