jacobi elliptic functions

Image:JacobiFunctionAbstract.png

There are twelve Jacobi elliptic functions denoted by $\backslash operatorname\{pq\}(u,\; m)$, where $\backslash mathrm\; p$ and $\backslash mathrm\; q$ are any of the letters $\backslash mathrm\; c$, $\backslash mathrm\; s$, $\backslash mathrm\; n$, and $\backslash mathrm\; d$. (Functions of the form $\backslash operatorname\{pp\}(u,m)$ are trivially set to unity for notational completeness.) $u$ is the argument, and $m$ is the parameter, both of which may be complex. In fact, the Jacobi elliptic functions are meromorphic in both $u$ and $m$.{{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=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 }} The distribution of the zeros and poles in the $u$-plane is well-known. However, questions of the distribution of the zeros and poles in the $m$-plane remain to be investigated.

In the complex plane of the argument $u$, the twelve functions form a repeating lattice of simple poles and zeroes.{{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 }} Depending on the function, one repeating parallelogram, or unit cell, will have sides of length $2K$ or $4K$ on the real axis, and $2K\text{'}$ or $4K\text{'}$ on the imaginary axis, where $K=K(m)$ and $K\text{'}=K(1-m)$ are known as the quarter periods with $K(\backslash cdot)$ 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 $(0,0)$ at one corner, and $(K,K\text{'})$ as the diagonally opposite corner. As in the diagram, the four corners of the auxiliary rectangle are named $\backslash mathrm\; s$, $\backslash mathrm\; c$, $\backslash mathrm\; d$, and $\backslash mathrm\; n$, going counter-clockwise from the origin. The function $\backslash operatorname\{pq\}(u,m)$ will have a zero at the $\backslash mathrm\; p$ corner and a pole at the $\backslash mathrm\; q$ corner. The twelve functions correspond to the twelve ways of arranging these poles and zeroes in the corners of the rectangle.

When the argument $u$ and parameter $m$ are real, with , $K$ and $K\text{'}$ 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 Jacobi elliptic functions are doubly periodic in $u$, 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 $4K$ and the second $4K\text{'}$, where $K$ and $K\text{'}$ are the quarter periods. Each function has two zeroes and two poles at opposite positions on the torus. Among the points {{nowrap|$0$, $K$, $K\; +\; iK\text{'}$, $iK\text{'}$}} there is one zero and one pole.

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

• There is a simple zero at the corner $\backslash mathrm\; p$, and a simple pole at the corner $\backslash mathrm\; q$.
• The complex number $\backslash mathrm\; p-\backslash mathrm\; q$ is equal to half the period of the function $\backslash operatorname\{pq\}\; u$; that is, the function $\backslash operatorname\{pq\}\; u$ is periodic in the direction $\backslash operatorname\{pq\}$, with the period being $2(\backslash mathrm\; p-\backslash mathrm\; q)$. The function $\backslash operatorname\{pq\}\; u$ is also periodic in the other two directions $\backslash mathrm\{pp\}\text{'}$ and $\backslash mathrm\{pq\}\text{'}$, with periods such that $\backslash mathrm\; p-\backslash mathrm\; p\text{'}$ and $\backslash mathrm\; p-\backslash mathrm\; q\text{'}$ are quarter periods.

{{multiple image

|align=center

|footer=Plots of four Jacobi Elliptic Functions in the complex plane of $u$, illustrating their double periodic behavior. Images generated using a version of the domain coloring method.{{Cite web|url=https://github.com/nschloe/cplot|title=cplot, Python package for plotting complex-valued functions|website=GitHub }} All have values of $k=\backslash sqrt\{m\}$ equal to $0.8$.

| image1 = Ellipj-sn-08.png

| alt1=Elliptic Jacobi function $\backslash operatorname\{sn\}$, $k=0.8$

| caption1=Jacobi elliptic function $\backslash operatorname\{sn\}$

| image2 = Ellipj-cn08.png

| alt2=Elliptic Jacobi function $\backslash operatorname\{cn\}$, $k=0.8$

| caption2=Jacobi elliptic function $\backslash operatorname\{cn\}$

| image3 = Ellipj-dn08.png

| alt3=Elliptic Jacobi function $\backslash operatorname\{dn\}$, $k=0.8$

| caption3=Jacobi elliptic function $\backslash operatorname\{dn\}$

| image4 = Ellipj-sc08.png

| alt4=Elliptic Jacobi function $\backslash operatorname\{sc\}$, $k=0.8$

| caption4=Jacobi elliptic function $\backslash operatorname\{sc\}$

}}

{{clear}}

The elliptic functions can be given in a variety of notations, which can make the subject unnecessarily confusing. Elliptic functions are functions of two variables. The first variable might be given in terms of the amplitude $\backslash varphi$, or more commonly, in terms of $u$ given below. The second variable might be given in terms of the parameter $m$, or as the elliptic modulus $k$, where $k^2=m$, or in terms of the modular angle $\backslash alpha$, where $m=\backslash sin^2\backslash alpha$. The complements of $k$ and $m$ are defined as $m\text{'}=1-m$ and $k\text{'}\; =\; \backslash sqrt\{m\text{'}\}$. These four terms are used below without comment to simplify various expressions.

The twelve Jacobi elliptic functions are generally written as $\backslash operatorname\{pq\}(u,\; m)$ where $\backslash mathrm\; p$ and $\backslash mathrm\; q$ are any of the letters $\backslash mathrm\; c$, $\backslash mathrm\; s$, $\backslash mathrm\; n$, and $\backslash mathrm\; d$. Functions of the form $\backslash operatorname\{pp\}(u,m)$ are trivially set to unity for notational completeness. The “major” functions are generally taken to be $\backslash operatorname\{cn\}(u,m)$, $\backslash operatorname\{sn\}(u,m)$ and $\backslash operatorname\{dn\}(u,m)$ from which all other functions can be derived and expressions are often written solely in terms of these three functions, however, various symmetries and generalizations are often most conveniently expressed using the full set. (This notation is due to Gudermann and Glaisher and is not Jacobi's original notation.)

Throughout this article, $\backslash operatorname\{pq\}(u,t^2)=\backslash operatorname\{pq\}(u;t)$.

The functions are notationally related to each other by the multiplication rule: (arguments suppressed)

:$\backslash operatorname\{pq\}\backslash cdot\; \backslash operatorname\{p\text{'}q\text{'}\}=\; \backslash operatorname\{pq\text{'}\}\backslash cdot\; \backslash operatorname\{p\text{'}q\}$

from which other commonly used relationships can be derived:

:$\backslash frac\{\backslash operatorname\{pr\}\}\{\backslash operatorname\{qr\}\}=\backslash operatorname\{pq\}$

:$\backslash operatorname\{pr\}\backslash cdot\; \backslash operatorname\{rq\}=\backslash operatorname\{pq\}$

:$\backslash frac\{1\}\{\backslash operatorname\{qp\}\}=\backslash operatorname\{pq\}$

The multiplication rule follows immediately from the identification of the elliptic functions with the Neville theta functions{{cite book |last=Neville |first=Eric Harold |date=1944 |title=Jacobian Elliptic Functions |url=https://archive.org/details/jacobianelliptic00neviuoft |location=Oxford |publisher=Oxford University Press |author-link=Eric Harold Neville}}

:$\backslash operatorname\{pq\}(u,m)=\backslash frac\{\backslash theta\_\backslash operatorname\{p\}(u,m)\}\{\backslash theta\_\backslash operatorname\{q\}(u,m)\}$

Also note that:

: $K(m)=K(k^2)=\backslash int\_0^1\backslash frac\{dt\}\{\backslash sqrt\{(1-t^2)(1-mt^2)\}\}=\backslash int\_0^1\backslash frac\{dt\}\{\backslash sqrt\{(1-t^2)(1-k^2t^2)\}\}.$

File:Modell der elliptischen Funktion φ=am (u, k) durch eine Fläche -Schilling V, 1 - 317-.jpg

There is a definition, relating the elliptic functions to the inverse of the incomplete elliptic integral of the first kind $F$. These functions take the parameters $u$ and $m$ as inputs. The $\backslash varphi$ that satisfies

