Editing Jacobi elliptic functions (section)

==Overview==
[[Image:JacobiFunctionAbstract.png|322px|thumb|The fundamental rectangle in the complex plane of <math>u</math>]]

There are twelve Jacobi elliptic functions denoted by <math>\operatorname{pq}(u, m)</math>, where <math>\mathrm p</math> and <math>\mathrm q</math> are any of the letters <math>\mathrm c</math>, <math>\mathrm s</math>, <math>\mathrm n</math>, and <math>\mathrm d</math>. (Functions of the form <math>\operatorname{pp}(u,m)</math> are trivially set to unity for notational completeness.) <math>u</math> is the argument, and <math>m</math> is the parameter, both of which may be complex. In fact, the Jacobi elliptic functions are [[meromorphic function|meromorphic]] in both <math>u</math> and <math>m</math>.<ref name="Walker">{{cite journal |last1=Walker |first1=Peter |date=2003 |title=The Analyticity of Jacobian Functions with Respect to the Parameter k |url=https://www.jstor.org/stable/3560143 |bibcode=2003RSPSA.459.2569W |journal=Proceedings of the Royal Society |volume=459 |issue=2038 |pages=2569–2574|doi=10.1098/rspa.2003.1157 |jstor=3560143 |s2cid=121368966 }}</ref> The distribution of the zeros and poles in the <math>u</math>-plane is well-known. However, questions of the distribution of the zeros and poles in the <math>m</math>-plane remain to be investigated.<ref name="Walker"/>

In the complex plane of the argument <math>u</math>, the twelve functions form a repeating lattice of simple [[Zeros and poles|poles and zeroes]].<ref name="DLMF22">{{cite web|url=http://dlmf.nist.gov/22|title=NIST Digital Library of Mathematical Functions (Release 1.0.17)|editor-last=Olver|editor-first=F. W. J.|display-editors=et al |date=2017-12-22|publisher=National Institute of Standards and Technology|access-date=2018-02-26 }}</ref> Depending on the function, one repeating parallelogram, or unit cell, will have sides of length <math>2K</math> or <math>4K</math> on the real axis, and <math>2K'</math> or <math>4K'</math> on the imaginary axis, where <math>K=K(m)</math> and <math>K'=K(1-m)</math> are known as the [[quarter period]]s with <math>K(\cdot)</math> being the [[elliptic integral]] of the first kind. The nature of the unit cell can be determined by inspecting the "auxiliary rectangle" (generally a parallelogram), which is a rectangle formed by the origin <math>(0,0)</math> at one corner, and  <math>(K,K')</math> as the diagonally opposite corner. As in the diagram, the four corners of the auxiliary rectangle are named <math>\mathrm s</math>, <math>\mathrm c</math>, <math>\mathrm d</math>, and <math>\mathrm n</math>, going counter-clockwise from the origin. The function <math>\operatorname{pq}(u,m)</math> will have a zero at the <math>\mathrm p</math> corner and a pole at the <math>\mathrm q</math> corner. The twelve functions correspond to the twelve ways of arranging these poles and zeroes in the corners of the rectangle.

When the argument <math>u</math> and parameter <math>m</math> are real, with <math>0 < m < 1</math>, <math>K</math> and <math>K'</math> will be real and the auxiliary parallelogram will in fact be a rectangle, and the Jacobi elliptic functions will all be real valued on the real line.

Since the Jacobi elliptic functions are doubly periodic in <math>u</math>, they factor through a [[torus]] – in effect, their domain can be taken to be a torus, just as cosine and sine are in effect defined on a circle. Instead of having only one circle, we now have the product of two circles, one real and the other imaginary. The complex plane can be replaced by a [[complex torus]]. The circumference of the first circle is <math>4K</math> and the second <math>4K'</math>, where <math>K</math> and <math>K'</math> are the [[quarter period]]s. Each function has two zeroes and two poles at opposite positions on the torus. Among the points {{nowrap|<math>0</math>, <math>K</math>, <math>K + iK'</math>, <math>iK'</math>}} there is one zero and one pole.

The Jacobi elliptic functions are then doubly periodic, meromorphic functions satisfying the following properties:
* There is a simple zero at the corner <math>\mathrm p</math>, and a simple pole at the corner&nbsp;<math>\mathrm q</math>.
* The complex number <math>\mathrm p-\mathrm q</math> is equal to half the period of the function <math>\operatorname{pq} u</math>; that is, the function <math>\operatorname{pq} u</math> is periodic in the direction <math>\operatorname{pq}</math>, with the period being <math>2(\mathrm p-\mathrm q)</math>. The function <math>\operatorname{pq} u</math> is also periodic in the other two directions <math>\mathrm{pp}'</math> and <math>\mathrm{pq}'</math>, with periods such that <math>\mathrm p-\mathrm p'</math> and <math>\mathrm p-\mathrm q'</math> are quarter periods.

 {{multiple image
 |align=center
 |footer=Plots of four Jacobi Elliptic Functions in the complex plane of <math>u</math>, illustrating their double periodic behavior. Images generated  using a version of the [[domain coloring]] method.<ref>{{Cite web|url=https://github.com/nschloe/cplot|title=cplot, Python package for plotting complex-valued functions|website=[[GitHub]] }}</ref> All have values of <math>k=\sqrt{m}</math> equal to <math>0.8</math>.
 | image1 = Ellipj-sn-08.png
 | alt1=Elliptic Jacobi function <math>\operatorname{sn}</math>, <math>k=0.8</math>
 | caption1=Jacobi elliptic function <math>\operatorname{sn}</math>

 | image2 = Ellipj-cn08.png
 | alt2=Elliptic Jacobi function <math>\operatorname{cn}</math>, <math>k=0.8</math>
 | caption2=Jacobi elliptic function <math>\operatorname{cn}</math>

 | image3 = Ellipj-dn08.png
 | alt3=Elliptic Jacobi function <math>\operatorname{dn}</math>, <math>k=0.8</math>
 | caption3=Jacobi elliptic function <math>\operatorname{dn}</math>

 | image4 = Ellipj-sc08.png
 | alt4=Elliptic Jacobi function <math>\operatorname{sc}</math>, <math>k=0.8</math>
 | caption4=Jacobi elliptic function <math>\operatorname{sc}</math>
 }}
{{clear}}