general elliptic integrals

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11: 2.6 Distributional Methods

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§2.6(i) Divergent Integrals

… ►Although divergent, these integrals may be interpreted in a generalized sense. … ►Corresponding results for the generalized Stieltjes transform …An application has been given by López (2000) to derive asymptotic expansions of standard symmetric elliptic integrals, complete with error bounds; see §19.27(vi). … ►However, in the theory of generalized functions (distributions), there is a method, known as “regularization”, by which these integrals can be interpreted in a meaningful manner. …

12: 22.14 Integrals

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22.14.2

\int\operatorname{cn}\left(x,k\right)\,\mathrm{d}x=k^{-1}\operatorname{Arccos}% \left(\operatorname{dn}\left(x,k\right)\right),

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Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{Arccos}\NVar{z}$ : general arccosine function, $x$ : real and $k$ : modulus
A&S Ref:: 16.24.2
Permalink:: http://dlmf.nist.gov/22.14.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §22.14(i), §22.14 and Ch.22

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22.14.3

\int\operatorname{dn}\left(x,k\right)\,\mathrm{d}x=\operatorname{Arcsin}\left(% \operatorname{sn}\left(x,k\right)\right)=\operatorname{am}\left(x,k\right).

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Symbols:: $\operatorname{am}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s amplitude function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{Arcsin}\NVar{z}$ : general arcsine function, $x$ : real and $k$ : modulus
A&S Ref:: 16.24.3
Permalink:: http://dlmf.nist.gov/22.14.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §22.14(i), §22.14 and Ch.22

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22.14.5

\int\operatorname{sd}\left(x,k\right)\,\mathrm{d}x=(kk^{\prime})^{-1}% \operatorname{Arcsin}\left(-k\operatorname{cd}\left(x,k\right)\right),

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Symbols:: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{Arcsin}\NVar{z}$ : general arcsine function, $x$ : real, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 16.24.5
Permalink:: http://dlmf.nist.gov/22.14.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §22.14(i), §22.14 and Ch.22

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22.14.6

\int\operatorname{nd}\left(x,k\right)\,\mathrm{d}x={k^{\prime}}^{-1}% \operatorname{Arccos}\left(\operatorname{cd}\left(x,k\right)\right).

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Symbols:: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{Arccos}\NVar{z}$ : general arccosine function, $x$ : real, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 16.24.6
Permalink:: http://dlmf.nist.gov/22.14.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §22.14(i), §22.14 and Ch.22

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13: 22.4 Periods, Poles, and Zeros

… ►Again, one member of each congruent set of zeros appears in the second row; all others are generated by translations of the form

2mK+2niK^{\prime}

, where

m,n\in\mathbb{Z}

. … ►The set of points

z=mK+niK^{\prime}

m,n\in\mathbb{Z}

, comprise the lattice for the 12 Jacobian functions; all other lattice unit cells are generated by translation of the fundamental unit cell by

mK+niK^{\prime}

, where again

m,n\in\mathbb{Z}

. …

14: Bibliography C

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B. C. Carlson (1999) Toward symbolic integration of elliptic integrals. J. Symbolic Comput. 28 (6), pp. 739–753.

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Notes:: Code written by J. FitzSimons to generate reduction tables for elliptic integrals symbolically in the Derive computer algebra system is available.
External Links:: ISSN 0747-7171, Document, Code, MathReview (Axel Riese), ZentralBlatt
Referenced by:: §19.29(i), §19.29(ii), §19.29(ii), §19.29(ii), §19.29(ii).
Encodings:: BibTeX

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15: 22.15 Inverse Functions

§22.15 Inverse Functions

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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. … ►For representations of the inverse functions as symmetric elliptic integrals see §19.25(v). …

16: 23.1 Special Notation

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$\mathbb{L}$	lattice in $\mathbb{C}$ .
…
$K\left(k\right)$ , $K'\left(k\right)$	complete elliptic integrals (§19.2(i)).
$2\omega_{1},2\omega_{3}$	lattice generators ( $\Im\left(\omega_{3}/\omega_{1}\right)>0$ ).
…
$\phantom{q}=e^{i\pi\tau}$	nome.
…

►The main functions treated in this chapter are the Weierstrass

\wp

-function

\wp\left(z\right)=\wp\left(z|\mathbb{L}\right)=\wp\left(z;g_{2},g_{3}\right)

; the Weierstrass zeta function

\zeta\left(z\right)=\zeta\left(z|\mathbb{L}\right)=\zeta\left(z;g_{2},g_{3}\right)

; the Weierstrass sigma function

\sigma\left(z\right)=\sigma\left(z|\mathbb{L}\right)=\sigma\left(z;g_{2},g_{3}\right)

; the elliptic modular function

\lambda\left(\tau\right)

; Klein’s complete invariant

J\left(\tau\right)

; Dedekind’s eta function

\eta\left(\tau\right)

. …

17: 36.1 Special Notation

… ►The main functions covered in this chapter are cuspoid catastrophes

\Phi_{K}\left(t;\mathbf{x}\right)

; umbilic catastrophes with codimension three

\Phi^{(\mathrm{E})}\left(s,t;\mathbf{x}\right)

\Phi^{(\mathrm{H})}\left(s,t;\mathbf{x}\right)

; canonical integrals

\Psi_{K}\left(\mathbf{x}\right)

\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)

\Psi^{(\mathrm{H})}\left(\mathbf{x}\right)

; diffraction catastrophes

\Psi_{K}(\mathbf{x};k)

\Psi^{(\mathrm{E})}(\mathbf{x};k)

\Psi^{(\mathrm{H})}(\mathbf{x};k)

generated by the catastrophes. …

18: 19.11 Addition Theorems

§19.11 Addition Theorems

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§19.11(i) General Formulas

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19.11.7

F\left(\phi,k\right)=K\left(k\right)-F\left(\theta,k\right),

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Symbols:: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\theta$ : amplitude
Permalink:: http://dlmf.nist.gov/19.11.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §19.11(ii), §19.11 and Ch.19

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19.11.8

E\left(\phi,k\right)=E\left(k\right)-E\left(\theta,k\right)+k^{2}\sin\theta% \sin\phi,

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Symbols:: $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\theta$ : amplitude
Permalink:: http://dlmf.nist.gov/19.11.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §19.11(ii), §19.11 and Ch.19

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§19.11(iii) Duplication Formulas

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19: 36.10 Differential Equations

§36.10 Differential Equations

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§36.10(i) Equations for $\Psi_{K}\left(\mathbf{x}\right)$

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K=2

, cusp: … ►

K=3

, swallowtail: … ►In terms of the normal forms (36.2.2) and (36.2.3), the

\Psi^{(\mathrm{U})}\left(\mathbf{x}\right)

satisfy the following operator equations …

20: 19.25 Relations to Other Functions

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19.25.31

u=R_{F}\left({\operatorname{ps}}^{2}\left(u,k\right),{\operatorname{qs}}^{2}% \left(u,k\right),{\operatorname{rs}}^{2}\left(u,k\right)\right);

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\operatorname{pq}\left(\NVar{z},\NVar{k}\right)$ : generic Jacobian elliptic function and $k$ : real or complex modulus
Referenced by:: §19.25(v)
Permalink:: http://dlmf.nist.gov/19.25.E31
Encodings:: pMML, png, TeX
See also:: Annotations for §19.25(v), §19.25 and Ch.19

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