Legendre%20elliptic%20integrals

SearchAdvancedHelp

(0.006 seconds)

11—16 of 16 matching pages

11: Bibliography O

… ►

Complete elliptic integrals resulting from infinite integrals of Bessel functions. J. Res. Nat. Bur. Standards Sect. B 78B (3), pp. 113–135.

ⓘ

External Links:

ISSN 0160-1741, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§10.22(vi), §10.43(vi).

Encodings:

BibTeX

►

Complete elliptic integrals resulting from infinite integrals of Bessel functions. II. J. Res. Nat. Bur. Standards Sect. B 79B (3-4), pp. 137–170.

ⓘ

External Links:

ISSN 0160-1741, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§10.22(vi), §10.43(vi).

Encodings:

BibTeX

… ►

An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

ⓘ

External Links:

Document, MathReview (David L. Elliott), ZentralBlatt

Referenced by:

§3.11(ii).

Encodings:

BibTeX

… ►

Associated Legendre functions on the cut. J. Comput. Phys. 51 (3), pp. 502–518.

ⓘ

Notes:

Includes single, double precision Fortran code.

External Links:

GAMS (module DXNRMP; package SLATEC), ISSN 0021-9991, Document, MathReview (G. P. Bhattacharjee), ZentralBlatt

Referenced by:

§14.32, 7th item.

Encodings:

BibTeX

… ►

Legendre functions with both parameters large. Philos. Trans. Roy. Soc. London Ser. A 278, pp. 175–185.

ⓘ

External Links:

FullText, MathReview (P. F. Hsieh), ZentralBlatt

Referenced by:

§14.15(v), §14.15(v).

Encodings:

BibTeX

…

12: Bibliography V

… ►

On the series expansion method for computing incomplete elliptic integrals of the first and second kinds. Math. Comp. 23 (105), pp. 61–69.

ⓘ

External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§19.12, §19.5.

Encodings:

BibTeX

… ►

Modular hypergeometric residue sums of elliptic Selberg integrals. Lett. Math. Phys. 58 (3), pp. 223–238.

ⓘ

External Links:

ISSN 0377-9017, Document, MathReview (Laurent Habsieger), ZentralBlatt

Referenced by:

§19.35(i).

Encodings:

BibTeX

… ►

Generalized Associated Legendre Functions and their Applications. World Scientific Publishing Co. Inc., Singapore.

ⓘ

External Links:

ISBN 981-02-4353-7, MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§14.29.

Encodings:

BibTeX

… ►

Integral relations for Lamé functions. SIAM J. Math. Anal. 13 (6), pp. 978–987.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.8, §29.8.

Encodings:

BibTeX

… ►

Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.

ⓘ

External Links:

ISSN 0176-4276, Document, MathReview, ZentralBlatt

Referenced by:

§30.16(i), §30.16(ii), §30.16(ii).

Encodings:

BibTeX

…

13: Bibliography D

… ►

Integral representations for elliptic functions. J. Math. Anal. Appl. 316 (1), pp. 142–160.

ⓘ

External Links:

ISSN 0022-247X, Document, MathReview Entry, ZentralBlatt

Referenced by:

§23.11.

Encodings:

BibTeX

… ►

Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

ⓘ

Notes:

For a numerical error in one of the zeros see Amos (1985).

External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

… ►

Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

ⓘ

Notes:

(This reference has several typographical errors.)

External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§13.21(ii), §13.21(iii), §13.21(iv).

Encodings:

BibTeX

… ►

Uniform asymptotic expansions for associated Legendre functions of large order. Proc. Roy. Soc. Edinburgh Sect. A 133 (4), pp. 807–827.

ⓘ

External Links:

ISSN 0308-2105, Document, MathReview, ZentralBlatt

Referenced by:

§14.15(i), §14.15(i), §14.15(ii), §14.15(ii), §14.26.

Encodings:

BibTeX

… ►

The Legendre Relation for Elliptic Integrals. In Paul Halmos: Celebrating 50 Years of Mathematics, J. H. Ewing and F. W. Gehring (Eds.), pp. 305–315.

ⓘ

External Links:

ISBN 0-387-97509-8, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§19.7(i).

