Difference between revisions of "Elliptic function"
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(Created page with "A function $f$ is called elliptic if it is a doubly periodic function and it is meromorphic.") |
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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= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> All constant functions are elliptic functions. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <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> | ||
Revision as of 21:20, 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: █
