Periodicity of Complex Exponential Function

From ProofWiki
Jump to: navigation, search

Theorem

For all $k \in \Z$:

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


Proof

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

$\blacksquare$


Sources