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
| \(\ds e^{i \paren {\theta + 2 k \pi} }\) | \(=\) | \(\ds \map \cos {\theta + 2 k \pi} + i \, \map \sin {\theta + 2 k \pi}\) | 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
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 7$: Periodicity of Exponential Functions: $7.23$
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: De Moivre's Theorem: $25$