Difference between revisions of "Elliptic function"

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=Properties=
 
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[[Constant functions are elliptic functions]]<br />
<strong>Theorem:</strong> All constant functions are elliptic functions.
 
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<strong>Proof:</strong> █
 
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Latest revision as of 00:00, 23 December 2016

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:

References

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