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 |
---|---|---|---|
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 | |||
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 |
non-continuous | ||
Magnitude of sine wave with amplitude, A, and period, p/2 |
[2]: p. 193 | non-continuous | ||
Cycloid |
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]- 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
[edit]Notes
[edit]- ^ Formulae are given as Taylor series or derived from other entries.
- ^ http://web.mit.edu/jorloff/www/18.03-esg/notes/fourier-tan.pdf [bare URL PDF]
- ^ Papula, Lothar (2009). Mathematische Formelsammlung: für Ingenieure und Naturwissenschaftler. Vieweg+Teubner Verlag. ISBN 978-3834807571.