# Definition:Periodic Function/Period

## Definition

Let $f: X \to X$ be a periodic function, where $X$ is either $\R$ or $\C$.

The period of $f$ is the smallest value $\cmod L \in \R_{\ne 0}$ such that:

$\forall x \in X: \map f x = \map f {x + L}$

where $\cmod L$ is the modulus of $L$.

## Also defined as

Some sources define a period of a periodic function $f$ as any $\cmod L \in \R_{\ne 0}$ such that $\forall x \in X: \map f x = \map f {x + L}$, not necessarily the smallest.

## Also known as

The period of a periodic function $f$ is also known as the principal period of $f$, particularly by those sources which define a period of $f$ as any $\cmod L \in \R_{\ne 0}$ such that $\forall x \in X: \map f x = \map f {x + L}$.