periodicity of exponential functionPeriodicityOfExponentialFunction2005-05-23 06:11:162008-05-02 15:05:55Theoremcomplex exponential functionone-periodic% this is the default PlanetMath preamble. as your knowledge
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\newtheorem*{thmplain}{Theorem}\begin{thmplain}
\, The only periods of the complex exponential function\, $z\mapsto e^z$\, are the multiples of $2\pi i$.\, Thus the function is {\em one-periodic}.
\end{thmplain}
{\em Proof.}\, Let $\omega$ be any period of the exponential function, i.e.\, $e^{z+\omega} = e^ze^\omega = e^z$\, for all\, $z\in\mathbb{C}$.\, Because $e^z$ is always $\neq 0$, we have
\begin{align}
e^\omega = 1.
\end{align}
If we set\, $\omega := a+ib$\, with $a$ and $b$ reals, (1) gets the form
\begin{align}
e^a\cos{b}+ie^a\sin{b} = 1,
\end{align}
which implies (see equality of complex numbers)
$$e^a\cos{b} = 1,\quad e^a\sin{b} = 0.$$
As these equations are squared and added, we obtain\, $e^{2a} = 1$\, which \PMlinkescapetext{means}, since $a$ is real, that\, $a = 0$.\, Thus the preceding equations get the form
$$\cos{b} = 1,\quad \sin{b} = 0.$$
These result that\, $b = n\cdot 2\pi$\, and therefore
$$\omega = n\cdot 2\pi i\quad (n \,=\, 0,\,\pm 1,\,\pm 2,\,\pm 3,\,\ldots)$$
Q.E.D.
\begin{thebibliography}{8}
\bibitem{lindelof}{\sc Ernst Lindel\"of}: {\em Johdatus funktioteoriaan} (`Introduction to function theory').\, Mercatorin kirjapaino, Helsinki (1936).
\end{thebibliography}