Jacobi elliptic functions

In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of electronic elliptic filters. While trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation ${\displaystyle \operatorname {sn} }$ for ${\displaystyle \sin }$. 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 Carl Gustav Jakob Jacobi (1829). Carl Friedrich Gauss had already studied special Jacobi elliptic functions in 1797, the lemniscate elliptic functions in particular,[1] but his work was published much later.

Overview

There are twelve Jacobi elliptic functions denoted by ${\displaystyle \operatorname {pq} (u,m)}$, where ${\displaystyle \mathrm {p} }$ and ${\displaystyle \mathrm {q} }$ are any of the letters ${\displaystyle \mathrm {c} }$, ${\displaystyle \mathrm {s} }$, ${\displaystyle \mathrm {n} }$, and ${\displaystyle \mathrm {d} }$. (Functions of the form ${\displaystyle \operatorname {pp} (u,m)}$ are trivially set to unity for notational completeness.) ${\displaystyle u}$ is the argument, and ${\displaystyle m}$ is the parameter, both of which may be complex. In fact, the Jacobi elliptic functions are meromorphic in both ${\displaystyle u}$ and ${\displaystyle m}$.[2] The distribution of the zeros and poles in the ${\displaystyle u}$-plane is well-known. However, questions of the distribution of the zeros and poles in the ${\displaystyle m}$-plane remain to be investigated.[2]

In the complex plane of the argument ${\displaystyle u}$, the twelve functions form a repeating lattice of simple poles and zeroes.[3] Depending on the function, one repeating parallelogram, or unit cell, will have sides of length ${\displaystyle 2K}$ or ${\displaystyle 4K}$ on the real axis, and ${\displaystyle 2K'}$ or ${\displaystyle 4K'}$ on the imaginary axis, where ${\displaystyle K=K(m)}$ and ${\displaystyle K'=K(1-m)}$ are known as the quarter periods with ${\displaystyle K(\cdot )}$ 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 ${\displaystyle (0,0)}$ at one corner, and ${\displaystyle (K,K')}$ as the diagonally opposite corner. As in the diagram, the four corners of the auxiliary rectangle are named ${\displaystyle \mathrm {s} }$, ${\displaystyle \mathrm {c} }$, ${\displaystyle \mathrm {d} }$, and ${\displaystyle \mathrm {n} }$, going counter-clockwise from the origin. The function ${\displaystyle \operatorname {pq} (u,m)}$ will have a zero at the ${\displaystyle \mathrm {p} }$ corner and a pole at the ${\displaystyle \mathrm {q} }$ corner. The twelve functions correspond to the twelve ways of arranging these poles and zeroes in the corners of the rectangle.

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

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

• There is a simple zero at the corner ${\displaystyle \mathrm {p} }$, and a simple pole at the corner ${\displaystyle \mathrm {q} }$.
• The complex number ${\displaystyle \mathrm {p} -\mathrm {q} }$ is equal to half the period of the function ${\displaystyle \operatorname {pq} u}$; that is, the function ${\displaystyle \operatorname {pq} u}$ is periodic in the direction ${\displaystyle \operatorname {pq} }$, with the period being ${\displaystyle 2(\mathrm {p} -\mathrm {q} )}$. The function ${\displaystyle \operatorname {pq} u}$ is also periodic in the other two directions ${\displaystyle \mathrm {pp} '}$ and ${\displaystyle \mathrm {pq} '}$, with periods such that ${\displaystyle \mathrm {p} -\mathrm {p} '}$ and ${\displaystyle \mathrm {p} -\mathrm {q} '}$ are quarter periods.
Jacobi elliptic function ${\displaystyle \operatorname {sn} }$
Jacobi elliptic function ${\displaystyle \operatorname {cn} }$
Jacobi elliptic function ${\displaystyle \operatorname {dn} }$
Jacobi elliptic function ${\displaystyle \operatorname {sc} }$
Plots of four Jacobi Elliptic Functions in the complex plane of ${\displaystyle u}$, illustrating their double periodic behavior. Images generated using a version of the domain coloring method.[4] All have values of ${\displaystyle k={\sqrt {m}}}$ equal to ${\displaystyle 0.8}$.

Notation

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 ${\displaystyle \varphi }$, or more commonly, in terms of ${\displaystyle u}$ given below. The second variable might be given in terms of the parameter ${\displaystyle m}$, or as the elliptic modulus ${\displaystyle k}$, where ${\displaystyle k^{2}=m}$, or in terms of the modular angle ${\displaystyle \alpha }$, where ${\displaystyle m=\sin ^{2}\alpha }$. The complements of ${\displaystyle k}$ and ${\displaystyle m}$ are defined as ${\displaystyle m'=1-m}$ and ${\textstyle k'={\sqrt {m'}}}$. These four terms are used below without comment to simplify various expressions.

The twelve Jacobi elliptic functions are generally written as ${\displaystyle \operatorname {pq} (u,m)}$ where ${\displaystyle \mathrm {p} }$ and ${\displaystyle \mathrm {q} }$ are any of the letters ${\displaystyle \mathrm {c} }$, ${\displaystyle \mathrm {s} }$, ${\displaystyle \mathrm {n} }$, and ${\displaystyle \mathrm {d} }$. Functions of the form ${\displaystyle \operatorname {pp} (u,m)}$ are trivially set to unity for notational completeness. The “major” functions are generally taken to be ${\displaystyle \operatorname {cn} (u,m)}$, ${\displaystyle \operatorname {sn} (u,m)}$ and ${\displaystyle \operatorname {dn} (u,m)}$ 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, ${\displaystyle \operatorname {pq} (u,t^{2})=\operatorname {pq} (u;t)}$.

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

${\displaystyle \operatorname {pq} \cdot \operatorname {p'q'} =\operatorname {pq'} \cdot \operatorname {p'q} }$

from which other commonly used relationships can be derived:

${\displaystyle {\frac {\operatorname {pr} }{\operatorname {qr} }}=\operatorname {pq} }$
${\displaystyle \operatorname {pr} \cdot \operatorname {rq} =\operatorname {pq} }$
${\displaystyle {\frac {1}{\operatorname {qp} }}=\operatorname {pq} }$

The multiplication rule follows immediately from the identification of the elliptic functions with the Neville theta functions[5]

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

Also note that:

${\displaystyle K(m)=K(k^{2})=\int _{0}^{1}{\frac {dt}{\sqrt {(1-t^{2})(1-mt^{2})}}}=\int _{0}^{1}{\frac {dt}{\sqrt {(1-t^{2})(1-k^{2}t^{2})}}}.}$

Definition in terms of inverses of elliptic integrals

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

${\displaystyle u=\int _{0}^{\varphi }{\frac {\mathrm {d} \theta }{\sqrt {1-m\sin ^{2}\theta }}}}$

is called the Jacobi amplitude:

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

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

${\displaystyle \operatorname {sn} (u,m)=\sin \operatorname {am} (u,m)}$

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

${\displaystyle \operatorname {cn} (u,m)=\cos \operatorname {am} (u,m)}$

and the delta amplitude dn u (Latin: delta amplitudinis)[note 1]

${\displaystyle \operatorname {dn} (u,m)={\frac {\mathrm {d} }{\mathrm {d} u}}\operatorname {am} (u,m).}$

In the above, the value ${\displaystyle m}$ is a free parameter, usually taken to be real such that ${\displaystyle 0\leq m\leq 1}$, and so the elliptic functions can be thought of as being given by two variables, ${\displaystyle u}$ and the parameter ${\displaystyle m}$. The remaining nine elliptic functions are easily built from the above three (${\displaystyle \operatorname {sn} }$, ${\displaystyle \operatorname {cn} }$, ${\displaystyle \operatorname {dn} }$), and are given in a section below.

In the most general setting, ${\displaystyle \operatorname {am} (u,m)}$ is a multivalued function (in ${\displaystyle u}$) with infinitely many logarithmic branch points (the branches differ by integer multiples of ${\displaystyle 2\pi }$), namely the points ${\displaystyle 2sK(m)+(4t+1)K(1-m)i}$ and ${\displaystyle 2sK(m)+(4t+3)K(1-m)i}$ where ${\displaystyle s,t\in \mathbb {Z} }$.[6] 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 ${\displaystyle \operatorname {am} (u,m)}$ analytic everywhere except on the branch cuts. In contrast, ${\displaystyle \sin \operatorname {am} (u,m)}$ and other elliptic functions have no branch points, give consistent values for every branch of ${\displaystyle \operatorname {am} }$, and are meromorphic in the whole complex plane. Since every elliptic function is meromorphic in the whole complex plane (by definition), ${\displaystyle \operatorname {am} (u,m)}$ (when considered as a single-valued function) is not an elliptic function.

However, the integral inversion above defines a unique single-valued real-analytic function in a real neighborhood of ${\displaystyle u=0}$ if ${\displaystyle m}$ is real. There is a unique analytic continuation of this function from that neighborhood to ${\displaystyle u\in \mathbb {R} }$. The analytic continuation of this function is periodic in ${\displaystyle u}$ if and only if ${\displaystyle m>1}$ (with the minimal period ${\displaystyle 4K(1/m)/{\sqrt {m}}}$), and it is denoted by ${\displaystyle \operatorname {am} (u,m)}$ in the rest of this article.

Jacobi also introduced the coamplitude function:

${\displaystyle \operatorname {coam} (u,m)=\operatorname {am} (K(m)-u,m)}$.

The Jacobi epsilon function can be defined as[7]

${\displaystyle {\mathcal {E}}(u,m)=\int _{0}^{u}\operatorname {dn} ^{2}(t,m)\,\mathrm {d} t}$

and relates the incomplete elliptic integral of the first kind to the incomplete elliptic integral of the second kind (with parameter ${\displaystyle m}$):

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

The Jacobi epsilon function is not an elliptic function. However, unlike the Jacobi amplitude and coamplitude, the Jacobi epsilon function is meromorphic in the whole complex plane (in both ${\displaystyle u}$ and ${\displaystyle m}$).