:$u=F(\backslash varphi,m)=\backslash int\_0^\backslash varphi\; \backslash frac\{\backslash mathrm\; d\backslash theta\}\; \{\backslash sqrt\; \{1-m\; \backslash sin^2\; \backslash theta\}\}$

is called the Jacobi amplitude:

:$\backslash operatorname\{am\}(u,m)=\backslash varphi.$

In this framework, the elliptic sine sn u (Latin: sinus amplitudinis) is given by

:$\backslash operatorname\; \{sn\}\; (u,m)\; =\; \backslash sin\; \backslash operatorname\{am\}(u,m)$

and the elliptic cosine cn u (Latin: cosinus amplitudinis) is given by

:$\backslash operatorname\; \{cn\}\; (u,m)\; =\; \backslash cos\; \backslash operatorname\{am\}(u,m)$

and the delta amplitude dn u (Latin: delta amplitudinis)If $u\backslash in\backslash mathbb\{R\}$ and $m$ is restricted to $[0,1]$, then $\backslash operatorname\{dn\}(u,m)$ can be also written as $\backslash sqrt\; \{1-m\backslash sin^2\; \backslash operatorname\{am\}(u,m)\}.$

:$\backslash operatorname\; \{dn\}\; (u,m)\; =\; \backslash frac\{\backslash mathrm\; d\}\{\backslash mathrm\; du\}\backslash operatorname\{am\}(u,m).$

In the above, the value $m$ is a free parameter, usually taken to be real such that $0\backslash leq\; m\; \backslash leq\; 1$ (but can be complex in general), and so the elliptic functions can be thought of as being given by two variables, $u$ and the parameter $m$. The remaining nine elliptic functions are easily built from the above three ($\backslash operatorname\{sn\}$, $\backslash operatorname\{cn\}$, $\backslash operatorname\{dn\}$), and are given in a section below. Note that when $\backslash varphi=\backslash pi/2$, that $u$ then equals the quarter period $K$.

In the most general setting, $\backslash operatorname\{am\}(u,m)$ is a multivalued function (in $u$) with infinitely many logarithmic branch points (the branches differ by integer multiples of $2\backslash pi$), namely the points $2sK(m)+(4t+1)K(1-m)i$ and $2sK(m)+(4t+3)K(1-m)i$ where $s,t\backslash in\backslash mathbb\{Z\}$.{{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 }} 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 $\backslash operatorname\{am\}(u,m)$ analytic everywhere except on the branch cuts. In contrast, $\backslash sin\backslash operatorname\{am\}(u,m)$ and other elliptic functions have no branch points, give consistent values for every branch of $\backslash operatorname\{am\}$, and are meromorphic in the whole complex plane. Since every elliptic function is meromorphic in the whole complex plane (by definition), $\backslash operatorname\{am\}(u,m)$ (when considered as a single-valued function) is not an elliptic function.

However, a particular cutting for $\backslash operatorname\{am\}(u,m)$ can be made in the $u$-plane by line segments from $2sK(m)+(4t+1)K\; (1-m)i$ to $2sK(m)+(4t+3)K(1-m)i$ with $s,t\backslash in\backslash mathbb\{Z\}$; then it only remains to define $\backslash operatorname\{am\}(u,m)$ at the branch cuts by continuity from some direction. Then $\backslash operatorname\{am\}(u,m)$ becomes single-valued and singly-periodic in $u$ with the minimal period $4iK(1-m)$ and it has singularities at the logarithmic branch points mentioned above. If $m\backslash in\backslash mathbb\{R\}$ and $m\backslash le\; 1$, $\backslash operatorname\{am\}(u,m)$ is continuous in $u$ on the real line. When $m>1$, the branch cuts of $\backslash operatorname\{am\}(u,m)$ in the $u$-plane cross the real line at $2(2s+1)K(1/m)/\backslash sqrt\{m\}$ for $s\backslash in\backslash mathbb\{Z\}$; therefore for $m>1$, $\backslash operatorname\{am\}(u,m)$ is not continuous in $u$ on the real line and jumps by $2\backslash pi$ on the discontinuities.

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

Let

:$E(\backslash varphi,m)=\backslash int\_0^\{\backslash varphi\}\backslash sqrt\{1-m\backslash sin^2\backslash theta\}\backslash ,\backslash mathrm\; d\backslash theta$

be the incomplete elliptic integral of the second kind with parameter $m$.

Then the Jacobi epsilon function can be defined as

:$\backslash mathcal\{E\}(u,m)=E(\backslash operatorname\{am\}(u,m),m)$

for $u\backslash in\backslash mathbb\{R\}$ and

:$\backslash mathcal\{E\}\; (u,m)=\backslash int\_0^u\; \backslash operatorname\{dn\}^2(t,m)\backslash ,\; \backslash mathrm\; dt;$

$\backslash mathcal\{E\}$ is well-defined in this way because all residues of $t\backslash mapsto\backslash operatorname\{dn\}(t,m)^2$ 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:

:$E(\backslash varphi,m)=\backslash mathcal\{E\}(F(\backslash varphi,m),m).$

The Jacobi epsilon function is not an elliptic function, but it appears when differentiating the Jacobi elliptic functions with respect to the parameter.

The Jacobi zn function is defined by

:$\backslash operatorname\{zn\}(u,m)=\backslash mathcal\{E\}(u,m)-\backslash frac\{E(m)\}\{K(m)\}u.$

It is a singly periodic function which is meromorphic in $u$, but not in $m$ (due to the branch cuts of $E$ and $K$). Its minimal period in $u$ is $2K(m)$. It is related to the Jacobi zeta function by $Z(\backslash varphi,m)=\backslash operatorname\{zn\}(F(\backslash varphi,m),m).$

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 $\backslash operatorname\{sn\}(F(\backslash varphi,m),m)=\backslash sin\backslash varphi$ (or $\backslash operatorname\{cn\}(F(\backslash varphi,m),m)=\backslash cos\backslash varphi$) instead of $\backslash operatorname\{am\}(F(\backslash varphi,m),m)=\backslash varphi$.

File:Jacobi Elliptic Functions (on Jacobi Ellipse).svg of the first kind (with parameter $m=k^2$). 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.]]

$\backslash cos\; \backslash varphi,\; \backslash sin\; \backslash varphi$ are defined on the unit circle, with radius r = 1 and angle $\backslash varphi\; =$ 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 = 1. Let

:$$

\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}

then:

:$r(\; \backslash varphi,m)\; =\; \backslash frac\{1\}\; \{\backslash sqrt\; \{1-m\; \backslash sin^2\; \backslash varphi\}\}\backslash ,\; .$

For each angle $\backslash varphi$ the parameter

:$u\; =\; u(\backslash varphi,m)=\backslash int\_0^\backslash varphi\; r(\backslash theta,m)\; \backslash ,\; d\backslash theta$

(the incomplete elliptic integral of the first kind) is computed.

On the unit circle ($a=b=1$), $u$ would be an arc length.

However, the relation of $u$ to the arc length of an ellipse is more complicated.{{dlmf|first=B. C.|last=Carlson|id=19.8.E13|title=Elliptic Integrals}}

Let $P=(x,y)=(r\; \backslash cos\backslash varphi,\; r\backslash sin\backslash varphi)$ be a point on the ellipse, and let $P\text{'}=(x\text{'},y\text{'})=(\backslash cos\backslash varphi,\backslash sin\backslash varphi)$ be the point where the unit circle intersects the line between $P$ and the origin $O$.

