Editing Jacobi elliptic functions (section)

== Jacobi transformations==

===The Jacobi imaginary transformations===

[[File:JacobiElliptic.HT.svg|right|thumb|upright=1.5|Plot of the degenerate Jacobi curve (''x''<sup>2</sup>&nbsp;+&nbsp;''y''<sup>2</sup>/''b''<sup>2</sup>&nbsp;=&nbsp;1, ''b''&nbsp;=&nbsp;&infin;) and the twelve Jacobi Elliptic functions pq(''u'',1) for a particular value of angle&nbsp;''&phi;''.  The solid curve is the degenerate ellipse (''x''<sup>2</sup>&nbsp;=&nbsp;1) with ''m''&nbsp;=&nbsp;1 and ''u''&nbsp;=&nbsp;''F''(''&phi;'',1) where ''F''(&sdot;,&sdot;) is the [[elliptic integral]] of the first kind. The dotted curve is the unit circle. Since these are the Jacobi functions for ''m''&nbsp;=&nbsp;0 (circular trigonometric functions) but with imaginary arguments, they correspond to the six hyperbolic trigonometric functions.]]

The Jacobi imaginary transformations relate various functions of the imaginary variable ''i u'' or, equivalently, relations between various values of the ''m'' parameter. In terms of the major functions:<ref name="W&W">{{cite book |last1=Whittaker |first1=E.T. |last2=Watson |first2=G.N.|date=1940 |title=A Course in Modern Analysis |url=https://archive.org/details/courseofmodernan00whit |location=New York, USA |publisher=The MacMillan Co.|isbn=978-0-521-58807-2|author-link=A Course of Modern Analysis}}</ref>{{rp|506}}

:<math>\operatorname{cn}(u, m)=    \operatorname{nc}(i\,u,1\!-\!m)</math>
:<math>\operatorname{sn}(u, m)= -i \operatorname{sc}(i\,u,1\!-\!m)</math>
:<math>\operatorname{dn}(u, m)=    \operatorname{dc}(i\,u,1\!-\!m)</math>

Using the multiplication rule, all other functions may be expressed in terms of the above three. The transformations may be generally written as <math>\operatorname{pq}(u,m)=\gamma_{\operatorname{pq}} \operatorname{pq}'(i\,u,1\!-\!m)</math>. The following table gives the <math>\gamma_{\operatorname{pq}} \operatorname{pq}'(i\,u,1\!-\!m)</math> for the specified pq(''u,m'').<ref name="WolframJE">{{cite web |url=http://functions.wolfram.com/EllipticFunctions/JacobiAmplitude/introductions/JacobiPQs/ShowAll.html |title=Introduction to the Jacobi elliptic functions |date=2018 |website=The Wolfram Functions Site |publisher=Wolfram Research, Inc. |access-date=January 7, 2018}}</ref> (The arguments <math>(i\,u,1\!-\!m)</math> are suppressed)
:{| class="wikitable" style="text-align:center"
|+ Jacobi Imaginary transformations <math>\gamma_{\operatorname{pq}}\operatorname{pq}'(i\,u,1\!-\!m)</math>
!colspan="2" rowspan="2"|
!colspan="4"|q
|-
! c
! s
! n
! d
|-
!rowspan="6"|p
|-
! c
| 1 || i ns || nc || nd
|-
! s
| −''i'' sn || 1 || −''i'' sc || −''i'' sd
|-
! n
| cn || ''i'' cs || 1 || cd
|-
! d
| dn || ''i'' ds || dc || 1
|}

Since the [[Hyperbolic functions|hyperbolic trigonometric functions]] are proportional to the circular trigonometric functions with imaginary arguments, it follows that the Jacobi functions will yield the hyperbolic functions for m=1.<ref name="Neville1944"/>{{rp|249}} In the figure, the Jacobi curve has degenerated to two vertical lines at ''x''&nbsp;=&nbsp;1 and ''x''&nbsp;=&nbsp;−1.