Encodings:

BibTeX

…

14: Errata

… ►

The material surrounding (19.2.8), (19.2.9) has been updated so that the complementary complete elliptic integrals of the first and second kind are defined with consistent multivalued properties and correct analytic continuation. In particular, (19.2.9) has been corrected to read

19.2.9

… ►

14.6.6 $${𝖯}_{\nu }^{-m}\left(x\right)={\left(1-x^2\right)}^{-m/2}{\int }_x^1\mathrm{\dots }{\int }_x^1{𝖯}_{\nu }\left(x\right){\left(dx\right)}^m$$

The right-hand side has been corrected by replacing the Legendre function $P_{\nu }\left(x\right)$ with the Ferrers function ${𝖯}_{\nu }\left(x\right)$.

… ►

22.14.16 $${\int }_0^{K\left(k\right)}\ln \left(\mathrm{sn}(t,k)\right)dt=-\frac{\pi }{4}{K}^{\prime }\left(k\right)-\frac{1}{2}K\left(k\right)\ln k,$$

22.14.17 $${\int }_0^{K\left(k\right)}\ln \left(\mathrm{cn}(t,k)\right)dt=-\frac{\pi }{4}{K}^{\prime }\left(k\right)+\frac{1}{2}K\left(k\right)\ln \left(k^{\prime }/k\right)$$

Originally, a factor of $\pi$ was missing from the terms containing the $-\frac{1}{4}{K}^{\prime }\left(k\right)$.

Reported by Fred Hucht on 2020-08-06

… ►

Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

… ►

36.10.14 $$3\left(\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{E})}}{{\partial x}^2}-\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{E})}}{{\partial y}^2}\right)+2\mathrm{i}z\frac{\partial {\mathrm{\Psi }}^{(\mathrm{E})}}{\partial x}-x{\mathrm{\Psi }}^{(\mathrm{E})}=0$$

Originally this equation appeared with $\frac{\partial {\mathrm{\Psi }}^{(\mathrm{H})}}{\partial x}$ in the second term, rather than $\frac{\partial {\mathrm{\Psi }}^{(\mathrm{E})}}{\partial x}$.

Reported 2010-04-02.

…

15: Bibliography R

… ►

On the definition and properties of generalized $6$-$j$ symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

ⓘ

External Links:

ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§16.4(iii), §16.4(iii).

Encodings:

BibTeX

… ►

Elliptic Integrals of the First and Second Kind – Comparison of Bulirsch’s and Carlson’s Algorithms for Numerical Calculation. In Special Functions (Hong Kong, 1999), C. Dunkl, M. Ismail, and R. Wong (Eds.), pp. 293–308.

ⓘ

External Links:

MathReview, ZentralBlatt

Referenced by:

§19.36(i), §19.36(ii).

Encodings:

BibTeX

… ►

Fonctions sphériques de Legendre et fonctions sphéroïdales. Tome I. Gauthier-Villars, Paris.

ⓘ

Notes:

Préface de H. Villat

External Links:

MathReview (R. Campbell), ZentralBlatt

Referenced by:

Ch.14.

Encodings:

BibTeX

►

Fonctions sphériques de Legendre et fonctions sphéroïdales. Tome II. Gauthier-Villars, Paris.

ⓘ

External Links:

MathReview (R. Campbell), ZentralBlatt

Referenced by:

Ch.14.

Encodings:

BibTeX

►

Fonctions sphériques de Legendre et fonctions sphéroïdales. Tome III. Collection Technique et Scientifique du C. N. E. T. Gauthier-Villars, Paris.

ⓘ

External Links:

MathReview (R. Campbell), ZentralBlatt

Referenced by:

Ch.14.

Encodings:

BibTeX

…

16: Bibliography G

… ►

Integrals of three Bessel functions and Legendre functions. I. J. Math. Phys. 26 (4), pp. 633–644.

ⓘ

External Links:

ISSN 0022-2488, Document, MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§10.22(iv), §10.43(iv).

Encodings:

BibTeX

►

Integrals of three Bessel functions and Legendre functions. II. J. Math. Phys. 26 (4), pp. 645–655.

ⓘ

External Links:

ISSN 0022-2488, Document, MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§10.22(iv), §10.43(iv).

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§19.13(i).

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

§19.29(ii).

Encodings:

BibTeX

… ►

Staircase polygons, elliptic integrals, Heun functions, and lattice Green functions. Phys. Rev. E 47 (4), pp. R2233–R2236.

ⓘ

External Links:

Document

Referenced by:

§19.35(ii).

Encodings:

BibTeX