Doubly periodic function

Definition

A meromorphic function $f:\mathbb {C} \to S^{2}$ is termed doubly periodic or elliptic if there exist two real-linearly independent nonzero complex numbers $\omega _{1}$ and $\omega _{2}$ such that:

$f(z+\omega _{1})=f(z),f(z+\omega _{2})=f(z)\ \forall \ z\in \mathbb {C}$

The additive subgroup generated by $\omega _{1}$ and $\omega _{2}$ is a lattice in $\mathbb {C}$, and an elliptic function on $\mathbb {C}$ can be defined as a function that descends to a function on the quotient of $\mathbb {C}$ by the lattice.