Periodic function - Knowlpedia

A function $f:\mathbb R^d\to Y$ has period $p\in\mathbb R^d\setminus\{0\}$ if $f(x+p)=f(x)$ for every $x$ . In one dimension a positive period $L$ gives an $L$ -periodic function. The least positive period, when one exists, is called the fundamental period.

Several coordinates

A function is $\mathbb Z^d$ -periodic if every standard basis vector is a period, equivalently $f(x+m)=f(x)$ for all $m\in\mathbb Z^d$ . It is determined by one unit cell, with compatible boundary values. For measurable functions, periodicity may instead be imposed almost everywhere; that convention must be stated.

Different variables of $f(\theta,H)$ may have different periods. Periodicity in an auxiliary coordinate $H$ does not imply periodicity of a physical field until the map used to evaluate $H$ is specified. A constant function has every nonzero period and no least positive one.