Period of Complex Exponential Function
Theorem
Let $z \in \C$, and let $k \in \Z$.
Then:
- $\map \exp {z + 2 k \pi i} = \map \exp z$
Proof $1$
| \(\ds \map \exp {z + 2 k \pi i}\) | \(=\) | \(\ds \map \exp z \, \map \exp {2 k \pi i}\) | Exponential of Sum: Complex Numbers | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \exp z \times 1\) | Euler's Formula Example: $e^{2 k i \pi}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \exp z\) |
$\blacksquare$
Proof $2$
| \(\ds e^{i \paren {\theta + 2 k \pi} }\) | \(=\) | \(\ds \map \cos {\theta + 2 k \pi} + i \, \map \sin {\theta + 2 k \pi}\) | 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
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations: $(2.21)$
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$: $\text{(ii)}$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 7$: Periodicity of Exponential Functions: $7.23$
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): complex exponential