# periodicity of exponential function

###### Theorem.

The only periods of the complex exponential function$z\mapsto e^{z}$  are the multiples of $2\pi i$.  Thus the function is one-periodic.

Proof.  Let $\omega$ be any period of the exponential function, i.e.  $e^{z+\omega}=e^{z}e^{\omega}=e^{z}$  for all  $z\in\mathbb{C}$.  Because $e^{z}$ is always $\neq 0$, we have

 $\displaystyle e^{\omega}\;=\;1.$ (1)

If we set  $\omega=:a\!+\!ib$  with $a$ and $b$ reals, (1) gets the form

 $\displaystyle e^{a}\cos{b}+ie^{a}\sin{b}\;=\;1,$ (2)

which implies (see equality of complex numbers)

 $e^{a}\cos{b}\;=\;1,\qquad e^{a}\sin{b}\;=\;0.$

As these equations are squared and added, we obtain  $e^{2a}=1$  which , since $a$ is real, that  $a=0$.  Thus the preceding equations get the form

 $\cos{b}\;=\;1,\qquad\sin{b}\;=\;0.$

These result that  $b=n\!\cdot\!2\pi$  and therefore

 $\omega\;=\;n\!\cdot\!2\pi i\qquad(n\;=\;0,\,\pm 1,\,\pm 2,\,\pm 3,\,\ldots)$

Q.E.D.

## References

• 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