reduction to basic elliptic integrals

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21: Bibliography G

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Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.

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External Links:

ISSN 1023-6198, Document, FullText, MathReview (Thomas Britz), ZentralBlatt

Referenced by:

§17.7(iii).

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Basic Hypergeometric Series. Encyclopedia of Mathematics and its Applications, Vol. 35, Cambridge University Press, Cambridge.

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Notes:

A revised and updated edition, with three new chapters, was published in 2004 by Cambridge University Press (Encyclopedia of Mathematics and its Applications, vol. 96).

External Links:

ISBN 0-521-35049-2, MathReview (Shaun Cooper), ZentralBlatt

Referenced by:

(17.9.3), Erratum (V1.0.23) for Equation (17.9.3).

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Definite integrals of the complete elliptic integral $K$ . J. Res. Nat. Bur. Standards Sect. B 80B (2), pp. 313–323.

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External Links:

MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§19.13(i).

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Automatic reduction of elliptic integrals using Carlson’s relations. Math. Comp. 71 (237), pp. 311–318.

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ISSN 0025-5718, Document, MathReview (David A. Richter), ZentralBlatt

Referenced by:

§19.29(ii).

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Multilateral summation theorems for ordinary and basic hypergeometric series in $\mathrm{U}(n)$ . SIAM J. Math. Anal. 18 (6), pp. 1576–1596.

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ISSN 0036-1410, Document, MathReview, ZentralBlatt

Referenced by:

§17.15.

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22: 19.1 Special Notation

… ►All derivatives are denoted by differentials, not by primes. … ►We use also the function $D(\varphi ,k)$, introduced by Jahnke et al. (1966, p. 43). … ►In Abramowitz and Stegun (1964, Chapter 17) the functions (19.1.1) and (19.1.2) are denoted, in order, by $K(\alpha )$, $E(\alpha )$, $\mathrm{\Pi }(n\backslash \alpha )$, $F(\varphi \backslash \alpha )$, $E(\varphi \backslash \alpha )$, and $\mathrm{\Pi }(n;\varphi \backslash \alpha )$, where $\alpha =\arcsin k$ and $n$ is the ${\alpha }^2$ (not related to $k$) in (19.1.1) and (19.1.2). …However, it should be noted that in Chapter 8 of Abramowitz and Stegun (1964) the notation used for elliptic integrals differs from Chapter 17 and is consistent with that used in the present chapter and the rest of the NIST Handbook and DLMF. … ► $R_{-a}(b_1,b_2,\mathrm{\dots },b_n;z_1,z_2,\mathrm{\dots },z_n)$ is a multivariate hypergeometric function that includes all the functions in (19.1.3). …

23: 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).

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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).

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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).

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

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

Referenced by:

§19.24(ii), §19.24(i), §19.9(i).

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Notes:

References and links to software for factorization and primality testing.

External Links:

Home Page

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6th item.

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24: 16.16 Transformations of Variables

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§16.16(i) Reduction Formulas

… ►See Erdélyi et al. (1953a, §5.10) for these and further reduction formulas. …

25: 15.11 Riemann’s Differential Equation

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15.11.1 $$\frac{d^{2}w}{{dz}^2}+\left(\frac{1-a_1-a_2}{z-\alpha }+\frac{1-b_1-b_2}{z-\beta }+\frac{1-c_1-c_2}{z-\gamma }\right)\frac{dw}{dz}+\left(\frac{(\alpha -\beta )(\alpha -\gamma )a_1{a}_2}{z-\alpha }+\frac{(\beta -\alpha )(\beta -\gamma )b_1{b}_2}{z-\beta }+\frac{(\gamma -\alpha )(\gamma -\beta )c_1{c}_2}{z-\gamma }\right)\frac{w}{(z-\alpha )(z-\beta )(z-\gamma )}=0,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $\alpha$: point, $\beta$: point, $\gamma$: point and $w$: solution

Referenced by:

