relation%20to%20Legendre%20elliptic%20integrals

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1: Software Index

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Research Software.

This is software of narrow scope developed as a byproduct of a research project and subsequently made available at no cost to the public. The software is often meant to demonstrate new numerical methods or software engineering strategies which were the subject of a research project. When developed, the software typically contains capabilities unavailable elsewhere. While the software may be quite capable, it is typically not professionally packaged and its use may require some expertise. The software is typically provided as source code or via a web-based service, and no support is provided.

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Open Source Collections and Systems.

These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

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Software Associated with Books.

An increasing number of published books have included digital media containing software described in the book. Often, the collection of software covers a fairly broad area. Such software is typically developed by the book author. While it is not professionally packaged, it often provides a useful tool for readers to experiment with the concepts discussed in the book. The software itself is typically not formally supported by its authors.

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Commercial Software.

Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

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Guide to Available Mathematical Software

A cross index of mathematical software in use at NIST.

2: Bibliography B

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S. Bielski (2013) Orthogonality relations for the associated Legendre functions of imaginary order. Integral Transforms Spec. Funct. 24 (4), pp. 331–337.

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External Links:: ISSN 1065-2469, Document, FullText, MathReview (Praveen Agarwal), ZentralBlatt
Referenced by:: §14.17(iii), §14.17(iii).
Encodings:: BibTeX

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S. Bochner (1952) Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.

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External Links:: MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §35.5(i).
Encodings:: BibTeX

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R. Bulirsch (1969a) An extension of the Bartky-transformation to incomplete elliptic integrals of the third kind. Numer. Math. 13 (3), pp. 266–284.

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External Links:: ISSN 0029-599X, Document, MathReview (Y. L. Luke), ZentralBlatt
Referenced by:: §19.1, §19.2(iii), §19.36(ii), §19.36(ii).
Encodings:: BibTeX

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R. Bulirsch (1965b) Numerical calculation of elliptic integrals and elliptic functions. Numer. Math. 7 (1), pp. 78–90.

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Notes:: Algol 60 programs are included for real and complex elliptic integrals, and for Jacobian elliptic functions of real argument.
External Links:: ISSN 0029-599X, Document, MathReview (M. Feliciano), ZentralBlatt
Referenced by:: §19.1, §19.2(iii), §19.36(ii), §19.39(iii), §22.22(ii).
Encodings:: BibTeX

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P. J. Bushell (1987) On a generalization of Barton’s integral and related integrals of complete elliptic integrals. Math. Proc. Cambridge Philos. Soc. 101 (1), pp. 1–5.

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External Links:: ISSN 0305-0041, MathReview (J. M. H. Peters), ZentralBlatt
Referenced by:: §19.13(i).
Encodings:: BibTeX

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3: 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). … ►Computation of Legendre’s integrals of all three kinds by quadratic transformation is described by Cazenave (1969, pp. 128–159, 208–230). … ►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). … ►

4: Bibliography

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H. Airault, H. P. McKean, and J. Moser (1977) Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem. Comm. Pure Appl. Math. 30 (1), pp. 95–148.

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External Links:: MathReview (Alexander A. Pankov), ZentralBlatt
Referenced by:: §23.21(ii).
Encodings:: BibTeX

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D. E. Amos (1989) Repeated integrals and derivatives of

K

Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

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External Links:: ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt
Referenced by:: §10.43(iii).
Encodings:: BibTeX

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G. D. Anderson and M. K. Vamanamurthy (1985) Inequalities for elliptic integrals. Publ. Inst. Math. (Beograd) (N.S.) 37(51), pp. 61–63.

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External Links:: ISSN 0350-1302, MathReview, ZentralBlatt
Referenced by:: §19.9(i).
Encodings:: BibTeX

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T. M. Apostol and T. H. Vu (1984) Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.

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External Links:: ISSN 0022-314X, Document, MathReview (Kai Man Tsang), ZentralBlatt
Referenced by:: (25.16.10), (25.16.11), (25.16.12), (25.16.14), (25.16.15), (25.16.5), (25.16.6), (25.16.7), (25.16.8), (25.16.9), §25.16(ii), §25.16(ii), §25.16.
Encodings:: BibTeX

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F. M. Arscott (1964a) Integral equations and relations for Lamé functions. Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.