The Jacobi zn function is defined by

${\displaystyle \operatorname {zn} (u,m)=\int _{0}^{u}\left(\operatorname {dn} (t,m)^{2}-{\frac {E(m)}{K(m)}}\right)\,\mathrm {d} t.}$

It is a singly periodic function which is meromorphic in ${\displaystyle u}$. Its minimal period is ${\displaystyle 2K(m)}$. It is related to the Jacobi zeta function by ${\displaystyle Z(\varphi ,m)=\operatorname {zn} (F(\varphi ,m),m).}$

Note that when ${\displaystyle \varphi =\pi /2}$, that ${\displaystyle u}$ then equals the quarter period ${\displaystyle K}$.

Definition as trigonometry: the Jacobi ellipse

${\displaystyle \cos \varphi ,\sin \varphi }$ are defined on the unit circle, with radius r = 1 and angle ${\displaystyle \varphi =}$ arc length of the unit circle measured from the positive x-axis. Similarly, Jacobi elliptic functions are defined on the unit ellipse,[citation needed] with a = 1. Let

{\displaystyle {\begin{aligned}&x^{2}+{\frac {y^{2}}{b^{2}}}=1,\quad b>1,\\&m=1-{\frac {1}{b^{2}}},\quad 0

then:

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

For each angle ${\displaystyle \varphi }$ the parameter

${\displaystyle u=u(\varphi ,m)=\int _{0}^{\varphi }r(\theta ,m)\,d\theta }$

(the incomplete elliptic integral of the first kind) is computed. On the unit circle (${\displaystyle a=b=1}$), ${\displaystyle u}$ would be an arc length. The quantity ${\displaystyle u[\varphi ,k]=u(\varphi ,k^{2})}$ is related to the incomplete elliptic integral of the second kind (with modulus ${\displaystyle k}$) by[8]

${\displaystyle u[\varphi ,k]={\frac {1}{\sqrt {1-k^{2}}}}\left({\frac {1+{\sqrt {1-k^{2}}}}{2}}\operatorname {E} \left(\varphi +\arctan \left({\sqrt {1-k^{2}}}\tan \varphi \right),{\frac {1-{\sqrt {1-k^{2}}}}{1+{\sqrt {1-k^{2}}}}}\right)-\operatorname {E} (\varphi ,k)+{\frac {k^{2}\sin \varphi \cos \varphi }{2{\sqrt {1-k^{2}\sin ^{2}\varphi }}}}\right),}$

and therefore is related to the arc length of an ellipse. Let ${\displaystyle P=(x,y)=(r\cos \varphi ,r\sin \varphi )}$ be a point on the ellipse, and let ${\displaystyle P'=(x',y')=(\cos \varphi ,\sin \varphi )}$ be the point where the unit circle intersects the line between ${\displaystyle P}$ and the origin ${\displaystyle O}$. Then the familiar relations from the unit circle:

${\displaystyle x'=\cos \varphi ,\quad y'=\sin \varphi }$

${\displaystyle x'=\operatorname {cn} (u,m),\quad y'=\operatorname {sn} (u,m).}$

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

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

For the ${\displaystyle x}$ and ${\displaystyle y}$ value of the point ${\displaystyle P}$ with ${\displaystyle u}$ and parameter ${\displaystyle m}$ we get, after inserting the relation:

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

into: ${\displaystyle x=r(\varphi ,m)\cos(\varphi ),y=r(\varphi ,m)\sin(\varphi )}$ that:

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

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

Jacobi elliptic functions pq[u,m] as functions of {x,y,r} and {φ,dn}
q
c s n d
p
c 1 ${\displaystyle x/y=\cot(\varphi )}$ ${\displaystyle x/r=\cos(\varphi )}$ ${\displaystyle x=\cos(\varphi )/\operatorname {dn} }$
s ${\displaystyle y/x=\tan(\varphi )}$ 1 ${\displaystyle y/r=\sin(\varphi )}$ ${\displaystyle y=\sin(\varphi )/\operatorname {dn} }$
n ${\displaystyle r/x=\sec(\varphi )}$ ${\displaystyle r/y=\csc(\varphi )}$ 1 ${\displaystyle r=1/\operatorname {dn} }$
d ${\displaystyle 1/x=\sec(\varphi )\operatorname {dn} }$ ${\displaystyle 1/y=\csc(\varphi )\operatorname {dn} }$ ${\displaystyle 1/r=\operatorname {dn} }$ 1

Definition in terms of Jacobi theta functions

Jacobi theta function description

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

{\displaystyle {\begin{aligned}\operatorname {sn} (u;k)&=-{\frac {\vartheta _{00}[q(k)]\,\vartheta _{11}[u\div {\bar {K}}(k);\,q(k)]}{\vartheta _{10}[q(k)]\,\vartheta _{01}[u\div {\bar {K}}(k);\,q(k)]}}\\[7pt]\operatorname {cn} (u;k)&={\frac {\vartheta _{01}[q(k)]\,\vartheta _{10}[u\div {\bar {K}}(k);\,q(k)]}{\vartheta _{10}[q(k)]\,\vartheta _{01}[u\div {\bar {K}}(k);\,q(k)]}}\\[7pt]\operatorname {dn} (u;k)&={\frac {\vartheta _{01}[q(k)]\,\vartheta _{00}[u\div {\bar {K}}(k);\,q(k)]}{\vartheta _{00}[q(k)]\,\vartheta _{01}[u\div {\bar {K}}(k);\,q(k)]}}\end{aligned}}}

where ${\displaystyle {\bar {K}}(k)={\frac {2}{\pi }}\,K(k)}$ and ${\displaystyle q=\exp[-\pi \,K'(k)/K(k)]}$.

Edmund Whittaker and George Watson defined the Jacobi theta functions this way in their textbook A Course of Modern Analysis:[9]

${\displaystyle \vartheta _{00}(v;w)=\prod _{n=1}^{\infty }(1-w^{2n})[1+2\cos(2v)w^{2n-1}+w^{4n-2}]}$
${\displaystyle \vartheta _{01}(v;w)=\prod _{n=1}^{\infty }(1-w^{2n})[1-2\cos(2v)w^{2n-1}+w^{4n-2}]}$
${\displaystyle \vartheta _{10}(v;w)=2w^{1/4}\cos(v)\prod _{n=1}^{\infty }(1-w^{2n})[1+2\cos(2v)w^{2n}+w^{4n}]}$
${\displaystyle \vartheta _{11}(v;w)=-2w^{1/4}\sin(v)\prod _{n=1}^{\infty }(1-w^{2n})[1-2\cos(2v)w^{2n}+w^{4n}]}$

Jacobi zn function

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

{\displaystyle {\begin{aligned}\operatorname {zn} (u;k)&={\frac {\pi }{2K}}{\frac {\vartheta _{01}'[u\div {\bar {K}}(k);\,q(k)]}{\vartheta _{01}[u\div {\bar {K}}(k);\,q(k)]}}\\&={\frac {\pi }{2K}}{\frac {\vartheta _{00}'[u\div {\bar {K}}(k);\,q(k)]}{\vartheta _{00}[u\div {\bar {K}}(k);\,q(k)]}}+k^{2}{\frac {\operatorname {sn} (u;k)\operatorname {cn} (u;k)}{\operatorname {dn} (u;k)}}\\[6pt]&={\frac {\pi }{2K}}{\frac {\vartheta _{10}'[u\div {\bar {K}}(k);\,q(k)]}{\vartheta _{10}[u\div {\bar {K}}(k);\,q(k)]}}+{\frac {\operatorname {dn} (u;k)\operatorname {sn} (u;k)}{\operatorname {cn} (u;k)}}\\[6pt]&={\frac {\pi }{2K}}{\frac {\vartheta _{11}'[u\div {\bar {K}}(k);\,q(k)]}{\vartheta _{11}[u\div {\bar {K}}(k);\,q(k)]}}-{\frac {\operatorname {cn} (u;k)\operatorname {dn} (u;k)}{\operatorname {sn} (u;k)}}\end{aligned}}}

where ${\displaystyle '}$ denotes the partial derivative with respect to the left bracket entry:

{\displaystyle {\begin{aligned}&\vartheta '_{00}(v;w)={\frac {\partial }{\partial v}}\vartheta _{00}(v;w)\\[6pt]&\vartheta '_{01}(v;w)={\frac {\partial }{\partial v}}\vartheta _{01}(v;w)\end{aligned}}}

and so on.

The following definition of the Jacobi zn function is identical to the now mentioned formulas:

${\displaystyle {\text{zn}}(u;k)=\sum _{n=1}^{\infty }{\frac {2\pi K(k)^{-1}\sin[\pi K(k)^{-1}u]q(k)^{2n-1}}{1-2\cos[\pi K(k)^{-1}u]q(k)^{2n-1}+q(k)^{4n-2}}}}$

In a successive way the amplitude sine sn can be generated as follows:

${\displaystyle \operatorname {sn} (u;k)={\frac {2\{\operatorname {zn} ({\tfrac {1}{2}}u;k)+\operatorname {zn} [K(k)-{\tfrac {1}{2}}u;k]\}}{k^{2}+\{\operatorname {zn} ({\tfrac {1}{2}}u;k)+\operatorname {zn} [K(k)-{\tfrac {1}{2}}u;k]\}^{2}}}}$

Comparison between sums and products

The reduced elliptic integral of first kind shall be defined as follows again:

${\displaystyle {\bar {K}}(\varepsilon )={\frac {2}{\pi }}K(\varepsilon )}$

And the reduced elliptic nome shall be defined after this pattern:

${\displaystyle {\bar {q}}(\varepsilon )={\sqrt[{4}]{\varepsilon ^{-2}q(\varepsilon )}}}$

The brothers Peter and Jonathan Borwein also gave these two following formulas for the amplitude sine in their work π and the AGM on page 60 ff:

 ${\displaystyle \operatorname {sn} (u;k)=\,{\frac {4}{{\bar {K}}(k)}}\,{\bar {q}}(k)^{2}\sin[u\div {\bar {K}}(k)]\sum _{n=1}^{\infty }{\frac {q(k)^{n-1}[1+q(k)^{2n-1}]}{1-2q(k)^{2n-1}\cos[2u\div {\bar {K}}(k)]+q(k)^{4n-2}}}}$ ${\displaystyle \operatorname {sn} (u;k)=2\,{\bar {q}}(k)\sin[u\div {\bar {K}}(k)]\prod _{n=1}^{\infty }{\frac {1-2q(k)^{2n}\cos[2u\div {\bar {K}}(k)]+q(k)^{4n}}{1-2q(k)^{2n-1}\cos[2u\div {\bar {K}}(k)]+q(k)^{4n-2}}}}$

This defining formula, which results directly from the inner substitution ${\displaystyle z\rightarrow K(k)-z}$, applies analogously to the cd function:

 ${\displaystyle \operatorname {cd} (u;k)=\,{\frac {4}{{\bar {K}}(k)}}\,{\bar {q}}(k)^{2}\cos[u\div {\bar {K}}(k)]\sum _{n=1}^{\infty }{\frac {q(k)^{n-1}[1+q(k)^{2n-1}]}{1+2q(k)^{2n-1}\cos[2u\div {\bar {K}}(k)]+q(k)^{4n-2}}}}$ ${\displaystyle \operatorname {cd} (u;k)=2\,{\bar {q}}(k)\cos[u\div {\bar {K}}(k)]\prod _{n=1}^{\infty }{\frac {1+2q(k)^{2n}\cos[2u\div {\bar {K}}(k)]+q(k)^{4n}}{1+2q(k)^{2n-1}\cos[2u\div {\bar {K}}(k)]+q(k)^{4n-2}}}}$

These formulas are based on Whittaker and Watson's definition of theta non-zero value functions.

