# 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

$\blacksquare$