Then the familiar relations from the unit circle:

:$x\text{'}\; =\; \backslash cos\; \backslash varphi,\; \backslash quad\; y\text{'}\; =\; \backslash sin\; \backslash varphi$

read for the ellipse:

:$x\text{'}\; =\; \backslash operatorname\{cn\}(u,m),\backslash quad\; y\text{'}\; =\; \backslash operatorname\{sn\}(u,m).$

So the projections of the intersection point $P\text{'}$ of the line $OP$ with the unit circle on the x- and y-axes are simply $\backslash operatorname\{cn\}(u,m)$ and $\backslash operatorname\{sn\}(u,m)$. These projections may be interpreted as 'definition as trigonometry'. In short:

:$\backslash operatorname\{cn\}(u,m)\; =\; \backslash frac\{x\}\{r(\backslash varphi,m)\},\; \backslash quad\; \backslash operatorname\{sn\}(u,m)\; =\; \backslash frac\{y\}\{r(\backslash varphi,m)\},\; \backslash quad\; \backslash operatorname\{dn\}(u,m)\; =\; \backslash frac\{1\}\{r(\backslash varphi,m)\}.$

For the $x$ and $y$ value of the point $P$ with

$u$ and parameter $m$ we get, after inserting the relation:

:$r(\backslash varphi,m)\; =\; \backslash frac\; 1\; \{\backslash operatorname\{dn\}(u,m)\}$

into: $x\; =\; r(\backslash varphi,m)\; \backslash cos\; (\backslash varphi),\; y\; =\; r(\backslash varphi,m)\; \backslash sin\; (\backslash varphi)$ that:

:$x\; =\; \backslash frac\{\backslash operatorname\{cn\}(u,m)\}\; \{\backslash operatorname\{dn\}(u,m)\},\backslash quad\; y\; =\; \backslash frac\{\backslash operatorname\{sn\}(u,m)\}\; \{\backslash operatorname\{dn\}(u,m)\}.$

The latter relations for the x- and y-coordinates of points on the unit ellipse may be considered as generalization of the relations $x\; =\; \backslash cos\; \backslash varphi,\; y\; =\; \backslash sin\; \backslash varphi$ 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 $r\; =\; \backslash sqrt\{x^2+y^2\}$

Equivalently, Jacobi's elliptic functions can be defined in terms of the theta functions.{{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}} With $z,\backslash tau\backslash in\backslash mathbb\{C\}$ such that $\backslash operatorname\{Im\}\backslash tau\; >0$, let

:$\backslash theta\_1(z|\backslash tau)=\backslash displaystyle\backslash sum\_\{n=-\backslash infty\}^\backslash infty\; (-1)^\{n-\backslash frac12\}e^\{(2n+1)iz+\backslash pi\; i\backslash tau\backslash left(n+\backslash frac12\backslash right)^2\},$

:$\backslash theta\_2(z|\backslash tau)=\backslash displaystyle\backslash sum\_\{n=-\backslash infty\}^\backslash infty\; e^\{(2n+1)iz+\backslash pi\; i\backslash tau\; \backslash left(n+\backslash frac12\backslash right)^2\},$

:$\backslash theta\_3(z|\backslash tau)=\backslash displaystyle\backslash sum\_\{n=-\backslash infty\}^\backslash infty\; e^\{2niz+\backslash pi\; i\backslash tau\; n^2\},$

:$\backslash theta\_4(z|\backslash tau)=\backslash displaystyle\backslash sum\_\{n=-\backslash infty\}^\backslash infty\; (-1)^n\; e^\{2niz+\backslash pi\; i\backslash tau\; n^2\}$

and let $\backslash theta\_2(\backslash tau)=\backslash theta\_2(0|\backslash tau)$, $\backslash theta\_3(\backslash tau)=\backslash theta\_3(0|\backslash tau)$, $\backslash theta\_4(\backslash tau)=\backslash theta\_4(0|\backslash tau)$. Then with $K=K(m)$, $K\text{'}=K(1-m)$, $\backslash zeta=\backslash pi\; u/(2K)$ and $\backslash tau=iK\text{'}/K$,

:

\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}

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

:

&=\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}

where $\text{'}$ denotes the partial derivative with respect to the first variable.

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: The function $\backslash lambda$, defined by

File:The region F1 for modular inversion.jpg

:$\backslash lambda\; (\backslash tau)=\backslash frac\{\backslash theta\_2(\backslash tau)^4\}\{\backslash theta\_3(\backslash tau)^4\},$

assumes every value in $\backslash mathbb\{C\}-\backslash \{0,1\backslash \}$ once and only once{{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}} in

:

where $\backslash mathbb\{H\}$ is the upper half-plane in the complex plane, $\backslash partial\; F\_1$ is the boundary of $F\_1$ and

:$F\_1=\backslash \{\backslash tau\backslash in\backslash mathbb\{H\}:\backslash left|\backslash operatorname\{Re\}\backslash tau\backslash right|\backslash le\; 1,\backslash left|\backslash operatorname\{Re\}(1/\backslash tau)\backslash right|\backslash le\; 1\backslash \}.$

In this way, each $m\backslash ,\backslash overset\{\backslash text\{def\}\}\{=\}\backslash ,\backslash lambda\; (\backslash tau)\backslash in\backslash mathbb\{C\}-\backslash \{0,1\backslash \}$ can be associated with one and only one $\backslash tau$. Then Whittaker & Watson define the Jacobi elliptic functions by

:

\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}

where $\backslash zeta=u/\backslash theta\_3(\backslash tau)^2$.

In the book, they place an additional restriction on $m$ (that $m\backslash notin\; (-\backslash infty,0)\backslash cup\; (1,\backslash infty)$), but it is in fact not a necessary restriction (see the Cox reference). Also, if $m=0$ or $m=1$, the Jacobi elliptic functions degenerate to non-elliptic functions which is described below.

The Jacobi elliptic functions can be defined very simply using the Neville theta functions:

:$\backslash operatorname\{pq\}(u,m)=\backslash frac\{\backslash theta\_\backslash operatorname\{p\}(u,m)\}\{\backslash theta\_\backslash operatorname\{q\}(u,m)\}$

Simplifications of complicated products of the Jacobi elliptic functions are often made easier using these identities.

File:JacobiElliptic.HT.svg of the first kind. The dotted curve is the unit circle. Since these are the Jacobi functions for m = 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:{{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}}{{rp|506}}

:$\backslash operatorname\{cn\}(u,\; m)=\; \backslash operatorname\{nc\}(i\backslash ,u,1\backslash !-\backslash !m)$

:$\backslash operatorname\{sn\}(u,\; m)=\; -i\; \backslash operatorname\{sc\}(i\backslash ,u,1\backslash !-\backslash !m)$

:$\backslash operatorname\{dn\}(u,\; m)=\; \backslash operatorname\{dc\}(i\backslash ,u,1\backslash !-\backslash !m)$

Using the multiplication rule, all other functions may be expressed in terms of the above three. The transformations may be generally written as $\backslash operatorname\{pq\}(u,m)=\backslash gamma\_\{\backslash operatorname\{pq\}\}\; \backslash operatorname\{pq\}\text{'}(i\backslash ,u,1\backslash !-\backslash !m)$. The following table gives the $\backslash gamma\_\{\backslash operatorname\{pq\}\}\; \backslash operatorname\{pq\}\text{'}(i\backslash ,u,1\backslash !-\backslash !m)$ for the specified pq(u,m).{{cite web |url=http://functions.wolfram.com/EllipticFunctions/JacobiAmplitude/introductions/JacobiPQs/ShowAll.html |title=Introduction to the Jacobi elliptic functions |date=2018 |website=The Wolfram Functions Site |publisher=Wolfram Research, Inc. |access-date=January 7, 2018}} (The arguments $(i\backslash ,u,1\backslash !-\backslash !m)$ are suppressed)

:

Since the hyperbolic trigonometric functions are proportional to the circular trigonometric functions with imaginary arguments, it follows that the Jacobi functions will yield the hyperbolic functions for m=1.{{rp|249}} In the figure, the Jacobi curve has degenerated to two vertical lines at x = 1 and x = −1.