These formulas[10] apply to the cosine amplitude:

 ${\displaystyle \operatorname {cn} (u;k)=\,{\frac {4}{{\bar {K}}(k)}}\,{\bar {q}}(k)^{2}\cos[u\div {\bar {K}}(k)]\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}q(k)^{n-1}[1-q(k)^{2n-1}]}{1-2q(k)^{2n-1}\cos[2u\div {\bar {K}}(k)]+q(k)^{4n-2}}}}$ ${\displaystyle \operatorname {cn} (u;k)=2\,{\sqrt[{4}]{1-k^{2}}}\,{\bar {q}}(k)\cos[u\div {\bar {K}}(k)]\prod _{n=1}^{\infty }{\frac {1+2q(k)^{2n}\cos[2u\div {\bar {K}}(k)]+q(k)^{4n}}{1-2q(k)^{2n-1}\cos[2u\div {\bar {K}}(k)]+q(k)^{4n-2}}}}$

According to the Whittaker-Watson product formulas, this formula also applies to the delta amplitude function:

${\displaystyle \operatorname {dn} (u;k)={\sqrt[{4}]{1-k^{2}}}\prod _{n=1}^{\infty }{\frac {1+2\cos[2u\div {\bar {K}}(k)]q(k)^{2n-1}+q(k)^{4n-2}}{1-2\cos[2u\div {\bar {K}}(k)]q(k)^{2n-1}+q(k)^{4n-2}}}}$

With a Hyperbolic secant sum is a definition[11] possible for the Delta Amplitudinis:

${\displaystyle \operatorname {dn} (z;k)={\frac {\pi }{2K({\sqrt {1-k^{2}}})}}\sum _{n=-\infty }^{\infty }\operatorname {sech} {\bigl \{}\pi K({\sqrt {1-k^{2}}})^{-1}{\bigl [}K(k)n+{\tfrac {1}{2}}z{\bigr ]}{\bigr \}}}$

The elliptic nome and its series

Elliptic integral and elliptic nome

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

${\displaystyle k'(\tau )={\sqrt {1-k^{2}}}={\biggl \{}{\vartheta _{01}[q(k)] \over \vartheta _{00}[q(k)]}{\biggr \}}^{2}}$

Let us define the elliptic nome and the complete elliptic integral of the first kind:

${\displaystyle q(k)=\exp {\biggl [}-\pi {\frac {K({\sqrt {1-k^{2}}})}{K(k)}}{\biggr ]}}$

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

${\displaystyle K(k)=\int _{0}^{\pi /2}{\frac {1}{\sqrt {1-k^{2}\sin(\varphi )^{2}}}}\partial \varphi }$
${\displaystyle K(k)={\frac {\pi }{2}}\sum _{a=0}^{\infty }{\frac {[(2a)!]^{2}}{16^{a}(a!)^{4}}}k^{2a}}$

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

${\displaystyle {\frac {1-{\sqrt[{4}]{1-k^{2}}}}{1+{\sqrt[{4}]{1-k^{2}}}}}={\frac {\vartheta _{00}[q(k)]-\vartheta _{01}[q(k)]}{\vartheta _{00}[q(k)]+\vartheta _{01}[q(k)]}}={\biggl [}\sum _{n=1}^{\infty }2\,q(k)^{(2n-1)^{2}}{\biggr ]}{\biggl [}1+\sum _{n=1}^{\infty }2\,q(k)^{4n^{2}}{\biggr ]}^{-1}}$

Since we may reduce to the case where the imaginary part of ${\displaystyle \tau }$ is greater than or equal to ${\displaystyle {\sqrt {3}}/2}$ (see Modular group), we can assume the absolute value of ${\displaystyle q}$ is less than or equal to ${\displaystyle \exp(-\pi {\sqrt {3}}/2)\approx 0.0658}$; for values this small the above series converges very rapidly and easily allows us to find the appropriate value for ${\displaystyle q}$. By solving this function after q we get this[12][13][14] result:

${\displaystyle q(k)=\sum _{n=1}^{\infty }{\frac {{\text{Sc}}(n)}{2^{4n-3}}}{\biggl (}{\frac {1-{\sqrt[{4}]{1-k^{2}}}}{1+{\sqrt[{4}]{1-k^{2}}}}}{\biggr )}^{4n-3}=k^{2}{\biggl \{}{\frac {1}{2}}+{\biggl [}\sum _{n=1}^{\infty }{\frac {{\text{Sc}}(n+1)}{2^{4n+1}}}k^{2n}{\biggr ]}{\biggr \}}^{4}}$

This table shows numbers of the Schwarz integer sequence A002103 accurately:

 Sc(1) Sc(2) Sc(3) Sc(4) Sc(5) Sc(6) Sc(7) Sc(8) 1 2 15 150 1707 20910 268616 3567400

Kneser integer sequence

The German mathematician Adolf Kneser researched on the integer sequence of the elliptic period ratio in his essay Neue Untersuchung einer Reihe aus der Theorie der elliptischen Funktionen and showed that the generating function of this sequence is an elliptic function. Also a further mathematician with the name Robert Fricke analyzed this integer sequence in his essay Die elliptischen Funktionen und ihre Anwendungen and described the accurate computing methods by using this mentioned sequence. The Kneser integer sequence Kn(n) can be constructed in this way:

 ${\displaystyle {\text{Kn}}(2n)=2^{4n-3}{\binom {4n}{2n}}+\sum _{m=1}^{n}4^{2n-2m}{\binom {4n}{2n-2m}}{\text{Kn}}(m)}$ ${\displaystyle {\text{Kn}}(2n+1)=2^{4n-1}{\binom {4n+2}{2n+1}}+\sum _{m=1}^{n}4^{2n-2m+1}{\binom {4n+2}{2n-2m+1}}{\text{Kn}}(m)}$

Executed examples:

 ${\displaystyle {\text{Kn}}(2)=2\times 6+1\times {\color {cornflowerblue}1}={\color {cornflowerblue}13}}$ ${\displaystyle {\text{Kn}}(3)=8\times 20+24\times {\color {cornflowerblue}1}={\color {cornflowerblue}184}}$ ${\displaystyle {\text{Kn}}(4)=32\times 70+448\times {\color {cornflowerblue}1}+1\times {\color {cornflowerblue}13}={\color {cornflowerblue}2701}}$ ${\displaystyle {\text{Kn}}(5)=128\times 252+7680\times {\color {cornflowerblue}1}+40\times {\color {cornflowerblue}13}={\color {cornflowerblue}40456}}$ ${\displaystyle {\text{Kn}}(6)=512\times 924+126720\times {\color {cornflowerblue}1}+1056\times {\color {cornflowerblue}13}+1\times {\color {cornflowerblue}184}={\color {cornflowerblue}613720}}$ ${\displaystyle {\text{Kn}}(7)=2048\times 3432+2050048\times {\color {cornflowerblue}1}+23296\times {\color {cornflowerblue}13}+56\times {\color {cornflowerblue}184}={\color {cornflowerblue}9391936}}$

The Kneser sequence appears in the Taylor series of the period ratio (half period ratio):

${\displaystyle {\frac {1}{4}}\ln {\bigl (}{\frac {16}{x^{2}}}{\bigr )}-{\frac {\pi \,K'(x)}{4\,K(x)}}=\sum _{n=1}^{\infty }{\frac {{\text{Kn}}(n)}{2^{4n-1}n}}\,x^{2n}}$

Schellbach Schwarz integer sequence

The mathematician Karl Heinrich Schellbach discovered the integer number sequence that appears in the MacLaurin series of the Elliptic Nome function. This scientist[15] constructed this sequence A002103 in his work Die Lehre von den elliptischen Integralen und den Thetafunktionen in detail. Especially on page 60 of this work a synthesis route of this sequence is written down in his work. Also the Silesian German mathematician Hermann Amandus Schwarz wrote in his work Formeln und Lehrsätze zum Gebrauche der elliptischen Funktionen in the chapter Berechnung der Grösse k on pages 54 to 56 that integer number sequence down. This Schellbach Schwarz number sequence Sc(n) (OEIS: A002103) was also analyzed by the mathematicians Karl Theodor Wilhelm Weierstrass and Louis Melville Milne-Thomson in the 20th century. The mathematician Adolf Kneser determined a synthesis method for this sequence based on the following pattern:

${\displaystyle {\text{Sc}}(n+1)={\frac {2}{n}}\sum _{m=1}^{n}{\text{Sc}}(m){\text{Kn}}(n+1-m)}$

The Schellbach Schwarz sequence Sc(n) is entered in the online encyclopedia of number sequences under the number A002103 and the Kneser sequence Kn(n) is entered under the number A227503. Adolf Kneser researched on this integer sequence in his essay Neue Untersuchung einer Reihe aus der Theorie der elliptischen Funktionen and showed that the generating function of this sequence is an elliptic function. Also Robert Fricke analyzed this integer sequence in his essay Die elliptischen Funktionen und ihre Anwendungen and described accurate computing methods by using this sequence. Following table[16][17] contains the Kneser numbers and the Schellbach Schwarz numbers:

