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 All trigonometric functions listed have period 2 π {\displaystyle 2\pi } , 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, q = e − π K ( 1 − m ) K ( m ) {\displaystyle q=e^{-\pi {\frac {K(1-m)}{K(m)}}}}
Non-smooth functions The following functions have period p {\displaystyle p} and take x {\displaystyle x} as their argument. The symbol ⌊ n ⌋ {\displaystyle \lfloor n\rfloor } is the floor function of n {\displaystyle n} and sgn {\displaystyle \operatorname {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
Notes