The Jacobi real transformations{{rp|308}} yield expressions for the elliptic functions in terms with alternate values of m. The transformations may be generally written as $\backslash operatorname\{pq\}(u,m)=\backslash gamma\_\{\backslash operatorname\{pq\}\}\; \backslash operatorname\{pq\}\text{'}(k\backslash ,u,1/m)$. The following table gives the $\backslash gamma\_\{\backslash operatorname\{pq\}\}\; \backslash operatorname\{pq\}\text{'}(k\backslash ,u,1/m)$ for the specified pq(u,m). (The arguments $(k\backslash ,u,1/m)$ are suppressed)

:

Jacobi's real and imaginary transformations can be combined in various ways to yield three more simple transformations

.{{rp|214}} The real and imaginary transformations are two transformations in a group (D₃ or anharmonic group) of six transformations. If

:$\backslash mu\_R(m)\; =\; 1/m$

is the transformation for the m parameter in the real transformation, and

:$\backslash mu\_I(m)\; =\; 1-m\; =\; m\text{'}$

is the transformation of m in the imaginary transformation, then the other transformations can be built up by successive application of these two basic transformations, yielding only three more possibilities:

:$$

\begin{align}

\mu_{IR}(m)&=&\mu_I(\mu_R(m))&=&-m'/m \\

\mu_{RI}(m)&=&\mu_R(\mu_I(m))&=&1/m' \\

\mu_{RIR}(m)&=&\mu_R(\mu_I(\mu_R(m)))&=&-m/m'

\end{align}

These five transformations, along with the identity transformation (μ_U(m) = m) yield the six-element group. With regard to the Jacobi elliptic functions, the general transformation can be expressed using just three functions:

:$\backslash operatorname\{cs\}(u,m)=\backslash gamma\_i\; \backslash operatorname\{cs\text{'}\}(\backslash gamma\_i\; u,\; \backslash mu\_i(m))$

:$\backslash operatorname\{ns\}(u,m)=\backslash gamma\_i\; \backslash operatorname\{ns\text{'}\}(\backslash gamma\_i\; u,\; \backslash mu\_i(m))$

:$\backslash operatorname\{ds\}(u,m)=\backslash gamma\_i\; \backslash operatorname\{ds\text{'}\}(\backslash gamma\_i\; u,\; \backslash mu\_i(m))$

where i = U, I, IR, R, RI, or RIR, identifying the transformation, γ_i is a multiplication factor common to these three functions, and the prime indicates the transformed function. The other nine transformed functions can be built up from the above three. The reason the cs, ns, ds functions were chosen to represent the transformation is that the other functions will be ratios of these three (except for their inverses) and the multiplication factors will cancel.

The following table lists the multiplication factors for the three ps functions, the transformed m{{'}}s, and the transformed function names for each of the six transformations.{{rp|214}} (As usual, k² = m, 1 − k² = k₁² = m′ and the arguments ($\backslash gamma\_i\; u,\; \backslash mu\_i(m)$) are suppressed)

:

Thus, for example, we may build the following table for the RIR transformation. The transformation is generally written $\backslash operatorname\{pq\}(u,m)=\backslash gamma\_\{\backslash operatorname\{pq\}\}\backslash ,\backslash operatorname\{pq\text{'}\}(k\text{'}\backslash ,u,-m/m\text{'})$ (The arguments $(k\text{'}\backslash ,u,-m/m\text{'})$ are suppressed)

:

The value of the Jacobi transformations is that any set of Jacobi elliptic functions with any real-valued parameter m can be converted into another set for which u, the function values will be real.{{rp|p. 215}}

In the following, the second variable is suppressed and is equal to $m$:

:$\backslash sin(\backslash operatorname\{am\}(u+v)+\backslash operatorname\{am\}(u-v))=\backslash frac\{2\backslash operatorname\{sn\}u\backslash operatorname\{cn\}u\backslash operatorname\{dn\}v\}\{1-m\backslash operatorname\{sn\}^2u\backslash operatorname\{sn\}^2v\},$

:$\backslash cos(\backslash operatorname\{am\}(u+v)-\backslash operatorname\{am\}(u-v))=\backslash dfrac\{\backslash operatorname\{cn\}^2v-\backslash operatorname\{sn\}^2v\backslash operatorname\{dn\}^2u\}\{1-m\backslash operatorname\{sn\}^2u\backslash operatorname\{sn\}^2v\}$

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

With

:$m\_1=\backslash left(\backslash frac\{1-\backslash sqrt\{m\text{'}\}\}\{1+\backslash sqrt\{m\text{'}\}\}\backslash right)^2,$

we have

:$\backslash cos\; (\backslash operatorname\{am\}(u,m)+\backslash operatorname\{am\}(K-u,m))=-\backslash operatorname\{sn\}((1-\backslash sqrt\{m\text{'}\})u,1/m\_1),$

:$\backslash sin(\backslash operatorname\{am\}(\backslash sqrt\{m\text{'}\}u,-m/m\text{'})+\backslash operatorname\{am\}((1-\backslash sqrt\{m\text{'}\})u,1/m\_1))=\backslash operatorname\{sn\}(u,m),$

:$\backslash sin(\backslash operatorname\{am\}((1+\backslash sqrt\{m\text{'}\})u,m\_1)+\backslash operatorname\{am\}((1-\backslash sqrt\{m\text{'}\})u,1/m\_1))=\backslash sin(2\backslash operatorname\{am\}(u,m))$

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

File:Jacobi Elliptic Functions (on Jacobi Hyperbola).svg of the first kind. The dotted curve is the unit circle. For the ds-dc triangle, σ =  sin(φ)cos(φ).]]

Introducing complex numbers, our ellipse has an associated hyperbola:

:$x^2\; -\; \backslash frac\{y^2\}\{b^2\}\; =\; 1$

from applying Jacobi's imaginary transformation to the elliptic functions in the above equation for x and y.

:$x\; =\; \backslash frac\{1\}\; \{\backslash operatorname\{dn\}(u,1-m)\},\backslash quad\; y\; =\; \backslash frac\{\; \backslash operatorname\{sn\}(u,1-m)\}\; \{\backslash operatorname\{dn\}(u,1-m)\}$

It follows that we can put $x=\backslash operatorname\{dn\}(u,1-m),\; y=\backslash operatorname\{sn\}(u,1-m)$. So our ellipse has a dual ellipse with m replaced by 1-m. This leads to the complex torus mentioned in the Introduction.{{Cite web|url=https://paramanands.blogspot.co.uk/2011/01/elliptic-functions-complex-variables.html#.WlHhTbp2t9A|title = Elliptic Functions: Complex Variables}} Generally, m may be a complex number, but when m is real and m<0, the curve is an ellipse with major axis in the x direction. At m=0 the curve is a circle, and for 0m = 1, the curve degenerates into two vertical lines at x = ±1. For m > 1, the curve is a hyperbola. When m is complex but not real, x or y or both are complex and the curve cannot be described on a real x-y diagram.

Reversing the order of the two letters of the function name results in the reciprocals of the three functions above:

:$$

\operatorname{ns}(u) = \frac{1}{\operatorname{sn}(u)}, \qquad \operatorname{nc}(u) = \frac{1}{\operatorname{cn}(u)}, \qquad

\operatorname{nd}(u) = \frac{1}{\operatorname{dn}(u)}.

Similarly, the ratios of the three primary functions correspond to the first letter of the numerator followed by the first letter of the denominator:

:$$

\begin{align}

\operatorname{sc}(u) = \frac{\operatorname{sn}(u)}{\operatorname{cn}(u)}, \qquad

\operatorname{sd}(u) = \frac{\operatorname{sn}(u)}{\operatorname{dn}(u)}, \qquad \operatorname{dc}(u) = \frac{\operatorname{dn}(u)}{\operatorname{cn}(u)}, \qquad \operatorname{ds}(u) = \frac{\operatorname{dn}(u)}{\operatorname{sn}(u)}, \qquad \operatorname{cs}(u) = \frac{\operatorname{cn}(u)}{\operatorname{sn}(u)}, \qquad

\operatorname{cd}(u) = \frac{\operatorname{cn}(u)}{\operatorname{dn}(u)}.

\end{align}

More compactly, we have

:$\backslash operatorname\{pq\}(u)=\backslash frac\{\backslash operatorname\{pn\}(u)\}\{\backslash operatorname\{qn\}(u)\}$

where p and q are any of the letters s, c, d.

File:JacobiEllipticFunctions.svg

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.

For the Jacobi amplitude and the Jacobi epsilon function:

:$\backslash operatorname\{am\}(u+2K,m)=\backslash operatorname\{am\}(u,m)+\backslash pi,$

:$\backslash operatorname\{am\}(u+4iK\text{'},m)=\backslash operatorname\{am\}(u,m),$

:$\backslash mathcal\{E\}(u+2K,m)=\backslash mathcal\{E\}(u,m)+2E,$

:$\backslash mathcal\{E\}(u+2iK\text{'},m)=\backslash mathcal\{E\}(u,m)+2iE\; \backslash frac\{K\text{'}\}\{K\}-\backslash frac\{\backslash pi\; i\}\{K\}$

where $E(m)$ is the complete elliptic integral of the second kind with parameter $m$.

