# Simply-periodic function

simple periodic function

A periodic function $f(z)$ of the complex variable $z$ all periods $p$ of which are integer multiples of a single unique fundamental, or primitive, period $2\omega\neq0$, i.e. $p=2n\omega$ ($n\in\mathbf Z$). For example, the exponential function $e^z$ is an entire simply-periodic function with fundamental period $2\omega=2\pi i$, and the trigonometric functions $\tan z$ and $\operatorname{cotan}z$ are meromorphic simply-periodic functions with fundamental period $2\omega=\pi$.

More generally, a simply-periodic function on a linear space $X$ is a periodic function whose periods are integer multiples of some basic period $2\omega\in X$. A non-constant continuous periodic function of a real variable is necessarily simply-periodic.