Difference between revisions of "Elliptic function"
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A function $f$ is called elliptic if it is a [[doubly periodic function]] and it is [[meromorphic]]. | A function $f$ is called elliptic if it is a [[doubly periodic function]] and it is [[meromorphic]]. | ||
| + | |||
| + | =Properties= | ||
| + | [[Constant functions are elliptic functions]]<br /> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> A nonconstant [[elliptic function]] has a [[fundamental pair of periods]]. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> If an [[elliptic function]] $f$ has no [[pole|poles]] in some [[period parallelogram]], then $f$ is a constant function. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> If an [[elliptic function]] $f$ has no [[zero|zeros]] in some [[period parallelogram]], then $f$ is a constant function. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The [[contour integral]] of an [[elliptic function]] taken along the boundary of any [[cell]] equals zero. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The sum of the [[residue|residues]] of an [[elliptic function]] at its [[pole|poles]] in any [[period parallelogram]] equals zero. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <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. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | =References= | ||
| + | [http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_629.htm]<br /> | ||
| + | [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: █