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

:$\backslash operatorname\{pq\}(u\; +\; 2\; \backslash alpha\; K(m)\; +\; 2\; i\; \backslash beta\; K(1-m)\backslash ,,\backslash ,m)=(-1)^\backslash gamma\; \backslash operatorname\{pq\}(u,m)$

where α and β are any pair of integers. K(⋅) is the complete elliptic integral of the first kind, also known as the quarter period. The power of negative unity (γ) is given in the following table:

:

When the factor (−1)^γ is equal to −1, the equation expresses quasi-periodicity. When it is equal to unity, it expresses full periodicity. It can be seen, for example, that for the entries containing only α when α is even, full periodicity is expressed by the above equation, and the function has full periods of 4K(m) and 2iK(1 − m). Likewise, functions with entries containing only β have full periods of 2K(m) and 4iK(1 − m), while those with α + β have full periods of 4K(m) and 4iK(1 − m).

In the diagram on the right, which plots one repeating unit for each function, indicating phase along with the location of poles and zeroes, a number of regularities can be noted: The inverse of each function is opposite the diagonal, and has the same size unit cell, with poles and zeroes exchanged. The pole and zero arrangement in the auxiliary rectangle formed by (0,0), (K,0), (0,K′) and (K,K′) are in accordance with the description of the pole and zero placement described in the introduction above. Also, the size of the white ovals indicating poles are a rough measure of the absolute value of the residue for that pole. The residues of the poles closest to the origin in the figure (i.e. in the auxiliary rectangle) are listed in the following table:

:

When applicable, poles displaced above by 2K or displaced to the right by 2K′ 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 characterize the Jacobi elliptic functions:{{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}}

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

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

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

Setting $m=-1$ gives the lemniscate elliptic functions $\backslash operatorname\{sl\}$ and $\backslash operatorname\{cl\}$:

:$\backslash operatorname\{sl\}u=\backslash operatorname\{sn\}(u,-1),\backslash quad\; \backslash operatorname\{cl\}u=\backslash operatorname\{cd\}(u,-1)=\backslash frac\{\backslash operatorname\{cn\}(u,-1)\}\{\backslash operatorname\{dn\}(u,-1)\}.$

When $m=0$ or $m=1$, the Jacobi elliptic functions are reduced to non-elliptic functions:

For the Jacobi amplitude, $\backslash operatorname\{am\}(u,0)=u$ and $\backslash operatorname\{am\}(u,1)=\backslash operatorname\{gd\}u$ where $\backslash operatorname\{gd\}$ is the Gudermannian function.

In general if neither of p,q is d then $\backslash operatorname\{pq\}(u,1)=\backslash operatorname\{pq\}(\backslash operatorname\{gd\}(u),0)$.

$$\backslash operatorname\{sn\}\backslash left(\backslash frac\{u\}\{2\},m\backslash right)=\backslash pm\backslash sqrt\{\backslash frac\{1-\backslash operatorname\{cn\}(u,m)\}\{1+\backslash operatorname\{dn\}(u,m)\}\}$$

$$\backslash operatorname\{cn\}\backslash left(\backslash frac\{u\}\{2\},m\backslash right)=\backslash pm\backslash sqrt\{\backslash frac\{\backslash operatorname\{cn\}(u,m)+\backslash operatorname\{dn\}(u,m)\}\{1+\backslash operatorname\{dn\}(u,m)\}\}$$

$$\backslash operatorname\{cn\}\backslash left(\backslash frac\{u\}\{2\},m\backslash right)=\backslash pm\backslash sqrt\{\backslash frac\{m\text{'}+\backslash operatorname\{dn\}(u,m)+m\backslash operatorname\{cn\}(u,m)\}\{1+\backslash operatorname\{dn\}(u,m)\}\}$$

Half K formula

$$\backslash operatorname\{sn\}\backslash left[\backslash tfrac\{1\}\{2\}K(k);\; k\backslash right]\; =\; \backslash frac\{\backslash sqrt\{2\}\}\{\backslash sqrt\{1+k\}\; +\; \backslash sqrt\{1-k\}\}$$

$$\backslash operatorname\{cn\}\backslash left[\backslash tfrac\{1\}\{2\}K(k);\; k\backslash right]\; =\; \backslash frac\{\backslash sqrt\{2\}\backslash ,\backslash sqrt[4]\{1-k^2\}\}\{\backslash sqrt\{1+k\}\; +\; \backslash sqrt\{1-k\}\}$$

$$\backslash operatorname\{dn\}\backslash left[\backslash tfrac\{1\}\{2\}K(k);\; k\backslash right]\; =\; \backslash sqrt[4]\{1-k^2\}$$

Third K formula

:$\backslash operatorname\{sn\}\backslash left[\backslash frac\{1\}\{3\}K\backslash left(\backslash frac\{x^3\}\{\backslash sqrt\{x^6+1\}+1\}\backslash right);\backslash frac\{x^3\}\{\backslash sqrt\{x^6+1\}+1\}\backslash right]\; =$

\frac{\sqrt{2\sqrt{x^4-x^2+1}-x^2+2}+\sqrt{x^2+1}-1}{\sqrt{2\sqrt{x^4-x^2+1}-x^2+2}+\sqrt{x^2+1}+1}

To get x³, we take the tangent of twice the arctangent of the modulus.

Also this equation leads to the sn-value of the third of K:

:$k^2s^4-2k^2s^3+2s-1\; =\; 0$

:$s\; =\; \backslash operatorname\{sn\}\backslash left[\backslash tfrac\{1\}\{3\}K(k);\; k\backslash right]$

These equations lead to the other values of the Jacobi-Functions:

:$\backslash operatorname\{cn\}\backslash left[\backslash tfrac\{2\}\{3\}K(k);\; k\backslash right]\; =\; 1\; -\; \backslash operatorname\{sn\}\backslash left[\backslash tfrac\{1\}\{3\}K(k);\; k\backslash right]$

:$\backslash operatorname\{dn\}\backslash left[\backslash tfrac\{2\}\{3\}K(k);\; k\backslash right]\; =\; 1/\backslash operatorname\{sn\}\backslash left[\backslash tfrac\{1\}\{3\}K(k);\; k\backslash right]\; -\; 1$

Fifth K formula

Following equation has following solution:

:$4k^2x^6+8k^2x^5+2(1-k^2)^2x-(1-k^2)^2\; =\; 0$

:$x\; =\; \backslash frac\{1\}\{2\}-\backslash frac\{1\}\{2\}k^2\backslash operatorname\{sn\}\backslash left[\backslash tfrac\{2\}\{5\}K(k);\; k\backslash right]^2\; \backslash operatorname\{sn\}\backslash left[\backslash tfrac\{4\}\{5\}K(k);\; k\backslash right]^2\; =\; \backslash frac\{\backslash operatorname\{sn\}\backslash left[\backslash frac\{4\}\{5\}K(k);\; k\backslash right]^2-\backslash operatorname\{sn\}\backslash left[\backslash frac\{2\}\{5\}K(k);\; k\backslash right]^2\}\{2\backslash operatorname\{sn\}\backslash left[\backslash frac\{2\}\{5\}K(k);\; k\backslash right]\backslash operatorname\{sn\}\backslash left[\backslash frac\{4\}\{5\}K(k);\; k\backslash right]\}$

To get the sn-values, we put the solution x into following expressions:

:$\backslash operatorname\{sn\}\backslash left[\backslash tfrac\{2\}\{5\}K(k);\; k\backslash right]\; =\; (1\; +\; k^2)^\{-1/2\}\backslash sqrt\{2(1-x-x^2)(x^2+1-x\backslash sqrt\{x^2+1\})\}$

:$\backslash operatorname\{sn\}\backslash left[\backslash tfrac\{4\}\{5\}K(k);\; k\backslash right]\; =\; (1\; +\; k^2)^\{-1/2\}\backslash sqrt\{2(1-x-x^2)(x^2+1+x\backslash sqrt\{x^2+1\})\}$