§15.11(i), §15.11(i), §15.17(v)

Permalink:

http://dlmf.nist.gov/15.11.E1

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pMML, png, TeX

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Annotations for §15.11(i), §15.11 and Ch.15

… ►These constants can be chosen to map any two sets of three distinct points $\{\alpha ,\beta ,\gamma \}$ and $\{\tilde{\alpha },\tilde{\beta },\tilde{\gamma }\}$ onto each other. …The reduction of a general homogeneous linear differential equation of the second order with at most three regular singularities to the hypergeometric differential equation is given by …

26: 19.36 Methods of Computation

… ►The reductions in §19.29(i) represent $x,y,z$ as squares, for example $x=U_{12}^2$ in (19.29.4). … ►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). … ►Descending Gauss transformations of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ (see (19.8.20)) are used in Fettis (1965) to compute a large table (see §19.37(iii)). … ►Quadratic transformations can be applied to compute Bulirsch’s integrals (§19.2(iii)). … ►

27: 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)). ►

§19.35(ii) Physical

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28: 20.7 Identities

… ►Also, in further development along the lines of the notations of Neville (§20.1) and of Glaisher (§22.2), the identities (20.7.6)–(20.7.9) have been recast in a more symmetric manner with respect to suffices $2,3,4$. The symmetry, applicable also to §§20.7(iii) and 20.7(vii), is obtained by modifying traditional theta functions in the manner recommended by Carlson (2011) and used for further purposes by Fukushima (2012). … ►

§20.7(iv) Reduction Formulas for Products

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§20.7(vi) Landen Transformations

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§20.7(ix) Addendum to 20.7(iv) Reduction Formulas for Products

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29: Bibliography W

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The irregular primes to $125000$ . Math. Comp. 32 (142), pp. 583–591.

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External Links:

FullText, Document, MathReview (Wells Johnson), ZentralBlatt

Referenced by:

§24.20.

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The distribution of the zeros of Jacobian elliptic functions with respect to the parameter $k$ . Comput. Methods Funct. Theory 9 (2), pp. 579–591.

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External Links:

ISSN 1617-9447, MathReview (M. K. Jain), ZentralBlatt

Referenced by:

§22.4(i).

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Reduction formulae for products of theta functions. J. Res. Nat. Inst. Standards and Technology 117, pp. 297–303.

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External Links:

Document

Referenced by:

(20.7.34), §20.7(ix).

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Introduction to Cyclotomic Fields. 2nd edition, Springer-Verlag, New York.

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External Links:

ISBN 0-387-94762-0, MathReview (T. Metsänkylä), ZentralBlatt

Referenced by:

§24.10(i), §24.10(ii), §24.16(ii), §24.17(iii).

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Elliptic Functions According to Eisenstein and Kronecker. Classics in Mathematics, Springer-Verlag, Berlin.

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Notes:

Reprint of the 1976 original

External Links:

ISBN 3-540-65036-9, MathReview, ZentralBlatt

Referenced by:

§22.12, §23.8(ii).

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30: Bibliography L

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Reduction of Elliptic Integrals to Legendre Normal Form. Technical report Technical Report 97-21, Department of Computer Science, University of Waterloo, Waterloo, Ontario.

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Referenced by:

§19.14(ii).

Encodings:

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Application of theta functions for numerical evaluation of complete elliptic integrals of the first and second kinds. Comput. Phys. Comm. 60 (3), pp. 319–327.

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ISSN 0010-4655, Document, MathReview, ZentralBlatt

Referenced by:

§19.36(iii), §19.5.

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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).

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Approximations for elliptic integrals. Math. Comp. 22 (103), pp. 627–634.

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FullText, MathReview, ZentralBlatt

Referenced by:

§19.38.

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Further approximations for elliptic integrals. Math. Comp. 24 (109), pp. 191–198.

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FullText, MathReview, ZentralBlatt

Referenced by:

§19.38.

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