# 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 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.

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

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

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

The following functions are also not smooth: