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List of periodic functions

From Wikipedia, the free encyclopedia

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[edit]

All trigonometric functions listed have period , unless otherwise stated. For the following trigonometric functions:

Un is the nth up/down number,
Bn is the nth Bernoulli number
in Jacobi elliptic functions,
Name Symbol Formula [nb 1] Fourier Series
cas (mathematics)
cis (mathematics) cos(x) + i sin(x)
Tangent [1]
Cotangent [citation needed]
Secant -
Cosecant -
Exsecant -
Excosecant -
Jacobi elliptic function sn
Jacobi elliptic function cn
Jacobi elliptic function dn
Jacobi elliptic function zn
Weierstrass elliptic function
Clausen function

Non-smooth functions[edit]

The following functions have period and take as their argument. The symbol is the floor function of and is the sign function.

K means Elliptic integral K(m)

Name Formula Limit Fourier Series Notes
Triangle wave non-continuous first derivative
Sawtooth wave non-continuous
Square wave non-continuous
Pulse wave

where is the Heaviside step function
t is how long the pulse stays at 1

Magnitude of sine wave
with amplitude, A, and period, p/2
[2]: p. 193  non-continuous

given and is

its real-valued inverse.

where is the Bessel Function of the first kind.

non-continuous first derivative
Dirac comb non-continuous
Dirichlet function - non-continuous

Vector-valued functions[edit]

Doubly periodic functions[edit]


  1. ^ Formulae are given as Taylor series or derived from other entries.
  1. ^ http://web.mit.edu/jorloff/www/18.03-esg/notes/fourier-tan.pdf [bare URL PDF]
  2. ^ Papula, Lothar (2009). Mathematische Formelsammlung: für Ingenieure und Naturwissenschaftler. Vieweg+Teubner Verlag. ISBN 978-3834807571.