Jacobian elliptic-function form

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1: 22.15 Inverse Functions

§22.15 Inverse Functions

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§22.15(i) Definitions

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§22.15(ii) Representations as Elliptic Integrals

… ►The integrals (22.15.12)–(22.15.14) can be regarded as normal forms for representing the inverse functions. … ►

2: 22.2 Definitions

§22.2 Definitions

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22.2.9 $$\mathrm{sc}(z,k)=\frac{{\theta }_3(0,q)}{{\theta }_4(0,q)}\frac{{\theta }_1(\zeta ,q)}{{\theta }_2(\zeta ,q)}=\frac{1}{\mathrm{cs}(z,k)}.$$

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Defines:

$\mathrm{cs}(z,k)$: Jacobian elliptic function and $\mathrm{sc}(z,k)$: Jacobian elliptic function

Symbols:

${\theta }_j(z,q)$: theta function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable

Referenced by:

§23.6(ii)

Permalink:

http://dlmf.nist.gov/22.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §22.2 and Ch.22

►As a function of $z$, with fixed $k$, each of the 12 Jacobian elliptic functions is doubly periodic, having two periods whose ratio is not real. … … ►The Jacobian functions are related in the following way. …

3: 1.13 Differential Equations

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§1.13(vii) Closed-Form Solutions

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§1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms

… ►This is the Sturm-Liouville form of a second order differential equation, where ^′ denotes $\frac{d}{dx}$. Assuming that $u(x)$ satisfies un-mixed boundary conditions of the form … ►

Transformation to Liouville normal Form

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4: 22.18 Mathematical Applications

§22.18 Mathematical Applications

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§22.18(i) Lengths and Parametrization of Plane Curves

… ► … ►The special case $y^2=(1-x^2)(1-k^2{x}^2)$ is in Jacobian normal form. …

5: 22.4 Periods, Poles, and Zeros

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§22.4(i) Distribution

… ► … ►For the distribution of the $k$-zeros of the Jacobian elliptic functions see Walker (2009). ►

§22.4(ii) Graphical Interpretation via Glaisher’s Notation

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6: 22.8 Addition Theorems

§22.8 Addition Theorems

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22.8.1 $$\mathrm{sn}(u+v)=\frac{\mathrm{sn}u\mathrm{cn}v\mathrm{dn}v+\mathrm{sn}v\mathrm{cn}u\mathrm{dn}u}{1-k^2{\mathrm{sn}}^{2}u{\mathrm{sn}}^{2}v},$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $u$: complex, $v$: complex and $k$: modulus

A&S Ref:

16.17.1

Referenced by:

§22.18(iv)

Permalink:

http://dlmf.nist.gov/22.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.8(i), §22.8 and Ch.22

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§22.8(ii) Alternative Forms for Sum of Two Arguments

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§22.8(iii) Special Relations Between Arguments

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7: 22.5 Special Values

§22.5 Special Values

… ►Table 22.5.1 gives the value of each of the 12 Jacobian elliptic functions, together with its $z$-derivative (or at a pole, the residue), for values of $z$ that are integer multiples of $K$, $\mathrm{i}{K}^{\prime }$. … ►

§22.5(ii) Limiting Values of $k$

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Table 22.5.3: Limiting forms of Jacobian elliptic functions as $k\to 0$.

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$\mathrm{sn}(z,k)$ $\to$	$\sin z$	$\mathrm{cd}(z,k)$ $\to$	$\cos z$	$\mathrm{dc}(z,k)$ $\to$	$\sec z$	$\mathrm{ns}(z,k)$ $\to$	$\csc z$
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8: Bille C. Carlson

… ►In Symmetry in c, d, n of Jacobian elliptic functions (2004) he found a previously hidden symmetry in relations between Jacobian elliptic functions, which can now take a form that remains valid when the letters c, d, and n are permuted. This invariance usually replaces sets of twelve equations by sets of three equations and applies also to the relation between the first symmetric elliptic integral and the Jacobian functions. …

9: 22.17 Moduli Outside the Interval [0,1]

§22.17 Moduli Outside the Interval [0,1]

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§22.17(i) Real or Purely Imaginary Moduli

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§22.17(ii) Complex Moduli

►When $z$ is fixed each of the twelve Jacobian elliptic functions is a meromorphic function of $k^2$. …For proofs of these results and further information see Walker (2003).

10: 22.9 Cyclic Identities

§22.9 Cyclic Identities

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§22.9(i) Notation

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