List of periodic functions

This is a list of some well-known periodic functions. The constant function {{math|{{var|f}}{{sub| }}({{var|x}}) {{=}} {{var|c}}}}, where {{mvar|c}} is independent of {{mvar|x}}, is periodic with any period, but lacks a fundamental period. A definition is given for some of the following functions, though each function may have many equivalent definitions.

Smooth functions

All trigonometric functions listed have period $2\pi$ , unless otherwise stated. For the following trigonometric functions:

: {{mvar|U_n}} is the {{mvar|n}}th up/down number,

: {{mvar|B_n}} is the {{mvar|n}}th Bernoulli number

: in Jacobi elliptic functions, $q=e^{-\pi \frac{K(1-m)}{K(m)}}$

Name	Symbol	Formula {{refn\|group=nb\|Formulae are given as Taylor series or derived from other entries.}}	Fourier Series
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Sine	$\sin(x)$	$\sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n + 1)!}$	$\sin(x)$
cas (mathematics)	$\operatorname{cas}(x)$	$\sin(x)+\cos(x)$	$\sin(x) + \cos(x)$
Cosine	$\cos(x)$	$\sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!}$	$\cos(x)$
cis (mathematics)	$e^{ix}, \operatorname{cis}(x)$	{{math\| cos(x) + i sin(x)}}	$\cos(x)+i\sin(x)$
Tangent	$\tan(x)$	$\frac{\sin x}{\cos x}=\sum_{n=0}^\infty \frac{U_{2n+1} x^{2n+1}}{(2n+1)!}$	$2\sum_{n=1}^\infty (-1)^{n-1}\sin(2nx)$ {{cite web\|url=http://web.mit.edu/jorloff/www/18.03-esg/notes/fourier-tan.pdf\|archive-url=https://web.archive.org/web/20190331130103/http://web.mit.edu/jorloff/www/18.03-esg/notes/fourier-tan.pdf\|archive-date=2019-03-31\|title=ES.1803 Fourier Expansion of tan(x)\|first=Jeremy\|last=Orloff\|publisher=Massachusetts Institute of Technology\|url-status=dead}}
Cotangent	$\cot(x)$	$\frac{\cos x}{\sin x}=\sum_{n=0}^\infty \frac{(-1)^n 2^{2n} B_{2n} x^{2n-1}}{(2n)!}$	$i+2i\sum_{n=1}^\infty(\cos2nx-i\sin2nx)$ {{citation needed\|date=March 2019}}
Secant	$\sec(x)$	$\frac1{\cos x}=\sum_{n=0}^\infty \frac{U_{2n} x^{2n}}{(2n)!}$	-
Cosecant	$\csc(x)$	$\frac1{\sin x}=\sum_{n=0}^\infty \frac{(-1)^{n+1} 2 \left(2^{2n-1}-1\right) B_{2n} x^{2n-1}}{(2n)!}$	-
Exsecant	$\operatorname{exsec}(x)$	$\sec(x)-1$	-
Excosecant	$\operatorname{excsc}(x)$	$\csc(x)-1$	-
Versine	$\operatorname{versin}(x)$	$1-\cos(x)$	$1-\cos(x)$
Vercosine	$\operatorname{vercosin}(x)$	$1+\cos(x)$	$1+\cos(x)$
Coversine	$\operatorname{coversin}(x)$	$1-\sin(x)$	$1-\sin(x)$
Covercosine	$\operatorname{covercosin}(x)$	$1+\sin(x)$	$1+\sin(x)$
Haversine	$\operatorname{haversin}(x)$	$\frac{1-\cos(x)}{2}$	$\frac{1}{2}-\frac12\cos(x)$
Havercosine	$\operatorname{havercosin}(x)$	$\frac{1+\cos(x)}{2}$	$\frac{1}{2}+\frac12\cos(x)$
Hacoversine	$\operatorname{hacoversin}(x)$	$\frac{1-\sin(x)}{2}$	$\frac{1}{2}-\frac12\sin(x)$
Hacovercosine	$\operatorname{hacovercosin}(x)$	$\frac{1+\sin(x)}{2}$	$\frac{1}{2}+\frac12\sin(x)$
Jacobi elliptic function sn	$\operatorname{sn}(x,m)$	$\sin \operatorname{am}(x,m)$	$\frac{2\pi}{K(m)\sqrt m} \sum_{n=0}^\infty \frac{q^{n+1/2}}{1-q^{2n+1}}~\sin \frac{(2n+1)\pi x}{2K(m)}$
Jacobi elliptic function cn	$\operatorname{cn}(x,m)$	$\cos \operatorname{am}(x,m)$	$\frac{2\pi}{K(m)\sqrt m} \sum_{n=0}^\infty \frac{q^{n+1/2}}{1+q^{2n+1}}~\cos\frac{(2n+1)\pi x}{2K(m)}$
Jacobi elliptic function dn	$\operatorname{dn}(x,m)$	$\sqrt{1-m\operatorname{sn}^2(x,m)}$	$\frac{\pi}{2K(m)} + \frac{2\pi}{K(m)} \sum_{n=1}^\infty \frac{q^{n}}{1+q^{2n}}~\cos\frac{n\pi x}{K(m)}$
Jacobi elliptic function zn	$\operatorname{zn}(x,m)$	$\int^x_0\left[\operatorname{dn}(t,m)^2-\frac{E(m)}{K(m)}\right]dt$	$\frac{2\pi}{K(m)}\sum_{n=1}^\infty \frac{q^n}{1-q^{2n}}~\sin\frac{n\pi x}{K(m)}$
Weierstrass elliptic function	$\weierp(x,\Lambda)$	$\frac1{x^2}+\sum_{\lambda\in\Lambda-\{0\}}\left[\frac1{(x-\lambda)^2}-\frac1{\lambda^2}\right]$
Clausen function \|Clausen function \| $\operatorname{Cl}_2(x)$ \| $-\int^x_0\ln\left\|2\sin\frac{t}{2}\right\|dt$ \| $\sum_{k=1}^\infty\frac{\sin kx}{k^2}$