Constructed sequences Kneser and Schellbach Schwarz
Index n Kn(n) (A227503) Sc(n) (A002103)
1 1 1
2 13 2
3 184 15
4 2701 150
5 40456 1707
6 613720 20910
7 9391936 268616
8 144644749 3567400

In the following, it will be shown as an example how the Schellbach Schwarz numbers are built up successively. For this, the examples with the numbers Sc(4) = 150, Sc(5) = 1707 and Sc(6) = 20910 are used:

${\displaystyle \mathrm {Sc} (4)={\frac {2}{3}}\sum _{m=1}^{3}\mathrm {Sc} (m)\,\mathrm {Kn} (4-m)={\frac {2}{3}}{\bigl [}{\color {navy}\mathrm {Sc} (1)}\,{\color {cornflowerblue}\mathrm {Kn} (3)}+{\color {navy}\mathrm {Sc} (2)}\,{\color {cornflowerblue}\mathrm {Kn} (2)}+{\color {navy}\mathrm {Sc} (3)}\,{\color {cornflowerblue}\mathrm {Kn} (1)}{\bigr ]}}$
${\displaystyle {\color {navy}\mathrm {Sc} (4)}={\frac {2}{3}}{\bigl (}{\color {navy}1}\times {\color {cornflowerblue}184}+{\color {navy}2}\times {\color {cornflowerblue}13}+{\color {navy}15}\times {\color {cornflowerblue}1}{\bigr )}={\color {navy}150}}$
${\displaystyle \mathrm {Sc} (5)={\frac {2}{4}}\sum _{m=1}^{4}\mathrm {Sc} (m)\,\mathrm {Kn} (5-m)={\frac {2}{4}}{\bigl [}{\color {navy}\mathrm {Sc} (1)}\,{\color {cornflowerblue}\mathrm {Kn} (4)}+{\color {navy}\mathrm {Sc} (2)}\,{\color {cornflowerblue}\mathrm {Kn} (3)}+{\color {navy}\mathrm {Sc} (3)}\,{\color {cornflowerblue}\mathrm {Kn} (2)}+{\color {navy}\mathrm {Sc} (4)}\,{\color {cornflowerblue}\mathrm {Kn} (1)}{\bigr ]}}$
${\displaystyle {\color {navy}\mathrm {Sc} (5)}={\frac {2}{4}}{\bigl (}{\color {navy}1}\times {\color {cornflowerblue}2701}+{\color {navy}2}\times {\color {cornflowerblue}184}+{\color {navy}15}\times {\color {cornflowerblue}13}+{\color {navy}150}\times {\color {cornflowerblue}1}{\bigr )}={\color {navy}1707}}$
${\displaystyle \mathrm {Sc} (6)={\frac {2}{5}}\sum _{m=1}^{5}\mathrm {Sc} (m)\,\mathrm {Kn} (6-m)={\frac {2}{5}}{\bigl [}{\color {navy}\mathrm {Sc} (1)}\,{\color {cornflowerblue}\mathrm {Kn} (5)}+{\color {navy}\mathrm {Sc} (2)}\,{\color {cornflowerblue}\mathrm {Kn} (4)}+{\color {navy}\mathrm {Sc} (3)}\,{\color {cornflowerblue}\mathrm {Kn} (3)}+{\color {navy}\mathrm {Sc} (4)}\,{\color {cornflowerblue}\mathrm {Kn} (2)}+{\color {navy}\mathrm {Sc} (5)}\,{\color {cornflowerblue}\mathrm {Kn} (1)}{\bigr ]}}$
${\displaystyle {\color {navy}\mathrm {Sc} (6)}={\frac {2}{5}}{\bigl (}{\color {navy}1}\times {\color {cornflowerblue}40456}+{\color {navy}2}\times {\color {cornflowerblue}2701}+{\color {navy}15}\times {\color {cornflowerblue}184}+{\color {navy}150}\times {\color {cornflowerblue}13}+{\color {navy}1707}\times {\color {cornflowerblue}1}{\bigr )}={\color {navy}20910}}$

And this sequence creates the MacLaurin series of the elliptic nome in exactly this way mentioned above:

${\displaystyle q(x)=\sum _{n=1}^{\infty }{\frac {{\text{Sc}}(n)}{2^{4n-3}}}{\biggl (}{\frac {1-{\sqrt[{4}]{1-x^{2}}}}{1+{\sqrt[{4}]{1-x^{2}}}}}{\biggr )}^{4n-3}=x^{2}{\biggl \{}{\frac {1}{2}}+{\biggl [}\sum _{n=1}^{\infty }{\frac {{\text{Sc}}(n+1)}{2^{4n+1}}}x^{2n}{\biggr ]}{\biggr \}}^{4}}$

Definition in terms of Neville theta functions

The Jacobi elliptic functions can be defined very simply using the Neville theta functions:[18]

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

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

Jacobi transformations

The Jacobi imaginary transformations

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:[19]: 506

${\displaystyle \operatorname {cn} (u,m)=\operatorname {nc} (i\,u,1\!-\!m)}$
${\displaystyle \operatorname {sn} (u,m)=-i\operatorname {sc} (i\,u,1\!-\!m)}$
${\displaystyle \operatorname {dn} (u,m)=\operatorname {dc} (i\,u,1\!-\!m)}$

Using the multiplication rule, all other functions may be expressed in terms of the above three. The transformations may be generally written as ${\displaystyle \operatorname {pq} (u,m)=\gamma _{\operatorname {pq} }\operatorname {pq} '(i\,u,1\!-\!m)}$. The following table gives the ${\displaystyle \gamma _{\operatorname {pq} }\operatorname {pq} '(i\,u,1\!-\!m)}$ for the specified pq(u,m).[18] (The arguments ${\displaystyle (i\,u,1\!-\!m)}$ are suppressed)

Jacobi Imaginary transformations ${\displaystyle \gamma _{\operatorname {pq} }\operatorname {pq} '(i\,u,1\!-\!m)}$
q
c s n d
p
c 1 i ns nc nd
s i sn 1 i sc i sd
n cn i cs 1 cd
d dn i ds dc 1

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.[5]: 249  In the figure, the Jacobi curve has degenerated to two vertical lines at x = 1 and x = −1.

The Jacobi real transformations

The Jacobi real transformations[5]: 308  yield expressions for the elliptic functions in terms with alternate values of m. The transformations may be generally written as ${\displaystyle \operatorname {pq} (u,m)=\gamma _{\operatorname {pq} }\operatorname {pq} '(k\,u,1/m)}$. The following table gives the ${\displaystyle \gamma _{\operatorname {pq} }\operatorname {pq} '(k\,u,1/m)}$ for the specified pq(u,m).[18] (The arguments ${\displaystyle (k\,u,1/m)}$ are suppressed)

Jacobi real transformations ${\displaystyle \gamma _{\operatorname {pq} }\operatorname {pq} '(k\,u,1/m)}$
q
c s n d
p
c ${\displaystyle 1}$ ${\displaystyle k\operatorname {ds} }$ ${\displaystyle \operatorname {dn} }$ ${\displaystyle \operatorname {dc} }$
s ${\displaystyle {\frac {1}{k}}\operatorname {sd} }$ ${\displaystyle 1}$ ${\displaystyle {\frac {1}{k}}\operatorname {sn} }$ ${\displaystyle {\frac {1}{k}}\operatorname {sc} }$
n ${\displaystyle \operatorname {nd} }$ ${\displaystyle k\operatorname {ns} }$ ${\displaystyle 1}$ ${\displaystyle \operatorname {nc} }$
d ${\displaystyle \operatorname {cd} }$ ${\displaystyle k\operatorname {cs} }$ ${\displaystyle \operatorname {cn} }$ ${\displaystyle 1}$

Other Jacobi transformations

Jacobi's real and imaginary transformations can be combined in various ways to yield three more simple transformations .[5]: 214  The real and imaginary transformations are two transformations in a group (D3 or anharmonic group) of six transformations. If

${\displaystyle \mu _{R}(m)=1/m}$

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

${\displaystyle \mu _{I}(m)=1-m=m'}$

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:

{\displaystyle {\begin{aligned}\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{aligned}}}

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

${\displaystyle \operatorname {cs} (u,m)=\gamma _{i}\operatorname {cs'} (\gamma _{i}u,\mu _{i}(m))}$
${\displaystyle \operatorname {ns} (u,m)=\gamma _{i}\operatorname {ns'} (\gamma _{i}u,\mu _{i}(m))}$
${\displaystyle \operatorname {ds} (u,m)=\gamma _{i}\operatorname {ds'} (\gamma _{i}u,\mu _{i}(m))}$

where i = U, I, IR, R, RI, or RIR, identifying the transformation, γi 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.[5]: 214  (As usual, k2 = m, 1 − k2 = k12 = m′ and the arguments (${\displaystyle \gamma _{i}u,\mu _{i}(m)}$) are suppressed)

Parameters for the six transformations
Transformation i ${\displaystyle \gamma _{i}}$ ${\displaystyle \mu _{i}(m)}$ cs' ns' ds'
U 1 m cs ns ds
I i m ns cs ds
IR i k m′/m ds cs ns
R k 1/m ds ns cs
RI i k1 1/m ns ds cs
RIR k1 m/m cs ds ns

Thus, for example, we may build the following table for the RIR transformation.[18] The transformation is generally written ${\displaystyle \operatorname {pq} (u,m)=\gamma _{\operatorname {pq} }\,\operatorname {pq'} (k'\,u,-m/m')}$ (The arguments ${\displaystyle (k'\,u,-m/m')}$ are suppressed)

The RIR transformation ${\displaystyle \gamma _{\operatorname {pq} }\,\operatorname {pq'} (k'\,u,-m/m')}$
q
c s n d
p
c 1 k' cs cd cn
s ${\displaystyle {\frac {1}{k'}}}$ sc 1 ${\displaystyle {\frac {1}{k'}}}$ sd ${\displaystyle {\frac {1}{k'}}}$ sn
n dc ${\displaystyle k'}$ ds 1 dn
d nc ${\displaystyle k'}$ ns nd 1

The value of the Jacobi transformations is that any set of Jacobi elliptic functions with any complex-valued parameter m can be converted into another set for which 0 ≤ m ≤ 1 and, for real values of u, the function values will be real.[5]: p. 215

The Jacobi hyperbola

Introducing complex numbers, our ellipse has an associated hyperbola:

${\displaystyle x^{2}-{\frac {y^{2}}{b^{2}}}=1}$

from applying Jacobi's imaginary transformation[18] to the elliptic functions in the above equation for x and y.