Relations between squares of the functions can be derived from two basic relationships (Arguments (u,m) suppressed):

$$\backslash operatorname\{cn\}^2+\backslash operatorname\{sn\}^2=1$$

$$\backslash operatorname\{cn\}^2+m\text{'}\; \backslash operatorname\{sn\}^2=\backslash operatorname\{dn\}^2$$

where m + m' = 1. Multiplying by any function of the form nq yields more general equations:

$$\backslash operatorname\{cq\}^2+\backslash operatorname\{sq\}^2=\backslash operatorname\{nq\}^2$$

$$\backslash operatorname\{cq\}^2\{\}+m\text{'}\; \backslash operatorname\{sq\}^2=\backslash operatorname\{dq\}^2$$

With q = d, these correspond trigonometrically to the equations for the unit circle ($x^2+y^2=r^2$) and the unit ellipse ($x^2\{\}+m\text{'}\; y^2=1$), with x = cd, y = sd and r = nd. Using the multiplication rule, other relationships may be derived. For example:

-\operatorname{dn}^2{}+m'= -m\operatorname{cn}^2 = m\operatorname{sn}^2-m

-m'\operatorname{nd}^2{}+m'= -mm'\operatorname{sd}^2 = m\operatorname{cd}^2-m

m'\operatorname{sc}^2{}+m'= m'\operatorname{nc}^2 = \operatorname{dc}^2-m

\operatorname{cs}^2{}+m'=\operatorname{ds}^2=\operatorname{ns}^2-m

The functions satisfy the two square relations (dependence on m suppressed)

$$\backslash operatorname\{cn\}^2(u)\; +\; \backslash operatorname\{sn\}^2(u)\; =\; 1,\backslash ,$$

$$\backslash operatorname\{dn\}^2(u)\; +\; m\; \backslash operatorname\{sn\}^2(u)\; =\; 1.\backslash ,$$

From this we see that (cn, sn, dn) parametrizes an elliptic curve which is the intersection of the two quadrics defined by the above two equations. We now may define a group law for points on this curve by the addition formulas for the Jacobi functions

\begin{align}

\operatorname{cn}(x+y) & =

{\operatorname{cn}(x) \operatorname{cn}(y)

- \operatorname{sn}(x) \operatorname{sn}(y) \operatorname{dn}(x) \operatorname{dn}(y)

\over {1 - m \operatorname{sn}^2 (x) \operatorname{sn}^2 (y)}}, \\[8pt]

\operatorname{sn}(x+y) & =

{\operatorname{sn}(x) \operatorname{cn}(y) \operatorname{dn}(y) +

\operatorname{sn}(y) \operatorname{cn}(x) \operatorname{dn}(x)

\over {1 - m \operatorname{sn}^2 (x) \operatorname{sn}^2 (y)}}, \\[8pt]

\operatorname{dn}(x+y) & =

{\operatorname{dn}(x) \operatorname{dn}(y) - m \operatorname{sn}(x) \operatorname{sn}(y) \operatorname{cn}(x) \operatorname{cn}(y)

\over {1 - m \operatorname{sn}^2 (x) \operatorname{sn}^2 (y)}}.

\end{align}

The Jacobi epsilon and zn functions satisfy a quasi-addition theorem:

\operatorname{zn}(x+y,m)&=\operatorname{zn}(x,m)+\operatorname{zn}(y,m)-m\operatorname{sn}(x,m)\operatorname{sn}(y,m)\operatorname{sn}(x+y,m).\end{align}

Double angle formulae can be easily derived from the above equations by setting x = y. Half angle formulae are all of the form:

$$\backslash operatorname\{pq\}(\backslash tfrac\{1\}\{2\}u,m)^2\; =\; f\_\{\backslash mathrm\; p\}/f\_\{\backslash mathrm\; q\}$$

where:

$$f\_\{\backslash mathrm\; c\}\; =\; \backslash operatorname\{cn\}(u,m)+\backslash operatorname\{dn\}(u,m)$$

$$f\_\{\backslash mathrm\; s\}\; =\; 1-\backslash operatorname\{cn\}(u,m)$$

$$f\_\{\backslash mathrm\; n\}\; =\; 1+\backslash operatorname\{dn\}(u,m)$$

$$f\_\{\backslash mathrm\; d\}\; =\; (1+\backslash operatorname\{dn\}(u,m))-m(1-\backslash operatorname\{cn\}(u,m))$$

The derivatives of the three basic Jacobi elliptic functions (with respect to the first variable, with $m$ fixed) are:

$$\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}z\}\; \backslash operatorname\{sn\}(z)\; =\; \backslash operatorname\{cn\}(z)\; \backslash operatorname\{dn\}(z),$$

$$\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}z\}\; \backslash operatorname\{cn\}(z)\; =\; -\backslash operatorname\{sn\}(z)\; \backslash operatorname\{dn\}(z),$$

$$\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}z\}\; \backslash operatorname\{dn\}(z)\; =\; -\; m\; \backslash operatorname\{sn\}(z)\; \backslash operatorname\{cn\}(z).$$

These can be used to derive the derivatives of all other functions as shown in the table below (arguments (u,m) suppressed):

Also

:$\backslash frac\{\backslash mathrm\; d\}\{\backslash mathrm\; dz\}\backslash mathcal\{E\}(z)=\backslash operatorname\{dn\}(z)^2.$

With the addition theorems above and for a given m with 0 < m < 1 the major functions are therefore solutions to the following nonlinear ordinary differential equations:

• $\backslash operatorname\{am\}(x)$ solves the differential equations $\backslash frac\{\backslash mathrm\; d^2y\}\{\backslash mathrm\; dx^2\}+m\backslash sin\; (y)\backslash cos\; (y)=0$ and

:$\backslash left(\backslash frac\{\backslash mathrm\; dy\}\{\backslash mathrm\; dx\}\backslash right)^2=1-m\backslash sin(y)^2$ (for $x$ not on a branch cut)

• $\backslash operatorname\{sn\}(x)$ solves the differential equations $\backslash frac\{\backslash mathrm\{d\}^2\; y\}\{\backslash mathrm\{d\}x^2\}\; +\; (1+m)\; y\; -\; 2\; m\; y^3\; =\; 0$ and $\backslash left(\backslash frac\{\backslash mathrm\{d\}\; y\}\{\backslash mathrm\{d\}x\}\backslash right)^2\; =\; (1-y^2)\; (1-m\; y^2)$
• $\backslash operatorname\{cn\}(x)$ solves the differential equations $\backslash frac\{\backslash mathrm\{d\}^2\; y\}\{\backslash mathrm\{d\}x^2\}\; +\; (1-2m)\; y\; +\; 2\; m\; y^3\; =\; 0$ and $\backslash left(\backslash frac\{\backslash mathrm\{d\}\; y\}\{\backslash mathrm\{d\}x\}\backslash right)^2\; =\; (1-y^2)\; (1-m\; +\; my^2)$
• $\backslash operatorname\{dn\}(x)$ solves the differential equations $\backslash frac\{\backslash mathrm\{d\}^2\; y\}\{\backslash mathrm\{d\}x^2\}\; -\; (2\; -\; m)\; y\; +\; 2\; y^3\; =\; 0$ and $\backslash left(\backslash frac\{\backslash mathrm\{d\}\; y\}\{\backslash mathrm\{d\}x\}\backslash right)^2\; =\; (y^2\; -\; 1)\; (1\; -\; m\; -\; y^2)$

The function which exactly solves the pendulum differential equation,

:$\backslash frac\{\backslash mathrm\; d^2\; \backslash theta\}\{\backslash mathrm\; dt^2\}+c\backslash sin\; \backslash theta=0,$

with initial angle $\backslash theta\_0$ and zero initial angular velocity is

:

&=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}

where $m=\backslash sin\; (\backslash theta\_0/2)^2$, $c>0$ and $t\backslash in\backslash mathbb\{R\}$.

