Weierstrass elliptic function

Description

Let $\omega_1, \omega_2 \in \C$ be linearly independent over $\R$, and let $\Lambda = \Z\omega_1 + \Z\omega_2$ be the lattice they span. The Weierstrass elliptic function of $\Lambda$ is $$\wp(z) = \frac{1}{z^2} + \sum_{\omega \in \Lambda \setminus \set{0}} \left( \frac{1}{(z - \omega)^2} - \frac{1}{\omega^2} \right), \qquad z \in \C \setminus \Lambda.$$ The series converges absolutely and uniformly on compact subsets of $\C \setminus \Lambda$, because $\sum_{\omega \ne 0} \abs{\omega}^{-3} < \infty$.

1. $\wp$ is a meromorphic function on $\C$ whose poles are exactly the points of $\Lambda$, each a pole of order $2$.
2. $\wp$ is even, and $\wp(z + \omega) = \wp(z)$ for all $\omega \in \Lambda$, so $\wp$ is an elliptic function for $\Lambda$.
3. $\wp$ satisfies the differential equation $(\wp')^2 = 4\wp^3 - g_2 \wp - g_3$, where $g_2 = 60 \sum_{\omega \ne 0} \omega^{-4}$ and $g_3 = 140 \sum_{\omega \ne 0} \omega^{-6}$.