= Periodic function
{title2=$f(x+T)=f(x)$}
{wiki}
= Periodic
{synonym}
A periodic <function> repeats exactly after a nonzero shift of its input. For $f:\mathbb R\to Y$, a positive <period of a function> $T$ satisfies
$$
\boxed{f(x+T)=f(x)\qquad\text{for every }x\in\mathbb R.}
$$
One interval of length $T$ therefore determines the whole <function>. For another <domain of a function> $D$, a period must also preserve the domain, $D+T=D$; equality only where two domains happen to overlap is weaker. Periodicity is an exact <translation of a function> symmetry, rather than merely two similar-looking pieces of its graph.
The <sine> $\sin(2\pi x)$ has <fundamental period> one. The <function> $\sin^2(2\pi x)$ repeats after half a unit: the <sine> changes sign under that shift, but squaring removes the sign. The mixed <function> $g(x)=2+\sin(2\pi x)+\tfrac12\cos(4\pi x)$ again has <fundamental period> one. All positive integer multiples of a <period> are periods, so specifying one period need not identify the shortest repetition.
\Image[/periodic-functions-repetition-and-fundamental-period.png]
{title=Mixed harmonics, a squared sine, and a constant have different sets of periods}
If the input is time and $T_0$ is the <fundamental period>, the <frequency> is $1/T_0$ and the <angular frequency> is $2\pi/T_0$. Repetition lets a <Fourier series> describe a cycle using <sines> and <cosines>; the <period average> separates the reference level from the oscillating terms. Strang develops this representation and its <Fourier coefficients> in §4.1 of <Computational Science and Engineering by Gilbert Strang>.<Computational Science and Engineering by Gilbert Strang/Fourier Series for Periodic Functions>[§4.1; pp. 317]{ref}
This connection grew out of a physical problem. In his 1822 book <Théorie analytique de la chaleur by Joseph Fourier>, <Joseph Fourier> used <periodic functions> to extend profiles beyond their initial interval while treating <heat conduction>. Chapter III, §§233–234 combines <even functions> and <odd functions> into sine–cosine expansions and rescales their period.<Théorie analytique de la chaleur by Joseph Fourier/Propagation de la chaleur dans un solide rectangle infini>[§§233–234; equation (p)]{ref} A modern <Fourier series> interpretation still requires the appropriate convergence conditions.
A <constant function> has every real shift as a period and no <fundamental period>. Without <continuity>, even a <nonconstant> <periodic function> can lack a smallest positive period; <periodic function without a least positive period> gives a concrete example. The <period group of a function> organizes all these possibilities.
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