Difference between revisions of "Elliptic function"
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<strong>Theorem:</strong> The [[contour integral]] of an [[elliptic function]] taken along the boundary of any [[cell]] equals zero. | <strong>Theorem:</strong> The [[contour integral]] of an [[elliptic function]] taken along the boundary of any [[cell]] equals zero. | ||
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| + | <strong>Proof:</strong> █ | ||
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| + | </div> | ||
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| + | <strong>Theorem:</strong> The sum of the [[residue|residues]] of an [[elliptic function]] at its [[pole|poles]] in any [[period parallelogram]] equals zero. | ||
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| + | <strong>Proof:</strong> █ | ||
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| + | <strong>Theorem:</strong> The number of [[zero|zeros]] of an [[elliptic function]] in and [[period parallelogram]] equals the number of [[pole|poles]], counted with multiplicity. | ||
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<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
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Revision as of 21:27, 6 June 2015
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: █
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: █
