Difference between revisions of "Elliptic function"
| Line 2: | Line 2: | ||
=Properties= | =Properties= | ||
| − | + | [[Constant functions are elliptic functions]]<br /> | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | </ | ||
<div class="toccolours mw-collapsible mw-collapsed"> | <div class="toccolours mw-collapsible mw-collapsed"> | ||
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: █
