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 ${\displaystyle 2\pi }$, 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, ${\displaystyle q=e^{-\pi {\frac {K(1-m)}{K(m)}}}}$

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

Non-smooth functions

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

K means Elliptic integral K(m)

Name	Formula	Limit	Fourier Series	Notes
Triangle wave	${\displaystyle {\frac {4}{p}}\left(x-{\frac {p}{2}}\left\lfloor {\frac {2x}{p}}+{\frac {1}{2}}\right\rfloor \right)(-1)^{\left\lfloor {\frac {2x}{p}}+{\frac {1}{2}}\right\rfloor }}$	${\displaystyle \lim _{m\rightarrow 1^{-}}\operatorname {zs} \left({\frac {4Kx}{p}}-K,m\right)}$	${\displaystyle {\frac {8}{\pi ^{2}}}\sum _{n\,\mathrm {odd} }^{\infty }{\frac {(-1)^{(n-1)/2}}{n^{2}}}\sin \left({\frac {2\pi nx}{p}}\right)}$	non-continuous first derivative
Sawtooth wave	${\displaystyle 2\left({\frac {x}{p}}-\left\lfloor {\frac {1}{2}}+{\frac {x}{p}}\right\rfloor \right)}$	${\displaystyle -\lim _{m\rightarrow 1^{-}}\operatorname {zn} \left({\frac {2Kx}{p}}+K,m\right)}$	${\displaystyle {\frac {2}{\pi }}\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{n}}\sin \left({\frac {2\pi nx}{p}}\right)}$	non-continuous
Square wave	${\displaystyle \operatorname {sgn} \left(\sin {\frac {2\pi x}{p}}\right)}$	${\displaystyle \lim _{m\rightarrow 1^{-}}\operatorname {sn} \left({\frac {4Kx}{p}},m\right)}$	${\displaystyle {\frac {4}{\pi }}\sum _{n\,\mathrm {odd} }^{\infty }{\frac {1}{n}}\sin \left({\frac {2\pi nx}{p}}\right)}$	non-continuous
Pulse wave	${\displaystyle H\left(\cos {\frac {2\pi x}{p}}-\cos {\frac {\pi t}{p}}\right)}$ where ${\displaystyle H}$ is the Heaviside step function t is how long the pulse stays at 1		${\displaystyle {\frac {t}{p}}+\sum _{n=1}^{\infty }{\frac {2}{n\pi }}\sin \left({\frac {\pi nt}{p}}\right)\cos \left({\frac {2\pi nx}{p}}\right)}$	non-continuous
Magnitude of sine wave with amplitude, A, and period, p/2	${\displaystyle A\left|\sin {\frac {\pi x}{p}}\right|}$		${\displaystyle {\frac {4A}{2\pi }}+\sum _{n=1}^{\infty }{\frac {4A}{\pi }}{\frac {1}{4n^{2}-1}}\cos {\frac {2\pi nx}{p}}}$ [2]: p. 193	non-continuous
Cycloid	${\displaystyle {\frac {p-p\cos \left(f^{(-1)}\left({\frac {2\pi x}{p}}\right)\right)}{2\pi }}}$ given ${\displaystyle f(x)=x-\sin(x)}$ and ${\displaystyle f^{(-1)}(x)}$ is its real-valued inverse.		${\displaystyle {\frac {p}{\pi }}{\biggl (}{\frac {3}{4}}+\sum _{n=1}^{\infty }{\frac {\operatorname {J} _{n}(n)-\operatorname {J} _{n-1}(n)}{n}}\cos {\frac {2\pi nx}{p}}{\biggr )}}$ where ${\displaystyle \operatorname {J} _{n}(x)}$ is the Bessel Function of the first kind.	non-continuous first derivative
Dirac comb	${\displaystyle \sum _{n=-\infty }^{\infty }\delta (x-np)}$	${\displaystyle \lim _{m\rightarrow 1^{-}}{\frac {2K(m)}{p\pi }}\operatorname {dn} \left({\frac {2Kx}{p}},m\right)}$	${\displaystyle {\frac {1}{p}}\sum _{n=-\infty }^{\infty }e^{\frac {2n\pi ix}{p}}}$	non-continuous
Dirichlet function	${\displaystyle {\displaystyle \mathbf {1} _{\mathbb {Q} }(x)={\begin{cases}1&x\in \mathbb {Q} \\0&x\notin \mathbb {Q} \end{cases}}}}$	${\displaystyle \lim _{m,n\rightarrow \infty }\cos ^{2m}(n!x\pi )}$	-	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

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

Notes