Non-smooth functions

The following functions have period $p$ and take $x$ as their argument. The symbol $\lfloor n \rfloor$ is the floor function of $n$ and $\sgn$ is the sign function.

K means Elliptic integral K(m)

Name	Formula	Limit	Fourier Series	Notes
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\| Triangle wave	$\frac{4}{p} \left (x-\frac{p}{2} \left \lfloor\frac{2 x}{p}+\frac{1}{2} \right \rfloor \right )(-1)^\left \lfloor\frac{2 x}{p}+\frac{1}{2} \right \rfloor$	$\lim_{m\rightarrow1^-}\operatorname{zs}\left(\frac{4Kx}p-K,m\right)$	$\frac8{\pi^2}\sum_{n\,\mathrm{odd}}^{\infty} \frac{(-1)^{(n-1)/2}}{n^2} \sin\left(\frac{2\pi n x}{p}\right)$	non-continuous first derivative
\| Sawtooth wave	$2 \left( {\frac x p} - \left \lfloor {\frac 1 2} + {\frac x p} \right \rfloor \right)$	$-\lim_{m\rightarrow1^-}\operatorname{zn}\left(\frac{2Kx}p+K,m\right)$	$\frac2\pi\sum_{n=1}^\infty\frac{(-1)^{n-1}}n\sin\left(\frac{2\pi nx}{p}\right)$	non-continuous
\| Square wave	$\sgn\left(\sin \frac{2\pi x}{p} \right)$	$\lim_{m\rightarrow1^-}\operatorname{sn}\left(\frac{4Kx}p,m\right)$	$\frac4\pi\sum_{n\,\mathrm{odd}}^\infty\frac1n\sin\left(\frac{2\pi nx}{p}\right)$	non-continuous
\| Pulse wave	$H \left( \cos\frac{2\pi x}{p}- \cos\frac{\pi t}{p}\right)$ where $H$ is the Heaviside step function t is how long the pulse stays at 1 \| \| $\frac{t}{p} + \sum_{n=1}^{\infty} \frac{2}{n\pi} \sin\left(\frac{\pi nt}{p}\right) \cos\left(\frac{2\pi n x}{p}\right)$	non-continuous
Magnitude of sine wave with amplitude, A, and period, p/2	$A\left\|\sin\frac{\pi x}p\right\|$		$\frac{4A}{2\pi}+\sum_{n=1}^{\infty} \frac{4A}{\pi}\frac{1}{4n^2-1}\cos\frac{2\pi nx}p$ {{cite book \| author=Papula, Lothar\| title=Mathematische Formelsammlung: für Ingenieure und Naturwissenschaftler\| publisher=Vieweg+Teubner Verlag \| year=2009 \| isbn=978-3834807571}}{{rp\|p. 193}}	non-continuous
\| Cycloid	$\frac{p - p\cos \left( f^{(-1)}\left( \frac{2\pi x}{p} \right) \right)}{2\pi}$ given $f(x)=x-\sin(x)$ and $f^{(-1)}(x)$ is its real-valued inverse.	\| $\frac{p}{\pi} \biggl(\frac{3}4 + \sum_{n=1}^\infty \frac{\operatorname{J}_n(n)-\operatorname{J}_{n-1}(n)}n \cos\frac{2\pi nx}p\biggr)$ where $\operatorname{J}_n(x)$ is the Bessel Function of the first kind. \| non-continuous first derivative
\| Dirac comb	$\sum_{n=-\infty}^{\infty}\delta(x-np)$ \| $\lim_{m\rightarrow1^-}\frac{2K(m)}{p\pi}\operatorname{dn}\left(\frac{2Kx}p,m\right)$ \| $\frac1p\sum_{n=-\infty}^{\infty}e^{\frac{2n\pi ix}p}$	non-continuous
Dirichlet function \| ${\displaystyle \mathbf {1} _{\mathbb {Q} }(x)={\begin{cases}1&x\in \mathbb {Q} \\0&x\notin \mathbb {Q} \end{cases}}}$ \| $\lim_{m,n\rightarrow\infty}\cos^{2m}(n!x\pi)$ \| - \|non-continuous

Vector-valued functions

Epitrochoid
Epicycloid (special case of the epitrochoid)
Limaçon (special case of the epitrochoid)
Hypotrochoid
Hypocycloid (special case of the hypotrochoid)
Spirograph (special case of the hypotrochoid)

Doubly periodic functions

Notes

Category:Mathematics-related lists

Category:Types of functions