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List of periodic functions

## Summary

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
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 ${\displaystyle \tan(x)}$  ${\displaystyle \sum _{n=0}^{\infty }{\frac {U_{2n+1}x^{2n+1}}{(2n+1)!}}}$  ${\displaystyle 2\sum _{n=1}^{\infty }(-1)^{n-1}\sin(2nx)}$  [1]
Cotangent ${\displaystyle \cot(x)}$  ${\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}2^{2n}B_{2n}x^{2n-1}}{(2n)!}}}$  ${\displaystyle i+2i\sum _{n=1}^{\infty }(\cos 2nx-i\sin 2nx)}$ [citation needed]
Secant ${\displaystyle \sec(x)}$  ${\displaystyle \sum _{n=0}^{\infty }{\frac {U_{2n}x^{2n}}{(2n)!}}}$  -
Cosecant ${\displaystyle \csc(x)}$  ${\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n+1}2\left(2^{2n-1}-1\right)B_{2n}x^{2n-1}}{(2n)!}}}$  -
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 ${\displaystyle \operatorname {hacovercosin} (x)}$  ${\displaystyle {\frac {1+\sin(x)}{2}}}$  ${\displaystyle {\frac {1}{2}}+{\frac {1}{2}}\sin(x)}$
Magnitude of sine wave
with amplitude, A, and period, T
- ${\displaystyle A|\sin \left({\frac {2\pi }{T}}x\right)|}$  ${\displaystyle {\frac {4A}{2\pi }}+\sum _{n\,\mathrm {even} }{\frac {-4A}{\pi }}{\frac {1}{1-n^{2}}}\cos({\frac {2\pi n}{T}}x)}$  [2]: p. 193
Clausen function ${\displaystyle \operatorname {Cl} _{2}(x)}$  ${\displaystyle -\int _{0}^{x}\ln \left|2\sin {\frac {x}{2}}\right|dx}$  ${\displaystyle \sum _{k=1}^{\infty }{\frac {\sin kx}{k^{2}}}}$

## 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.

Name Formula Fourier Series Notes
Triangle wave ${\displaystyle {\frac {4}{p}}\left(x-{\frac {p}{2}}\left\lfloor {\frac {2x}{p}}+{\frac {1}{2}}\right\rfloor \right)(-1)^{\left\lfloor {\frac {2x}{p}}+{\frac {1}{2}}\right\rfloor }}$  ${\displaystyle {\frac {8}{\pi ^{2}}}\sum _{n\,\mathrm {odd} }^{\infty }{\frac {(-1)^{(n-1)/2}}{n^{2}}}\sin \left({\frac {2\pi nx}{p}}\right)}$  non-continuous first derivative
Sawtooth wave ${\displaystyle 2\left({\frac {x}{p}}-\left\lfloor {\frac {1}{2}}+{\frac {x}{p}}\right\rfloor \right)}$  ${\displaystyle {\frac {2}{\pi }}\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{n}}\sin \left({\frac {2\pi nx}{p}}\right)}$  non-continuous
Square wave ${\displaystyle \operatorname {sgn} \left(\sin {\frac {2\pi x}{p}}\right)}$  ${\displaystyle {\frac {4}{\pi }}\sum _{n\,\mathrm {odd} }^{\infty }{\frac {1}{n}}\sin \left({\frac {2\pi nx}{p}}\right)}$  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.

${\displaystyle {\frac {p}{\pi }}{\biggl (}{\frac {3}{4}}+\sum _{n=1}^{\infty }{\frac {\operatorname {J} _{n}(n)-\operatorname {J} _{n-1}(n)}{n}}\cos {\Bigl (}{\frac {2\pi nx}{p}}{\Bigr )}{\biggr )}}$

where ${\displaystyle \operatorname {J} _{n}(x)}$  is the Bessel Function of the first kind.

non-continuous first derivative
Pulse wave ${\displaystyle H\left(\cos \left({\frac {2\pi x}{p}}\right)-\cos \left({\frac {\pi t}{p}}\right)\right)}$

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
Dirichlet function ${\displaystyle {\displaystyle \mathbf {1} _{\mathbb {Q} }(x)={\begin{cases}1&x\in \mathbb {Q} \\0&x\notin \mathbb {Q} \end{cases}}}}$  - non-continuous

## Notes

1. ^ Formulae are given as Taylor series or derived from other entries.
1. ^ http://web.mit.edu/jorloff/www/18.03-esg/notes/fourier-tan.pdf[bare URL PDF]
2. ^ Papula, Lothar (2009). Mathematische Formelsammlung: für Ingenieure und Naturwissenschaftler. Vieweg+Teubner Verlag. ISBN 978-3834807571.