With the first argument $z$ fixed, the derivatives with respect to the second variable $m$ are as follows:

:

\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}

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

:$\backslash operatorname\{am\}(u,m)=\backslash frac\{\backslash pi\; u\}\{2K(m)\}+2\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{q^n\}\{n(1+q^\{2n\})\}\backslash sin\; (2nv),$

:$\backslash operatorname\{sn\}(u,m)=\backslash frac\{2\backslash pi\}\{kK(m)\}$

\sum_{n=0}^\infty \frac{q^{n+1/2}}{1-q^{2n+1}} \sin ((2n+1)v),

:$\backslash operatorname\{cn\}(u,m)=\backslash frac\{2\backslash pi\}\{kK(m)\}$

\sum_{n=0}^\infty \frac{q^{n+1/2}}{1+q^{2n+1}} \cos ((2n+1)v),

:$\backslash operatorname\{dn\}(u,m)=\backslash frac\{\backslash pi\}\{2K(m)\}\; +\; \backslash frac\{2\backslash pi\}\{K(m)\}$

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

:$\backslash operatorname\{zn\}(u,m)=\backslash frac\{2\backslash pi\}\{K(m)\}\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{q^n\}\{1-q^\{2n\}\}\backslash sin\; (2nv)$

when

Bivariate power series expansions have been published by Schett.{{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 }}

The theta function ratios provide an efficient way of computing the Jacobi elliptic functions. There is an alternative method, based on the arithmetic-geometric mean and Landen's transformations:

Initialize

:$a\_0=1,\backslash ,\; b\_0=\backslash sqrt\{1-m\}$

where

Define

:$a\_n=\backslash frac\{a\_\{n-1\}+b\_\{n-1\}\}\{2\},\backslash ,\; b\_n=\backslash sqrt\{a\_\{n-1\}b\_\{n-1\}\},\backslash ,\; c\_n=\backslash frac\{a\_\{n-1\}-b\_\{n-1\}\}\{2\}$

where $n\backslash ge\; 1$.

Then define

:$\backslash varphi\_N=2^N\; a\_N\; u$

for $u\backslash in\backslash mathbb\{R\}$ and a fixed $N\backslash in\backslash mathbb\{N\}$. If

:$\backslash varphi\_\{n-1\}=\backslash frac\{1\}\{2\}\backslash left(\backslash varphi\_n+\backslash arcsin\; \backslash left(\backslash frac\{c\_n\}\{a\_n\}\backslash sin\; \backslash varphi\_n\backslash right)\backslash right)$

for $n\backslash ge\; 1$, then

:$\backslash operatorname\{am\}(u,m)=\backslash varphi\_0,\backslash quad\; \backslash operatorname\{zn\}(u,m)=\backslash sum\_\{n=1\}^N\; c\_n\backslash sin\backslash varphi\_n$

as $N\backslash to\backslash infty$. This is notable for its rapid convergence. It is then trivial to compute all Jacobi elliptic functions from the Jacobi amplitude $\backslash operatorname\{am\}$ on the real line.For the $\backslash operatorname\{dn\}$ function,

$\backslash operatorname\{dn\}(u,m)=\backslash frac\{\backslash operatorname\{cn\}(u,m)\}\{\backslash operatorname\{sn\}(K(m)-u,m)\}$ can be used.

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:{{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 }}

Let

:$0\backslash le\; m\backslash le\; 1,\backslash ,0\backslash le\; u\backslash le\; K(m),\backslash ,\; a\_0=1,\backslash ,\; b\_0=\backslash sqrt\{1-m\},$

:$a\_\{n+1\}=\backslash frac\{a\_n+b\_n\}\{2\},\backslash ,\; b\_\{n+1\}=\backslash sqrt\{a\_n\; b\_n\},\backslash ,c\_\{n+1\}=\backslash frac\{a\_n-b\_n\}\{2\}.$

Set

:

y_{N-1}&=y_N+\frac{a_Nc_N}{y_N}\\

y_{N-2}&=y_{N-1}+\frac{a_{N-1}c_{N-1}}{y_{N-1}}\\

\vdots&=\vdots\\

y_0&=y_1+\frac{m}{4y_1}.\end{align}

Then

:

\operatorname{cn}(u,m)&=\sqrt{1-\frac{1}{y_0^2}}\\

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

as $N\backslash to\backslash infty$.

Yet, another method for a rapidly converging fast computation of the Jacobi elliptic sine function found in the literature is shown below.{{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 |url=https://ieeexplore.ieee.org/document/1127543 |journal=IEEE Transactions on Microwave Theory and Techniques |volume=MTT-19 |issue=5 |pages=430 |doi=10.1109/TMTT.1971.1127543 |bibcode=1971ITMTT..19..424S |via=IEEE Xplore}}

Let:

:$\backslash begin\{align\}$

&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}

Then set:

:$\backslash begin\{align\}$

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}

Then:

:$\backslash operatorname\{sn\}(u,m)\; =\; y\_0\; \backslash text\{\; as\; \}n\; \backslash rightarrow\backslash infty$.

The Jacobi elliptic functions can be expanded in terms of the hyperbolic functions. When $m$ is close to unity, such that $m\text{'}^2$ and higher powers of $m\text{'}$ can be neglected, we have:{{dlmf|first1=W. P.|last1=Reinhardt|first2=P. L.|last2=Walker|id=22.10|title=Jacobian Elliptic Functions}}{{dlmf|first1=W. P.|last1=Reinhardt|first2=P. L.|last2=Walker|id=22.16.E8|title=Jacobian Elliptic Functions}}

• sn(u): $$\backslash operatorname\{sn\}\; (u,m)\backslash approx\; \backslash tanh\; (u)+\backslash frac\{1\}\{4\}m\text{'}(\backslash sinh\; (u)\backslash cosh\; (u)\; -u)\backslash operatorname\{sech\}^2\; (u).$$
• cn(u): $$\backslash operatorname\{cn\}\; (u,m)\backslash approx\; \backslash operatorname\{sech\}\; (u)-\backslash frac\{1\}\{4\}\; m\text{'}(\backslash sinh\; (u)\backslash cosh\; (u)\; -u)\backslash tanh\; (u)\; \backslash operatorname\{sech\}\; (u).$$
• dn(u): $$\backslash operatorname\{dn\}\; (u,m)\; \backslash approx\; \backslash operatorname\{sech\}\; (u)+\backslash frac\{1\}\{4\}\; m\text{'}(\backslash sinh\; (u)\backslash cosh(u)\; +u)\backslash tanh\; (u)\; \backslash operatorname\{sech\}\; (u)\; .$$

For the Jacobi amplitude,

$$\backslash operatorname\{am\}\; (u,m)\; \backslash approx\; \backslash operatorname\{gd\}\; (u)+\backslash frac\{1\}\{4\}m\text{'}(\backslash sinh\; (u)\backslash cosh\; (u)\; -u)\backslash operatorname\{sech\}\; (u)\; .$$

Assuming real numbers $a,p$ with

:$$

\begin{align}

&\frac{\textrm{dn}\left((p/2-a)\tau K\left[\frac{p\tau}{2}\right];k\left(\frac{p\tau}{2}\right)\right)}{\sqrt{k'\left(\frac{p\tau}{2}\right)}} = \frac{\sum^\infty_{n=-\infty}q^{p/2 n^2+(p/2-a)n}}{\sum^\infty_{n=-\infty}(-1)^nq^{p/2 n^2+(p/2-a)n}}\\[4pt]

={}&-1+\frac{2}{1-{}} \, \frac{q^a+q^{p-a}}{1-q^p+{}} \, \frac{(q^a+q^{2p-a})(q^{a+p}+q^{p-a})}{1-q^{3p}+{}} \, \frac{q^p(q^a+q^{3p-a})(q^{a+2p}+q^{p-a})}{1-q^{5p}+{}} \, \frac{q^{2p}(q^a+q^{4p-a})(q^{a+3p}+q^{p-a})}{1-q^{7p}+{}}\cdots

\end{align}

Known continued fractions involving $\backslash textrm\{sn\}(t),\backslash textrm\{cn\}(t)$ and $\backslash textrm\{dn\}(t)$ with elliptic modulus $k$ are

For $z\backslash in\backslash Complex$, :H.S. Wall. (1948). "Analytic Theory of Continued Fractions", Van Nostrand, New York. pg. 374

