reduction to symmetric elliptic integrals

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1: 19.29 Reduction of General Elliptic Integrals

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§19.29(i) Reduction Theorems

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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

Encodings:

pMML, png, TeX

See also:

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

2: 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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3: 19.15 Advantages of Symmetry

§19.15 Advantages of Symmetry

… ► … ►Symmetry makes possible the reduction theorems of §19.29(i), permitting remarkable compression of tables of integrals while generalizing the interval of integration. … ►For the many properties of ellipses and triaxial ellipsoids that can be represented by elliptic integrals, any symmetry in the semiaxes remains obvious when symmetric integrals are used (see (19.30.5) and §19.33). …

4: 19.36 Methods of Computation

§19.36 Methods of Computation

… ►Because of cancellations in (19.26.21) it is advisable to compute $R_G$ from $R_F$ and $R_D$ by (19.21.10) or else to use §19.36(ii). ►Legendre’s integrals can be computed from symmetric integrals by using the relations in §19.25(i). … ►Complete cases of Legendre’s integrals and symmetric integrals can be computed with quadratic convergence by the AGM method (including Bartky transformations), using the equations in §19.8(i) and §19.22(ii), respectively. … ►

5: Bibliography N

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Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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External Links:

FullText, MathReview (I. A. Stegun), ZentralBlatt

Referenced by:

§19.37(iv).

Encodings:

BibTeX

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Elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 99–102.

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Notes:

Includes Algol program.

External Links:

MathReview

Referenced by:

§19.39(iii).

Encodings:

BibTeX

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On the calculation of elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 91–94.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§19.36(ii).

Encodings:

BibTeX

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Bounds for symmetric elliptic integrals. J. Approx. Theory 122 (2), pp. 249–259.

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External Links:

ISSN 0021-9045, Document, MathReview (C. M. Joshi), ZentralBlatt

Referenced by:

§19.24(ii), §19.24(i), §19.9(i).

Encodings:

BibTeX

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Jacobian Elliptic Functions. 2nd edition, Clarendon Press, Oxford.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§20.1, Notations T, Notations T, Notations T, Notations T.

Encodings:

BibTeX

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