This is a list of some well-known periodic functions. 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 , unless otherwise stated. For the following trigonometric functions: is the th up/down number, is the th Bernoulli number in Jacobi elliptic functions,
Name Symbol Formula Fourier SeriesSine cas Cosine cis Tangent Cotangent Secant - Cosecant - Exsecant - Versine Coversine Haversine Jacobi elliptic function sn Jacobi elliptic function cn Jacobi elliptic function dn Jacobi elliptic function zn Weierstrass elliptic function Clausen function
Non-smooth functions
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 functiont is how long the pulse stays at 1 non-continuous Magnitude of sine wave with amplitude, A, and period, p/2 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-continuousDirichlet function -non-continuous
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