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External Links:: Document, MathReview (I. Marx), ZentralBlatt
Referenced by:: §29.8.
Encodings:: BibTeX

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5: Bibliography S

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D. Schmidt and G. Wolf (1979) A method of generating integral relations by the simultaneous separability of generalized Schrödinger equations. SIAM J. Math. Anal. 10 (4), pp. 823–838.

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External Links:: ISSN 0036-1410, Document, MathReview (Ernest G. Kalnins), ZentralBlatt
Referenced by:: §28.32(i).
Encodings:: BibTeX

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S. Yu. Slavyanov and N. A. Veshev (1997) Structure of avoided crossings for eigenvalues related to equations of Heun’s class. J. Phys. A 30 (2), pp. 673–687.

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External Links:: ISSN 0305-4470, Document, MathReview (Éric Delabaere), ZentralBlatt
Referenced by:: §31.13.
Encodings:: BibTeX

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B. D. Sleeman (1968a) Integral equations and relations for Lamé functions and ellipsoidal wave functions. Proc. Cambridge Philos. Soc. 64, pp. 113–126.

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External Links:: Document, MathReview (Y. L. Luke), ZentralBlatt
Referenced by:: §29.8.
Encodings:: BibTeX

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D. M. Smith (2011) Algorithm 911: multiple-precision exponential integral and related functions. ACM Trans. Math. Software 37 (4), pp. Art. 46, 16.

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External Links:: Document, ZentralBlatt
Referenced by:: 5th item, §4.48(iii), 3rd item, §6.21(ii).
Encodings:: BibTeX

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C. Snow (1952) Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory. National Bureau of Standards Applied Mathematics Series, No. 19, U. S. Government Printing Office, Washington, D.C..

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External Links:: MathReview, ZentralBlatt
Referenced by:: §13.23(iv), Ch.14, §31.3(iii).
Encodings:: BibTeX

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6: Bibliography V

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G. Valent (1986) An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.

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External Links:: ISSN 0036-1410, Document, MathReview (William L. Perry), ZentralBlatt
Referenced by:: §31.10(i).
Encodings:: BibTeX

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H. Van de Vel (1969) On the series expansion method for computing incomplete elliptic integrals of the first and second kinds. Math. Comp. 23 (105), pp. 61–69.

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External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §19.12, §19.5.
Encodings:: BibTeX

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J. F. Van Diejen and V. P. Spiridonov (2001) Modular hypergeometric residue sums of elliptic Selberg integrals. Lett. Math. Phys. 58 (3), pp. 223–238.

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External Links:: ISSN 0377-9017, Document, MathReview (Laurent Habsieger), ZentralBlatt
Referenced by:: §19.35(i).
Encodings:: BibTeX

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H. Volkmer (1982) Integral relations for Lamé functions. SIAM J. Math. Anal. 13 (6), pp. 978–987.

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External Links:: ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt
Referenced by:: §29.8, §29.8.
Encodings:: BibTeX

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H. Volkmer (2004a) Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.

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External Links:: ISSN 0176-4276, Document, MathReview, ZentralBlatt
Referenced by:: §30.16(i), §30.16(ii), §30.16(ii).
Encodings:: BibTeX

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7: Bibliography D

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A. Dienstfrey and J. Huang (2006) Integral representations for elliptic functions. J. Math. Anal. Appl. 316 (1), pp. 142–160.

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External Links:: ISSN 0022-247X, Document, MathReview Entry, ZentralBlatt
Referenced by:: §23.11.
Encodings:: BibTeX

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B. Döring (1966) Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

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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

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T. M. Dunster (2004) Convergent expansions for solutions of linear ordinary differential equations having a simple pole, with an application to associated Legendre functions. Stud. Appl. Math. 113 (3), pp. 245–270.

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External Links:: ISSN 0022-2526, Document, MathReview (Wolfgang Bühring), ZentralBlatt
Referenced by:: §14.15(iii), §2.8(i).
Encodings:: BibTeX

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L. Durand (1975) Nicholson-type Integrals for Products of Gegenbauer Functions and Related Topics. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), R. A. Askey (Ed.), pp. 353–374. Math. Res. Center, Univ. Wisconsin, Publ. No. 35.

