List of periodic functions explained

This is a list of some well-known periodic functions. The constant function, where is independent of, 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:

is the th up/down number,

is the th Bernoulli number

in Jacobi elliptic functions,

q = e ^{- \pi (K( 1- m))/(K(m))}

Name	Symbol	Formula	Fourier Series
	\sin(x)	\sum ^{inf ty} _{n =0} (( -1 ) ^{n} x ^{2n+1})/(( 2 n +1 )!)	\sin(x)
	\operatorname{cas}(x)	\sin(x) + \cos(x)	\sin(x) + \cos(x)
	\cos(x)	\sum ^{inf ty} _{n =0} (( -1 ) ^{n} x ^{2n})/(( 2 n)!)	\cos(x)
	e ^{ix} , \operatorname{cis}(x)		\cos(x) + i\sin(x)
	\tan(x)	(\sin x)/(\cos x) = \sum ^{inf ty} _{n =0} (U _{2n+1} x ^{2n+1})/(( 2 n +1 )!)	2 \sum ^{inf ty} _{n =1} ( -1 ) ^{n-1} \sin( 2 nx) ^[1]
	\cot(x)	(\cos x)/(\sin x) = \sum ^{inf ty} _{n =0} (( -1 ) ^{n} 2 ^{2n} B _{2n} x ^{2n-1})/(( 2 n)!)	i +2 i\sum ^{inf ty(\cos 2 nx - i\sin 2 nx)} _{n =1}
	\sec(x)	(1 {\cos)/(x} = \sum) ^{inf ty} ^{n =0} (U _{2n} x ^{2n})/(( 2 n)!)	-
	\csc(x)	(1 {\sin)/(x} = \sum) ^{inf ty} ^{n =0} (( -1 ) ^{n+1} 2 \left( 2 ^{2n-1} -1 \right) B _{2n} x ^{2n-1})/(( 2 n)!)	-
	\operatorname{exsec}(x)	\sec(x) -1	-
	\operatorname{excsc}(x)	\csc(x) -1	-
	\operatorname{versin}(x)	1- \cos(x)	1- \cos(x)
	\operatorname{vercosin}(x)	1+ \cos(x)	1+ \cos(x)
	\operatorname{coversin}(x)	1- \sin(x)	1- \sin(x)
	\operatorname{covercosin}(x)	1+ \sin(x)	1+ \sin(x)
	\operatorname{haversin}(x)	(1- \cos(x))/(2)	(1)/(- 2) (12 \cos(x))/()
	\operatorname{havercosin}(x)	(1+ \cos(x))/(2)	(1)/(+ 2) (12 \cos(x))/()
	\operatorname{hacoversin}(x)	(1- \sin(x))/(2)	(1)/(- 2) (12 \sin(x))/()
	\operatorname{hacovercosin}(x)	(1+ \sin(x))/(2)	(1)/(+ 2) (12 \sin(x))/()
Jacobi elliptic function sn	\operatorname{sn}(x,m)	\sin \operatorname{am}(x,m)	(2 \pi)/(K(m)\sqrt m) \sum ^{inf ty} _{n =0} (q ^{n+1/2})/(1- q ^{2n+1}) ~\sin (( 2 n +1 )\pi x)/(2 K(m))
Jacobi elliptic function cn	\operatorname{cn}(x,m)	\cos \operatorname{am}(x,m)	(2 \pi)/(K(m)\sqrt m) \sum ^{inf ty} _{n =0} (q ^{n+1/2})/(~\cos 1+ q ^{2n+1}) (( 2 n +1 )\pi x)/(2 K(m))
Jacobi elliptic function dn	\operatorname{dn}(x,m)	\sqrt{ 1- m\operatorname{sn} ^{2(x,m)}}	(\pi)/(2 K(m)) + (2 \pi)/(K(m)) \sum ^{inf ty} _{n =1} (q ^{n})/(~\cos 1+ q ^{2n}) (n\pi x)/(K(m))
Jacobi elliptic function zn	\operatorname{zn}(x,m)	\int ^{2- (E(m))/(K(m))} _{0 \left[\operatorname{dn}(t,m)} \right]dt	(2 \pi)/(K(m)) \sum ^{inf ty} _{n =1} (q ^{n})/(~\sin 1- q ^{2n}) (n\pi x)/(K(m))
	\weierp(x, Λ )	(1 {x)/(^{2}+\sum}) _{λ\inΛ-\{0\} }\left[\frac1{(x-\lambda)^2}-\frac1{\lambda^2}\right]
Clausen function	\operatorname{Cl} _{2(x)}	- \int ^{x} _{0ln \left}	2\sin\frac\right	dt	\sum ^{inf ty (\sin kx)/(k ^{2})} _{k =1}

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 sgn is the sign function.

