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  $z\in ℂ$.  Because $e^z$ is always $\ne 0$, we have

$e^{\omega }=\mathrm{\hspace{0.33em}1}.$ (1)

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

$e^a\cos b+i{e}^a\sin b=\mathrm{\hspace{0.33em}1},$ (2)

which implies (see equality of complex numbers)

$$e^a\cos b=\mathrm{\hspace{0.33em}1},e^a\sin b=\mathrm{\hspace{0.33em}0}.$$

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

$$\cos b=\mathrm{\hspace{0.33em}1},\sin b=\mathrm{\hspace{0.33em}0}.$$

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

$$\omega =n\cdot 2\pi i\mathit{\hspace{1em}\hspace{1em}}(n=\mathrm{\hspace{0.33em}0},\pm 1,\pm 2,\pm 3,\mathrm{\dots })$$

Q.E.D.

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