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External Links:: MathReview (K. M. Saksena), ZentralBlatt
Referenced by:: (12.12.4), §12.12, §18.17(iii), §18.17(iii).
Encodings:: BibTeX

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P. L. Duren (1991) 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.

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External Links:: ISBN 0-387-97509-8, MathReview (J. M. H. Peters), ZentralBlatt
Referenced by:: §19.7(i).
Encodings:: BibTeX

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8: Bibliography C

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B. C. Carlson and J. FitzSimons (2000) 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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B. C. Carlson (1999) 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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B. C. Carlson (2006b) 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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H. S. Cohl (2010) Derivatives with respect to the degree and order of associated Legendre functions for

|z|>1

using modified Bessel functions. Integral Transforms Spec. Funct. 21 (7-8), pp. 581–588.

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External Links:: ISSN 1065-2469, Document, FullText, MathReview (Dmitriy Karp), ZentralBlatt
Referenced by:: §14.11, §14.11.
Encodings:: BibTeX

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D. Cvijović and J. Klinowski (1999) Integrals involving complete elliptic integrals. J. Comput. Appl. Math. 106 (1), pp. 169–175.

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External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: §19.13(i).
Encodings:: BibTeX

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9: Bibliography O

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O. M. Ogreid and P. Osland (1998) Summing one- and two-dimensional series related to the Euler series. J. Comput. Appl. Math. 98 (2), pp. 245–271.

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External Links:: ISSN 0377-0427, Document, MathReview (Boris Petrovich Osilenker), ZentralBlatt
Referenced by:: §25.8.
Encodings:: BibTeX

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S. Okui (1974) Complete elliptic integrals resulting from infinite integrals of Bessel functions. J. Res. Nat. Bur. Standards Sect. B 78B (3), pp. 113–135.

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External Links:: ISSN 0160-1741, MathReview (Y. L. Luke), ZentralBlatt
Referenced by:: §10.22(vi), §10.43(vi).
Encodings:: BibTeX

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S. Okui (1975) Complete elliptic integrals resulting from infinite integrals of Bessel functions. II. J. Res. Nat. Bur. Standards Sect. B 79B (3-4), pp. 137–170.

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External Links:: ISSN 0160-1741, MathReview (Y. L. Luke), ZentralBlatt
Referenced by:: §10.22(vi), §10.43(vi).
Encodings:: BibTeX

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J. Oliver (1977) An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

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External Links:: Document, MathReview (David L. Elliott), ZentralBlatt
Referenced by:: §3.11(ii).
Encodings:: BibTeX

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F. W. J. Olver and J. M. Smith (1983) Associated Legendre functions on the cut. J. Comput. Phys. 51 (3), pp. 502–518.

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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

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10: Bibliography R

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J. Raynal (1979) On the definition and properties of generalized

6

j

symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

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External Links:: ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt
Referenced by:: §16.4(iii), §16.4(iii).
Encodings:: BibTeX

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K. Reinsch and W. Raab (2000) 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.

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External Links:: MathReview, ZentralBlatt
Referenced by:: §19.36(i), §19.36(ii).
Encodings:: BibTeX

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L. Robin (1957) Fonctions sphériques de Legendre et fonctions sphéroïdales. Tome I. Gauthier-Villars, Paris.

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Notes:: Préface de H. Villat
External Links:: MathReview (R. Campbell), ZentralBlatt
Referenced by:: Ch.14.
Encodings:: BibTeX

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L. Robin (1958) Fonctions sphériques de Legendre et fonctions sphéroïdales. Tome II. Gauthier-Villars, Paris.

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External Links:: MathReview (R. Campbell), ZentralBlatt
Referenced by:: Ch.14.
Encodings:: BibTeX

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R. Roy (2017) Elliptic and modular functions from Gauss to Dedekind to Hecke. Cambridge University Press, Cambridge.

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External Links:: ISBN 978-1-107-15938-9, Document, Link, MathReview (Paul M. Jenkins), ZentralBlatt
Referenced by:: Profile Ranjan Roy.
Encodings:: BibTeX

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