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

1. Formulae are given as Taylor series or derived from other entries.

1. Orloff, Jeremy. "ES.1803 Fourier Expansion of tan(x)" (PDF). Massachusetts Institute of Technology. Archived from the original (PDF) on 2019-03-31.
2. Papula, Lothar (2009). Mathematische Formelsammlung: für Ingenieure und Naturwissenschaftler. Vieweg+Teubner Verlag. ISBN   978-3834807571.