List of periodic functions

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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 [math]\displaystyle{ 2\pi }[/math], 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, [math]\displaystyle{ q=e^{-\pi \frac{K(1-m)}{K(m)}} }[/math]

Name	Symbol	Formula	Fourier Series
Sine	[math]\displaystyle{ \sin(x) }[/math]	[math]\displaystyle{ \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n + 1)!} }[/math]	[math]\displaystyle{ \sin(x) }[/math]
cas (mathematics)	[math]\displaystyle{ \operatorname{cas}(x) }[/math]	[math]\displaystyle{ \sin(x)+\cos(x) }[/math]	[math]\displaystyle{ \sin(x) + \cos(x) }[/math]
Cosine	[math]\displaystyle{ \cos(x) }[/math]	[math]\displaystyle{ \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!} }[/math]	[math]\displaystyle{ \cos(x) }[/math]
cis (mathematics)	[math]\displaystyle{ e^{ix}, \operatorname{cis}(x) }[/math]	cos(x) + i sin(x)	[math]\displaystyle{ \cos(x)+i\sin(x) }[/math]
Tangent	[math]\displaystyle{ \tan(x) }[/math]	[math]\displaystyle{ \frac{\sin x}{\cos x}=\sum_{n=0}^\infty \frac{U_{2n+1} x^{2n+1}}{(2n+1)!} }[/math]	[math]\displaystyle{ 2\sum_{n=1}^\infty (-1)^{n-1}\sin(2nx) }[/math]
Cotangent	[math]\displaystyle{ \cot(x) }[/math]	[math]\displaystyle{ \frac{\cos x}{\sin x}=\sum_{n=0}^\infty \frac{(-1)^n 2^{2n} B_{2n} x^{2n-1}}{(2n)!} }[/math]	[math]\displaystyle{ i+2i\sum_{n=1}^\infty(\cos2nx-i\sin2nx) }[/math]
Secant	[math]\displaystyle{ \sec(x) }[/math]	[math]\displaystyle{ \frac1{\cos x}=\sum_{n=0}^\infty \frac{U_{2n} x^{2n}}{(2n)!} }[/math]	-
Cosecant	[math]\displaystyle{ \csc(x) }[/math]	[math]\displaystyle{ \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]\displaystyle{ \operatorname{exsec}(x) }[/math]	[math]\displaystyle{ \sec(x)-1 }[/math]	-
Excosecant	[math]\displaystyle{ \operatorname{excsc}(x) }[/math]	[math]\displaystyle{ \csc(x)-1 }[/math]	-
Versine	[math]\displaystyle{ \operatorname{versin}(x) }[/math]	[math]\displaystyle{ 1-\cos(x) }[/math]	[math]\displaystyle{ 1-\cos(x) }[/math]
Vercosine	[math]\displaystyle{ \operatorname{vercosin}(x) }[/math]	[math]\displaystyle{ 1+\cos(x) }[/math]	[math]\displaystyle{ 1+\cos(x) }[/math]
Coversine	[math]\displaystyle{ \operatorname{coversin}(x) }[/math]	[math]\displaystyle{ 1-\sin(x) }[/math]	[math]\displaystyle{ 1-\sin(x) }[/math]
Covercosine	[math]\displaystyle{ \operatorname{covercosin}(x) }[/math]	[math]\displaystyle{ 1+\sin(x) }[/math]	[math]\displaystyle{ 1+\sin(x) }[/math]
Haversine	[math]\displaystyle{ \operatorname{haversin}(x) }[/math]	[math]\displaystyle{ \frac{1-\cos(x)}{2} }[/math]	[math]\displaystyle{ \frac{1}{2}-\frac12\cos(x) }[/math]
Havercosine	[math]\displaystyle{ \operatorname{havercosin}(x) }[/math]	[math]\displaystyle{ \frac{1+\cos(x)}{2} }[/math]	[math]\displaystyle{ \frac{1}{2}+\frac12\cos(x) }[/math]
Hacoversine	[math]\displaystyle{ \operatorname{hacoversin}(x) }[/math]	[math]\displaystyle{ \frac{1-\sin(x)}{2} }[/math]	[math]\displaystyle{ \frac{1}{2}-\frac12\sin(x) }[/math]
Hacovercosine	[math]\displaystyle{ \operatorname{hacovercosin}(x) }[/math]	[math]\displaystyle{ \frac{1+\sin(x)}{2} }[/math]	[math]\displaystyle{ \frac{1}{2}+\frac12\sin(x) }[/math]
Jacobi elliptic function sn	[math]\displaystyle{ \operatorname{sn}(x,m) }[/math]	[math]\displaystyle{ \sin \operatorname{am}(x,m) }[/math]	[math]\displaystyle{ \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]\displaystyle{ \operatorname{cn}(x,m) }[/math]	[math]\displaystyle{ \cos \operatorname{am}(x,m) }[/math]	[math]\displaystyle{ \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]\displaystyle{ \operatorname{dn}(x,m) }[/math]	[math]\displaystyle{ \sqrt{1-m\operatorname{sn}^2(x,m)} }[/math]	[math]\displaystyle{ \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]\displaystyle{ \operatorname{zn}(x,m) }[/math]	[math]\displaystyle{ \int^x_0\left[\operatorname{dn}(t,m)^2-\frac{E(m)}{K(m)}\right]dt }[/math]	[math]\displaystyle{ \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]\displaystyle{ \weierp(x,\Lambda) }[/math]	[math]\displaystyle{ \frac1{x^2}+\sum_{\lambda\in\Lambda-\{0\}}\left[\frac1{(x-\lambda)^2}-\frac1{\lambda^2}\right] }[/math]	[math]\displaystyle{ }[/math]
Clausen function	[math]\displaystyle{ \operatorname{Cl}_2(x) }[/math]	[math]\displaystyle{ -\int^x_0\ln\left|2\sin\frac{t}{2}\right|dt }[/math]	[math]\displaystyle{ \sum_{k=1}^\infty\frac{\sin kx}{k^2} }[/math]

