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 motion of a [[pendulum]] (seemechanics)|motion alsoof a [[pendulum (mathematics)]]), 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 |s2cid=121368966 }}</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.

Since the JacobianJacobi elliptic functions are doubly periodic in <math>u</math>, they factor through a [[torus]] – in effect, their ___domain can be taken to be a torus, just as cosine and sine are in effect defined on a circle. Instead of having only one circle, we now have the product of two circles, one real and the other imaginary. The complex plane can be replaced by a [[complex torus]]. The circumference of the first circle is <math>4K</math> and the second <math>4K'</math>, where <math>K</math> and <math>K'</math> are the [[quarter period]]s. Each function has two zeroes and two poles at opposite positions on the torus. Among the points {{nowrap|<math>0</math>, <math>K</math>, <math>K + iK'</math>, <math>iK'</math>}} there is one zero and one pole.

The JacobianJacobi elliptic functions are then doubly periodic, meromorphic functions satisfying the following properties:

* There is a simple zero at the corner <math>\mathrm p</math>, and a simple pole at the corner&nbsp;<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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[[File:Modell der elliptischen Funktion φ=am (u, k) durch eine Fläche -Schilling V, 1 - 317-.jpg|thumb|Model of the Jacobi amplitude (measured along vertical axis) as a function of independent variables ''u'' and the modulus ''k'']]

There is a definition, relating the elliptic functions to the inverse of the incomplete [[Elliptic integral#Incomplete elliptic integral]] of the first kind|incomplete elliptic integral of the first kind]] <math>F</math>. These functions take the parameters <math>u</math> and <math>m</math> as inputs. The <math>\varphi</math> that satisfies

:<math>u=F(\varphi,m)=\int_0^\varphi \frac{\mathrm d\theta} {\sqrt {1-m \sin^2 \theta}}</math>

is called the '''Jacobi amplitude''':

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:<math>\operatorname {dn} (u,m) = \frac{\mathrm d}{\mathrm du}\operatorname{am}(u,m).</math>

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&nbsp;<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]]&nbsp;<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|url=https://epubs.siam.org/doi/abs/10.1137/0520100 |journal=SIAM Journal on Mathematical Analysis|volume=20|issue=6|pages=1514–1528|doi=10.1137/0520100 }}</ref> This multivalued function can be made single-valued by cutting the complex plane along the line segments joining these branch points (the cutting can be done in non-equivalent ways, giving non-equivalent single-valued functions), thus making <math>\operatorname{am}(u,m)</math> [[Analytic function|analytic]] everywhere except on the [[Branch point#Branch cuts|branch cuts]]. In contrast, <math>\sin\operatorname{am}(u,m)</math> and other elliptic functions have no branch points, give consistent values for every branch of <math>\operatorname{am}</math>, and are [[meromorphic function|meromorphic]] in the whole complex plane. Since every elliptic function is meromorphic in the whole complex plane (by definition), <math>\operatorname{am}(u,m)</math> (when considered as a single-valued function) is not an elliptic function.

However, a particular cutting for <math>\operatorname{am}(u,m)</math> can be made in the integral<math>u</math>-plane inversionby aboveline definessegments afrom unique<math>2sK(m)+(4t+1)K single(1-valuedm)i</math> realto <math>2sK(m)+(4t+3)K(1-analyticm)i</math> functionwith <math>s,t\in\mathbb{Z}</math>; athen realit [[Neighbourhoodonly remains to define <math>\operatorname{am}(mathematicsu,m)|neighborhood]]</math> ofat the branch cuts by continuity from some direction. Then <math>\operatorname{am}(u=0,m)</math> ifbecomes single-valued and singly-periodic in <math>mu</math> iswith real.the Thereminimal isperiod a<math>4iK(1-m)</math> uniqueand [[analyticit continuation]]has ofsingularities thisat functionthe fromlogarithmic thatbranch neighborhoodpoints tomentioned above. If <math>um\in\mathbb{R}</math>. Theand analytic<math>m\le continuation1</math>, of this function<math>\operatorname{am}(u,m)</math> is periodiccontinuous in <math>u</math> ifon andthe onlyreal line. ifWhen <math>m>1</math>, the branch cuts of <math>\operatorname{am}(withu,m)</math> in the minimal<math>u</math>-plane periodcross the real line at <math>4K2(2s+1)K(1/m)/\sqrt{m}</math>), andfor it<math>s\in\mathbb{Z}</math>; istherefore denotedfor by<math>m>1</math>, <math>\operatorname{am}(u,m)</math> is not continuous in <math>u</math> on the restreal ofline thisand jumps by <math>2\pi</math> on the articlediscontinuities.

But defining <math>\operatorname{am}(u,m)</math> this way gives rise to very complicated branch cuts in the <math>m</math>-plane (''not'' the <math>u</math>-plane); they have not been fully described as of yet.

