WeierstrassP (Weierstrass elliptic function), WeierstrassPPrime, WeierstrassZeta, and WeierstrassSigma are defined by

where sums and products range over  such that  is in . WeierstrassP and WeierstrassPPrime are elliptic functions (also known as doubly periodic functions) with periods  and .

Quantities g2 and g3 are known as the invariants and are related to  and  by

where sums range over  such that  is in .

An important property of the invariants g2 and g3 is that WeierstrassP satisfies the differential equation

A special case of WeierstrassP happens when the discriminant  is equal to zero, in which case  and  are related, can be expressed in terms of a single parameter, say , and the function is given by

Refer to Chapter 18, "Weierstrass Elliptic and Related Functions" of Handbook of Mathematical Functions edited by Abramowitz and Stegun for more extensive information.