=== The Jacobi real transformations ===

The Jacobi real transformations<ref name="Neville1944"/>{{rp|308}} yield expressions for the elliptic functions in terms with alternate values of ''m''. The transformations may be generally written as <math>\operatorname{pq}(u,m)=\gamma_{\operatorname{pq}} \operatorname{pq}'(k\,u,1/m)</math>. The following table gives the <math>\gamma_{\operatorname{pq}} \operatorname{pq}'(k\,u,1/m)</math> for the specified pq(''u,m'').<ref name="WolframJE"/> (The arguments <math>(k\,u,1/m)</math> are suppressed)

:{| class="wikitable" style="text-align:center"
|+ Jacobi real transformations <math>\gamma_{\operatorname{pq}}\operatorname{pq}'(k\,u,1/m)</math>
!colspan="2" rowspan="2"|
!colspan="4"|q
|-
! c
! s
! n
! d
|-
!rowspan="6"|p
|-
! c
| <math>1 </math> ||<math>k\operatorname{ds} </math> || <math> \operatorname{dn} </math> || <math> \operatorname{dc} </math>
|-
! s
|<math>\frac 1 k \operatorname{sd} </math> || <math>1</math> || <math>\frac 1 k \operatorname{sn} </math> || <math>\frac 1 k \operatorname{sc} </math>
|-
! n
| <math> \operatorname{nd} </math> || <math>k \operatorname{ns} </math> || <math>1</math> || <math> \operatorname{nc} </math>
|-
! d
| <math> \operatorname{cd} </math> || <math>k \operatorname{cs} </math> || <math> \operatorname{cn} </math> || <math> 1 </math>
|}

=== Other Jacobi transformations ===

Jacobi's real and imaginary transformations can be combined in various ways to yield three more simple transformations
.<ref name="Neville1944"/>{{rp|214}} The real and imaginary transformations are two transformations in a group ([[Dihedral group of order 6|D<sub>3</sub>]] or [[anharmonic group]]) of six transformations. If

:<math>\mu_R(m) = 1/m</math>

is the transformation for the ''m'' parameter in the real transformation, and

:<math>\mu_I(m) = 1-m = m'</math>

is the transformation of ''m'' in the imaginary transformation, then the other transformations can be built up by successive application of these two basic transformations, yielding only three more possibilities:

:<math>
\begin{align}
\mu_{IR}(m)&=&\mu_I(\mu_R(m))&=&-m'/m \\
\mu_{RI}(m)&=&\mu_R(\mu_I(m))&=&1/m' \\
\mu_{RIR}(m)&=&\mu_R(\mu_I(\mu_R(m)))&=&-m/m'
\end{align}
</math>

These five transformations, along with the identity transformation (''&mu;''<sub>''U''</sub>(''m'')&nbsp;=&nbsp;''m'') yield the six-element group. With regard to the Jacobi elliptic functions, the general transformation can be expressed using just three functions:

:<math>\operatorname{cs}(u,m)=\gamma_i \operatorname{cs'}(\gamma_i u, \mu_i(m))</math>
:<math>\operatorname{ns}(u,m)=\gamma_i \operatorname{ns'}(\gamma_i u, \mu_i(m))</math>
:<math>\operatorname{ds}(u,m)=\gamma_i \operatorname{ds'}(\gamma_i u, \mu_i(m))</math>

where ''i'' = U, I, IR, R, RI, or RIR, identifying the transformation, &gamma;<sub>i</sub> is a multiplication factor common to these three functions, and the prime indicates the transformed function. The other nine transformed functions can be built up from the above three. The reason the cs, ns, ds  functions were chosen to represent the transformation is that the other functions will be ratios of these three (except for their inverses) and the multiplication factors will cancel.

