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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==
[[Image:JacobiFunctionAbstract.png|322px|thumb|The fundamental rectangle in the complex plane of <math>u</math>]]
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
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.
Since the
The
* There is a simple zero at the corner <math>\mathrm p</math>, and a simple pole at the corner <math>\mathrm q</math>.
* The complex number <math>\mathrm p-\mathrm q</math> is equal to half the period of the function <math>\operatorname{pq} u</math>; that is, the function <math>\operatorname{pq} u</math> is periodic in the direction <math>\operatorname{pq}</math>, with the period being <math>2(\mathrm p-\mathrm q)</math>. The function <math>\operatorname{pq} u</math> is also periodic in the other two directions <math>\mathrm{pp}'</math> and <math>\mathrm{pq}'</math>, with periods such that <math>\mathrm p-\mathrm p'</math> and <math>\mathrm p-\mathrm q'</math> are quarter periods.
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In the above, the value <math>m</math> is a free parameter, usually taken to be real such that <math>0\leq m \leq 1</math> (but can be complex in general), and so the elliptic functions can be thought of as being given by two variables, <math>u</math> and the parameter <math>m</math>. The remaining nine elliptic functions are easily built from the above three (<math>\operatorname{sn}</math>, <math>\operatorname{cn}</math>, <math>\operatorname{dn}</math>), and are given in a section below. Note that when <math>\varphi=\pi/2</math>, that <math>u</math> then equals the [[quarter period]] <math>K</math>.
In the most general setting, <math>\operatorname{am}(u,m)</math> is a [[multivalued function]] (in <math>u</math>) with infinitely many [[Branch point|logarithmic branch points]] (the branches differ by integer multiples of <math>2\pi</math>), namely the points <math>2sK(m)+(4t+1)K(1-m)i</math> and <math>2sK(m)+(4t+3)K(1-m)i</math> where <math>s,t\in\mathbb{Z}</math>.<ref name="sala">{{cite journal |last=Sala |first=Kenneth L. |date=November 1989 |title=Transformations of the Jacobian Amplitude Function and Its Calculation via the Arithmetic-Geometric Mean
However, a particular cutting for <math>\operatorname{am}(u,m)</math> can be made in the <math>u</math>-plane by line segments from <math>2sK(m)+(4t+1)K (1-m)i</math> to <math>2sK(m)+(4t+3)K(1-m)i</math> with <math>s,t\in\mathbb{Z}</math>; then it only remains to define <math>\operatorname{am}(u,m)</math> at the branch cuts by continuity from some direction. Then <math>\operatorname{am}(u,m)</math> becomes single-valued and singly-periodic in <math>u</math> with the minimal period <math>4iK(1-m)</math> and it has singularities at the logarithmic branch points mentioned above. If <math>m\in\mathbb{R}</math> and <math>m\le 1</math>, <math>\operatorname{am}(u,m)</math> is continuous in <math>u</math> on the real line. When <math>m>1</math>, the branch cuts of <math>\operatorname{am}(u,m)</math> in the <math>u</math>-plane cross the real line at <math>2(2s+1)K(1/m)/\sqrt{m}</math> for <math>s\in\mathbb{Z}</math>; therefore for <math>m>1</math>, <math>\operatorname{am}(u,m)</math> is not continuous in <math>u</math> on the real line and jumps by <math>2\pi</math> on the discontinuities.
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[[File:Jacobi Elliptic Functions (on Jacobi Ellipse).svg|right|thumb|upright=1.5|Plot of the Jacobi ellipse (''x''<sup>2</sup> + ''y''<sup>2</sup>/''b''<sup>2</sup> = 1, ''b'' real) and the twelve Jacobi elliptic functions ''pq''(''u'',''m'') for particular values of angle ''φ'' and parameter ''b''. The solid curve is the ellipse, with ''m'' = 1 − 1/''b''<sup>2</sup> and ''u'' = ''F''(''φ'',''m'') where ''F''(⋅,⋅) 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'' = cd crossing the ''x''-axis at dc are shown in light grey.]]