:$\backslash int^\{\backslash infty\}\_\{0\}\backslash textrm\{sn\}(t)e^\{-tz\}\backslash ,\; \backslash mathrm\; dt=\backslash frac\{1\}\{1^2(1+k^2)+z^2-\{\}\}\; \backslash ,\; \backslash frac\{1\backslash cdot\; 2^2\backslash cdot\; 3k^2\}\{3^2(1+k^2)+z^2-\{\}\}\; \backslash ,\; \backslash frac\{3\backslash cdot\; 4^2\backslash cdot\; 5k^2\}\{5^2(1+k^2)+z^2-\{\}\}\backslash cdots$

For $z\; \backslash in\; \backslash Complex\backslash setminus\backslash \{0\backslash \}$, : pg. 375

:$\backslash int^\{\backslash infty\}\_\{0\}\backslash textrm\{sn\}^2(t)e^\{-t\; z\}\backslash ,\backslash mathrm\; dt=\backslash frac\{2z^\{-1\}\}\{2^2(1+k^2)+z^2-\{\}\}\; \backslash ,\; \backslash frac\{2\backslash cdot\; 3^2\backslash cdot\; 4k^2\}\{4^2(1+k^2)+z^2-\{\}\}\; \backslash ,\; \backslash frac\{4\backslash cdot\; 5^2\backslash cdot\; 6k^2\}\{6^2(1+k^2)+z^2-\{\}\}\backslash cdots$

For $z\backslash in\backslash Complex\backslash setminus\backslash \{0\backslash \}$, :Perron, O. (1957). "Die Lehre von den Kettenbruchen", Band II, B.G. Teubner, Stuttgart. pg. 220

:$\backslash int^\backslash infty\_0\; \backslash textrm\{cn\}(t)e^\{-t\; z\}\backslash ,\; \backslash mathrm\; dt=\backslash frac\{1\}\{z+\{\}\}\; \backslash ,\; \backslash frac\{1^2\}\{z+\{\}\}\; \backslash ,\; \backslash frac\{2^2k^2\}\{z+\{\}\}\backslash ,\; \backslash frac\{3^2\}\{z+\{\}\}\; \backslash ,\; \backslash frac\{4^2k^2\}\{z+\{\}\}\; \backslash ,\; \backslash frac\{5^2\}\{z+\{\}\}\; \backslash cdots$

For $z\backslash in\backslash Complex\backslash setminus\backslash \{0\backslash \}$, : pg. 374

:$\backslash int^\backslash infty\_0\backslash textrm\{dn\}(t)e^\{-t\; z\}\backslash ,\; \backslash mathrm\; dt=\backslash frac\{1\}\{z+\{\}\}\; \backslash ,\; \backslash frac\{1^2k^2\}\{z+\{\}\}\; \backslash ,\; \backslash frac\{2^2\}\{z+\{\}\}\; \backslash ,\; \backslash frac\{3^2k^2\}\{z+\{\}\}\; \backslash ,\; \backslash frac\{4^2\}\{z+\{\}\}\; \backslash ,\; \backslash frac\{5^2k^2\}\{z+\{\}\}\; \backslash cdots$

For $z\backslash in\backslash Complex$, : pg. 375

:$\backslash int^\{\backslash infty\}\_\{0\}\backslash frac\{\backslash textrm\{sn\}(t)\backslash textrm\{cn\}(t)\}\{\backslash textrm\{dn\}(t)\}e^\{-tz\}\backslash ,\; \backslash mathrm\; dt=\backslash frac\{1\}\{2\backslash cdot\; 1^2(2-k^2)+z^2-\{\}\}\; \backslash ,\; \backslash frac\{1\backslash cdot\; 2^2\backslash cdot\; 3k^4\}\{2\backslash cdot\; 3^2(2-k^2)+z^2-\{\}\}\; \backslash ,\; \backslash frac\{3\backslash cdot\; 4^2\backslash cdot\; 5k^4\}\{2\backslash cdot\; 5^2(2-k^2)+z^2-\{\}\}\backslash cdots$

The inverses of the Jacobi elliptic functions can be defined similarly to the inverse trigonometric functions; if $x=\backslash operatorname\{sn\}(\backslash xi,\; m)$, $\backslash xi=\backslash operatorname\{arcsn\}(x,\; m)$. They can be represented as elliptic integrals,{{dlmf|title=§22.15 Inverse Functions|first1=W. P.|last1=Reinhardt|first2=P. L.|last2=Walker|id=22.15}}{{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}}{{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}} and power series representations have been found.{{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=2008MaCom..77.1615C|doi-access=free}}

• $\backslash operatorname\{arcsn\}(x,m)\; =\; \backslash int\_0^x\; \backslash frac\{\backslash mathrm\{d\}t\}\{\backslash sqrt\{(1-t^2)(1-mt^2)\}\}$
• $\backslash operatorname\{arccn\}(x,m)\; =\backslash int\_x^1\; \backslash frac\{\backslash mathrm\{d\}t\}\{\backslash sqrt\{(1-t^2)(1-m+mt^2)\}\}$
• $\backslash operatorname\{arcdn\}(x,m)\; =\; \backslash int\_x^1\; \backslash frac\{\backslash mathrm\{d\}t\}\{\backslash sqrt\{(1-t^2)(t^2+m-1)\}\}$

The Peirce quincuncial projection is a map projection based on Jacobian elliptic functions.

• Elliptic curve
• Schwarz–Christoffel mapping
• Carlson symmetric form
• Jacobi theta function
• Ramanujan theta function
• Dixon elliptic functions
• Abel elliptic functions
• Weierstrass elliptic function
• Lemniscate elliptic functions

{{reflist|group=note}}

{{Reflist}}

• {{AS ref|16|569}}
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• A. C. Dixon [https://archive.org/details/117736039 The elementary properties of the elliptic functions, with examples] (Macmillan, 1894)
• Alfred George Greenhill [https://archive.org/details/applicationselli00greerich The applications of elliptic functions] (London, New York, Macmillan, 1892)
• Edmund T. Whittaker, George Neville Watson: A Course in Modern Analysis. 4th ed. Cambridge, England: Cambridge University Press, 1990. S. 469–470.
• H. Hancock [https://archive.org/details/lecturestheorell00hancrich Lectures on the theory of elliptic functions] (New York, J. Wiley & sons, 1910)
• {{Citation | last1=Jacobi | first1=C. G. J. | title=Fundamenta nova theoriae functionum ellipticarum | url=https://archive.org/details/fundamentanovat00jacogoog | publisher=Königsberg | language=la | isbn=978-1-108-05200-9 | id=Reprinted by Cambridge University Press 2012 | year=1829}}
• {{dlmf|first=William P. |last=Reinhardt|first2=Peter L. |last2=Walker|id=22|title=Jacobian Elliptic Functions}}
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• {{in lang|fr}} J. Tannery and J. Molk [http://gallica.bnf.fr/notice?N=FRBNF37258246 Eléments de la théorie des fonctions elliptiques. Tome IV, Calcul intégral. IIe partie, Applications] (Paris : Gauthier-Villars et fils, 1893)
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• Toshio Fukushima: Fast Computation of Complete Elliptic Integrals and Jacobian Elliptic Functions. 2012, National Astronomical Observatory of Japan (国立天文台)
• Lowan, Blanch und Horenstein: On the Inversion of the q-Series Associated with Jacobian Elliptic Functions. Bull. Amer. Math. Soc. 48, 1942
• H. Ferguson, D. E. Nielsen, G. Cook: A partition formula for the integer coefficients of the theta function nome. Mathematics of computation, Volume 29, Nummer 131, Juli 1975
• J. D. Fenton and R. S. Gardiner-Garden: Rapidly-convergent methods for evaluating elliptic integrals and theta and elliptic functions. J. Austral. Math. Soc. (Series B) 24, 1982, S. 57
• Adolf Kneser: Neue Untersuchung einer Reihe aus der Theorie der elliptischen Funktionen. J. reine u. angew. Math. 157, 1927. pages 209 – 218

• {{springer|title=Jacobi elliptic functions|id=p/j054050}}
• {{MathWorld|JacobiEllipticFunctions|Jacobi Elliptic Functions}}

{{DEFAULTSORT:Jacobi Elliptic Functions}}

Category:Elliptic functions

Category:Special functions