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

It follows that we can put ${\displaystyle x=\operatorname {dn} (u,1-m),y=\operatorname {sn} (u,1-m)}$. So our ellipse has a dual ellipse with m replaced by 1-m. This leads to the complex torus mentioned in the Introduction.[20] 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 0<m<1, the curve is an ellipse with major axis in the y direction. At m = 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.

Minor functions

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

${\displaystyle \operatorname {ns} (u)={\frac {1}{\operatorname {sn} (u)}},\qquad \operatorname {nc} (u)={\frac {1}{\operatorname {cn} (u)}},\qquad \operatorname {nd} (u)={\frac {1}{\operatorname {dn} (u)}}.}$

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

{\displaystyle {\begin{aligned}\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{aligned}}}

More compactly, we have

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

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

Periodicity, poles, and residues

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.

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

${\displaystyle \operatorname {am} (u+2K,m)=\operatorname {am} (u,m)+\pi ,}$
${\displaystyle {\mathcal {E}}(u+2K,m)={\mathcal {E}}(u,m)+2E}$

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

Also

${\displaystyle \operatorname {zn} (u+2iK',m)=\operatorname {zn} (u,m)-{\frac {\pi i}{K}}}$.

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

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

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:

${\displaystyle \gamma }$
q
c s n d
p
c 0 β α + β α
s β 0 α α + β
n α + β α 0 β
d α α + β β 0

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:

Residues of Jacobi elliptic functions
q
c s n d
p
c 1 ${\displaystyle -{\frac {i}{k}}}$ ${\displaystyle -{\frac {1}{k}}}$
s ${\displaystyle -{\frac {1}{k'}}}$ ${\displaystyle {\frac {1}{k}}}$ ${\displaystyle -{\frac {i}{k\,k'}}}$
n ${\displaystyle -{\frac {1}{k'}}}$ 1 ${\displaystyle -{\frac {i}{k'}}}$
d -1 1 ${\displaystyle -i}$

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.

Special values

Lemniscatic values

The Values along with the modulus ${\displaystyle k={\tfrac {1}{2}}{\sqrt {2}}=\lambda ^{*}(1)}$ are called lemniscatic values:

Values for the thirds of the integral K:

${\displaystyle \operatorname {sn} {\bigl [}{\tfrac {1}{3}}K({\tfrac {1}{2}}{\sqrt {2}}\,);{\tfrac {1}{2}}{\sqrt {2}}\,{\bigr ]}={\tfrac {1}{2}}({\sqrt[{4}]{12}}-{\sqrt {3}}+1)}$
${\displaystyle \operatorname {cn} {\bigl [}{\tfrac {2}{3}}K({\tfrac {1}{2}}{\sqrt {2}}\,);{\tfrac {1}{2}}{\sqrt {2}}\,{\bigr ]}={\tfrac {1}{2}}({\sqrt {3}}-{\sqrt[{4}]{12}}+1)}$
${\displaystyle \operatorname {cn} {\bigl [}{\tfrac {1}{3}}K({\tfrac {1}{2}}{\sqrt {2}}\,);{\tfrac {1}{2}}{\sqrt {2}}\,{\bigr ]}={\sqrt[{4}]{2{\sqrt {3}}-3}}}$
${\displaystyle \operatorname {sn} {\bigl [}{\tfrac {2}{3}}K({\tfrac {1}{2}}{\sqrt {2}}\,);{\tfrac {1}{2}}{\sqrt {2}}\,{\bigr ]}={\tfrac {1}{2}}({\sqrt[{4}]{27}}+{\sqrt[{4}]{3}}-{\sqrt {2}}){\sqrt[{4}]{2{\sqrt {3}}-3}}}$

Values for the fifths of the integral K:

${\displaystyle \operatorname {sn} {\bigl [}{\tfrac {1}{5}}K({\tfrac {1}{2}}{\sqrt {2}}\,);{\tfrac {1}{2}}{\sqrt {2}}\,{\bigr ]}={\tfrac {1}{2}}{\bigl [}\tan({\tfrac {1}{5}}\pi )+1-{\sqrt {2\tan({\tfrac {3}{20}}\pi )}}{\bigr ]}}$
${\displaystyle \operatorname {sn} {\bigl [}{\tfrac {3}{5}}K({\tfrac {1}{2}}{\sqrt {2}}\,);{\tfrac {1}{2}}{\sqrt {2}}\,{\bigr ]}={\tfrac {1}{2}}{\bigl [}\tan({\tfrac {1}{5}}\pi )-1+{\sqrt {2\cot({\tfrac {3}{20}}\pi )}}{\bigr ]}}$
${\displaystyle \operatorname {cn} {\bigl [}{\tfrac {2}{5}}K({\tfrac {1}{2}}{\sqrt {2}}\,);{\tfrac {1}{2}}{\sqrt {2}}\,{\bigr ]}={\tfrac {1}{2}}{\bigl [}\cot({\tfrac {1}{10}}\pi )-1-{\sqrt {2\tan({\tfrac {1}{20}}\pi )}}{\bigr ]}}$
${\displaystyle \operatorname {cn} {\bigl [}{\tfrac {4}{5}}K({\tfrac {1}{2}}{\sqrt {2}}\,);{\tfrac {1}{2}}{\sqrt {2}}\,{\bigr ]}={\tfrac {1}{2}}{\bigl [}\cot({\tfrac {1}{10}}\pi )+1-{\sqrt {2\cot({\tfrac {1}{20}}\pi )}}{\bigr ]}}$

For the corresponding Pythagorean opposites these formulas are valid:

${\displaystyle \operatorname {cn} {\bigl [}{\tfrac {1}{5}}K({\tfrac {1}{2}}{\sqrt {2}}\,);{\tfrac {1}{2}}{\sqrt {2}}\,{\bigr ]}=2{\sqrt[{4}]{{\sqrt {5}}-2}}\,{\sqrt {\sin({\tfrac {3}{20}}\pi )\cos({\tfrac {1}{20}}\pi )}}}$
${\displaystyle \operatorname {cn} {\bigl [}{\tfrac {3}{5}}K({\tfrac {1}{2}}{\sqrt {2}}\,);{\tfrac {1}{2}}{\sqrt {2}}\,{\bigr ]}=2{\sqrt[{4}]{{\sqrt {5}}-2}}\,{\sqrt {\sin({\tfrac {1}{20}}\pi )\cos({\tfrac {3}{20}}\pi )}}}$
${\displaystyle \operatorname {sn} {\bigl [}{\tfrac {2}{5}}K({\tfrac {1}{2}}{\sqrt {2}}\,);{\tfrac {1}{2}}{\sqrt {2}}\,{\bigr ]}={\tfrac {1}{2}}{\sqrt[{4}]{2\tan({\tfrac {1}{20}}\pi )}}\,({\sqrt {\cot({\tfrac {1}{10}}\pi )+1}}-{\sqrt {\cot({\tfrac {1}{10}}\pi )-3}})}$
${\displaystyle \operatorname {sn} {\bigl [}{\tfrac {4}{5}}K({\tfrac {1}{2}}{\sqrt {2}}\,);{\tfrac {1}{2}}{\sqrt {2}}\,{\bigr ]}={\tfrac {1}{2}}{\sqrt[{4}]{2\cot({\tfrac {1}{20}}\pi )}}\,({\sqrt {\cot({\tfrac {1}{10}}\pi )+3}}-{\sqrt {\cot({\tfrac {1}{10}}\pi )-1}})}$

Values for the sevenths of the integral K:

${\displaystyle \operatorname {cn} [{\tfrac {2}{7}}K({\tfrac {1}{2}}{\sqrt {2}});{\tfrac {1}{2}}{\sqrt {2}}]=\tanh {\bigl \{}{\tfrac {1}{2}}\operatorname {arcoth} {\bigl [}{\tfrac {1}{2}}{\sqrt {2\sin({\tfrac {1}{7}}\pi )\cot({\tfrac {3}{28}}\pi )}}+\sin({\tfrac {1}{14}}\pi ){\bigr ]}{\bigr \}}}$
${\displaystyle \operatorname {cn} [{\tfrac {4}{7}}K({\tfrac {1}{2}}{\sqrt {2}});{\tfrac {1}{2}}{\sqrt {2}}]=\tanh {\bigl \{}{\tfrac {1}{2}}\operatorname {arcoth} {\bigl [}{\tfrac {1}{2}}{\sqrt {2\cos({\tfrac {1}{14}}\pi )\tan({\tfrac {5}{28}}\pi )}}+\sin({\tfrac {3}{14}}\pi ){\bigr ]}{\bigr \}}}$
${\displaystyle \operatorname {cn} [{\tfrac {6}{7}}K({\tfrac {1}{2}}{\sqrt {2}});{\tfrac {1}{2}}{\sqrt {2}}]=\tanh {\bigl \{}{\tfrac {1}{2}}\operatorname {arcoth} {\bigl [}{\tfrac {1}{2}}{\sqrt {2\cos({\tfrac {3}{14}}\pi )\cot({\tfrac {1}{28}}\pi )}}+\cos({\tfrac {1}{7}}\pi ){\bigr ]}{\bigr \}}}$

For the corresponding Pythagorean opposites these formulas are valid:

