View source for List of periodic functions

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{{incomplete list|date=December 2012}}

This is a list of some well-known [[Periodic function|periodic function]]s. The constant function {{math|{{var|f}}{{sub| }}({{var|x}}) {{=}} {{var|c}}}}, where {{mvar|c}} is independent of {{mvar|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 ==
<!-- I've included power series for functions unless they are a trivial linear combination of other functions -Jamgoodman 2019-->

All trigonometric functions listed have period <math>2\pi</math>, unless otherwise stated. For the following trigonometric functions:
: {{mvar|U<sub>n</sub>}} is the {{mvar|n}}th up/down number,
: {{mvar|B<sub>n</sub>}} is the {{mvar|n}}th [[Bernoulli number]]
: in Jacobi elliptic functions, <math>q=e^{-\pi \frac{K(1-m)}{K(m)}}</math>

{| class="wikitable sortable"
|-
! Name !! Symbol !! Formula {{refn|group=nb|Formulae are given as Taylor series or derived from other entries.}} !! Fourier Series
|-
|[[Sine]] || <math> \sin(x) </math> || <math>\sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n + 1)!}</math> || <math> \sin(x) </math>
|-
|[[Cas (mathematics)|cas (mathematics)]] || <math> \operatorname{cas}(x) </math> || <math>\sin(x)+\cos(x)</math> || <math> \sin(x) + \cos(x) </math>
|-
| [[Cosine]] || <math> \cos(x) </math> || <math>\sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!}</math> ||  <math> \cos(x) </math>
|-
| [[Cis (mathematics)|cis (mathematics)]] || <math> e^{ix}, \operatorname{cis}(x) </math> || {{math| cos(''x'') + ''i'' sin(''x'')}} || <math>\cos(x)+i\sin(x)</math>
|-
| [[Tangent (trigonometry)|Tangent]] || <math> \tan(x) </math> || <math>\frac{\sin x}{\cos x}=\sum_{n=0}^\infty \frac{U_{2n+1} x^{2n+1}}{(2n+1)!}</math> || <math>2\sum_{n=1}^\infty (-1)^{n-1}\sin(2nx)</math> <ref>http://web.mit.edu/jorloff/www/18.03-esg/notes/fourier-tan.pdf {{Bare URL PDF|date=March 2022}}</ref>
|-
| [[Cotangent]] || <math> \cot(x) </math> || <math>\frac{\cos x}{\sin x}=\sum_{n=0}^\infty \frac{(-1)^n 2^{2n} B_{2n} x^{2n-1}}{(2n)!}</math> || <math>i+2i\sum_{n=1}^\infty(\cos2nx-i\sin2nx)</math> {{citation needed|date=March 2019}}
|-
| [[Secant (trigonometry)|Secant]] || <math> \sec(x) </math> || <math>\frac1{\cos x}=\sum_{n=0}^\infty \frac{U_{2n} x^{2n}}{(2n)!}</math> || -
|-
| [[Cosecant]] || <math> \csc(x) </math> || <math>\frac1{\sin x}=\sum_{n=0}^\infty \frac{(-1)^{n+1} 2 \left(2^{2n-1}-1\right) B_{2n} x^{2n-1}}{(2n)!}</math>  || -
|-
| [[Exsecant]] || <math> \operatorname{exsec}(x) </math> || <math>\sec(x)-1</math> || -
|-
| [[Excosecant]] || <math> \operatorname{excsc}(x) </math> || <math>\csc(x)-1</math> || -
|-
| [[Versine]] || <math> \operatorname{versin}(x) </math> || <math>1-\cos(x)</math> || <math>1-\cos(x)</math>
|-
| [[Vercosine]] || <math> \operatorname{vercosin}(x) </math> || <math>1+\cos(x)</math> || <math>1+\cos(x)</math>
|-
| [[Coversine]] || <math> \operatorname{coversin}(x) </math> || <math>1-\sin(x)</math> || <math>1-\sin(x)</math>
|-
| [[Covercosine]] || <math> \operatorname{covercosin}(x) </math> || <math>1+\sin(x)</math> || <math>1+\sin(x)</math>
|-
| [[Haversine]] || <math> \operatorname{haversin}(x) </math> || <math>\frac{1-\cos(x)}{2}</math> || <math>\frac{1}{2}-\frac12\cos(x)</math>
|-
| [[Havercosine]] || <math> \operatorname{havercosin}(x) </math> || <math>\frac{1+\cos(x)}{2}</math> || <math>\frac{1}{2}+\frac12\cos(x)</math>
|-
| [[Hacoversine]] || <math> \operatorname{hacoversin}(x) </math> || <math>\frac{1-\sin(x)}{2}</math> || <math>\frac{1}{2}-\frac12\sin(x)</math>
|-
| [[Hacovercosine]] || <math> \operatorname{hacovercosin}(x) </math> || <math>\frac{1+\sin(x)}{2}</math> || <math>\frac{1}{2}+\frac12\sin(x)</math>
|-
| Jacobi elliptic function sn || <math> \operatorname{sn}(x,m) </math> || <math>\sin \operatorname{am}(x,m)</math> || <math>\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)} </math>
|-
| Jacobi elliptic function cn || <math> \operatorname{cn}(x,m) </math> || <math>\cos \operatorname{am}(x,m)</math> || <math>\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)}</math>
|-
| Jacobi elliptic function dn || <math> \operatorname{dn}(x,m) </math> || <math>\sqrt{1-m\operatorname{sn}^2(x,m)}</math> || <math>\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)} </math>
|-
| Jacobi elliptic function zn || <math> \operatorname{zn}(x,m) </math> || <math>\int^x_0\left[\operatorname{dn}(t,m)^2-\frac{E(m)}{K(m)}\right]dt </math> || <math>\frac{2\pi}{K(m)}\sum_{n=1}^\infty \frac{q^n}{1-q^{2n}}~\sin\frac{n\pi x}{K(m)} </math>
|-
| [[Weierstrass elliptic function]] || <math> \weierp(x,\Lambda) </math> || <math>\frac1{x^2}+\sum_{\lambda\in\Lambda-\{0\}}\left[\frac1{(x-\lambda)^2}-\frac1{\lambda^2}\right] </math> || <math> </math>
|-[[Clausen function]]
|[[Clausen function]]
|<math>\operatorname{Cl}_2(x)</math>
|<math>-\int^x_0\ln\left|2\sin\frac{t}{2}\right|dt</math>
|<math>\sum_{k=1}^\infty\frac{\sin kx}{k^2}</math>
|}

