---
date: 2022-04-05 23:42
modification date: Thursday 12th May 2022 15:35:45
title: "Weierstrass p function"
aliases: [Weierstrass p function]
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# Weierstrass p function

$$
\wp\left(z, \omega_{1}, \omega_{2}\right):=\wp(z, \Lambda):=\frac{1}{z^{2}}+\sum_{\lambda \in \Lambda \backslash\{0\}}\left(\frac{1}{(z-\lambda)^{2}}-\frac{1}{\lambda^{2}}\right)
.$$

An entire elliptic function thatt satisfies the following differential equation, which has a solution by Liouville's theorem:
$$
\wp^{\prime 2}(z)=4 \wp^{3}(z)-g_{2} \wp(z)-g_{3}, \qquad g_2 = 60G_4, g_3 = 140 G_6
$$