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.

Trigonometric 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
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[edit]

Non-smooth functions[edit]

The following functions take the variable , period and have range to . The symbol is the floor function of n and is the sign function.

Name Formula Fourier Series Notes
Triangle wave - non-continuous first derivative
Sawtooth wave [3] non-continuous
Square wave - non-continuous
Cycloid No closed form[citation needed]. - non-continuous first derivative
Pulse wave - - non-continuous

The following functions are also not smooth:

Vector-valued functions[edit]

Doubly periodic functions[edit]

Notes[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
  2. ^ Papula, Lothar (2009). Mathematische Formelsammlung: für Ingenieure und Naturwissenschaftler. Vieweg+Teubner Verlag. ISBN 3834807575.
  3. ^ http://mathworld.wolfram.com/FourierSeries.html