Jacobi also introduced the '''coamplitude''' function:

:<math>\operatorname{coam}(u,m)=\operatorname{am}(K(m)-u,m)</math>.

Let

The '''Jacobi epsilon''' function can be defined as<ref>{{dlmf|first1=W. P.|last1=Reinhardt|first2=P. L.|last2=Walker|id=22.16.E17|title=Jacobian Elliptic Functions}}</ref>

:<math>\mathcal{E} (u\varphi,m)=\int_0^u {\varphi}\operatornamesqrt{dn}1-m\sin^2(t,m)\theta}\, \mathrm dtd\theta</math>

and relates the [[Elliptic integral#Incomplete elliptic integral of the first kind|incomplete elliptic integral of the first kind]] tobe the [[Elliptic integral#Incomplete elliptic integral of the second kind|incomplete elliptic integral of the second kind]] (with parameter <math>m</math>):.

Then the '''Jacobi epsilon''' function can be defined as

:<math>\mathcal{E}(u,m)=E(\operatorname{am}(u,m),m)</math>

for <math>u\in\mathbb{R}</math> and <math>0<m<1</math> and by [[analytic continuation]] in each of the variables otherwise: the Jacobi epsilon function is meromorphic in the whole complex plane (in both <math>u</math> and <math>m</math>). Alternatively, throughout both the <math>u</math>-plane and <math>m</math>-plane,<ref>{{dlmf|first1=W. P.|last1=Reinhardt|first2=P. L.|last2=Walker|id=22.16.E17|title=Jacobian Elliptic Functions}}</ref>

:<math>\mathcal{E} (u,m)=\int_0^u \operatorname{dn}^2(t,m)\, \mathrm dt;</math>

<math>\mathcal{E}</math> is well-defined in this way because all [[Residue (complex analysis)|residues]] of <math>t\mapsto\operatorname{dn}(t,m)^2</math> are zero, so the integral is path-independent. So the Jacobi epsilon relates the incomplete elliptic integral of the first kind to the incomplete elliptic integral of the second kind:

:<math>E(\varphi,m)=\mathcal{E}(F(\varphi,m),m).</math>

The Jacobi epsilon function is not an elliptic function. However, unlikebut theit Jacobiappears amplitudewhen and coamplitude,differentiating the Jacobi epsilonelliptic functionfunctions iswith meromorphicrespect into the whole complex plane (in both <math>u</math> and <math>m</math>)parameter.

The '''Jacobi zn''' function is defined by

:<math>\operatorname{zn}(u,m)=\int_0^u \left(\operatornamemathcal{dnE}(tu,m)^2-\frac{E(m)}{K(m)}\right)\,\mathrm dtu.</math>

It is a singly periodic function which is meromorphic in <math>u</math>, but not in <math>m</math> (due to the branch cuts of <math>E</math> and <math>K</math>). Its minimal period in <math>u</math> is <math>2K(m)</math>. It is related to the [[Elliptic integrals#Jacobi zeta function|Jacobi zeta function]] by <math>Z(\varphi,m)=\operatorname{zn}(F(\varphi,m),m).</math>

Historically, the Jacobi elliptic functions were first defined by using the amplitude. In more modern texts on elliptic functions, the Jacobi elliptic functions are defined by other means, for example by ratios of theta functions (see below), and the amplitude is ignored.

In modern terms, the relation to elliptic integrals would be expressed by <math>\operatorname{sn}(F(\varphi,m),m)=\sin\varphi</math> (or <math>\operatorname{cn}(F(\varphi,m),m)=\cos\varphi</math>) instead of <math>\operatorname{am}(F(\varphi,m),m)=\varphi</math>.

Note that when <math>\varphi=\pi/2</math>, that <math>u</math> then equals the [[quarter period]]&nbsp;<math>K</math>.

==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''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''(&middotsdot;,&middotsdot;) 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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(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.

TheHowever, the quantityrelation of <math>u[\varphi,k]=u(\varphi,k^2)</math> is related to the incomplete[[Ellipse#Arc ellipticlength|arc integrallength of thean secondellipse]] kindis (with modulus <math>k</math>)more bycomplicated.<ref>{{dlmf|first=B. C.|last=Carlson|id=19.8.E13|title=Elliptic Integrals}}</ref>

:<math>u[\varphi,k]=\frac{1}{\sqrt{1-k^2}}\left(\frac{1+\sqrt{1-k^2}}{2}\operatorname{E}\left(\varphi+\arctan\left(\sqrt{1-k^2}\tan \varphi\right),\frac{1-\sqrt{1-k^2}}{1+\sqrt{1-k^2}}\right)-\operatorname{E}(\varphi,k)+\frac{k^2 \sin\varphi\cos\varphi}{2\sqrt{1-k^2 \sin ^2\varphi}}\right),</math>

and therefore is related to the [[Ellipse#Arc length|arc length of an ellipse]]. 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:

