Editing Jacobi elliptic functions (section)

==Definition as trigonometry: the Jacobi ellipse==
[[File:Jacobi Elliptic Functions (on Jacobi Ellipse).svg|right|thumb|upright=1.5|Plot of the Jacobi ellipse (''x''<sup>2</sup>&nbsp;+&nbsp;''y''<sup>2</sup>/''b''<sup>2</sup>&nbsp;=&nbsp;1, ''b''&nbsp;real) and the twelve Jacobi elliptic functions ''pq''(''u'',''m'') for particular values of angle ''&phi;'' and  parameter&nbsp;''b''.  The solid curve is the ellipse, with  ''m''&nbsp;=&nbsp;1&nbsp;−&nbsp;1/''b''<sup>2</sup> and ''u''&nbsp;=&nbsp;''F''(''&phi;'',''m'') where ''F''(&sdot;,&sdot;) is the [[elliptic integral]] of the first kind (with parameter <math>m=k^2</math>). The dotted curve is the unit circle. Tangent lines from the circle and ellipse at ''x''&nbsp;=&nbsp;cd crossing the ''x''-axis at dc are shown in light grey.]]

<math> \cos \varphi, \sin \varphi </math> are defined on the unit circle, with radius ''r''&nbsp;=&nbsp;1 and angle <math>\varphi =</math> arc length of the unit circle measured from the positive ''x''-axis. Similarly, Jacobi elliptic functions are defined on the unit ellipse,{{citation needed|reason= see talk of this section.|date=July 2016}} with ''a''&nbsp;=&nbsp;1.  Let

:<math>
\begin{align}
& x^2 + \frac{y^2}{b^2} = 1, \quad  b >  1, \\
& m = 1 - \frac{1}{b^2}, \quad 0 < m < 1, \\
& x = r \cos \varphi, \quad y = r \sin \varphi
\end{align}
</math>

then:

:<math> r( \varphi,m) =  \frac{1} {\sqrt {1-m \sin^2 \varphi}}\, . </math>

For each angle <math>\varphi</math> the parameter
:<math>u = u(\varphi,m)=\int_0^\varphi r(\theta,m) \, d\theta</math>
(the incomplete elliptic integral of the first kind) is computed. 
On the unit circle (<math>a=b=1</math>), <math>u</math> would be an arc length.
However, the relation of <math>u</math> to the [[Ellipse#Arc length|arc length of an ellipse]] is more complicated.<ref>{{dlmf|first=B. C.|last=Carlson|id=19.8.E13|title=Elliptic Integrals}}</ref>

Let <math>P=(x,y)=(r \cos\varphi, r\sin\varphi)</math> be a point on the ellipse, and let <math>P'=(x',y')=(\cos\varphi,\sin\varphi)</math> be the point where the unit circle intersects the line between <math>P</math> and the origin <math>O</math>.
Then the familiar relations from the unit circle:  
:<math> x' = \cos \varphi, \quad y' = \sin \varphi</math>
read for the ellipse:
:<math>x' = \operatorname{cn}(u,m),\quad y' = \operatorname{sn}(u,m).</math>
So the projections of the intersection point <math>P'</math> of the line <math>OP</math> with the unit circle on the ''x''- and ''y''-axes are simply <math>\operatorname{cn}(u,m)</math> and <math>\operatorname{sn}(u,m)</math>. These projections may be interpreted as 'definition as trigonometry'. In short:

:<math> \operatorname{cn}(u,m) = \frac{x}{r(\varphi,m)}, \quad \operatorname{sn}(u,m) = \frac{y}{r(\varphi,m)}, \quad \operatorname{dn}(u,m) = \frac{1}{r(\varphi,m)}. </math>

For the <math>x</math> and <math>y</math> value of the point <math>P</math> with 
<math>u</math> and parameter <math>m</math> we get, after inserting the relation:
:<math>r(\varphi,m) = \frac 1 {\operatorname{dn}(u,m)} </math>
into: <math>x = r(\varphi,m) \cos (\varphi), y = r(\varphi,m) \sin (\varphi)</math> that:
:<math> x = \frac{\operatorname{cn}(u,m)} {\operatorname{dn}(u,m)},\quad y = \frac{\operatorname{sn}(u,m)} {\operatorname{dn}(u,m)}.</math>
The latter relations for the ''x''- and ''y''-coordinates of points on the unit ellipse may be considered as generalization of the relations <math> x = \cos \varphi, y = \sin \varphi</math> for the coordinates of points on the unit circle.

The following table summarizes the expressions for all Jacobi elliptic functions pq(u,m) in the variables (''x'',''y'',''r'') and (''φ'',dn) with <math display="inline">r = \sqrt{x^2+y^2}</math>
{| class="wikitable" style="text-align:center"
|+ Jacobi elliptic functions pq[''u'',''m''] as functions of {''x'',''y'',''r''} and {''&phi;'',dn}
!colspan="2" rowspan="2"|
!colspan="4"|q
|-
! c
! s
! n
! d
|-
!rowspan="6"|p
|-
! c
|1 || <math>x/y=\cot(\varphi)</math> || <math>x/r=\cos(\varphi)</math> ||  <math>x=\cos(\varphi)/\operatorname{dn}</math>
|-
! s
|<math>y/x=\tan(\varphi)</math> || 1 ||<math>y/r=\sin(\varphi)</math> || <math>y=\sin(\varphi)/\operatorname{dn}</math>
|-
! n
|<math>r/x=\sec(\varphi)</math> || <math>r/y=\csc(\varphi)</math> || 1 || <math>r=1/\operatorname{dn}</math>
|-
! d
| <math>1/x=\sec(\varphi)\operatorname{dn}</math> || <math>1/y=\csc(\varphi)\operatorname{dn}</math> ||  <math>1/r=\operatorname{dn} </math> || 1
|}