Elliptic function

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

Properties[edit]

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:

References[edit]

[1]
A chapter in elliptic functions - J.W.L. Glaisher