reduction to basic elliptic integrals

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11: 7.2 Definitions

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§7.2(ii) Dawson’s Integral

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§7.2(iii) Fresnel Integrals

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Values at Infinity

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§7.2(iv) Auxiliary Functions

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§7.2(v) Goodwin–Staton Integral

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12: 7.18 Repeated Integrals of the Complementary Error Function

§7.18 Repeated Integrals of the Complementary Error Function

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Hermite Polynomials

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Confluent Hypergeometric Functions

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Parabolic Cylinder Functions

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Probability Functions

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13: 22.16 Related Functions

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§22.16(i) Jacobi’s Amplitude ($\mathrm{am}$) Function

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Integral Representation

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Relation to Elliptic Integrals

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Relation to the Elliptic Integral $E(\varphi ,k)$

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Definition

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14: 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

… ►Partial fractions provide a reduction to integrals in which $𝐦$ has at most one nonzero component, and these are then reduced to basic integrals by the recurrence relations. … ►For an implementation by James FitzSimons of the method for reducing $I(𝐦)$ to basic integrals and extensive tables of such reductions, see Carlson (1999) and Carlson and FitzSimons (2000). …

15: 19.14 Reduction of General Elliptic Integrals

§19.14 Reduction of General Elliptic Integrals

… ►It then improves the classical method by first applying Hermite reduction to (19.2.3) to arrive at integrands without multiple poles and uses implicit full partial-fraction decomposition and implicit root finding to minimize computing with algebraic extensions. The choice among 21 transformations for final reduction to Legendre’s normal form depends on inequalities involving the limits of integration and the zeros of the cubic or quartic polynomial. A similar remark applies to the transformations given in Erdélyi et al. (1953b, §13.5) and to the choice among explicit reductions in the extensive table of Byrd and Friedman (1971), in which one limit of integration is assumed to be a branch point of the integrand at which the integral converges. …

16: 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. …These reduction theorems, unknown in the Legendre theory, allow symbolic integration without imposing conditions on the parameters and the limits of integration (see §19.29(ii)). … ►

17: 32.13 Reductions of Partial Differential Equations

§32.13 Reductions of Partial Differential Equations

… ►has the scaling reduction … ►has the scaling reduction … ►Equation (32.13.3) also has the similarity reduction … ►has the scaling reduction …

18: 31.7 Relations to Other Functions

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§31.7(i) Reductions to the Gauss Hypergeometric Function

… ►Other reductions of $H\mathrm{\ell }$ to a ${}_2{}^{}F_1^{}$, with at least one free parameter, exist iff the pair $(a,p)$ takes one of a finite number of values, where $q=\alpha \beta p$. Below are three such reductions with three and two parameters. … ►For additional reductions, see Maier (2005). Joyce (1994) gives a reduction in which the independent variable is transformed not polynomially or rationally, but algebraically. …

19: Peter A. Clarkson

… ► Kruskal, he developed the “direct method” for determining symmetry solutions of partial differential equations in New similarity reductions of the Boussinesq equation (with M. …He is also coauthor of the book From Nonlinearity to Coherence: Universal Features of Nonlinear Behaviour in Many-Body Physics (with J. …

20: 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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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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Three improvements in reduction and computation of elliptic integrals. J. Res. Nat. Inst. Standards Tech. 107 (5), pp. 413–418.

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External Links:

FullText

Referenced by:

§19.22(ii), §19.29(ii), §19.36(i), §19.8(i).

Encodings:

BibTeX

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Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric $R$-functions. Math. Comp. 75 (255), pp. 1309–1318.

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External Links:

ISSN 0025-5718, Document, MathReview, ZentralBlatt

Referenced by:

§19.25(v), §19.29(i), §22.14(ii).

Encodings:

BibTeX

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New similarity reductions of the Boussinesq equation. J. Math. Phys. 30 (10), pp. 2201–2213.

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External Links:

ISSN 0022-2488, Document, Link, MathReview (B. V. Baby), ZentralBlatt

Referenced by:

Profile Peter A. Clarkson.

Encodings:

BibTeX

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