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:

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), y = r(\varphi,m) \sin (\varphi)</math> that:

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

Line 172 ⟶ 186:

|}

==Definition in terms of the Jacobi theta functions==

===Using elliptic integrals===

===Jacobi theta function description===

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> With <math>z,\tau\in\mathbb{C}</math> such that <math>\operatorname{Im}\tau >0</math>, let

:<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>

Equivalently, Jacobi's elliptic functions can be defined in terms of his [[theta function]]s. If we abbreviate <math>\vartheta_{00}(0;q)</math> as <math>\vartheta_{00}(q)</math>, and <math>\vartheta_{01}(0;q), \vartheta_{10}(0;q), \vartheta_{11}(0;q)</math> respectively as <math>\vartheta_{01}(q), \vartheta_{10}(q), \vartheta_{11}(q)</math> (the ''theta constants'') then the [[Theta function|theta function elliptic modulus]] ''k'' is <math>k=\biggl\{{\vartheta_{10}[q(k)] \over \vartheta_{00}[q(k)]}\biggr\}^2</math>. We define the [[nome (mathematics)|nome]] as <math>q = \exp (\pi i \tau)</math> in relation to the period ratio. We have

:<math>\theta_2(z|\tau)=\displaystyle\sum_{n=-\infty}^\infty e^{(2n+1)iz+\pi i\tau \left(n+\frac12\right)^2},</math>

:<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)},\\

: <math id="def.theta">

\operatorname{cn}(u,m)&=\frac{\theta_4(\tau)\theta_2(\zeta|\tau)}{\theta_2(\tau)\theta_4(\zeta|\tau)},\\

\begin{align}

\operatorname{sndn}(u; k,m) & = -\frac{\vartheta_{00}theta_4(q\tau) \,\vartheta_{11}theta_3(\zeta;q|\tau)}{\vartheta_{10}theta_3(q\tau) \,\vartheta_{01}theta_4(\zeta;q|\tau)} .\\[7pt]end{align}</math>

\operatorname{cn}(u; k) & = \frac{\vartheta_{01}(q) \,\vartheta_{10}(\zeta;q)}{\vartheta_{10}(q) \,\vartheta_{01}(\zeta;q)} \\[7pt]

\operatorname{dn}(u; k) & = \frac{\vartheta_{01}(q) \,\vartheta_{00}(\zeta;q)}{\vartheta_{00}(q) \,\vartheta_{01}(\zeta;q)}

\end{align}

</math>

where <math>\zeta=\pi u/(2K)</math>.

[[E. T. Whittaker|Edmund Whittaker]] and [[G. N. Watson|George Watson]] defined the [[theta function|Jacobi theta functions]] this way in their textbook ''[[A Course of Modern Analysis]]'':<ref>Whittaker and Watson (1990) pp. 469–470</ref>

:<math>\vartheta_{00}(v;w) = \prod_{n = 1}^\infty (1-w^{2n})[1+2\cos(2v)w^{2n-1}+w^{4n-2}]</math>

:<math>\vartheta_{01}(v;w) = \prod_{n = 1}^\infty (1-w^{2n})[1-2\cos(2v)w^{2n-1}+w^{4n-2}]</math>

:<math>\vartheta_{10}(v;w) = 2 w^{1/4}\cos(v)\prod_{n = 1}^\infty (1-w^{2n})[1+2\cos(2v)w^{2n}+w^{4n}]</math>

:<math>\vartheta_{11}(v;w) = -2 w^{1/4}\sin(v)\prod_{n = 1}^\infty (1-w^{2n})[1-2\cos(2v)w^{2n}+w^{4n}]</math>

The Jacobi zn function can be expressed by theta functions as well:

:<math>\begin{align}\operatorname{zn}(u;k,m)&=\frac{\pi}{2K}\frac{\vartheta_theta_{014}'(\zeta;q|\tau)}{\vartheta_theta_{014}(\zeta;q|\tau)}\\ &=\frac{\pi}{2K}\frac{\vartheta_theta_{003}'(\zeta;q|\tau)}{\vartheta_theta_{003}(\zeta;q|\tau)}+k^2m\frac{\operatorname{sn}(u;k,m)\operatorname{cn}(u;k,m)}{\operatorname{dn}(u;k,m)}\\

&=\frac{\pi}{2K}\frac{\vartheta_theta_{102}'(\zeta;q|\tau)}{\vartheta_theta_{102}(\zeta;q|\tau)}+\frac{\operatorname{dn}(u;k,m)\operatorname{sn}(u;k,m)}{\operatorname{cn}(u;k,m)}\\

