Exponential of Complex Number plus 2 pi i

Theorem

Let $z \in \C$ be a complex number.

Then:

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

Proof

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

\(=\)

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

Exponential of Sum: Complex Numbers

\(\ds \)

\(=\)

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

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

\(\ds \)

\(=\)

\(\ds \map \exp z\)

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations: $(2.20)$