K means Elliptic integral K(m)

Name	Formula	Limit	Fourier Series	Notes
	(4)/(p) \left (x - (p)/(2) \left \lfloor (2 x)/(+ p) (1)/(2) \right \rfloor \right )( -1 ) ^{\left} \lfloor (2 x)/(+ p) (1)/(2) \right \rfloor	\lim _{m → 1 ^{-}} \operatorname{zs}\left( (4 Kx)/(p - K,m\right))	(8 {\pi)/(^{2}\sum}) ^{inf ty} ^{n odd} (( -1 ) ^{(n-1)/2})/(\sin\left( n ^{2}) (2 \pi n x)/(p) \right)	non-continuous first derivative
	2 \left( { (x)/(p}) - \left \lfloor { (1)/(2 }) + { (x)/(p}) \right \rfloor \right)	- \lim _{m → 1 ^{-}} \operatorname{zn}\left( (2 Kx)/(p + K,m\right))	(2 \pi\sum)/(_{n=1}) ^{inf ty (( -1 ) ^{n-1})/(n\sin\left( (2 \pi nx)/(p))} \right)	non-continuous
	sgn \left(\sin (2 \pi x)/(p) \right)	\lim _{m → 1 ^{-}} \operatorname{sn}\left( (4 Kx)/(p,m\right))	(4 \pi\sum)/(_{nodd}) ^{inf ty (1 n\sin\left( (2 \pi nx)/(p))/(\right))}	non-continuous
	H \left( \cos (2 \pi x)/(p) - \cos (\pi t)/(p) \right) where H is the Heaviside step function t is how long the pulse stays at 1		(t)/(p) + \sum ^{inf ty} _{n =1} (2)/(\sin\left( n\pi) (\pi nt)/(p) \right) \cos\left( (2 \pi n x)/(p) \right)	non-continuous
Magnitude of sine wave with amplitude, A, and period, p/2	A\left	\sin\fracp\right			(4 A)/(2 \pi) + \sum ^{inf ty} _{n =1} (4 A)/(\pi) (1)/(\cos 4 n ^{2-1}) (2 \pi nx)/(p) ^[2]	non-continuous
	(p - p\cos \left( f ^{(-1)} \left( (2 \pi x)/(p) \right) \right))/(2 \pi) given f(x) = x - \sin(x) and f ^{(-1)} (x) is its real-valued inverse.		(p)/(l ( \pi) (3)/(4) + \sum ^{inf ty} _{n =1} (\operatorname{J)/(_{n(n)-\operatorname{J}}) _{n-1} (n)}n \cos (2 \pi nx)/(p r )) where \operatorname{J} _{n(x)} is the Bessel Function of the first kind.	non-continuous first derivative
	\sum ^{inf ty} _{n =-inf ty} \delta(x - np)	\lim _{m → 1 ^{-}} (2 K(m))/(\operatorname{dn}\left( p\pi) (2 Kx)/(p,m\right))	(1 p\sum)/(_{n=-infty}) ^{infty} e ^{(2 n\pi ix)/(p)}	non-continuous
Dirichlet function	{\displaystyle 1 _{Q} (x) = {\begin{cases} 1 &x\in Q \\ 0 &x\notin Q \end{cases}}}	\lim _{m,n → 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

Notes

1. http://web.mit.edu/jorloff/www/18.03-esg/notes/fourier-tan.pdf
2. Book: Papula, Lothar. Mathematische Formelsammlung: für Ingenieure und Naturwissenschaftler. Vieweg+Teubner Verlag . 2009 . 978-3834807571.