# 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.

### Trigonometric 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

### Non-smooth functions

The following functions take the variable ${\displaystyle x}$, period ${\displaystyle p}$ and have range ${\displaystyle -1}$ to ${\displaystyle 1}$. The symbol ${\displaystyle \lfloor n\rfloor }$ is the floor function of n and ${\displaystyle \operatorname {sgn} }$ is the sign function.

Name Formula Fourier Series Notes
Triangle wave ${\displaystyle {\frac {4}{p}}\left(x-{\frac {p}{2}}\left\lfloor {\frac {2x}{p}}+{\frac {1}{2}}\right\rfloor \right)(-1)^{\left\lfloor {\frac {2x}{p}}+{\frac {1}{2}}\right\rfloor }}$ - non-continuous first derivative
Sawtooth wave ${\displaystyle 2\left({\frac {x}{p}}-\left\lfloor {\frac {1}{2}}+{\frac {x}{p}}\right\rfloor \right)}$ ${\displaystyle {\frac {4}{\pi }}\sum _{n\,\mathrm {odd} }^{\infty }{\frac {1}{n}}\sin \left({\frac {n\pi x}{p/2}}\right)}$ [3] non-continuous
Square wave ${\displaystyle \operatorname {sgn} \left(\sin {\frac {2\pi x}{p}}\right)}$ - non-continuous
Cycloid No closed form[citation needed]. - non-continuous first derivative
Pulse wave - - non-continuous

The following functions are also not smooth:

## 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 3834807575.
3. ^