&=\frac{\pi}{2K}\frac{\vartheta_theta_{111}'(\zeta;q|\tau)}{\vartheta_theta_{111}(\zeta;q|\tau)}-\frac{\operatorname{cn}(u;k,m)\operatorname{dn}(u;k,m)}{\operatorname{sn}(u;k,m)}\end{align}</math>

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

===Elliptic integral and elliptic nome===

Since the Jacobi functions are defined in terms of the elliptic modulus <math>k(\tau)</math>, we need to invert this and find <math>\tau</math> in terms of <math>k</math>. We start from <math>k' = \sqrt{1-k^2}</math>, the ''complementary modulus''. As a function of <math>\tau</math> it is

:<math>k'(\tau) = \sqrt{1 - k^2} = \biggl\{{\vartheta_{01}[q(k)] \over \vartheta_{00}[q(k)]}\biggr\}^2</math>

Let us define the [[Nome (mathematics)|elliptic nome]] and the [[Elliptic Integral#Complete elliptic integral of the first kind|complete elliptic integral of the first kind]]:

:<math>q(k) = \exp\biggl[-\pi\frac{K(\sqrt{1 - k^2})}{K(k)}\biggr]</math>

These are two identical definitions of the complete elliptic integral of the first kind:

:<math>K(k) = \int_{0}^{\pi/2} \frac{1}{\sqrt{1 - k^2\sin(\varphi)^2}} \partial\varphi</math>

:<math>K(k) = \frac{\pi}{2}\sum_{a = 0}^{\infty} \frac{[(2a)!]^2}{16^{a}(a!)^4} k^{2a}</math>

An identical definition of the nome function can me produced by using a series. Following function has this identity:

:<math>\frac{1 - \sqrt[4]{1 - k^2}}{1 + \sqrt[4]{1 - k^2}} = \frac{\vartheta_{00}[q(k)] - \vartheta_{01}[q(k)]}{\vartheta_{00}[q(k)] + \vartheta_{01}[q(k)]} = \biggl[\sum_{n = 1}^{\infty} 2\,q(k)^{(2n - 1)^2}\biggr] \biggl[1 + \sum_{n = 1}^{\infty} 2\,q(k)^{4n^2}\biggr]^{-1}</math>

Since we may reduce to the case where the imaginary part of <math>\tau</math> is greater than or equal to <math>\sqrt{3}/2</math> (see [[Modular group#Relationship to hyperbolic geometry|Modular group]]), we can assume the absolute value of <math>q</math> is less than or equal to <math>\exp(-\pi\sqrt{3}/2) \approx 0.0658 </math>; for values this small the above series converges very rapidly and easily allows us to find the appropriate value for <math>q</math>. By solving this function after q we get:<ref>{{cite web|access-date=2023-05-28|title=A002103 - OEIS|url=https://oeis.org/A002103}}<!-- auto-translated by Module:CS1 translator --></ref><ref>{{cite web|access-date=2023-05-28|language=en|title=Series Expansion of EllipticNomeQ differs from older Mathematica Version|url=https://mathematica.stackexchange.com/questions/269455/series-expansion-of-ellipticnomeq-differs-from-older-mathematica-version}}<!-- auto-translated by Module:CS1 translator --></ref><ref>{{citation|access-date=2023-05-28|author=R. B. King, E. R. Canfield|date=1992-08-01|doi=10.1016/0898-1221(92)90210-9|issn=0898-1221|issue=3|pages=13–28|periodical=Computers & Mathematics with Applications|title=Icosahedral symmetry and the quintic equation|url=https://www.sciencedirect.com/science/article/pii/0898122192902109|volume=24|doi-access=free}}<!-- auto-translated by Module:CS1 translator --></ref>

:<math>q(k) = \sum_{n = 1}^{\infty} \frac{\text{Sw}(n)}{2^{4n - 3}} \biggl(\frac{1 - \sqrt[4]{1 - k^2}}{1 + \sqrt[4]{1 - k^2}}\biggr)^{4n - 3} = k^2\biggl\{\frac{1}{2} + \biggl[\sum_{n = 1}^{\infty} \frac{\text{Sw}(n + 1)}{2^{4n + 1}} k^{2n}\biggr]\biggr\}^4</math>

===Using modular inversion===

Where SW(n) is sequence [[OEIS:A002103|A002103]] in the [[On-Line Encyclopedia of Integer Sequences|OEIS]].

