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:

Un is the nth up/down number,
Bn 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

Notes

1. ^ Formulae are given as Taylor series or derived from other entries.
1. ^
2. ^ Papula, Lothar (2009). Mathematische Formelsammlung: für Ingenieure und Naturwissenschaftler. Vieweg+Teubner Verlag. ISBN 978-3834807571.