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.

Trigonometric functionsEdit

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
Name Symbol Formula [nb 1] Fourier Series
Sine      
cas (mathematics)      
Cosine      
cis (mathematics)   cos(x) + i sin(x)  
Tangent       [1]
Cotangent      [citation needed]
Secant     -
Cosecant     -
Exsecant     -
Excosecant     -
Versine      
Vercosine      
Coversine      
Covercosine      
Haversine      
Havercosine      
Hacoversine      
Hacovercosine      
Magnitude of sine wave
with amplitude, A, and period, T
-     [2]:p. 193

Sinus-like functionsEdit

Non-smooth functionsEdit

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

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


given   and   is

its real-valued inverse.

- non-continuous first derivative
Pulse wave  

where H is the Heaviside step function

t is how long the pulse stays at 1

  non-continuous

The following functions are also not smooth:

Vector-valued functionsEdit

Doubly periodic functionsEdit

NotesEdit

  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
  2. ^ Papula, Lothar (2009). Mathematische Formelsammlung: für Ingenieure und Naturwissenschaftler. Vieweg+Teubner Verlag. ISBN 978-3834807571.