List of periodic functions

This is a list of some well-known periodic functions. The constant function $fundefined(x) = c$, 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
All trigonometric functions listed have period $$2\pi$$, unless otherwise stated. For the following trigonometric functions:
 * $U_{n}$ is the $n$th up/down number,
 * $B_{n}$ is the $n$th Bernoulli number
 * in Jacobi elliptic functions, $$q=e^{-\pi \frac{K(1-m)}{K(m)}}$$

Non-smooth functions
The following functions have period $$p$$ and take $$x$$ as their argument. The symbol $$\lfloor n \rfloor$$ is the floor function of $$n$$ and $$\sgn$$ is the sign function.

K means Elliptic integral K(m)

Vector-valued functions

 * 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

 * Jacobi's elliptic functions
 * Weierstrass's elliptic function