= Periodicity
{title2=$f(x+P)=f(x)$}
Periodicity is invariance under translation by a nonzero period $P$: a <periodic function> satisfies $f(x+P)=f(x)$ at every point in its domain, with translation preserving the domain. For an angular coordinate on a full <circle>, single valuedness requires $2\pi$-periodicity. The <sine> and <cosine> factors $\sin(k\theta)$ and $\cos(k\theta)$ are $2\pi$-periodic for nonzero real $k$ precisely when $k$ is an integer. Their <Fourier series> supply the angular modes for the <harmonic expansion on a disk>.
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