# 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 $2\pi$, unless otherwise stated. For the following trigonometric functions:
    U$_n$ is the nth up/down number,
    B$_n$ is the nth Bernoulli number
    in Jacobi elliptic functions, $q=e^{-\pi {\frac {K(1-m)}{K(m)}}}$
| Name | Symbol | Formula | 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 | $\tan(x)$ | ${\frac {\sin x}{\cos x}}=\sum _{n=0}^{\infty }{\frac {U_{2n+1}x^{2n+1}}{(2n+1)!}}$ | $2\sum _{n=1}^{\infty }(-1)^{n-1}\sin(2nx)$ |
| Cotangent | $\cot(x)$ | ${\frac {\cos x}{\sin x}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}2^{2n}B_{2n}x^{2n-1}}{(2n)!}}$ | $i+2i\sum _{n=1}^{\infty }(\cos 2nx-i\sin 2nx)$ |
| Secant | $\sec(x)$ | ${\frac {1}{\cos x}}=\sum _{n=0}^{\infty }{\frac {U_{2n}x^{2n}}{(2n)!}}$ | - |
| Cosecant | $\csc(x)$ | ${\frac {1}{\sin x}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n+1}2\left(2^{2n-1}-1\right)B_{2n}x^{2n-1}}{(2n)!}}$ | - |
| Exsecant | $\operatorname {exsec} (x)$ | $\sec(x)-1$ | - |
| 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 | $\operatorname {hacovercosin} (x)$ | ${\frac {1+\sin(x)}{2}}$ | ${\frac {1}{2}}+{\frac {1}{2}}\sin(x)$ |
| Jacobi elliptic function sn | $\operatorname {sn} (x,m)$ | $\sin \operatorname {am} (x,m)$ | ${\frac {2\pi }{K(m){\sqrt {m}}}}\sum _{n=0}^{\infty }{\frac {q^{n+1/2}}{1-q^{2n+1}}}~\sin {\frac {(2n+1)\pi x}{2K(m)}}$ |
| Jacobi elliptic function cn | $\operatorname {cn} (x,m)$ | $\cos \operatorname {am} (x,m)$ | ${\frac {2\pi }{K(m){\sqrt {m}}}}\sum _{n=0}^{\infty }{\frac {q^{n+1/2}}{1+q^{2n+1}}}~\cos {\frac {(2n+1)\pi x}{2K(m)}}$ |
| Jacobi elliptic function dn | $\operatorname {dn} (x,m)$ | ${\sqrt {1-m\operatorname {sn} ^{2}(x,m)}}$ | ${\frac {\pi }{2K(m)}}+{\frac {2\pi }{K(m)}}\sum _{n=1}^{\infty }{\frac {q^{n}}{1+q^{2n}}}~\cos {\frac {n\pi x}{K(m)}}$ |
| Jacobi elliptic function zn | $\operatorname {zn} (x,m)$ | $\int _{0}^{x}\left[\operatorname {dn} (t,m)^{2}-{\frac {E(m)}{K(m)}}\right]dt$ | ${\frac {2\pi }{K(m)}}\sum _{n=1}^{\infty }{\frac {q^{n}}{1-q^{2n}}}~\sin {\frac {n\pi x}{K(m)}}$ |
| Weierstrass elliptic function | $\wp (x,\Lambda )$ | ${\frac {1}{x^{2}}}+\sum _{\lambda \in \Lambda -\{0\}}\left[{\frac {1}{(x-\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right]$ | $$ |
| Clausen function | $\operatorname {Cl} _{2}(x)$ | $-\int _{0}^{x}\ln \left|2\sin {\frac {t}{2}}\right|dt$ | $\sum _{k=1}^{\infty }{\frac {\sin kx}{k^{2}}}$ |

## 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.
K means Elliptic integral K(m)
| Name | Formula | Limit | Fourier Series | Notes |
| Triangle wave | ${\frac {4}{p}}\left(x-{\frac {p}{2}}\left\lfloor {\frac {2x}{p}}+{\frac {1}{2}}\right\rfloor \right)(-1)^{\left\lfloor {\frac {2x}{p}}+{\frac {1}{2}}\right\rfloor }$ | $\lim _{m\rightarrow 1^{-}}\operatorname {zs} \left({\frac {4Kx}{p}}-K,m\right)$ | ${\frac {8}{\pi ^{2}}}\sum _{n\,\mathrm {odd} }^{\infty }{\frac {(-1)^{(n-1)/2}}{n^{2}}}\sin \left({\frac {2\pi nx}{p}}\right)$ | non-continuous first derivative |
| Sawtooth wave | $2\left({\frac {x}{p}}-\left\lfloor {\frac {1}{2}}+{\frac {x}{p}}\right\rfloor \right)$ | $-\lim _{m\rightarrow 1^{-}}\operatorname {zn} \left({\frac {2Kx}{p}}+K,m\right)$ | ${\frac {2}{\pi }}\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{n}}\sin \left({\frac {2\pi nx}{p}}\right)$ | non-continuous |
| Square wave | $\operatorname {sgn} \left(\sin {\frac {2\pi x}{p}}\right)$ | $\lim _{m\rightarrow 1^{-}}\operatorname {sn} \left({\frac {4Kx}{p}},m\right)$ | ${\frac {4}{\pi }}\sum _{n\,\mathrm {odd} }^{\infty }{\frac {1}{n}}\sin \left({\frac {2\pi nx}{p}}\right)$ | non-continuous |
| Pulse wave | $H\left(\cos {\frac {2\pi x}{p}}-\cos {\frac {\pi t}{p}}\right)$ 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 |
| Magnitude of sine wave with amplitude, A, and period, p/2 | $A\left|\sin {\frac {\pi x}{p}}\right|$ | | ${\frac {4A}{2\pi }}+\sum _{n=1}^{\infty }{\frac {4A}{\pi }}{\frac {1}{4n^{2}-1}}\cos {\frac {2\pi nx}{p}}$$^{:p.193}$ | 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. | | ${\frac {p}{\pi }}{\biggl{\frac {3}{4}}+\sum _{n=1}^{\infty }{\frac {\operatorname {J} _{n}(n)-\operatorname {J} _{n-1}(n)}{n}}\cos {\frac {2\pi nx}{p}}{\biggr )}$ where $\operatorname {J} _{n}(x)$ is the Bessel Function of the first kind. | non-continuous first derivative |
| Dirac comb | $\sum _{n=-\infty }^{\infty }\delta (x-np)$ | $\lim _{m\rightarrow 1^{-}}{\frac {2K(m)}{p\pi }}\operatorname {dn} \left({\frac {2Kx}{p}},m\right)$ | ${\frac {1}{p}}\sum _{n=-\infty }^{\infty }e^{\frac {2n\pi ix}{p}}$ | non-continuous |
| Dirichlet function | ${\displaystyle \mathbf {1} _{\mathbb {Q} }(x)={\begin{cases}1&x\in \mathbb {Q} \\0&x\notin \mathbb {Q} \end{cases}}}$ | $\lim _{m,n\rightarrow \infty }\cos ^{2m}(n!x\pi )$ | - | non-continuous |

## Vector-valued functions

* Epitrochoid
* Epicycloid (special case of the epitrochoid)
* Limaçon (special case of the epitrochoid)
* Hypotrochoid
* Hypocycloid (special case of the hypotrochoid)
* Spirograph (special case of the hypotrochoid)

## Doubly periodic functions

* Jacobi's elliptic functions
* Weierstrass's elliptic function