<math> \cos \varphi, \, \sin \varphi </math> are defined on the unit circle
:<math>
\begin{align}
& x^2 + \frac{y^2}{b^2} = 1, \quad m
&
\end{align}
</math>
Then <math>0 \le m < 1</math> and
:<math> r(
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>
<math>u</math> and parameter <math>m</math> we get,
:<math>r(\varphi,m) = \frac 1 {\operatorname{dn}(u,m)} </math>
into
:<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
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 {''φ'',dn}
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==Definition in terms of the Jacobi theta functions==
===Using elliptic integrals===
Equivalently, Jacobi's elliptic functions can be defined in terms of the [[theta function]]s.<ref>{{cite book |last1=Whittaker |first1=Edmund Taylor |authorlink1=Edmund T. Whittaker |last2=Watson |first2=George Neville |authorlink2=George N. Watson |date= 1927 |page=492 |edition=4th |title=A Course of Modern Analysis |title-link=A Course of Modern Analysis |publisher= Cambridge University Press}}</ref>
:<math>\theta_1(z|\tau)=\displaystyle\sum_{n=-\infty}^\infty (-1)^{n-\frac12}e^{(2n+1)iz+\pi i\tau\left(n+\frac12\right)^2},</math>
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:<math>\theta_3(z|\tau)=\displaystyle\sum_{n=-\infty}^\infty e^{2niz+\pi i\tau n^2},</math>
:<math>\theta_4(z|\tau)=\displaystyle\sum_{n=-\infty}^\infty (-1)^n e^{2niz+\pi i\tau n^2}</math>
and let <math>\theta_2(\tau)=\theta_2(0|\tau)</math>, <math>\theta_3(\tau)=\theta_3(0|\tau)</math>, <math>\theta_4(\tau)=\theta_4(0|\tau)</math>. Then with <math>K=K(m)</math>, <math>K'=K(1-m)</math>, <math>\zeta=\pi u/(2K)</math> and <math>\tau=iK'/K</math>,
:<math>\begin{align}\operatorname{sn}(u,m)&=\frac{\theta_3(\tau)\theta_1(\zeta|\tau)}{\theta_2(\tau)\theta_4(\zeta|\tau)},\\
\operatorname{cn}(u,m)&=\frac{\theta_4(\tau)\theta_2(\zeta|\tau)}{\theta_2(\tau)\theta_4(
\operatorname{dn}(u,m)&=\frac{\theta_4(\tau)\theta_3(\zeta|\tau)}{\theta_3(\tau)\theta_4(\zeta|\tau)}.\end{align}</math>
The Jacobi zn function can be expressed by theta functions as well:
:<math>\begin{align}\operatorname{zn}(u,m)&=\frac{\pi}{2K}\frac{\theta_{4}'(\zeta|\tau)}{\theta_{4}(\zeta|\tau)}\\ &=\frac{\pi}{2K}\frac{\theta_{3}'(\zeta|\tau)}{\theta_{3}(\zeta|\tau)}+
&=\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===
In fact, the definition of the Jacobi elliptic functions in Whittaker & Watson is stated a little bit differently than the one given above (but it's equivalent to it) and relies on modular inversion: [[Modular lambda function|The function]] <math>\lambda</math>, defined by
[[File:The region F1 for modular inversion.jpg|thumb|The region <math>F_1</math> in the complex plane. It is bounded by two semicircles from below, by a ray from the left and by a ray from the right.]]
:<math>\lambda (\tau)=\frac{\theta_2(\tau)^4}{\theta_3(\tau)^4},</math>
assumes every value in <math>\mathbb{C}-\{0,1\}</math> ''once and only once''<ref>{{cite journal |last=Cox |first=David Archibald |authorlink1=David A. Cox |date=January 1984 |title=The Arithmetic-Geometric Mean of Gauss|url=https://www.researchgate.net/publication/248675540 |journal=L'Enseignement Mathématique|volume=30|issue=2|pages=290}}</ref> in
:<math>F_1-(\partial F_1\cap\{\tau\in\mathbb{H}:\operatorname{Re}\tau <0\})</math>
where <math>\mathbb{H}</math> is the upper half-plane in the complex plane, <math>\partial F_1</math> is the boundary of <math>F_1</math> and
:<math>F_1=\{\tau\in\mathbb{H}:\left|\operatorname{Re}\tau\right|\le 1,\left|\operatorname{Re}(1/\tau)\right|\le 1\}.</math>
In this way, each <math>m\,\overset{\text{def}}{=}\,\lambda (\tau)\in\mathbb{C}-\{0,1\}</math> can be associated with ''one and only one'' <math>\tau</math>. Then Whittaker & Watson define the Jacobi elliptic functions by
:<math>\begin{align}\operatorname{sn}(u,m)&=\frac{\theta_3(\tau)\theta_1(\zeta |\tau)}{\theta_2(\tau)\theta_4(\zeta|\tau)},\\
\operatorname{cn}(u,m)&=\frac{\theta_4(\tau)\theta_2(\zeta |\tau)}{\theta_2(\tau)\theta_4(\zeta|\tau)},\\
\operatorname{dn}(u,m)&=\frac{\theta_4(\tau)\theta_3(\zeta |\tau)}{\theta_3(\tau)\theta_4(\zeta|\tau)}\end{align}</math>
where <math>\zeta=u/\theta_3(\tau)^2</math>.
