Periodicity of Complex Exponential Function

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Theorem

For all $k \in \Z$:

$e^{i \paren {\theta + 2 k \pi} } = e^{i \theta}$


Proof

\(\displaystyle e^{i \paren {\theta + 2 k \pi} }\) \(=\) \(\displaystyle \map \cos {\theta + 2 k \pi} + i \, \map \sin {\theta + 2 k \pi}\) Euler's Formula
\(\displaystyle \) \(=\) \(\displaystyle \cos \theta + i \sin \theta\) Sine and Cosine are Periodic on Reals
\(\displaystyle \) \(=\) \(\displaystyle e^{i \theta}\) Euler's Formula

$\blacksquare$


Sources