The following table lists the multiplication factors for the three ps functions, the transformed ''m''{{'}}s, and the transformed function names for each of the six transformations.<ref name="Neville1944"/>{{rp|214}} (As usual, ''k''<sup>2</sup>&nbsp;=&nbsp;''m'', 1&nbsp;−&nbsp;''k''<sup>2</sup>&nbsp;=&nbsp;''k''<sub>1</sub><sup>2</sup>&nbsp;=&nbsp;''m''&prime; and the arguments (<math>\gamma_i u, \mu_i(m)</math>) are suppressed)

:{| class="wikitable" style="text-align:center"
|+ Parameters for the six transformations
!Transformation i||<math>\gamma_i</math>||<math>\mu_i(m)</math>||cs'||ns'||ds'
|-
! U
| 1              || m                || cs || ns || ds
|-
! I
| i              || m'    || ns || cs || ds
|-
! IR
| i k           || −m'/m || ds || cs || ns
|-
! R
| k              || 1/m              || ds || ns || cs
|-
! RI
|i k<sub>1</sub>|| 1/m'  || ns || ds || cs
|-
! RIR
| k<sub>1</sub>  || −m/m' || cs || ds || ns
|-
|}

Thus, for example, we may build the following table for the RIR transformation.<ref name="WolframJE"/> The transformation is generally written <math>\operatorname{pq}(u,m)=\gamma_{\operatorname{pq}}\,\operatorname{pq'}(k'\,u,-m/m')</math> (The arguments <math>(k'\,u,-m/m')</math> are suppressed)

:{| class="wikitable" style="text-align:center"
|+ The RIR transformation <math>\gamma_{\operatorname{pq}}\,\operatorname{pq'}(k'\,u,-m/m')</math>
!colspan="2" rowspan="2"|
!colspan="4"|q
|-
! c
! s
! n
! d
|-
!rowspan="6"|p
|-
! c
|1|| k' cs || cd || cn
|-
! s
|<math>\frac{1}{k'}</math> sc|| 1 ||<math>\frac{1}{k'}</math> sd ||<math> \frac{1}{k'}</math> sn
|-
! n
| dc || <math>k'</math> ds || 1 || dn
|-
! d
| nc || <math>k'</math> ns || nd || 1
|}

The value of the Jacobi transformations is that any set of Jacobi elliptic functions with any real-valued parameter ''m'' can be converted into another set for which <math>0<m\le 1/2</math> and, for real values of ''u'', the function values will be real.<ref name="Neville1944"/>{{rp|p. 215}}

===Amplitude transformations===

In the following, the second variable is suppressed and is equal to <math>m</math>:

:<math>\sin(\operatorname{am}(u+v)+\operatorname{am}(u-v))=\frac{2\operatorname{sn}u\operatorname{cn}u\operatorname{dn}v}{1-m\operatorname{sn}^2u\operatorname{sn}^2v},</math>
:<math>\cos(\operatorname{am}(u+v)-\operatorname{am}(u-v))=\dfrac{\operatorname{cn}^2v-\operatorname{sn}^2v\operatorname{dn}^2u}{1-m\operatorname{sn}^2u\operatorname{sn}^2v}</math>
where both identities are valid for all <math>u,v,m\in\mathbb{C}</math> such that both sides are well-defined.

With

:<math>m_1=\left(\frac{1-\sqrt{m'}}{1+\sqrt{m'}}\right)^2,</math>

we have

:<math>\cos (\operatorname{am}(u,m)+\operatorname{am}(K-u,m))=-\operatorname{sn}((1-\sqrt{m'})u,1/m_1),</math>
:<math>\sin(\operatorname{am}(\sqrt{m'}u,-m/m')+\operatorname{am}((1-\sqrt{m'})u,1/m_1))=\operatorname{sn}(u,m),</math>
:<math>\sin(\operatorname{am}((1+\sqrt{m'})u,m_1)+\operatorname{am}((1-\sqrt{m'})u,1/m_1))=\sin(2\operatorname{am}(u,m))</math>

where all the identities are valid for all <math>u,m\in\mathbb{C}</math> such that both sides are well-defined.