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

\(\ds \map \exp {z + 2 k \pi i}\)

\(=\)

\(\ds \map \exp z \, \map \exp {2 k \pi i}\)

Exponential of Sum: Complex Numbers

\(\ds \)

\(=\)

\(\ds \map \exp z \times 1\)

Euler's Formula Example: $e^{2 k i \pi}$

\(\ds \)

\(=\)

\(\ds \map \exp z\)

$\blacksquare$

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 $e^z$, $\cos z$, $\sin z$: $\text{(ii)}$
• 2001: Christian Berg: Kompleks funktionsteori: $\S 1.5$