generalized elliptic integrals

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1: 19.2 Definitions

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§19.2(i) General Elliptic Integrals

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2: 19.35 Other Applications

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§19.35(i) Mathematical

►Generalizations of elliptic integrals appear in analysis of modular theorems of Ramanujan (Anderson et al. (2000)); analysis of Selberg integrals (Van Diejen and Spiridonov (2001)); use of Legendre’s relation (19.7.1) to compute $\pi$ to high precision (Borwein and Borwein (1987, p. 26)). …

3: 19.14 Reduction of General Elliptic Integrals

§19.14 Reduction of General Elliptic Integrals

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4: 19.29 Reduction of General Elliptic Integrals

§19.29 Reduction of General Elliptic Integrals

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§19.29(i) Reduction Theorems

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§19.29(ii) Reduction to Basic Integrals

►(19.2.3) can be written … ►

19.29.33 $${(x-y)}^{2}U={\left(\sqrt{x^4+a^4}+\sqrt{y^4+a^4}\right)}^2-{(x^2-y^2)}^2.$$

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Symbols:

$U$

Permalink:

http://dlmf.nist.gov/19.29.E33

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See also:

Annotations for §19.29(iii), §19.29 and Ch.19

5: 19.1 Special Notation

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$l,m,n$	nonnegative integers.
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6: 19.20 Special Cases

… ►The general lemniscatic case is … ►The general lemniscatic case is …

7: Bibliography

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Generalized elliptic integrals and modular equations. Pacific J. Math. 192 (1), pp. 1–37.

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External Links:

ISSN 0030-8730, FullText, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§19.35(i).

Encodings:

BibTeX

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8: Bibliography M

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An improved calculation of the general elliptic integral of the second kind in the neighbourhood of $x=0$ . Numer. Math. 25 (1), pp. 99–101.

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External Links:

ISSN 0029-599X, Document, MathReview (Roland Bulirsch), ZentralBlatt

Referenced by:

§19.36(ii).

Encodings:

BibTeX

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9: Bibliography B

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Numerical calculation of a generalized complete elliptic integral. Rev. Mod. Phys. 10, pp. 264–269.

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External Links:

Document, ZentralBlatt

Referenced by:

§19.2(iii).

Encodings:

BibTeX

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Numerical calculation of elliptic integrals and elliptic functions. III. Numer. Math. 13 (4), pp. 305–315.

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Notes:

Algol 60 programs for the incomplete elliptic integral of the third kind, and for a general complete elliptic integral, are included.

External Links:

ISSN 0029-599X, Document, MathReview, ZentralBlatt

Referenced by:

§19.2(iii), §19.36(ii), §19.36(ii), §19.39(ii), §19.39(iii), R. Bulirsch (1965a).

Encodings:

BibTeX

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On a generalization of Barton’s integral and related integrals of complete elliptic integrals. Math. Proc. Cambridge Philos. Soc. 101 (1), pp. 1–5.

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External Links:

ISSN 0305-0041, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§19.13(i).

Encodings:

BibTeX

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10: 23.6 Relations to Other Functions

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23.6.18 $$e_1=\frac{K^2}{3{\omega }_1^2}(1+k_{}^{\prime }{}_{}{}^2),$$

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Symbols:

$e_j$: Weierstrass lattice roots, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $𝕃$: lattice, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

18.9.1

Referenced by:

§23.6(ii)

Permalink:

http://dlmf.nist.gov/23.6.E18

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Annotations for §23.6(ii), §23.6 and Ch.23

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23.6.19 $$e_2=\frac{K^2}{3{\omega }_1^2}(k^2-k_{}^{\prime }{}_{}{}^2),$$

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Symbols:

$e_j$: Weierstrass lattice roots, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $𝕃$: lattice, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

18.9.2

Permalink:

http://dlmf.nist.gov/23.6.E19

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Annotations for §23.6(ii), §23.6 and Ch.23

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23.6.20 $$e_3=-\frac{K^2}{3{\omega }_1^2}(1+k^2).$$

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Symbols:

$e_j$: Weierstrass lattice roots, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $𝕃$: lattice, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and $k$: modulus

A&S Ref:

18.9.3

Referenced by:

§23.6(ii)

Permalink:

http://dlmf.nist.gov/23.6.E20

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Annotations for §23.6(ii), §23.6 and Ch.23

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23.6.21 $$\mathrm{\wp }\left(z\right)-e_1=\frac{K^2}{{\omega }_1^2}{\mathrm{cs}}^2(\frac{Kz}{{\omega }_1},k),$$

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Symbols:

$\mathrm{cs}(z,k)$: Jacobian elliptic function, $\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $𝕃$: lattice, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and $k$: modulus

Referenced by:

§23.6(ii)

Permalink:

http://dlmf.nist.gov/23.6.E21

Encodings:

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See also:

Annotations for §23.6(ii), §23.6 and Ch.23

… ►Also, ${𝕃}_1$, ${𝕃}_2$, ${𝕃}_3$ are the lattices with generators $(4K,2\mathrm{i}{K}^{\prime })$, $(2K-2\mathrm{i}{K}^{\prime },2K+2\mathrm{i}{K}^{\prime })$, $(2K,4\mathrm{i}{K}^{\prime })$, respectively. …