Elliptic function

A function $f$ is called elliptic if it is a doubly periodic function and it is meromorphic.

Properties

Theorem: All constant functions are elliptic functions.

Proof: █

Theorem: A nonconstant elliptic function has a fundamental pair of periods.

Proof: █

Theorem: If an elliptic function $f$ has no poles in some period parallelogram, then $f$ is a constant function.

Proof: █

Theorem: If an elliptic function $f$ has no zeros in some period parallelogram, then $f$ is a constant function.

Proof: █

Theorem: The contour integral of an elliptic function taken along the boundary of any cell equals zero.

Proof: █