List of periodic functions

This is a list of some well-known periodic functions. The constant function f (x) = c, where c is independent of 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:

U_n is the nth up/down number,

B_n is the nth Bernoulli number

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

Name	Symbol	Formula	Fourier Series
Sine	$\sin (x)$	$\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n{x}^{2n+1}}{(2n+1)!}$	$\sin (x)$
cas (mathematics)	$\mathrm{cas}(x)$	$\sin (x)+\cos (x)$	$\sin (x)+\cos (x)$
Cosine	$\cos (x)$	$\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n{x}^{2n}}{(2n)!}$	$\cos (x)$
cis (mathematics)	$e^{ix},\mathrm{cis}(x)$	cos(x) + i sin(x)	$\cos (x)+i\sin (x)$
Tangent	$\tan (x)$	$\frac{\sin x}{\cos x}=\sum _{n=0}^{\mathrm{\infty }}\frac{U_{2n+1}{x}^{2n+1}}{(2n+1)!}$	$2\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n-1}\sin (2nx)$
Cotangent	$\cot (x)$	$\frac{\cos x}{\sin x}=\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n{2}^{2n}{B}_{2n}{x}^{2n-1}}{(2n)!}$	$i+2i\sum _{n=1}^{\mathrm{\infty }}(\cos 2nx-i\sin 2nx)$
Secant	$\sec (x)$	$\frac{1}{\cos x}=\sum _{n=0}^{\mathrm{\infty }}\frac{U_{2n}{x}^{2n}}{(2n)!}$	-
Cosecant	$\csc (x)$	$\frac{1}{\sin x}=\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^{n+1}2(2^{2n-1}-1)B_{2n}{x}^{2n-1}}{(2n)!}$	-
Exsecant	$\mathrm{exsec}(x)$	$\sec (x)-1$	-
Excosecant	$\mathrm{excsc}(x)$	$\csc (x)-1$	-
Versine	$\mathrm{versin}(x)$	$1-\cos (x)$	$1-\cos (x)$
Vercosine	$\mathrm{vercosin}(x)$	$1+\cos (x)$	$1+\cos (x)$
Coversine	$\mathrm{coversin}(x)$	$1-\sin (x)$	$1-\sin (x)$
Covercosine	$\mathrm{covercosin}(x)$	$1+\sin (x)$	$1+\sin (x)$
Haversine	$\mathrm{haversin}(x)$	$\frac{1-\cos (x)}{2}$	$\frac{1}{2}-\frac{1}{2}\cos (x)$
Havercosine	$\mathrm{havercosin}(x)$	$\frac{1+\cos (x)}{2}$	$\frac{1}{2}+\frac{1}{2}\cos (x)$
Hacoversine	$\mathrm{hacoversin}(x)$	$\frac{1-\sin (x)}{2}$	$\frac{1}{2}-\frac{1}{2}\sin (x)$
Hacovercosine	$\mathrm{hacovercosin}(x)$	$\frac{1+\sin (x)}{2}$	$\frac{1}{2}+\frac{1}{2}\sin (x)$
Jacobi elliptic function sn	$\mathrm{sn}(x,m)$	$\sin \mathrm{am}(x,m)$	$\frac{2\pi }{K(m)\sqrt{m}}\sum _{n=0}^{\mathrm{\infty }}\frac{q^{n+1/2}}{1-q^{2n+1}}\sin \frac{(2n+1)\pi x}{2K(m)}$
Jacobi elliptic function cn	$\mathrm{cn}(x,m)$	$\cos \mathrm{am}(x,m)$	$\frac{2\pi }{K(m)\sqrt{m}}\sum _{n=0}^{\mathrm{\infty }}\frac{q^{n+1/2}}{1+q^{2n+1}}\cos \frac{(2n+1)\pi x}{2K(m)}$
Jacobi elliptic function dn	$\mathrm{dn}(x,m)$	$\sqrt{1-m{\mathrm{sn}}^2(x,m)}$	$\frac{\pi }{2K(m)}+\frac{2\pi }{K(m)}\sum _{n=1}^{\mathrm{\infty }}\frac{q^n}{1+q^{2n}}\cos \frac{n\pi x}{K(m)}$
Jacobi elliptic function zn	$\mathrm{zn}(x,m)$	$\underset{0}{\overset{x}{\int }}[\mathrm{dn}(t,m{)}^2-\frac{E(m)}{K(m)}]dt$	$\frac{2\pi }{K(m)}\sum _{n=1}^{\mathrm{\infty }}\frac{q^n}{1-q^{2n}}\sin \frac{n\pi x}{K(m)}$
Weierstrass elliptic function	$\mathrm{\wp }(x,\mathrm{\Lambda })$	$\frac{1}{x^2}+\sum _{\lambda \in \mathrm{\Lambda }-\{0\}}[\frac{1}{(x-\lambda {)}^2}-\frac{1}{{\lambda }^2}]$	${\displaystyle }$
Clausen function	${\mathrm{Cl}}_2(x)$	$-\underset{0}{\overset{x}{\int }}\ln |2\sin \frac{t}{2}|dt$	$\sum _{k=1}^{\mathrm{\infty }}\frac{\sin kx}{k^2}$

