Period of Complex Exponential Function/Proof 2

Theorem

$\map \exp {z + 2 k \pi i} = \map \exp z$


Proof

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

$\blacksquare$


Sources

  • 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: De Moivre's Theorem: $25$
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