# Elliptic function

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

# Properties

Constant functions are elliptic functions

**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:** █

**Theorem:** The sum of the residues of an **elliptic function** at its poles in any period parallelogram equals zero.

**Proof:** █

**Theorem:** The number of zeros of an **elliptic function** in and period parallelogram equals the number of poles, counted with multiplicity.

**Proof:** █