== Non-smooth functions ==

The following functions have period <math>p</math> and take <math>x</math> as their argument. The symbol <math>\lfloor n \rfloor</math> is the [[Floor and ceiling functions|floor function]] of <math>n</math> and <math>\sgn</math> is the [[Sign function|sign function]].


K means [[Elliptic integral]] K(m)

{| class="wikitable sortable"
|-
! Name !! Formula !! Limit !! Fourier Series !! Notes
|-
||  [[Triangle wave]] || <math> \frac{4}{p} \left (x-\frac{p}{2} \left \lfloor\frac{2 x}{p}+\frac{1}{2} \right \rfloor \right )(-1)^\left \lfloor\frac{2 x}{p}+\frac{1}{2} \right \rfloor</math>
||<math>\lim_{m\rightarrow1^-}\operatorname{zs}\left(\frac{4Kx}p-K,m\right)</math> ||<math>\frac8{\pi^2}\sum_{n\,\mathrm{odd}}^{\infty} \frac{(-1)^{(n-1)/2}}{n^2} \sin\left(\frac{2\pi n x}{p}\right)
</math>|| non-continuous first derivative
|-
||  [[Sawtooth wave]] || <math>2 \left( {\frac x p} - \left \lfloor {\frac 1 2} + {\frac x p} \right \rfloor \right)</math> ||<math>-\lim_{m\rightarrow1^-}\operatorname{zn}\left(\frac{2Kx}p+K,m\right)</math> ||<math> \frac2\pi\sum_{n=1}^\infty\frac{(-1)^{n-1}}n\sin\left(\frac{2\pi nx}{p}\right) </math>|| non-continuous
|-
||  [[Square wave]] || <math> \sgn\left(\sin \frac{2\pi x}{p} \right) </math> ||<math>\lim_{m\rightarrow1^-}\operatorname{sn}\left(\frac{4Kx}p,m\right)</math> ||<math> \frac4\pi\sum_{n\,\mathrm{odd}}^\infty\frac1n\sin\left(\frac{2\pi nx}{p}\right) </math>|| non-continuous
|-
|| [[Pulse wave]] ||<math>H \left( \cos\frac{2\pi x}{p}- \cos\frac{\pi t}{p}\right)</math>
<small>where <math>H</math> is the [[Heaviside step function]]<br />t is how long the pulse stays at 1</small>
|
|<math>\frac{t}{p} + \sum_{n=1}^{\infty} \frac{2}{n\pi} \sin\left(\frac{\pi nt}{p}\right) \cos\left(\frac{2\pi n x}{p}\right)</math>|| non-continuous
|-
| Magnitude of sine wave <br /> with amplitude, A, and period, p/2 || <math> A\left|\sin\frac{\pi x}p\right| </math> || || <math>\frac{4A}{2\pi}+\sum_{n=1}^{\infty} \frac{4A}{\pi}\frac{1}{4n^2-1}\cos\frac{2\pi nx}p</math> <ref name=Papula>{{cite book | author=Papula, Lothar| title=Mathematische Formelsammlung: für Ingenieure und Naturwissenschaftler| publisher=Vieweg+Teubner Verlag | year=2009 | isbn=978-3834807571}}</ref>{{rp|p. 193}}|| non-continuous
|-
|| [[Cycloid]] ||<math>\frac{p - p\cos \left( f^{(-1)}\left( \frac{2\pi x}{p} \right) \right)}{2\pi}</math>

<small>given <math>f(x)=x-\sin(x)</math> and <math>f^{(-1)}(x)</math> is</small>

<small>its real-valued inverse</small><small>.</small>
||
| <math>\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)</math>

<small>where <math>\operatorname{J}_n(x)</math> is the [[Bessel function|Bessel Function of the first kind]].</small>
| non-continuous first derivative
|-
|| [[Dirac comb]] ||<math>\sum_{n=-\infty}^{\infty}\delta(x-np) </math>
|<math>\lim_{m\rightarrow1^-}\frac{2K(m)}{p\pi}\operatorname{dn}\left(\frac{2Kx}p,m\right)</math>
|<math>\frac1p\sum_{n=-\infty}^{\infty}e^{\frac{2n\pi ix}p}</math>|| non-continuous
|-
|[[Dirichlet function]]
|<math>{\displaystyle \mathbf {1} _{\mathbb {Q} }(x)={\begin{cases}1&x\in \mathbb {Q} \\0&x\notin \mathbb {Q} \end{cases}}}</math>
|<math>\lim_{m,n\rightarrow\infty}\cos^{2m}(n!x\pi)</math>
| -
|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

==Notes==
{{reflist|group=nb}}

<references />

{{DEFAULTSORT:Periodic functions}}
[[Category:Mathematics-related lists]]
[[Category:Types of functions]]

{{Sourceattribution|List of periodic functions}}