Period of a function

$f$ with domain $X$

A number $T \ne 0$ such that for any $x \in X \subset \mathbf{R}$ (or $x \in X \subset \mathbf{C}$) the numbers $x+T$ and $x-T$ also belong to $X$ and such that the following equality holds: $$ f(x \pm T) = f(x) \ . $$

The numbers $\pm nT$, where $n$ is a natural number, are also periods of $f$. For a function $f=\text{const.}$ on an axis or on a plane, any number $T\ne0$ is a period; for the Dirichlet function $$ D(x) = \begin{cases} 1 &\text{if}\ x\ \text{is rational} \\ 0 & \text{if}\ x\ \text{is irrational} \end{cases} \ , $$ any rational number $T\ne0$ is a period. If a function $f$ has period $T$, then the function $\psi(x) = f(ax+b)$, where $a$ and $b$ are constants and $a\ne0$, has period $T/a$. If a real-valued function $f$ of a real argument is continuous and periodic on $X$ (and is not identically equal to a constant), then it has a least period $T_0 > 0$ and any other real period is an integer multiple of $T_0$. There exist non-constant double-periodic functions of a complex argument, having two periods with non-real quotient, such as for example the elliptic functions.

Similarly one defines the period of a function defined on an Abelian group.

Comments

Cf. also Periodic function.