Jacobi elliptic functions: Difference between revisions

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{{Short description|Mathematical function}}

In [[mathematics]], the '''Jacobi elliptic functions''' are a set of basic [[elliptic function]]s. They are found in the description of the [[pendulum (mechanics)|motion of a pendulum]], as well as in the design of electronic [[elliptic filter]]s. While [[trigonometry|trigonometric functions]] are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other [[conic section]]s, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation <math>\operatorname{sn}</math> for <math>\sin</math>. The Jacobi elliptic functions are used more often in practical problems than the [[Weierstrass elliptic functions]] as they do not require notions of [[complex analysis]] to be defined and/or understood. They were introduced by {{harvs|txt|first=Carl Gustav Jakob |last=Jacobi|authorlink=Carl Gustav Jakob Jacobi|year=1829}}. [[Carl Friedrich Gauss]] had already studied special Jacobi elliptic functions in 1797, the [[lemniscate elliptic functions]] in particular,<ref>{{Cite book |last1=Armitage |first1=J. V. |last2=Eberlein| first2=W. F. |title=Elliptic Functions |publisher=Cambridge University Press |year=2006 |edition=First |isbn=978-0-521-78078-0}} p. 48</ref> but his work was published much later.

==Overview==

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There are twelve Jacobi elliptic functions denoted by <math>\operatorname{pq}(u, m)</math>, where <math>\mathrm p</math> and <math>\mathrm q</math> are any of the letters <math>\mathrm c</math>, <math>\mathrm s</math>, <math>\mathrm n</math>, and <math>\mathrm d</math>. (Functions of the form <math>\operatorname{pp}(u,m)</math> are trivially set to unity for notational completeness.) <math>u</math> is the argument, and <math>m</math> is the parameter, both of which may be complex. In fact, the Jacobi elliptic functions are [[meromorphic function|meromorphic]] in both <math>u</math> and <math>m</math>.<ref name="Walker">{{cite journal |last1=Walker |first1=Peter |date=2003 |title=The Analyticity of Jacobian Functions with Respect to the Parameter k |url=https://www.jstor.org/stable/3560143 |bibcode=<!-- useless bibcode 2003RSPSA.459.2569W --> |journal=Proceedings of the Royal Society |volume=459 |issue=2038 |pages=2569–2574|doi=10.1098/rspa.2003.1157 |jstor=3560143 }}</ref> The distribution of the zeros and poles in the <math>u</math>-plane is well-known. However, questions of the distribution of the zeros and poles in the <math>m</math>-plane remain to be investigated.<ref name="Walker"/>

In the complex plane of the argument <math>u</math>, the twelve functions form a repeating lattice of simple [[Zeros and poles|poles and zeroes]].<ref name="DLMF22">{{cite web|url=http://dlmf.nist.gov/22|title=NIST Digital Library of Mathematical Functions (Release 1.0.17)|editor-last=Olver|editor-first=F. W. J.|display-editors=et al |date=2017-12-22|publisher=National Institute of Standards and Technology|access-date=2018-02-26 }}</ref> Depending on the function, one repeating [[parallelogram]], or unit cell, will have sides of length <math>2K</math> or <math>4K</math> on the real axis, and <math>2K'</math> or <math>4K'</math> on the imaginary axis, where <math>K=K(m)</math> and <math>K'=K(1-m)</math> are known as the [[quarter period]]s with <math>K(\cdot)</math> being the [[elliptic integral]] of the first kind. The nature of the unit cell can be determined by inspecting the "auxiliary rectangle" (generally a parallelogram), which is a rectangle formed by the origin <math>(0,0)</math> at one corner, and <math>(K,K')</math> as the diagonally opposite corner. As in the diagram, the four corners of the auxiliary rectangle are named <math>\mathrm s</math>, <math>\mathrm c</math>, <math>\mathrm d</math>, and <math>\mathrm n</math>, going counter-clockwise from the origin. The function <math>\operatorname{pq}(u,m)</math> will have a zero at the <math>\mathrm p</math> corner and a pole at the <math>\mathrm q</math> corner. The twelve functions correspond to the twelve ways of arranging these poles and zeroes in the corners of the rectangle.

When the argument <math>u</math> and parameter <math>m</math> are real, with <math>0 < m < 1</math>, <math>K</math> and <math>K'</math> will be real and the auxiliary parallelogram will in fact be a rectangle, and the Jacobi elliptic functions will all be real valued on the real line.

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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}

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&=\frac{\pi}{2K}\frac{\theta_{2}'(\zeta|\tau)}{\theta_{2}(\zeta|\tau)}+\frac{\operatorname{dn}(u,m)\operatorname{sn}(u,m)}{\operatorname{cn}(u,m)}\\

&=\frac{\pi}{2K}\frac{\theta_{1}'(\zeta|\tau)}{\theta_{1}(\zeta|\tau)}-\frac{\operatorname{cn}(u,m)\operatorname{dn}(u,m)}{\operatorname{sn}(u,m)}\end{align}</math>

where <math>'</math> denotes the [[partial derivative]] with respect to the first variable.

===Using modular inversion===

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<math display="block">\operatorname{sn}\left(\frac{u}{2},m\right)=\pm\sqrt{\frac{1-\operatorname{cn}(u,m)}{1+\operatorname{dn}(u,m)}}</math>

<math display="block">\operatorname{cn}\left(\frac{u}{2},m\right)=\pm\sqrt{\frac{\operatorname{cn}(u,m)+\operatorname{dn}(u,m)}{1+\operatorname{dn}(u,m)}}</math>

<math display="block">\operatorname{cndn}\left(\frac{u}{2},m\right)=\pm\sqrt{\frac{m'+\operatorname{dn}(u,m)+m\operatorname{cn}(u,m)}{1+\operatorname{dn}(u,m)}}</math>

===K formulas===