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

Non-smooth functions

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

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

The following functions are also not smooth: