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: $e^{2 k i \pi}$ $\ds$ $=$ $\ds \map \exp z$

$\blacksquare$