Difference between revisions of "Elliptic function"
| Line 50: | Line 50: | ||
</div> | </div> | ||
</div> | </div> | ||
| + | |||
| + | =References= | ||
| + | [http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_629.htm] | ||
Revision as of 17:57, 25 July 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: █
