Weierstrass p-function

Definition

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

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

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