List of periodic functions

Last updated January 09, 2024

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

n

th up/down number,

B n

is the

n

th Bernoulli number

in Jacobi elliptic functions,

q=e^{-\pi {\frac {K(1-m)}{K(m)}}}

Name	Symbol	Formula ^{[nb 1]}	Fourier Series
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)	$e^{ix},\operatorname {cis} (x)$	$cos(x) + i sin(x)$	$\cos(x)+i\sin(x)$
Tangent	$\tan(x)$	${\frac {\sin x}{\cos x}}=\sum _{n=0}^{\infty }{\frac {U_{2n+1}x^{2n+1}}{(2n+1)!}}$	$2\sum _{n=1}^{\infty }(-1)^{n-1}\sin(2nx)$ ^[1]
Cotangent	$\cot(x)$	${\frac {\cos x}{\sin x}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}2^{2n}B_{2n}x^{2n-1}}{(2n)!}}$	$i+2i\sum _{n=1}^{\infty }(\cos 2nx-i\sin 2nx)$ ^{[ citation needed ]}
Secant	$\sec(x)$	${\frac {1}{\cos x}}=\sum _{n=0}^{\infty }{\frac {U_{2n}x^{2n}}{(2n)!}}$	-
Cosecant	$\csc(x)$	${\frac {1}{\sin x}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n+1}2\left(2^{2n-1}-1\right)B_{2n}x^{2n-1}}{(2n)!}}$	-
Exsecant	$\operatorname {exsec} (x)$	$\sec(x)-1$	-
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	$\operatorname {haversin} (x)$	${\frac {1-\cos(x)}{2}}$	${\frac {1}{2}}-{\frac {1}{2}}\cos(x)$
Havercosine	$\operatorname {havercosin} (x)$	${\frac {1+\cos(x)}{2}}$	${\frac {1}{2}}+{\frac {1}{2}}\cos(x)$
Hacoversine	$\operatorname {hacoversin} (x)$	${\frac {1-\sin(x)}{2}}$	${\frac {1}{2}}-{\frac {1}{2}}\sin(x)$
Hacovercosine	$\operatorname {hacovercosin} (x)$	${\frac {1+\sin(x)}{2}}$	${\frac {1}{2}}+{\frac {1}{2}}\sin(x)$
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	$\operatorname {zn} (x,m)$	$\int _{0}^{x}\left[\operatorname {dn} (t,m)^{2}-{\frac {E(m)}{K(m)}}\right]dt$	${\frac {2\pi }{K(m)}}\sum _{n=1}^{\infty }{\frac {q^{n}}{1-q^{2n}}}~\sin {\frac {n\pi x}{K(m)}}$
Weierstrass elliptic function	$\wp (x,\Lambda )$	${\frac {1}{x^{2}}}+\sum _{\lambda \in \Lambda -\{0\}}\left[{\frac {1}{(x-\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right]$
Clausen function	$\operatorname {Cl} _{2}(x)$	$-\int _{0}^{x}\ln \left\|2\sin {\frac {t}{2}}\right\|dt$	$\sum _{k=1}^{\infty }{\frac {\sin kx}{k^{2}}}$

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 $\operatorname {sgn}$ is the sign function.

K means Elliptic integral K(m)

Name	Formula	Limit	Fourier Series	Notes
Triangle wave	${\frac {4}{p}}\left(x-{\frac {p}{2}}\left\lfloor {\frac {2x}{p}}+{\frac {1}{2}}\right\rfloor \right)(-1)^{\left\lfloor {\frac {2x}{p}}+{\frac {1}{2}}\right\rfloor }$	$\lim _{m\rightarrow 1^{-}}\operatorname {zs} \left({\frac {4Kx}{p}}-K,m\right)$	${\frac {8}{\pi ^{2}}}\sum _{n\,\mathrm {odd} }^{\infty }{\frac {(-1)^{(n-1)/2}}{n^{2}}}\sin \left({\frac {2\pi nx}{p}}\right)$	non-continuous first derivative
Sawtooth wave	$2\left({\frac {x}{p}}-\left\lfloor {\frac {1}{2}}+{\frac {x}{p}}\right\rfloor \right)$	$-\lim _{m\rightarrow 1^{-}}\operatorname {zn} \left({\frac {2Kx}{p}}+K,m\right)$	${\frac {2}{\pi }}\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{n}}\sin \left({\frac {2\pi nx}{p}}\right)$	non-continuous
Square wave	$\operatorname {sgn} \left(\sin {\frac {2\pi x}{p}}\right)$	$\lim _{m\rightarrow 1^{-}}\operatorname {sn} \left({\frac {4Kx}{p}},m\right)$	${\frac {4}{\pi }}\sum _{n\,\mathrm {odd} }^{\infty }{\frac {1}{n}}\sin \left({\frac {2\pi nx}{p}}\right)$	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		${\frac {t}{p}}+\sum _{n=1}^{\infty }{\frac {2}{n\pi }}\sin \left({\frac {\pi nt}{p}}\right)\cos \left({\frac {2\pi nx}{p}}\right)$	non-continuous
Magnitude of sine wave with amplitude, A, and period, p/2	$A\left\|\sin {\frac {\pi x}{p}}\right\|$		${\frac {4A}{2\pi }}+\sum _{n=1}^{\infty }{\frac {4A}{\pi }}{\frac {1}{4n^{2}-1}}\cos {\frac {2\pi nx}{p}}$ ^[2]^{: 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	$\sum _{n=-\infty }^{\infty }\delta (x-np)$	$\lim _{m\rightarrow 1^{-}}{\frac {2K(m)}{p\pi }}\operatorname {dn} \left({\frac {2Kx}{p}},m\right)$	${\frac {1}{p}}\sum _{n=-\infty }^{\infty }e^{\frac {2n\pi ix}{p}}$	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)

Notes

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

↑ http://web.mit.edu/jorloff/www/18.03-esg/notes/fourier-tan.pdf ^{[ bare URL PDF ]}
↑ Papula, Lothar (2009). Mathematische Formelsammlung: für Ingenieure und Naturwissenschaftler. Vieweg+Teubner Verlag. ISBN 978-3834807571.

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

[2] ttp://web.mit.edu/jorloff/www/18.03-esg/notes/fourier-tan.pdf ^{[ bare URL PDF ]}

[Papula-3] Papula, Lothar (2009). Mathematische Formelsammlung: für Ingenieure und Naturwissenschaftler. Vieweg+Teubner Verlag. ISBN 978-3834807571.

[nb 1]

[1]

[2]