Elliptic function

A function f(z) which is doubly periodic in the complex plane with periods 2ω₁ and 2ω₂ such that

f(z + 2ω₁) = f(z + 2ω₂) = f(z)

which is an analytic function which has no singularities except for poles in the finite part of the complex plane. The ratio of the periods must also not be real or else the function becomes either singly periodic or a constant.

A cell of an elliptic function is defined as a parallelogram region in the complex plane where the function is not multi-valued. Elliptic functions have many interesting properties inside their cells:

1. The number of poles and zeroes inside a cell is always finite.
2. The sum of the residues in any cell is always 0
3. Liouville's elliptic function theorem states that an elliptic function with no poles in a cell is a constant.
4. Elliptic functions must have at least two poles, or a single pole of order two, as a function with only one simple pole would have to have a nonzero residue.
5. Elliptic functions that have a single second-order pole are called Weierstrass elliptic functions.
6. Elliptic functions with two simple poles with residues that are negatives of each other are called Jacobi elliptic functions.
7. Any elliptic function may be expressed in terms of either Weierstrass elliptic functions or Jacobi elliptic functions.
8. Two elliptic functions with the same periods are related algebraically.

Elliptic functions are obtained by inverting elliptic integrals, and can be thought of as generalizations of the trigonometric functions.