Introduction to Elliptic Functions

1. Doubly Periodic Functions

Let $\omega_1,\omega_2$ be any two numbers(real or complex) whose ratio is not real. A function which satisfies the equations $$f(z+2\omega_1)=f(z+2\omega_2)=f(z)$$ for all values of $z$ for which $f(z)$ exists is called a doubly periodic function with periods $2\omega_1,2\omega_2$. A doubly periodic function which is analytic(except at poles) and which has no singularities other than poles in the finite part of the plane is called an elliptic function.

2. Period Parallelograms

Suppose that in the plane of $z$ we mark the points $0,\omega_1,2\omega_2,2\omega_1+2\omega_2$, we obtain a parallelogram. If there is no point $\omega$ inside or on the boundary of the parallelogram(except the vertices) such that $$f(z+\omega)=f(z)$$ for all values of $z$ then this parallelogram is called the fundamental period parallelogram for an elliptic function with periods $\omega_1,2\omega_2$. The $z$ plane may be covered with a network of parallelograms equal to the fundamental period-paralellogram and similarly situated, each of the points $2m\omega_1+2n\omega_2$ being a vertex of four parallelograms.

3. Liouville's Theorem An elliptic function with no poles is merely a constant.