Periodic function

Definition

An entire function $f:\mathbb{C}\to \mathbb{C}$ is termed periodic with period $h\in {\mathbb{C}}^{*}$ if for any $z\in \mathbb{C}$, we have:

$f(z+h)=f(z)$

More generally, an entire meromorphic function is termed periodic if for any $z$ that is not a pole:

$f(z+h)=f(z)$

And moreover, $z$ is a pole iff $z+h$ is a pole.

Periodic functions are sometimes termed singly periodic to distinguish from an elliptic function, which is doubly periodic: it has two $\mathbb{R}$-linearly independent periods.

Examples

• The complex exponential is entire and periodic with period $2\pi i$
• The sine function and cosine function are periodic with period $2\pi$
• The tangent function and cotangent function are entire meromorphic and periodic with period $\pi$