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==

Line 238 ⟶ 233:

===The Jacobi imaginary transformations===

[[File:JacobiElliptic.HT.svg|right|thumb|upright=1.5|Plot of the degenerate Jacobi curve (''x''²&nbsp;+&nbsp;''y''²/''b''²&nbsp;=&nbsp;1, ''b''&nbsp;=&nbsp;&infin;) and the twelve Jacobi Elliptic functions pq(''u'',1) for a particular value of angle&nbsp;''&phi;''. The solid curve is the degenerate ellipse (''x''²&nbsp;=&nbsp;1) with ''m''&nbsp;=&nbsp;1 and ''u''&nbsp;=&nbsp;''F''(''&phi;'',1) where ''F''(&middotsdot;,&middotsdot;) is the [[elliptic integral]] of the first kind. The dotted curve is the unit circle. Since these are the Jacobi functions for ''m''&nbsp;=&nbsp;0 (circular trigonometric functions) but with imaginary arguments, they correspond to the six hyperbolic trigonometric functions.]]

The Jacobi imaginary transformations relate various functions of the imaginary variable ''i u'' or, equivalently, relations between various values of the ''m'' parameter. In terms of the major functions:<ref name="W&W">{{cite book |last1=Whittaker |first1=E.T. |last2=Watson |first2=G.N.|date=1940 |title=A Course in Modern Analysis |url=https://archive.org/details/courseofmodernan00whit |___location=New York, USA |publisher=The MacMillan Co.|isbn=978-0-521-58807-2|author-link=A Course of Modern Analysis}}</ref>{{rp|506}}

Line 385 ⟶ 380:

|}

The value of the Jacobi transformations is that any set of Jacobi elliptic functions with any complexreal-valued parameter ''m'' can be converted into another set for which <math>0&nbsp;≤&nbsp;''<m''&nbsp;≤&nbsp;\le 1/2</math> and, for real values of ''u'', the function values will be real.<ref name="Neville1944"/>{{rp|p. 215}}

===Amplitude transformations===

In the following, the second variable is suppressed and is equal to <math>m</math>:

:<math>\sin(\operatorname{am}(u+v)+\operatorname{am}(u-v))=\frac{2\operatorname{sn}u\operatorname{cn}u\operatorname{dn}v}{1-m\operatorname{sn}^2u\operatorname{sn}^2v},</math>

:<math>\cos(\operatorname{am}(u+v)-\operatorname{am}(u-v))=\dfrac{\operatorname{cn}^2v-\operatorname{sn}^2v\operatorname{dn}^2u}{1-m\operatorname{sn}^2u\operatorname{sn}^2v}</math>

where both identities are valid for all <math>u,v,m\in\mathbb{C}</math> such that both sides are well-defined.

With

:<math>m_1=\left(\frac{1-\sqrt{m'}}{1+\sqrt{m'}}\right)^2,</math>

we have

:<math>\cos (\operatorname{am}(u,m)+\operatorname{am}(K-u,m))=-\operatorname{sn}((1-\sqrt{m'})u,1/m_1),</math>

:<math>\sin(\operatorname{am}(\sqrt{m'}u,-m/m')+\operatorname{am}((1-\sqrt{m'})u,1/m_1))=\operatorname{sn}(u,m),</math>

:<math>\sin(\operatorname{am}((1+\sqrt{m'})u,m_1)+\operatorname{am}((1-\sqrt{m'})u,1/m_1))=\sin(2\operatorname{am}(u,m))</math>

where all the identities are valid for all <math>u,m\in\mathbb{C}</math> such that both sides are well-defined.

==The Jacobi hyperbola==

[[File:Jacobi Elliptic Functions (on Jacobi Hyperbola).svg|right|thumb|upright=1.5|Plot of the Jacobi hyperbola (''x''²&nbsp;+&nbsp;''y''²/''b''²&nbsp;=&nbsp;1, ''b'' imaginary) and the twelve Jacobi Elliptic functions pq(''u'',''m'') for particular values of angle ''&phi;'' and parameter ''b''. The solid curve is the hyperbola, with ''m''&nbsp;=&nbsp;1&nbsp;−&nbsp;1/''b''² and ''u''&nbsp;=&nbsp;''F''(''&phi;'',''m'') where ''F''(&middotsdot;,&middotsdot;) is the [[elliptic integral]] of the first kind. The dotted curve is the unit circle. For the ds-dc triangle, ''&sigma;''&nbsp;=&nbsp; sin(''&phi;'')cos(''&phi;'').]]

Introducing complex numbers, our ellipse has an associated hyperbola:

Line 425 ⟶ 440:

== Periodicity, poles, and residues ==

[[File:JacobiEllipticFunctions.svg|thumb|Plots of the phase for the twelve Jacobi Elliptic functions pq(u,m) as a function complex argument u, with poles and zeroes indicated. The plots are over one full cycle in the real and imaginary directions with the colored portion indicating phase according to the color wheel at the lower right (which replaces the trivial dd function). Regions with absolute value below 1/3 are colored black, roughly indicating the ___location of a zero, while regions with absolute value above 3 are colored white, roughly indicating the position of a pole. All plots use ''m''&nbsp;=&nbsp;2/3 with ''K''&nbsp;=&nbsp;''K''(''m''), ''K''&prime;&nbsp;=&nbsp;''K''(1&nbsp;−&nbsp;''m''), ''K''(&middotsdot;) being the complete elliptic integral of the first kind. Arrows at the poles point in direction of zero phase. Right and left arrows imply positive and negative real residues respectively. Up and down arrows imply positive and negative imaginary residues respectively.]]