${\displaystyle \operatorname {sn} [{\tfrac {2}{7}}K({\tfrac {1}{2}}{\sqrt {2}});{\tfrac {1}{2}}{\sqrt {2}}]=\operatorname {sech} {\bigl \{}{\tfrac {1}{2}}\operatorname {arcoth} {\bigl [}{\tfrac {1}{2}}{\sqrt {2\sin({\tfrac {1}{7}}\pi )\cot({\tfrac {3}{28}}\pi )}}+\sin({\tfrac {1}{14}}\pi ){\bigr ]}{\bigr \}}}$
${\displaystyle \operatorname {sn} [{\tfrac {4}{7}}K({\tfrac {1}{2}}{\sqrt {2}});{\tfrac {1}{2}}{\sqrt {2}}]=\operatorname {sech} {\bigl \{}{\tfrac {1}{2}}\operatorname {arcoth} {\bigl [}{\tfrac {1}{2}}{\sqrt {2\cos({\tfrac {1}{14}}\pi )\tan({\tfrac {5}{28}}\pi )}}+\sin({\tfrac {3}{14}}\pi ){\bigr ]}{\bigr \}}}$
${\displaystyle \operatorname {sn} [{\tfrac {6}{7}}K({\tfrac {1}{2}}{\sqrt {2}});{\tfrac {1}{2}}{\sqrt {2}}]=\operatorname {sech} {\bigl \{}{\tfrac {1}{2}}\operatorname {arcoth} {\bigl [}{\tfrac {1}{2}}{\sqrt {2\cos({\tfrac {3}{14}}\pi )\cot({\tfrac {1}{28}}\pi )}}+\cos({\tfrac {1}{7}}\pi ){\bigr ]}{\bigr \}}}$

Important identities:

Setting ${\displaystyle k={\tfrac {1}{2}}{\sqrt {2}}}$ gives the lemniscate elliptic functions ${\displaystyle \operatorname {sl} }$ and ${\displaystyle \operatorname {cl} }$ in this way:

${\displaystyle \operatorname {sl} (u)={\tfrac {1}{2}}{\sqrt {2}}\,\operatorname {sd} ({\sqrt {2}}\,u;{\tfrac {1}{2}}{\sqrt {2}}\,)={\tfrac {1}{2}}{\sqrt {2}}\,{\frac {\operatorname {sn} ({\sqrt {2}}\,u;{\tfrac {1}{2}}{\sqrt {2}}\,)}{\operatorname {dn} ({\sqrt {2}}\,u;{\tfrac {1}{2}}{\sqrt {2}}\,)}},\quad \operatorname {cl} (u)=\operatorname {cn} ({\sqrt {2}}\,u;{\tfrac {1}{2}}{\sqrt {2}}\,).}$

Setting ${\displaystyle k=i}$ gives the lemniscate elliptic functions ${\displaystyle \operatorname {sl} }$ and ${\displaystyle \operatorname {cl} }$ in the following way:

${\displaystyle \operatorname {sl} (u)=\operatorname {sn} (u;i),\quad \operatorname {cl} (u)=\operatorname {cd} (u;i)={\frac {\operatorname {cn} (u;i)}{\operatorname {dn} (u;i)}}.}$

Equianharmonic values

The Values along with the modulus ${\displaystyle k=\sin({\tfrac {1}{12}}\pi )=\lambda ^{*}(3)}$ and ${\displaystyle k=\cos({\tfrac {1}{12}}\pi )=\lambda ^{*}({\tfrac {1}{3}})}$ are called equianharmonic values or also values of the Equianharmonic Case:

${\displaystyle \operatorname {cn} \{{\tfrac {2}{5}}K{\bigl [}\sin({\tfrac {1}{12}}\pi ){\bigr ]};\sin({\tfrac {1}{12}}\pi )\}=\tan {\bigl \{}\arctan {\bigl [}{\sqrt[{3}]{10}}\tan({\tfrac {1}{10}}\pi )+{\tfrac {1}{3}}{\sqrt {3}}({\sqrt[{3}]{10}}-1){\bigr ]}-{\tfrac {1}{12}}\pi {\bigr \}}}$
${\displaystyle \operatorname {cn} \{{\tfrac {4}{5}}K{\bigl [}\sin({\tfrac {1}{12}}\pi ){\bigr ]};\sin({\tfrac {1}{12}}\pi )\}=\tan {\bigl \{}\arctan {\bigl [}{\sqrt[{3}]{10}}\tan({\tfrac {1}{10}}\pi )-{\tfrac {1}{3}}{\sqrt {3}}({\sqrt[{3}]{10}}-1){\bigr ]}+{\tfrac {1}{12}}\pi {\bigr \}}}$
${\displaystyle \operatorname {cn} \{{\tfrac {2}{5}}K{\bigl [}\cos({\tfrac {1}{12}}\pi ){\bigr ]};\cos({\tfrac {1}{12}}\pi )\}=\cot {\bigl \{}\arctan {\bigl [}{\sqrt[{3}]{10}}\cot({\tfrac {1}{5}}\pi )+{\tfrac {1}{3}}{\sqrt {3}}({\sqrt[{3}]{10}}-1){\bigr ]}-{\tfrac {1}{12}}\pi {\bigr \}}}$
${\displaystyle \operatorname {cn} \{{\tfrac {4}{5}}K{\bigl [}\cos({\tfrac {1}{12}}\pi ){\bigr ]};\cos({\tfrac {1}{12}}\pi )\}=\cot {\bigl \{}\arctan {\bigl [}{\sqrt[{3}]{10}}\cot({\tfrac {1}{5}}\pi )-{\tfrac {1}{3}}{\sqrt {3}}({\sqrt[{3}]{10}}-1){\bigr ]}+{\tfrac {1}{12}}\pi {\bigr \}}}$

The flower marks the Elliptic lambda star function!

Elementary functions

When ${\displaystyle k=0}$ or ${\displaystyle k=1}$, the Jacobi elliptic functions are reduced to non-elliptic functions:

Function k = 0 k = 1
${\displaystyle \operatorname {sn} (u;k)}$ ${\displaystyle \sin u}$ ${\displaystyle \tanh u}$
${\displaystyle \operatorname {cn} (u;k)}$ ${\displaystyle \cos u}$ ${\displaystyle \operatorname {sech} u}$
${\displaystyle \operatorname {dn} (u;k)}$ ${\displaystyle 1}$ ${\displaystyle \operatorname {sech} u}$
${\displaystyle \operatorname {ns} (u;k)}$ ${\displaystyle \csc u}$ ${\displaystyle \coth u}$
${\displaystyle \operatorname {nc} (u;k)}$ ${\displaystyle \sec u}$ ${\displaystyle \cosh u}$
${\displaystyle \operatorname {nd} (u;k)}$ ${\displaystyle 1}$ ${\displaystyle \cosh u}$
${\displaystyle \operatorname {sd} (u;k)}$ ${\displaystyle \sin u}$ ${\displaystyle \sinh u}$
${\displaystyle \operatorname {cd} (u;k)}$ ${\displaystyle \cos u}$ ${\displaystyle 1}$
${\displaystyle \operatorname {cs} (u;k)}$ ${\displaystyle \cot u}$ ${\displaystyle \operatorname {csch} u}$
${\displaystyle \operatorname {ds} (u;k)}$ ${\displaystyle \csc u}$ ${\displaystyle \operatorname {csch} u}$
${\displaystyle \operatorname {dc} (u;k)}$ ${\displaystyle \sec u}$ ${\displaystyle 1}$
${\displaystyle \operatorname {sc} (u;k)}$ ${\displaystyle \tan u}$ ${\displaystyle \sinh u}$

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

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

Identities

The functions satisfy the two square relations in dependency of the modulus ${\displaystyle k}$ in Legendre form:

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

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

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[3][21]

{\displaystyle {\begin{aligned}\operatorname {sn} (x+y;k)&={\operatorname {sn} (x;k)\operatorname {cn} (y;k)\operatorname {dn} (y;k)+\operatorname {sn} (y;k)\operatorname {cn} (x;k)\operatorname {dn} (x;k) \over {1-k^{2}\operatorname {sn} ^{2}(x;k)\operatorname {sn} ^{2}(y;k)}},\\[8pt]\operatorname {cn} (x+y;k)&={\operatorname {cn} (x;k)\operatorname {cn} (y;k)-\operatorname {sn} (x;k)\operatorname {sn} (y;k)\operatorname {dn} (x;k)\operatorname {dn} (y;k) \over {1-k^{2}\operatorname {sn} ^{2}(x;k)\operatorname {sn} ^{2}(y;k)}},\\[8pt]\operatorname {dn} (x+y;k)&={\operatorname {dn} (x;k)\operatorname {dn} (y;k)-k^{2}\operatorname {sn} (x;k)\operatorname {sn} (y;k)\operatorname {cn} (x;k)\operatorname {cn} (y;k) \over {1-k^{2}\operatorname {sn} ^{2}(x;k)\operatorname {sn} ^{2}(y;k)}}.\end{aligned}}}

By only treating the functions sn and cd, the following pair of addition theorems can be established in which the two theorems are antisymmetric to each other:

 ${\displaystyle {\color {blueviolet}\operatorname {sn} }(x+y;k)={\frac {{\color {blueviolet}\operatorname {sn} }(x;k)\,{\color {blue}\operatorname {cd} }(y;k)+{\color {blue}\operatorname {cd} }(x;k)\,{\color {blueviolet}\operatorname {sn} }(y;k)}{1+k^{2}\,{\color {blueviolet}\operatorname {sn} }(x;k)\,{\color {blue}\operatorname {cd} }(x;k)\,{\color {blueviolet}\operatorname {sn} }(y;k)\,{\color {blue}\operatorname {cd} }(y;k)}}}$ ${\displaystyle {\color {blue}\operatorname {cd} }(x+y;k)={\frac {{\color {blue}\operatorname {cd} }(x;k)\,{\color {blue}\operatorname {cd} }(y;k)-{\color {blueviolet}\operatorname {sn} }(x;k)\,{\color {blueviolet}\operatorname {sn} }(y;k)}{1-k^{2}\,{\color {blueviolet}\operatorname {sn} }(x;k)\,{\color {blue}\operatorname {cd} }(x;k)\,{\color {blueviolet}\operatorname {sn} }(y;k)\,{\color {blue}\operatorname {cd} }(y;k)}}}$

