Periodicity ($f(x+P)=f(x)$)

Periodicity is invariance of a function under a nonzero shift $P$. Its domain of a function must be preserved by the shift and its inverse, and $f(x+P)=f(x)$ at every allowed function input. On the real line, both $\sin x$ and $\cos x$ have period $2\pi$.

For an angular coordinate on a full circle, single-valuedness requires $2\pi$-periodicity. For nonzero real $k$, the functions $\sin (k\theta )$ and $\cos (k\theta )$ are $2\pi$-periodic precisely when $k$ is an integer: the phase must change by an integer multiple of $2\pi$. The zero-frequency case instead gives constant functions. Their Fourier series supply angular modes for the harmonic expansion on a disk.