In the complex plane of the argument ''u'', the Jacobi elliptic functions form a repeating pattern of poles (and zeroes). The residues of the poles all have the same absolute value, differing only in sign. Each function pq(''u'',''m'') has an "inverse function" (in the multiplicative sense) qp(''u'',''m'') in which the positions of the poles and zeroes are exchanged. The periods of repetition are generally different in the real and imaginary directions, hence the use of the term "doubly periodic" to describe them.

TheFor the Jacobi amplitude and the Jacobi epsilon function are quasi-periodic:

:<math>\operatorname{am}(u+2K,m)=\operatorname{am}(u,m)+\pi,</math>

:<math>\mathcaloperatorname{Eam}(u+2K4iK',m)=\mathcaloperatorname{Eam}(u,m)+2E,</math>

:<math>\mathcal{E}(u+2K,m)=\mathcal{E}(u,m)+2E,</math>

:<math>\mathcal{E}(u+2iK',m)=\mathcal{E}(u,m)+2iE \frac{K'}{K}-\frac{\pi i}{K}</math>

where <math>E(m)</math> is the [[Elliptic integral#Complete elliptic integral of the second kind|complete elliptic integral of the second kind]] with parameter <math>m</math>.

Also

:<math>\operatorname{zn}(u+2iK',m)=\operatorname{zn}(u,m)-\frac{\pi i}{K}</math>.

The double periodicity of the Jacobi elliptic functions may be expressed as:

Line 441 ⟶ 455:

:<math>\operatorname{pq}(u + 2 \alpha K(m) + 2 i \beta K(1-m)\,,\,m)=(-1)^\gamma \operatorname{pq}(u,m)</math>

where ''&alpha;'' and ''&beta;'' are any pair of integers. ''K''(&middotsdot;) is the complete elliptic integral of the first kind, also known as the [[quarter period]]. The power of negative unity (''&gamma;'') is given in the following table:

:{| class="wikitable" style="text-align:center"

Line 499 ⟶ 513:

When applicable, poles displaced above by 2''K'' or displaced to the right by 2''K''&prime; have the same value but with signs reversed, while those diagonally opposite have the same value. Note that poles and zeroes on the left and lower edges are considered part of the unit cell, while those on the upper and right edges are not.

The information about poles can in fact be used to [[Characterization (mathematics)|characterize]] the Jacobi elliptic functions:<ref>{{cite book |last1=Whittaker |first1=Edmund Taylor |authorlink1=Edmund T. Whittaker |last2=Watson |first2=George Neville |authorlink2=George N. Watson |date= 1927 |pages=504–505 |edition=4th |title=A Course of Modern Analysis |title-link=A Course of Modern Analysis |publisher= Cambridge University Press}}</ref>

The function <math>u\mapsto\operatorname{sn}(u,m)</math> is the unique elliptic function having simple poles at <math>2rK+(2s+1)iK'</math> (with <math>r,s\in\mathbb{Z}</math>) with residues <math>(-1)^r/\sqrt{m}</math> taking the value <math>0</math> at <math>0</math>.

The function <math>u\mapsto\operatorname{cn}(u,m)</math> is the unique elliptic function having simple poles at <math>2rK+(2s+1)iK'</math> (with <math>r,s\in\mathbb{Z}</math>) with residues <math>(-1)^{r+s-1}i/\sqrt{m}</math> taking the value <math>1</math> at <math>0</math>.

The function <math>u\mapsto\operatorname{dn}(u,m)</math> is the unique elliptic function having simple poles at <math>2rK+(2s+1)iK'</math> (with <math>r,s\in\mathbb{Z}</math>) with residues <math>(-1)^{s-1}i</math> taking the value <math>1</math> at <math>0</math>.

==Special values==

Line 567 ⟶ 589:

==Identities==

===Half Angleangle formula===

<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===

Line 681 ⟶ 703:

==Jacobi elliptic functions as solutions of nonlinear ordinary differential equations==

===Derivatives with respect to the first variable===

The [[derivative]]s of the three basic Jacobi elliptic functions are:

The [[derivative]]s of the three basic Jacobi elliptic functions (with respect to the first variable, with <math>m</math> fixed) are:

<math display=block>\frac{\mathrm{d}}{\mathrm{d}z} \operatorname{sn}(z) = \operatorname{cn}(z) \operatorname{dn}(z),</math>

<math display=block>\frac{\mathrm{d}}{\mathrm{d}z} \operatorname{cn}(z) = -\operatorname{sn}(z) \operatorname{dn}(z),</math>