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

{\displaystyle {\begin{aligned}{\mathcal {E}}(x+y;k)&={\mathcal {E}}(x;k)+{\mathcal {E}}(y;k)-k^{2}\operatorname {sn} (x;k)\operatorname {sn} (y;k)\operatorname {sn} (x+y;k),\\\operatorname {zn} (x+y;k)&=\operatorname {zn} (x;k)+\operatorname {zn} (y;k)-k^{2}\operatorname {sn} (x;k)\operatorname {sn} (y;k)\operatorname {sn} (x+y;k).\end{aligned}}}

Double angle formulae can be easily derived from the above equations by setting x = y.[3] Half angle formulae[18][3] are all of the form:

${\displaystyle \operatorname {pq} ({\tfrac {1}{2}}u;k)^{2}=f_{\mathrm {p} }/f_{\mathrm {q} }}$

where:

${\displaystyle f_{\mathrm {c} }=\operatorname {cn} (u;k)+\operatorname {dn} (u;k)}$
${\displaystyle f_{\mathrm {s} }=1-\operatorname {cn} (u;k)}$
${\displaystyle f_{\mathrm {n} }=1+\operatorname {dn} (u;k)}$
${\displaystyle f_{\mathrm {d} }=(1+\operatorname {dn} (u;k))-m(1-\operatorname {cn} (u;k))}$

Half Angle formula

These three formulas describe the halving theorem:

${\displaystyle \operatorname {sn} \left({\frac {u}{2}};k\right)^{2}={\frac {1-\operatorname {cn} (u;k)}{1+\operatorname {dn} (u;k)}}}$
${\displaystyle \operatorname {cn} \left({\frac {u}{2}};k\right)^{2}={\frac {\operatorname {cn} (u;k)+\operatorname {dn} (u;k)}{1+\operatorname {dn} (u;k)}}}$
${\displaystyle \operatorname {dn} \left({\frac {u}{2}};k\right)^{2}={\frac {1-k^{2}+\operatorname {dn} (u;k)+k^{2}\operatorname {cn} (u;k)}{1+\operatorname {dn} (u;k)}}}$

And the theorem of the arithmetic mean is described by this formula:

${\displaystyle \operatorname {sn} \left({\frac {1}{2}}\,x+{\frac {1}{2}}\,y;\,k\right)^{2}={\frac {1+\operatorname {sn} (x;k)\operatorname {sn} (y;k)-\operatorname {cn} (x;k)\operatorname {cn} (y;k)}{1+k^{2}\operatorname {sn} (x;k)\operatorname {sn} (y;k)+\operatorname {dn} (x;k)\operatorname {dn} (y;k)}}}$

K formulas

Half K formula

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

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

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

Third K formula

Also this equation[21] leads to the sn-value of the third of K:

${\displaystyle k^{2}s^{4}-2k^{2}s^{3}+2s-1=0}$
${\displaystyle s=\operatorname {sn} \left[{\tfrac {1}{3}}K(k);k\right]}$

These equations lead to the other values of the Jacobi functions:

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

On the basis of that now mentioned quartic equation we can a simplified formula by using the solution algorithm for the General case of the quartic equation. This parametrized formula can be generated in that way:

${\displaystyle \operatorname {sn} {\biggl \langle }{\frac {1}{3}}K{\bigl \{}\tan {\bigl [}{\frac {1}{2}}\arctan {\bigl (}x^{3}{\bigr )}{\bigr ]}{\bigr \}};\tan {\bigl [}{\frac {1}{2}}\arctan {\bigl (}x^{3}{\bigr )}{\bigr ]}{\biggr \rangle }=\tanh {\bigl [}{\frac {1}{2}}\ln {\bigl (}{\sqrt {2{\sqrt {x^{4}-x^{2}+1}}-x^{2}+2}}+{\sqrt {x^{2}+1}}{\bigr )}{\bigr ]}}$

The Pythagorean opposite gives that formula successively:

${\displaystyle \operatorname {cn} {\biggl \langle }{\frac {1}{3}}K{\bigl \{}\tan {\bigl [}{\frac {1}{2}}\arctan {\bigl (}x^{3}{\bigr )}{\bigr ]}{\bigr \}};\tan {\bigl [}{\frac {1}{2}}\arctan {\bigl (}x^{3}{\bigr )}{\bigr ]}{\biggr \rangle }=\operatorname {sech} {\bigl [}{\frac {1}{2}}\ln {\bigl (}{\sqrt {2{\sqrt {x^{4}-x^{2}+1}}-x^{2}+2}}+{\sqrt {x^{2}+1}}{\bigr )}{\bigr ]}}$

These two now mentioned formulas are valid for all real values of x, the criterion ${\displaystyle x\in \mathbb {R} }$ is fulfilled in both equations.

To get x, we take the tangent of twice the arctangent of the modulus and then we take the cube root so that x appears.

And then we insert the generated value x into the right side of the balance of the shown parametrized equations.

In relation to this calculation algorithm, three examples are given below:

 First example: ${\displaystyle {\frac {1}{2}}{\sqrt {2}}=\tan {\bigl [}{\frac {1}{2}}\arctan {\bigl (}x^{3}{\bigr )}{\bigr ]}(x={\sqrt {2}}\,)}$ If we insert the value ${\displaystyle x={\sqrt {2}}\,}$ into the fraction showed above, we get this result: ${\displaystyle \operatorname {sn} {\bigl [}{\frac {1}{3}}K{\bigl (}{\frac {1}{2}}{\sqrt {2}}\,{\bigr )};{\frac {1}{2}}{\sqrt {2}}\,{\bigr ]}=\tanh {\bigl [}{\frac {1}{2}}\ln {\bigl (}{\sqrt[{4}]{12}}+{\sqrt {3}}{\bigr )}{\bigr ]}={\frac {1}{2}}({\sqrt[{4}]{12}}-{\sqrt {3}}+1)}$ ${\displaystyle \operatorname {cn} {\bigl [}{\frac {1}{3}}K{\bigl (}{\frac {1}{2}}{\sqrt {2}}\,{\bigr )};{\frac {1}{2}}{\sqrt {2}}\,{\bigr ]}=\operatorname {sech} {\bigl [}{\frac {1}{2}}\ln {\bigl (}{\sqrt[{4}]{12}}+{\sqrt {3}}{\bigr )}{\bigr ]}={\sqrt[{4}]{2{\sqrt {3}}-3}}}$
 Second example: ${\displaystyle {\frac {1}{2}}=\tan {\bigl [}{\frac {1}{2}}\arctan {\bigl (}x^{3}{\bigr )}{\bigr ]}(x={\sqrt[{3}]{4/3}}\,)}$ If we insert the value ${\displaystyle x={\sqrt[{3}]{4/3}}\,}$ into the fraction showed above, we get this result: ${\displaystyle \operatorname {sn} {\bigl [}{\frac {1}{3}}K{\bigl (}{\frac {1}{2}}{\bigr )};{\frac {1}{2}}{\bigr ]}=\tanh {\bigl [}{\frac {1}{2}}\ln {\bigl (}{\sqrt {2{\sqrt {({\tfrac {4}{3}})^{4/3}-({\tfrac {4}{3}})^{2/3}+1}}-({\tfrac {4}{3}})^{2/3}+2}}+{\sqrt {({\tfrac {4}{3}})^{2/3}+1}}{\bigr )}{\bigr ]}}$ ${\displaystyle \operatorname {cn} {\bigl [}{\frac {1}{3}}K{\bigl (}{\frac {1}{2}}{\bigr )};{\frac {1}{2}}{\bigr ]}=\operatorname {sech} {\bigl [}{\frac {1}{2}}\ln {\bigl (}{\sqrt {2{\sqrt {({\tfrac {4}{3}})^{4/3}-({\tfrac {4}{3}})^{2/3}+1}}-({\tfrac {4}{3}})^{2/3}+2}}+{\sqrt {({\tfrac {4}{3}})^{2/3}+1}}{\bigr )}{\bigr ]}}$
 Third example: ${\displaystyle {\frac {1}{4}}=\tan {\bigl [}{\frac {1}{2}}\arctan {\bigl (}x^{3}{\bigr )}{\bigr ]}(x={\sqrt[{3}]{8/15}}\,)}$ If we insert the value ${\displaystyle x={\sqrt[{3}]{8/15}}\,}$ into the fraction showed above, we get this result: ${\displaystyle \operatorname {sn} {\bigl [}{\frac {1}{3}}K{\bigl (}{\frac {1}{4}}{\bigr )};{\frac {1}{4}}{\bigr ]}=\tanh {\bigl [}{\frac {1}{2}}\ln {\bigl (}{\sqrt {2{\sqrt {({\tfrac {8}{15}})^{4/3}-({\tfrac {8}{15}})^{2/3}+1}}-({\tfrac {8}{15}})^{2/3}+2}}+{\sqrt {({\tfrac {8}{15}})^{2/3}+1}}{\bigr )}{\bigr ]}}$ ${\displaystyle \operatorname {cn} {\bigl [}{\frac {1}{3}}K{\bigl (}{\frac {1}{4}}{\bigr )};{\frac {1}{4}}{\bigr ]}=\operatorname {sech} {\bigl [}{\frac {1}{2}}\ln {\bigl (}{\sqrt {2{\sqrt {({\tfrac {8}{15}})^{4/3}-({\tfrac {8}{15}})^{2/3}+1}}-({\tfrac {8}{15}})^{2/3}+2}}+{\sqrt {({\tfrac {8}{15}})^{2/3}+1}}{\bigr )}{\bigr ]}}$

Fifth K formula

Following equation has following solution:

${\displaystyle 4k^{2}t^{6}+8k^{2}t^{5}+2(1-k^{2})^{2}t-(1-k^{2})^{2}=0}$
${\displaystyle t={\frac {1}{2}}-{\frac {1}{2}}k^{2}\operatorname {sn} \left[{\tfrac {2}{5}}K(k);k\right]^{2}\operatorname {sn} \left[{\tfrac {4}{5}}K(k);k\right]^{2}={\frac {\operatorname {sn} \left[{\frac {4}{5}}K(k);k\right]^{2}-\operatorname {sn} \left[{\frac {2}{5}}K(k);k\right]^{2}}{2\operatorname {sn} \left[{\frac {2}{5}}K(k);k\right]\operatorname {sn} \left[{\frac {4}{5}}K(k);k\right]}}}$

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

${\displaystyle \operatorname {sn} {\bigl [}{\tfrac {2}{5}}K(k);k{\bigr ]}=(1+k^{2})^{-1/2}{\sqrt {2(1-t-t^{2})(t^{2}+1-t{\sqrt {t^{2}+1}})}}}$
${\displaystyle \operatorname {sn} {\bigl [}{\tfrac {4}{5}}K(k);k{\bigr ]}=(1+k^{2})^{-1/2}{\sqrt {2(1-t-t^{2})(t^{2}+1+t{\sqrt {t^{2}+1}})}}}$

The t solution between zero and a half is this solution which has to be used for these expressions.