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 $\mathrm{sgn}$ is the sign function. K means Elliptic integral K(m)

Name	Formula	Limit	Fourier Series	Notes
Triangle wave	$\frac{4}{p}(x-\frac{p}{2}\lfloor \frac{2x}{p}+\frac{1}{2}\rfloor )(-1{)}^{\lfloor \frac{2x}{p}+\frac{1}{2}\rfloor }$	$\underset{m\to 1^{-}}{\lim }\mathrm{zs}(\frac{4Kx}{p}-K,m)$	$\frac{8}{{\pi }^2}\sum _{n\mathrm{o}\mathrm{d}\mathrm{d}}^{\mathrm{\infty }}\frac{(-1{)}^{(n-1)/2}}{n^2}\sin \left(\frac{2\pi nx}{p}\right)$	non-continuous first derivative
Sawtooth wave	$2(\frac{x}{p}-\lfloor \frac{1}{2}+\frac{x}{p}\rfloor )$	$-\underset{m\to 1^{-}}{\lim }\mathrm{zn}(\frac{2Kx}{p}+K,m)$	$\frac{2}{\pi }\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^{n-1}}{n}\sin \left(\frac{2\pi nx}{p}\right)$	non-continuous
Square wave	$\mathrm{sgn}(\sin \frac{2\pi x}{p})$	$\underset{m\to 1^{-}}{\lim }\mathrm{sn}(\frac{4Kx}{p},m)$	$\frac{4}{\pi }\sum _{n\mathrm{o}\mathrm{d}\mathrm{d}}^{\mathrm{\infty }}\frac{1}{n}\sin \left(\frac{2\pi nx}{p}\right)$	non-continuous
Pulse wave	$H(\cos \frac{2\pi x}{p}-\cos \frac{\pi t}{p})$ where $H$ is the Heaviside step function t is how long the pulse stays at 1		$\frac{t}{p}+\sum _{n=1}^{\mathrm{\infty }}\frac{2}{n\pi }\sin \left(\frac{\pi nt}{p}\right)\cos \left(\frac{2\pi nx}{p}\right)$	non-continuous
Magnitude of sine wave with amplitude, A, and period, p/2	$A|\sin \frac{\pi x}{p}|$		$\frac{4A}{2\pi }+\sum _{n=1}^{\mathrm{\infty }}\frac{4A}{\pi }\frac{1}{4n^2-1}\cos \frac{2\pi nx}{p}$	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.		$\frac{p}{\pi }(\frac{3}{4}+\sum _{n=1}^{\mathrm{\infty }}\frac{{\mathrm{J}}_n(n)-{\mathrm{J}}_{n-1}(n)}{n}\cos \frac{2\pi nx}{p})$ where ${\mathrm{J}}_n(x)$ is the Bessel Function of the first kind.	non-continuous first derivative
Dirac comb	$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\delta (x-np)$	$\underset{m\to 1^{-}}{\lim }\frac{2K(m)}{p\pi }\mathrm{dn}(\frac{2Kx}{p},m)$	$\frac{1}{p}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{\frac{2n\pi ix}{p}}$	non-continuous
Dirichlet function	${\mathbf{1}}_{\mathbb{\mathbb{Q}}}(x)=\{\begin{array}{ll}1& x\in \mathbb{\mathbb{Q}}\\ 0& x\notin \mathbb{\mathbb{Q}}\end{array}$	$\underset{m,n\to \mathrm{\infty }}{\lim }{\cos }^{2m}(n!x\pi )$	-	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

• Jacobi's elliptic functions
• Weierstrass's elliptic function