Line 713 ⟶ 735:

|m' nc sc || −cs ns || −''m'' cn sn ||0

|}

Also

:<math>\frac{\mathrm d}{\mathrm dz}\mathcal{E}(z)=\operatorname{dn}(z)^2.</math>

With the [[#Addition theorems|addition theorems above]] and for a given ''m'' with 0&nbsp;<&nbsp;''m''&nbsp;<&nbsp;1 the major functions are therefore solutions to the following nonlinear [[ordinary differential equation]]s:

* <math>\operatorname{am}(x)</math> solves the differential equations <math>\frac{\mathrm d^2y}{\mathrm dx^2}+m\sin (y)\cos (y)=0</math> and

:<math>\left(\frac{\mathrm dy}{\mathrm dx}\right)^2=1-m\sin(y)^2</math> (for <math>x</math> not on a branch cut)

* <math>\operatorname{sn}(x)</math> solves the differential equations <math>\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + (1+m) y - 2 m y^3 = 0</math> and <math> \left(\frac{\mathrm{d} y}{\mathrm{d}x}\right)^2 = (1-y^2) (1-m y^2)</math>

* <math>\operatorname{cn}(x)</math> solves the differential equations <math>\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + (1-2m) y + 2 m y^3 = 0</math> and <math> \left(\frac{\mathrm{d} y}{\mathrm{d}x}\right)^2 = (1-y^2) (1-m + my^2)</math>

* <math>\operatorname{dn}(x)</math> solves the differential equations <math>\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} - (2 - m) y + 2 y^3 = 0</math> and <math> \left(\frac{\mathrm{d} y}{\mathrm{d}x}\right)^2 = (y^2 - 1) (1 - m - y^2)</math>

The Jacobifunction amplitudewhich providesexactly a non-trivial solution ofsolves the differential equation describing the exact motion of a [[Pendulum (mechanics)#Simple gravity pendulum|simplependulum pendulumdifferential equation]]. In particular,

:<math>\frac{\mathrm d^2 \theta}{\mathrm dt^2}+c\sin \theta=0,</math>

with initial angle <math>\theta_0</math> and zero initial angular velocity is

:<math>\begin{align}\theta &=2\arcsin (\sqrt{m}\operatorname{amcd}\left(\frac{t\sqrt{2c}}{2c}t,2\rightm))\rightarrow \frac{\mathrm d^2 \theta}{\mathrm dt^2}+c\sin \theta=0.</math>

&=2\operatorname{am}\left(\frac{1+\sqrt{m}}{2}(\sqrt{c}t+K),\frac{4\sqrt{m}}{(1+\sqrt{m})^2}\right)-2\operatorname{am}\left(\frac{1+\sqrt{m}}{2}(\sqrt{c}t-K),\frac{4\sqrt{m}}{(1+\sqrt{m})^2}\right)-\pi\end{align}</math>

where <math>m=\sin (\theta_0/2)^2</math>, <math>c>0</math> and <math>t\in\mathbb{R}</math>.

===Derivatives with respect to the second variable===

A function which solves the above pendulum differential equation with initial angle <math>\theta_0</math> is

With the first argument <math>z</math> fixed, the derivatives with respect to the second variable <math>m</math> are as follows:

:<math>\theta=2\arcsin (k\operatorname{cd}(\sqrt{c}t,k^2)),\quad k=\sin\frac{\theta_0}{2}.</math>

:<math>\begin{align}\frac{\mathrm d}{\mathrm dm}\operatorname{sn}(z)&=\frac{\operatorname{dn}(z)\operatorname{cn}(z)((1-m)z-\mathcal{E}(z)+m\operatorname{cd}(z)\operatorname{sn}(z))}{2m(1-m)},\\

\frac{\mathrm d}{\mathrm dm}\operatorname{cn}(z)&=\frac{\operatorname{sn}(z)\operatorname{dn}(z)((m-1)z+\mathcal{E}(z)-m\operatorname{sn}(z)\operatorname{cd}(z))}{2m(1-m)},\\

\frac{\mathrm d}{\mathrm dm}\operatorname{dn}(z)&=\frac{\operatorname{sn}(z)\operatorname{cn}(z)((m-1)z+\mathcal{E}(z)-\operatorname{dn}(z)\operatorname{sc}(z))}{2(1-m)},\\

\frac{\mathrm d}{\mathrm dm}\mathcal{E}(z)&=\frac{\operatorname{cn}(z)(\operatorname{sn}(z)\operatorname{dn}(z)-\operatorname{cn}(z)\mathcal{E}(z))}{2(1-m)}-\frac{z}{2}\operatorname{sn}(z)^2.\end{align}</math>