In relation to the now mentioned calculation algorithm, an important example are given below:

 This equation of sixth degree is given: ${\displaystyle 4k^{2}t^{6}+8k^{2}t^{5}+2(1-k^{2})^{2}t-(1-k^{2})^{2}=0}$ The value ${\displaystyle k={\tfrac {1}{2}}{\sqrt {2}}}$ shall be entered in this example. By entering this modulus value and multiplying the whole equation with the factor 4 we get that equation: ${\displaystyle 8t^{6}+16t^{5}+2t-1=0}$ This is the solution of that equation for t that is located between zero and a half: ${\displaystyle t={\tfrac {1}{4}}({\sqrt {5}}+1)({\sqrt[{4}]{5}}-1)}$ This solution t and the corresponding modulus k will be inserted into the formulas for the amplitude sine values: ${\displaystyle \operatorname {sn} {\bigl [}{\tfrac {2}{5}}K({\tfrac {1}{2}}{\sqrt {2}}\,);{\tfrac {1}{2}}{\sqrt {2}}\,{\bigr ]}=(1+k^{2})^{-1/2}{\sqrt {2(1-t-t^{2})(t^{2}+1-t{\sqrt {t^{2}+1}})}}\,{\bigl [}k={\tfrac {1}{2}}{\sqrt {2}}\,\cap \,t={\tfrac {1}{4}}({\sqrt {5}}+1)({\sqrt[{4}]{5}}-1){\bigr ]}}$ ${\displaystyle \operatorname {sn} {\bigl [}{\tfrac {4}{5}}K({\tfrac {1}{2}}{\sqrt {2}}\,);{\tfrac {1}{2}}{\sqrt {2}}\,{\bigr ]}=(1+k^{2})^{-1/2}{\sqrt {2(1-t-t^{2})(t^{2}+1+t{\sqrt {t^{2}+1}})}}\,{\bigl [}k={\tfrac {1}{2}}{\sqrt {2}}\,\cap \,t={\tfrac {1}{4}}({\sqrt {5}}+1)({\sqrt[{4}]{5}}-1){\bigr ]}}$ By doing this, following results appear, that are mentioned already in the section of the special values: ${\displaystyle \operatorname {sn} {\bigl [}{\tfrac {2}{5}}K({\tfrac {1}{2}}{\sqrt {2}}\,);{\tfrac {1}{2}}{\sqrt {2}}\,{\bigr ]}={\tfrac {1}{2}}{\sqrt[{4}]{2\tan({\tfrac {1}{20}}\pi )}}\,({\sqrt {\cot({\tfrac {1}{10}}\pi )+1}}-{\sqrt {\cot({\tfrac {1}{10}}\pi )-3}})}$ ${\displaystyle \operatorname {sn} {\bigl [}{\tfrac {4}{5}}K({\tfrac {1}{2}}{\sqrt {2}}\,);{\tfrac {1}{2}}{\sqrt {2}}\,{\bigr ]}={\tfrac {1}{2}}{\sqrt[{4}]{2\cot({\tfrac {1}{20}}\pi )}}\,({\sqrt {\cot({\tfrac {1}{10}}\pi )+3}}-{\sqrt {\cot({\tfrac {1}{10}}\pi )-1}})}$ By using the hyperbolic functions, these equivalent expressions can be produced: ${\displaystyle \operatorname {sn} {\bigl [}{\tfrac {2}{5}}K({\tfrac {1}{2}}{\sqrt {2}}\,);{\tfrac {1}{2}}{\sqrt {2}}\,{\bigr ]}=\operatorname {sech} {\bigl \{}{\tfrac {1}{2}}\operatorname {arcoth} {\bigl [}{\tfrac {1}{2}}\cot {\bigl (}{\tfrac {1}{10}}\pi {\bigr )}-{\tfrac {1}{2}}{\bigr ]}{\bigr \}}}$ ${\displaystyle \operatorname {sn} {\bigl [}{\tfrac {4}{5}}K({\tfrac {1}{2}}{\sqrt {2}}\,);{\tfrac {1}{2}}{\sqrt {2}}\,{\bigr ]}=\operatorname {sech} {\bigl \{}{\tfrac {1}{2}}\operatorname {arcoth} {\bigl [}{\tfrac {1}{2}}\cot {\bigl (}{\tfrac {1}{10}}\pi {\bigr )}+{\tfrac {1}{2}}{\bigr ]}{\bigr \}}}$

According to Abel–Ruffini theorem, it is completely impossible to set up elementary expressions for the amplitude sine values of the fifths of the complete first kind elliptic integral for the regular case of the elliptic modulus.

Relations between squares of the functions

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

${\displaystyle \operatorname {cn} ^{2}+\operatorname {sn} ^{2}=1}$
${\displaystyle \operatorname {cn} ^{2}+m'\operatorname {sn} ^{2}=\operatorname {dn} ^{2}}$
where m + m' = 1. Multiplying by any function of the form nq yields more general equations:

${\displaystyle \operatorname {cq} ^{2}+\operatorname {sq} ^{2}=\operatorname {nq} ^{2}}$
${\displaystyle \operatorname {cq} ^{2}{}+m'\operatorname {sq} ^{2}=\operatorname {dq} ^{2}}$

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

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

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

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

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

Representations of the function values via Theta functions

Given are the product definitions of the Jacobi Theta Nullwert Functions as the mathematicians Edmund Taylor Whittaker and George Neville Watson[22][23][24] set them up:

${\displaystyle \vartheta _{00}(w)=\prod _{n=1}^{\infty }(1-w^{2n})(1+w^{2n-1})^{2}}$
${\displaystyle \vartheta _{01}(w)=\prod _{n=1}^{\infty }(1-w^{2n})(1-w^{2n-1})^{2}}$

Identities of thirds of the integral ${\displaystyle K}$:

With the so-called theta zero-value functions of the elliptic noun of the module, many Jacobian function values can be represented:

${\displaystyle \operatorname {sc} [{\tfrac {2}{3}}K(k);k]={\frac {{\sqrt {3}}\,\vartheta _{01}[q(k)^{6}]}{{\sqrt {1-k^{2}}}\,\vartheta _{01}[q(k)^{2}]}}}$
${\displaystyle \operatorname {sn} [{\tfrac {1}{3}}K(k);k]={\frac {2\,\vartheta _{00}[q(k)]^{2}}{3\,\vartheta _{00}[q(k)^{3}]^{2}+\vartheta _{00}[q(k)]^{2}}}={\frac {3\,\vartheta _{01}[q(k)^{3}]^{2}-\vartheta _{01}[q(k)]^{2}}{3\,\vartheta _{01}[q(k)^{3}]^{2}+\vartheta _{01}[q(k)]^{2}}}}$
${\displaystyle \operatorname {cn} [{\tfrac {2}{3}}K(k);k]={\frac {3\,\vartheta _{00}[q(k)^{3}]^{2}-\vartheta _{00}[q(k)]^{2}}{3\,\vartheta _{00}[q(k)^{3}]^{2}+\vartheta _{00}[q(k)]^{2}}}={\frac {2\,\vartheta _{01}[q(k)]^{2}}{3\,\vartheta _{01}[q(k)^{3}]^{2}+\vartheta _{01}[q(k)]^{2}}}}$

Identities of the fifths of the integral ${\displaystyle K}$:

${\displaystyle \operatorname {sn} [{\tfrac {1}{5}}K(k);k]={\biggl \{}{\frac {{\sqrt {5}}\,\vartheta _{01}[q(k)^{5}]}{\vartheta _{01}[q(k)]}}-1{\biggr \}}{\biggl \{}{\frac {5\,\vartheta _{01}[q(k)^{10}]^{2}}{\vartheta _{01}[q(k)^{2}]^{2}}}-1{\biggr \}}^{-1}}$
${\displaystyle \operatorname {sn} [{\tfrac {3}{5}}K(k);k]={\biggl \{}{\frac {{\sqrt {5}}\,\vartheta _{01}[q(k)^{5}]}{\vartheta _{01}[q(k)]}}+1{\biggr \}}{\biggl \{}{\frac {5\,\vartheta _{01}[q(k)^{10}]^{2}}{\vartheta _{01}[q(k)^{2}]^{2}}}-1{\biggr \}}^{-1}}$
${\displaystyle \operatorname {cn} [{\tfrac {2}{5}}K(k);k]={\biggl \{}{\frac {{\sqrt {5}}\,\vartheta _{00}[q(k)^{5}]}{\vartheta _{00}[q(k)]}}+1{\biggr \}}{\biggl \{}{\frac {5\,\vartheta _{01}[q(k)^{10}]^{2}}{\vartheta _{01}[q(k)^{2}]^{2}}}-1{\biggr \}}^{-1}}$
${\displaystyle \operatorname {cn} [{\tfrac {4}{5}}K(k);k]={\biggl \{}{\frac {{\sqrt {5}}\,\vartheta _{00}[q(k)^{5}]}{\vartheta _{00}[q(k)]}}-1{\biggr \}}{\biggl \{}{\frac {5\,\vartheta _{01}[q(k)^{10}]^{2}}{\vartheta _{01}[q(k)^{2}]^{2}}}-1{\biggr \}}^{-1}}$

The elementary combinations of theta zero-value functions and the elliptic nome are not sufficient for the representation of the Jacobian function values of left-hand parenthesis entries beyond rationally broken K-integrals. This requires the theta non-zero functions of the pattern described above.

Jacobi elliptic functions as solutions of nonlinear ordinary differential equations

The derivatives of the three basic Jacobi elliptic functions are:

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