Jacobi elliptic functions: Difference between revisions

Revision as of 18:36, 23 June 2025

Revision as of 10:20, 4 July 2025

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[[File:Jacobi Elliptic Functions (on Jacobi Ellipse).svg|right|thumb|upright=1.5|Plot of the Jacobi ellipse (''x''²&nbsp;+&nbsp;''y''²/''b''²&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''² 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 ''<math>r''&nbsp; =&nbsp; 1</math> 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 ''<math>a''&nbsp; =&nbsp; 1.</math> and <math>b \ge 1</math>. Let

:<math>

\begin{align}

& x^2 + \frac{y^2}{b^2} = 1, \quad m b= >1 - \frac{1}{b^2}, \\

& mx = 1r -\cos \frac{1}{b^2}varphi, \quad 0y <= m < 1,r \sin \varphi.

& x = r \cos \varphi, \quad y = r \sin \varphi

\end{align}

</math>

Then <math>0 \le m < 1</math> and

then:

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

For each angle <math>\varphi</math> the parameter

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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> valuevalues of the point <math>P</math> with

<math>u</math> and parameter <math>m</math> we get, afterby inserting the relation:

:<math>r(\varphi,m) = \frac 1 {\operatorname{dn}(u,m)} </math>

into: <math>x = r(\varphi,m) \cos (\varphi),</math> and <math>y = r(\varphi,m) \sin (\varphi)</math>, we that:find

:<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 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,</math> and <math>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>

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}