Editing Jacobi elliptic functions (section)

== Periodicity, poles, and residues ==

[[File:JacobiEllipticFunctions.svg|thumb|Plots of the phase for the twelve Jacobi Elliptic functions pq(u,m) as a function complex argument u, with poles and zeroes indicated. The plots are over one full cycle in the real and imaginary directions with the colored portion indicating phase according to the color wheel at the lower right (which replaces the trivial dd function). Regions with absolute value below 1/3 are colored black, roughly indicating the location of a zero, while regions with absolute value above 3 are colored white, roughly indicating the position of a pole. All plots use ''m''&nbsp;=&nbsp;2/3 with ''K''&nbsp;=&nbsp;''K''(''m''), ''K''&prime;&nbsp;=&nbsp;''K''(1&nbsp;−&nbsp;''m''), ''K''(&sdot;) being the complete elliptic integral of the first kind. Arrows at the poles point in direction of zero phase. Right and left arrows imply positive and negative real residues respectively. Up and down arrows imply positive and negative imaginary residues respectively.]]

In the complex plane of the argument ''u'', the Jacobi elliptic functions form a repeating pattern of poles (and zeroes). The residues of the poles all have the same absolute value, differing only in sign. Each function pq(''u'',''m'') has an "inverse function" (in the multiplicative sense) qp(''u'',''m'') in which the positions of the poles and zeroes are exchanged. The periods of repetition are generally different in the real and imaginary directions, hence the use of the  term "doubly periodic" to describe them.

For the Jacobi amplitude and the Jacobi epsilon function:
:<math>\operatorname{am}(u+2K,m)=\operatorname{am}(u,m)+\pi,</math>
:<math>\operatorname{am}(u+4iK',m)=\operatorname{am}(u,m),</math>
:<math>\mathcal{E}(u+2K,m)=\mathcal{E}(u,m)+2E,</math>
:<math>\mathcal{E}(u+2iK',m)=\mathcal{E}(u,m)+2iE \frac{K'}{K}-\frac{\pi i}{K}</math>
where <math>E(m)</math> is the [[Elliptic integral#Complete elliptic integral of the second kind|complete elliptic integral of the second kind]] with parameter <math>m</math>.

The double periodicity of the Jacobi elliptic functions may be expressed as:

:<math>\operatorname{pq}(u + 2 \alpha K(m) + 2 i \beta K(1-m)\,,\,m)=(-1)^\gamma \operatorname{pq}(u,m)</math>

where ''&alpha;'' and ''&beta;'' are any pair of integers. ''K''(&sdot;) is the complete elliptic integral of the first kind, also known as the [[quarter period]]. The power of negative unity (''&gamma;'') is given in the following table:

:{| class="wikitable" style="text-align:center"
|+ <math>\gamma</math>
!colspan="2" rowspan="2"|
!colspan="4"|q
|-
! c
! s
! n
! d
|-
!rowspan="6"|p
|-
! c
|0||&beta; || &alpha;&nbsp;+&nbsp;&beta; || &alpha;
|-
! s
|&beta; || 0 || &alpha; || &alpha;&nbsp;+&nbsp;&beta;
|-
! n
| &alpha;&nbsp;+&nbsp;&beta; || &alpha; || 0 || &beta;
|-
! d
| &alpha; || &alpha;&nbsp;+&nbsp;&beta; || &beta; || 0
|}

When the factor (−1)<sup>''&gamma;''</sup> is equal to −1, the equation expresses quasi-periodicity.  When it is equal to unity, it expresses full periodicity. It can be seen, for example, that for the entries containing only &alpha; when &alpha; is even, full periodicity is expressed by the above equation, and the function has full periods of 4''K''(''m'') and 2''iK''(1&nbsp;−&nbsp;''m'').  Likewise, functions with entries containing only ''&beta;'' have full periods of 2K(m) and 4''iK''(1&nbsp;−&nbsp;''m''), while those with &alpha; + &beta; have full periods of 4''K''(''m'') and 4''iK''(1&nbsp;−&nbsp;''m'').

In the diagram on the right, which plots one repeating unit for each function, indicating phase along with the location of poles and zeroes, a number of regularities can be noted: The inverse of each function is opposite the diagonal, and has the same size unit cell, with poles and zeroes exchanged. The pole and zero arrangement in the auxiliary rectangle formed by (0,0), (''K'',0), (0,''K''&prime;) and (''K'',''K''&prime;) are in accordance with the description of the pole and zero placement described in the introduction above. Also, the size of the white ovals indicating poles are a rough measure of the absolute value of the residue for that pole. The residues of the poles closest to the origin in the figure (i.e. in the auxiliary rectangle) are listed in the following table:

:{| class="wikitable" style="text-align:center; width:200px”"
|+ Residues of Jacobi Elliptic Functions
!colspan="2" rowspan="2"|
!colspan="4"|q
|-
! width="40pt"|c
! width="40pt"|s
! width="40pt"|n
! width="40pt"|d
|-
!rowspan="6"|p
|-
! height="40pt" |c
|  ||1||<math>-\frac{i}{k}</math>||<math>-\frac{1}{k}</math>
|-
! height="40pt" |s
| <math>-\frac{1}{k'}</math>|| ||<math>\frac{1}{k}</math>||<math>-\frac{i}{k\,k'}</math>
|-
! height="40pt" |n
|<math>-\frac{1}{k'}</math>||1|| ||<math>-\frac{i}{k'}</math>
|-
! height="40pt" | d
| -1 || 1 || <math>-i</math> ||  
|-
|}

When applicable, poles displaced above by 2''K'' or displaced to the right by 2''K''&prime; have the same value but with signs reversed, while those diagonally opposite have the same value. Note that poles and zeroes on the left and lower edges are considered part of the unit cell, while those on the upper and right edges are not.

The information about poles can in fact be used to [[Characterization (mathematics)|characterize]] the Jacobi elliptic functions:<ref>{{cite book |last1=Whittaker |first1=Edmund Taylor |authorlink1=Edmund T. Whittaker |last2=Watson |first2=George Neville |authorlink2=George N. Watson |date= 1927 |pages=504–505 |edition=4th |title=A Course of Modern Analysis |title-link=A Course of Modern Analysis |publisher= Cambridge University Press}}</ref>

The function <math>u\mapsto\operatorname{sn}(u,m)</math> is the unique elliptic function having simple poles at <math>2rK+(2s+1)iK'</math> (with <math>r,s\in\mathbb{Z}</math>) with residues <math>(-1)^r/\sqrt{m}</math> taking the value <math>0</math> at <math>0</math>.

The function <math>u\mapsto\operatorname{cn}(u,m)</math> is the unique elliptic function having simple poles at <math>2rK+(2s+1)iK'</math> (with <math>r,s\in\mathbb{Z}</math>) with residues <math>(-1)^{r+s-1}i/\sqrt{m}</math> taking the value <math>1</math> at <math>0</math>.

The function <math>u\mapsto\operatorname{dn}(u,m)</math> is the unique elliptic function having simple poles at <math>2rK+(2s+1)iK'</math> (with <math>r,s\in\mathbb{Z}</math>) with residues <math>(-1)^{s-1}i</math> taking the value <math>1</math> at <math>0</math>.