==Expansion in terms of the nome==

Let the [[nome (mathematics)|nome]] be <math>q=\exp(-\pi K'(m)/K(m))=e^{i\pi \tau}</math>, <math>\operatorname{Im}(\tau)>0</math>, <math>m=k^2</math> and let <math>v=\pi u /(2K(m))</math>. Then the functions have expansions as [[Lambert series]]

:<math>\operatorname{am}(u,m)=\frac{\pi u}{2K(m)}+2\sum_{n=1}^\infty \frac{q^n}{n(1+q^{2n})}\sin (2nv),</math>

:<math>\operatorname{sn}(u,m)=\frac{2\pi}{kK(m)}

Line 739 ⟶ 777:

:<math>\operatorname{dn}(u,m)=\frac{\pi}{2K(m)} + \frac{2\pi}{K(m)}

\sum_{n=1}^\infty \frac{q^{n}}{1+q^{2n}} \cos (2nv),</math>

:<math>\operatorname{zn}(u,m)=\frac{2\pi}{K(m)}\sum_{n=1}^\infty \frac{q^n}{1-q^{2n}}\sin (2nv)</math>

when <math>q\exp (2\left|\operatorname{Im} (vu/K)\right|<\operatorname{Im}(iK'/K)<1.</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 |doi-access=free }}</ref>

For the Jacobi amplitude,

:<math>\operatorname{am}(u,m)=\frac{\pi u}{2K(m)}+2\sum_{n=1}^\infty \frac{q^n}{n(1+q^{2n})}\sin (2nv)</math>

where <math>0<m<1</math> and <math>u\in\mathbb{R}</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|s2cid=120666361 }}</ref>

==Fast computation==

Line 757 ⟶ 789:

Initialize

:<math>a_0=1,\, b_0=\sqrt{1-m}, c_0=\sqrt{1-b_0^2}</math>

where <math>0<m<1</math>.

Define

Line 767 ⟶ 799:

:<math>\varphi_{n-1}=\frac{1}{2}\left(\varphi_n+\arcsin \left(\frac{c_n}{a_n}\sin \varphi_n\right)\right)</math>

for <math>n\ge 1</math>, then

:<math>\operatorname{am}(u,m)=\varphi_0,\quad \operatorname{zn}(u,m)=\sum_{n=1}^N c_n\sin\varphi_n</math>

as <math>N\to\infty</math>. This is notable for its rapid convergence. It is then trivial to compute all Jacobi elliptic functions from the Jacobi amplitude <math>\operatorname{am}</math> on the real line.<ref group="note">For the <math>\operatorname{dn}</math> function,

<math>\operatorname{dn}(u,m)=\frac{\operatorname{cn}(u,m)}{\operatorname{sn}(K(m)-u,m)}</math> can be used.</ref>

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 |s2cid=44953400 |doi-access=free }}</ref>

Let

Line 788 ⟶ 821:

\operatorname{dn}(u,m)&=\sqrt{1-\frac{m}{y_0^2}}\end{align}</math>

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 |journal=IEEE Transactions on Microwave Theory and Techniques |volume=MTT-19 |issue=5 |pages=430 |doi=10.1109/TMTT.1971.1127543 |bibcode=<!-- useless bibcode 1971ITMTT..19..424S --> }}</ref>

Let:

:<math>\begin{align}

&a_0 = u &b_0 = \frac{1-\sqrt{1-m}}{1+\sqrt{1-m}} \\

&a_1 = \frac{a_0}{1+b_0} &b_1 = \frac{1-\sqrt{1-b^2_0 }}{1+\sqrt{1-b^2_0}}\\

&\vdots = \vdots &\vdots = \vdots \\

&a_n = \frac{a_{n-1}}{1+b_{n-1}} &b_n = \frac{1-\sqrt{1-b^2_{n-1}}}{1+\sqrt{1-b^2_{n-1}}}\\

\end{align}</math>

Then set:

:<math>\begin{align}

y_{n+1} &= \sin(a_n) \\

y_{n} &= \frac{y_{n+1}(1+b_n)}{1+y^2_{n+1}b_n} \\

\vdots &= \vdots\\

y_0 &= \frac{y_1(1+b_0)}{1+y^2_1b_0} \\

\end{align}</math>

Then:

:<math>\operatorname{sn}(u,m) = y_0 \text{ as }n \rightarrow\infty</math>.

==Approximation in terms of hyperbolic functions==

Line 800 ⟶ 857:

==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.(2020)." |year=2018 |title=Evaluations of series related to Jacobi elliptic functions".function preprint https://www.researchgate|arxiv=1803.net/publication/331370071_Evaluations_of_Series_Related_to_Jacobi_Elliptic_Functions09445}}</ref>

:<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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* [[Dixon elliptic functions]]

* [[Abel elliptic functions]]

* [[Weierstrass elliptic functionsfunction]]

* [[Lemniscate elliptic functions]]