Weierstrass p-function

Definition

Let $\mathrm{\Lambda }$ denote a lattice in $\mathbb{C}$. The Weierstrass p-function corresponding to $\mathrm{\Lambda }$, denoted $\mathrm{\wp }$, is a doubly periodic function with period lattice $\mathrm{\Lambda }$, defined as:

$\mathrm{\wp }(z)=\frac{1}{z^2}+{\sum }_{w\in \mathrm{\Lambda }}(\frac{1}{(z+w{)}^2}-\frac{1}{w^2})$

It turns out that all even $\mathrm{\Lambda }$-periodic functions are rational functions in $\mathrm{\wp }$, and all periodic functions in $\mathrm{\Lambda }$ can be expressed in the form $A+{\mathrm{\wp }}^{\prime }B$ where $A,B$ are rational functions in $\mathrm{\wp }$.