Editing Jacobi elliptic functions (section)

=== 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}}