List of periodic functions

From Wikipedia
https://en.wikipedia.org/wiki/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 functions

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 functions

Non-smooth functions

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.


where is the Bessel Function of the first kind.

non-continuous first derivative
Pulse wave

where is the Heaviside step function

t is how long the pulse stays at 1

non-continuous

Vector-valued functions

Doubly periodic functions

Notes

  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.