Non-smooth functions

The following functions have period [math]\displaystyle{ p }[/math] and take [math]\displaystyle{ x }[/math] as their argument. The symbol [math]\displaystyle{ \lfloor n \rfloor }[/math] is the floor function of [math]\displaystyle{ n }[/math] and [math]\displaystyle{ \sgn }[/math] is the sign function.

K means Elliptic integral K(m)

Name	Formula	Limit	Fourier Series	Notes
Triangle wave	[math]\displaystyle{ \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]\displaystyle{ \lim_{m\rightarrow1^-}\operatorname{zs}\left(\frac{4Kx}p-K,m\right) }[/math]	[math]\displaystyle{ \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]\displaystyle{ 2 \left( {\frac x p} - \left \lfloor {\frac 1 2} + {\frac x p} \right \rfloor \right) }[/math]	[math]\displaystyle{ -\lim_{m\rightarrow1^-}\operatorname{zn}\left(\frac{2Kx}p+K,m\right) }[/math]	[math]\displaystyle{ \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]\displaystyle{ \sgn\left(\sin \frac{2\pi x}{p} \right) }[/math]	[math]\displaystyle{ \lim_{m\rightarrow1^-}\operatorname{sn}\left(\frac{4Kx}p,m\right) }[/math]	[math]\displaystyle{ \frac4\pi\sum_{n\,\mathrm{odd}}^\infty\frac1n\sin\left(\frac{2\pi nx}{p}\right) }[/math]	non-continuous
Pulse wave	[math]\displaystyle{ H \left( \cos\frac{2\pi x}{p}- \cos\frac{\pi t}{p}\right) }[/math] where [math]\displaystyle{ H }[/math] is the Heaviside step function t is how long the pulse stays at 1		[math]\displaystyle{ \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 with amplitude, A, and period, p/2	[math]\displaystyle{ A\left|\sin\frac{\pi x}p\right| }[/math]		[math]\displaystyle{ \frac{4A}{2\pi}+\sum_{n=1}^{\infty} \frac{4A}{\pi}\frac{1}{4n^2-1}\cos\frac{2\pi nx}p }[/math]	non-continuous
Cycloid	[math]\displaystyle{ \frac{p - p\cos \left( f^{(-1)}\left( \frac{2\pi x}{p} \right) \right)}{2\pi} }[/math] given [math]\displaystyle{ f(x)=x-\sin(x) }[/math] and [math]\displaystyle{ f^{(-1)}(x) }[/math] is its real-valued inverse.		[math]\displaystyle{ \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] where [math]\displaystyle{ \operatorname{J}_n(x) }[/math] is the Bessel Function of the first kind.	non-continuous first derivative
Dirac comb	[math]\displaystyle{ \sum_{n=-\infty}^{\infty}\delta(x-np) }[/math]	[math]\displaystyle{ \lim_{m\rightarrow1^-}\frac{2K(m)}{p\pi}\operatorname{dn}\left(\frac{2Kx}p,m\right) }[/math]	[math]\displaystyle{ \frac1p\sum_{n=-\infty}^{\infty}e^{\frac{2n\pi ix}p} }[/math]	non-continuous
Dirichlet function	[math]\displaystyle{ {\displaystyle \mathbf {1} _{\mathbb {Q} }(x)={\begin{cases}1&x\in \mathbb {Q} \\0&x\notin \mathbb {Q} \end{cases}}} }[/math]	[math]\displaystyle{ \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