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

where sums and products range over $\mathrm{\omega }=2m_1{\mathrm{\omega }}_1+2m_2{\mathrm{\omega }}_2$ such that $m_1,m_2$ is in $\left(ZxZ\right)-\left(0,0\right)$. WeierstrassP and WeierstrassPPrime are elliptic functions (also known as doubly periodic functions) with periods $2{\mathrm{\omega }}_1$ and $2{\mathrm{\omega }}_2$.

Quantities g2 and g3 are known as the invariants and are related to ${\mathrm{\omega }}_1$ and ${\mathrm{\omega }}_2$ by

where sums range over $\mathrm{\omega }=2m_1{\mathrm{\omega }}_1+2m_2{\mathrm{\omega }}_2$ such that $m_1,m_2$ is in $\left(ZxZ\right)-\left(0,0\right)$.

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 ${\mathrm{g2}}^3-27{\mathrm{g3}}^2$ is equal to zero, in which case $\mathrm{g2}$ and $\mathrm{g3}$ are related, can be expressed in terms of a single parameter, say $t$, 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.

$\mathrm{WeierstrassP}\left(1.0\,2.0\,3.0\right)$

$1.214433709$

$\mathrm{WeierstrassPPrime}\left(1.0\,2.0\,3.0\right)$

$\mathrm{-1.317406195}$

$\mathrm{WeierstrassZeta}\left(1.0\,2.0\,3.0\right)$

$0.9443449465$

$\mathrm{WeierstrassSigma}\left(1.0\,2.0\,3.0\right)$

$0.9880674335$