This is a list of some well-known

periodic function A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the t ...

s. The constant function , where is independent of , 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

All trigonometric functions listed have period

2\pi

, unless otherwise stated. For the following trigonometric functions: : is the th up/down number, : is the th

Bernoulli number In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent function ...

: in Jacobi elliptic functions,

q=e^

Non-smooth functions

The following functions have period

p

and take

x

as their argument. The symbol

\lfloor n \rfloor

is the

floor function In mathematics, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least integer greater than or eq ...

n

and

\sgn

is the

sign function In mathematics, the sign function or signum function (from '' signum'', Latin for "sign") is a function that has the value , or according to whether the sign of a given real number is positive or negative, or the given number is itself zer ...

. K means

Elliptic integral In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising i ...

K(m)

Vector-valued functions

* Epitrochoid * Epicycloid (special case of the epitrochoid) * Limaçon (special case of the epitrochoid) *

Hypotrochoid In geometry, a hypotrochoid is a roulette (curve), roulette traced by a point attached to a circle of radius rolling around the inside of a fixed circle of radius , where the point is a distance from the center of the interior circle. The par ...

Hypocycloid In geometry, a hypocycloid is a special plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle. As the radius of the larger circle is increased, the hypocycloid becomes more like the cycloid creat ...

(special case of the hypotrochoid) * Spirograph (special case of the hypotrochoid)

Doubly periodic functions

Jacobi's elliptic functions In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum, as well as in the design of electronic elliptic filters. While trigonometric functions are define ...

* Weierstrass's elliptic function

Notes

{{DEFAULTSORT:Periodic functions Mathematics-related lists Types of functions