Periodicity of Complex Functions

Definition: Periodicity of Complex Functions

A complex function $f:\mathcal{D}\subseteq \mathbb{C}\to \mathbb{C}$ is periodic if there exists some $P\in \mathbb{C}$ such that

$$f(z+P)=f(z)\forall z\in \mathcal{D}.$$

The number $P$ is called a period of $f$.