Difference between revisions of "Elliptic function"
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| + | [http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN600494829_0017%7CLOG_0012 A chapter in elliptic functions - J.W.L. Glaisher]<br /> | ||
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: █
