periodicity of exponential function


The only periods of the complex exponential functionzez  are the multiples of 2πi.  Thus the functionMathworldPlanetmath is one-periodic.

Proof.  Let ω be any period of the exponential functionDlmfDlmfMathworldPlanetmathPlanetmath, i.e.  ez+ω=ezeω=ez  for all  z.  Because ez is always 0, we have

eω= 1. (1)

If we set  ω=:a+ib  with a and b reals, (1) gets the form

eacosb+ieasinb= 1, (2)

which implies (see equality of complex numbers)

eacosb= 1,easinb= 0.

As these equations are squared and added, we obtain  e2a=1  which , since a is real, that  a=0.  Thus the preceding equations get the form

cosb= 1,sinb= 0.

These result that  b=n2π  and therefore

ω=n2πi  (n= 0,±1,±2,±3,)



  • 1 Ernst Lindelöf: Johdatus funktioteoriaan (‘Introduction to function theory’).  Mercatorin kirjapaino, Helsinki (1936).
Title periodicity of exponential function
Canonical name PeriodicityOfExponentialFunction
Date of creation 2014-02-20 14:29:59
Last modified on 2014-02-20 14:29:59
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 16
Author pahio (2872)
Entry type Theorem
Classification msc 32A05
Classification msc 30D20
Synonym period of exponential function
Related topic PeriodicFunctions
Related topic AnalyticContinuationOfRiemannZetaUsingIntegral
Related topic ExamplesOfPeriodicFunctions
Related topic ExponentialFunctionNeverVanishes
Defines one-periodic