symmetric elliptic integrals

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31: Bibliography C

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Reduction theorems for elliptic integrands with the square root of two quadratic factors. J. Comput. Appl. Math. 118 (1-2), pp. 71–85.

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Notes:

Code written by the second author to compute symmetric elliptic integrals numerically in the Derive computer algebra system is available.

External Links:

ISSN 0377-0427, Document, Code, MathReview, ZentralBlatt

Referenced by:

§19.29(ii), §19.36(i), §19.36(i), §19.39(iv).

Encodings:

BibTeX

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Asymptotic approximations for symmetric elliptic integrals. SIAM J. Math. Anal. 25 (2), pp. 288–303.

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External Links:

ISSN 0036-1410, Document, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§19.12, §19.27(ii), §19.27(iii), §19.27(iv), §19.27(v), §19.27(vi).

Encodings:

BibTeX

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Inequalities for a symmetric elliptic integral. Proc. Amer. Math. Soc. 25 (3), pp. 698–703.

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External Links:

FullText, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§19.24(ii), §19.9(ii).

Encodings:

BibTeX

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32: Bibliography F

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Series expansions of symmetric elliptic integrals. Math. Comp. 81 (278), pp. 957–990.

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External Links:

Document, ZentralBlatt

Referenced by:

§20.7(ii), §20.7(ii).

Encodings:

BibTeX

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33: Bibliography L

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Uniform asymptotic expansions of symmetric elliptic integrals. Constr. Approx. 17 (4), pp. 535–559.

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External Links:

ISSN 0176-4276, Document, MathReview (Valdir A. Menegatto), ZentralBlatt

Referenced by:

§19.27(vi).

Encodings:

BibTeX

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Asymptotic expansions of symmetric standard elliptic integrals. SIAM J. Math. Anal. 31 (4), pp. 754–775.

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External Links:

ISSN 1095-7154, Document, MathReview (P. Anandani), ZentralBlatt

Referenced by:

§19.27(vi), §2.6(ii).

Encodings:

BibTeX

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34: 22.14 Integrals

… ►For indefinite integrals of squares and products of even powers of Jacobian functions in terms of symmetric elliptic integrals, see Carlson (2006b). …

35: 2.6 Distributional Methods

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2.6.32 $${\int }_0^{\mathrm{\infty }}\frac{f(t)}{{(t+z)}^{\rho }}dt,$$ $\rho >0$,

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Symbols:

$dx$: differential of $x$ , $\int$: integral and $f(t)$: locally integrable function

Permalink:

http://dlmf.nist.gov/2.6.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §2.6(ii), §2.6 and Ch.2

… ►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). …

36: 23.6 Relations to Other Functions

… ►For relations to symmetric elliptic integrals see §19.25(vi). …

37: Mathematical Introduction

… ►Other examples are: (a) the notation for the Ferrers functions—also known as associated Legendre functions on the cut—for which existing notations can easily be confused with those for other associated Legendre functions (§14.1); (b) the spherical Bessel functions for which existing notations are unsymmetric and inelegant (§§10.47(i) and 10.47(ii)); and (c) elliptic integrals for which both Legendre’s forms and the more recent symmetric forms are treated fully (Chapter 19). …

38: 19.7 Connection Formulas

§19.7 Connection Formulas

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Reciprocal-Modulus Transformation

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Imaginary-Modulus Transformation

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Imaginary-Argument Transformation

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§19.7(iii) Change of Parameter of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$

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39: 19.39 Software

… ►Unless otherwise stated, the functions are $K\left(k\right)$ and $E\left(k\right)$, with $0\le k^2(=m)\le 1$. … ►For other software, sometimes with $\mathrm{\Pi }({\alpha }^2,k)$ and complex variables, see the Software Index. … ►Unless otherwise stated, the variables are real, and the functions are $F(\varphi ,k)$ and $E(\varphi ,k)$. ►For research software see Bulirsch (1965b, function $\mathrm{el2}$), Bulirsch (1969b, function $\mathrm{el3}$), Jefferson (1961), and Neuman (1969a, functions $E(\varphi ,k)$ and $\mathrm{\Pi }(\varphi ,k^2,k)$). … ►

§19.39(iv) Symmetric Integrals

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40: 19 Elliptic Integrals

Chapter 19 Elliptic Integrals

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