In the book, they place an additional restriction on <math>m</math> (that <math>m\notin (-\infty,0)\cup (1,\infty)</math>), but it is in fact not a necessary restriction (see the Cox reference). Also, if <math>m=0</math> or <math>m=1</math>, the Jacobi elliptic functions degenerate to non-elliptic functions which is described below.
==Definition in terms of Neville theta functions==
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==Identities==
===Half
<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{
===K formulas===
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when <math>\left|\operatorname{Im}(u/K)\right|<\operatorname{Im}(iK'/K).</math>
Bivariate power series expansions have been published by Schett.<ref>{{cite journal|first1=Alois|last1=Schett |title=Properties of the Taylor series expansion coefficients of the Jacobian Elliptic Functions|year=1976|journal=Math. Comp.|volume=30|number=133|pages=143–147|doi=10.1090/S0025-5718-1976-0391477-3| mr=0391477 |
==Fast computation==
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In conjunction with the addition theorems for elliptic functions (which hold for complex numbers in general) and the Jacobi transformations, the method of computation described above can be used to compute all Jacobi elliptic functions in the whole complex plane.
Another method of fast computation of the Jacobi elliptic functions via the arithmetic–geometric mean, avoiding the computation of the Jacobi amplitude, is due to Herbert E. Salzer:<ref>{{cite journal |last=Salzer |first=Herbert E. |date=July 1962 |title=Quick calculation of Jacobian elliptic functions|journal=Communications of the ACM|volume=5|issue=7|pages=399|doi=10.1145/368273.368573
Let
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as <math>N\to\infty</math>.
Yet, another method for a rapidly converging fast computation of the Jacobi elliptic sine function found in the literature is shown below.<ref>{{Cite journal |last=Smith |first=John I. |date=May 5, 1971 |title=The Even- and Odd-Mode Capacitance Parameters for Coupled Lines in Suspended Substrate
Let:
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==Continued fractions==
Assuming real numbers <math>a,p</math> with <math>0<a<p</math> and the [[Nome (mathematics)|nome]] <math>q=e^{\pi i \tau}</math>, <math>\operatorname{Im}(\tau)>0</math> with [[Theta function|elliptic modulus]] <math display="inline">k(\tau)=\sqrt{1-k'(\tau)^2}=(\vartheta_{10}(0;\tau)/\vartheta_{00}(0;\tau))^2</math>. If <math>K[\tau]=K(k(\tau))</math>, where <math>K(x)=\pi/2\cdot {}_2F_1(1/2,1/2;1;x^2)</math> is the [[complete elliptic integral of the first kind]], then holds the following [[Continued fraction|continued fraction expansion]]<ref name="bagis-evaluations">{{citation |mode=cs1 |type=Preprint |first=N. |last=Bagis
:<math>
\begin{align}
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== {{anchor|arcsn|arccn|arcdn}}Inverse functions ==
The inverses of the Jacobi elliptic functions can be defined similarly to the [[inverse trigonometric functions]]; if <math>x=\operatorname{sn}(\xi, m)</math>, <math>\xi=\operatorname{arcsn}(x, m)</math>. They can be represented as elliptic integrals,<ref>{{dlmf|title=§22.15 Inverse Functions|first1=W. P.|last1=Reinhardt|first2=P. L.|last2=Walker|id=22.15}}</ref><ref>{{cite web|last=Ehrhardt|first=Wolfgang|title=The AMath and DAMath Special Functions: Reference Manual and Implementation Notes|url=http://www.wolfgang-ehrhardt.de/specialfunctions.pdf|access-date=17 July 2013|page=42|archive-url=https://web.archive.org/web/20160731033441/http://www.wolfgang-ehrhardt.de/specialfunctions.pdf|archive-date=31 July 2016|url-status=dead}}</ref><ref>{{cite book|last1=Byrd|first1=P.F.|last2=Friedman|first2=M.D.|title=Handbook of Elliptic Integrals for Engineers and Scientists|date=1971|publisher=Springer-Verlag|___location=Berlin|edition=2nd}}</ref> and power series representations have been found.<ref>{{cite journal|last=Carlson|first=B. C.|title=Power series for inverse Jacobian elliptic functions|journal=Mathematics of Computation|year=2008|volume=77|issue=263|pages=1615–1621|url=https://www.ams.org/journals/mcom/2008-77-263/S0025-5718-07-02049-2/S0025-5718-07-02049-2.pdf|access-date=17 July 2013|doi=10.1090/s0025-5718-07-02049-2|bibcode=<!-- useless bibcode 2008MaCom..77.1615C -->|doi-access=free}}</ref><ref name="DLMF22"/>
*<math>\operatorname{arcsn}(x,m) = \int_0^x \frac{\mathrm{d}t}{\sqrt{(1-t^2)(1-mt^2)}}</math>
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