relation to associated Legendre functions

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21: Bibliography C

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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).

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Performance evaluation of programs related to the real gamma function. ACM Trans. Math. Software 17 (1), pp. 46–54.

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ISSN 0098-3500, Document, MathReview, ZentralBlatt

Referenced by:

§5.24(ii), §5.24(iii).

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Multi-Integral Representations for Associated Legendre and Ferrers Functions. Symmetry 12 (10).

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External Links:

Link, ISSN 2073-8994, Document, ZentralBlatt

Referenced by:

§14.6(ii), §14.6(ii), Erratum (V1.1.1) for Subsection 14.6(ii).

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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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ISSN 1065-2469, Document, FullText, MathReview (Dmitriy Karp), ZentralBlatt

Referenced by:

§14.11, §14.11.

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On parameter differentiation for integral representations of associated Legendre functions. SIGMA Symmetry Integrability Geom. Methods Appl. 7, pp. Paper 050, 16.

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ISSN 1815-0659, Document, FullText, MathReview (Michael Doschoris), ZentralBlatt

Referenced by:

§14.11, §14.11.

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22: Bille C. Carlson

… ► 2013) was Professor Emeritus in the Department of Mathematics and Associate of the Ames Laboratory (U. … ►This symmetry led to the development of symmetric elliptic integrals, which are free from the transformations of modulus and amplitude that complicate the Legendre theory. …Also, the homogeneity of the $R$-function has led to a new type of mean value for several variables, accompanied by various inequalities. … ►In Symmetry in c, d, n of Jacobian elliptic functions (2004) he found a previously hidden symmetry in relations between Jacobian elliptic functions, which can now take a form that remains valid when the letters c, d, and n are permuted. This invariance usually replaces sets of twelve equations by sets of three equations and applies also to the relation between the first symmetric elliptic integral and the Jacobian functions. …

23: 14.8 Behavior at Singularities

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§14.8(i) $x\to 1-$ or $x\to -1+$

►As $x\to 1-$, …In the next three relations $\mathrm{\Re }\mu >0$. … ►The behavior of ${𝖯}_{\nu }^{\mu }\left(x\right)$ and ${𝖰}_{\nu }^{\mu }\left(x\right)$ as $x\to -1+$ follows from the above results and the connection formulas (14.9.8) and (14.9.10). …

24: 18.30 Associated OP’s

… ►Associated polynomials and the related corecursive polynomials appear in Ismail (2009, §§2.3, 2.6, and 2.10), where the relationship of OP’s to continued fractions is made evident. … ►

§18.30(i) Associated Jacobi Polynomials

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§18.30(ii) Associated Legendre Polynomials

… ►Defining associated orthogonal polynomials and their relationship to their corecursive counterparts is particularly simple via use of the recursion relations for the monic, rather than via those for the traditional polynomials. … ►More generally, the $k$th corecursive monic polynomials (defined with the initialization of (18.30.28) followed by the $c=k$ recurrence of (18.30.27)) are related to the $(k+1)$st monic associated polynomials by …

25: Bibliography V

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

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ISSN 0036-1410, Document, MathReview (William L. Perry), ZentralBlatt

Referenced by:

§31.10(i).

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Airy Functions and Applications to Physics. Second edition, Imperial College Press, London.

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External Links:

ISBN 978-1-84816-548-9; 1-84816-548-X, MathReview Entry, ZentralBlatt

Referenced by:

§1.17(ii), §9.10(ii), §9.10(ix), §9.11(v), §9.16, §9.16, §9.5(i).

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Integrating products of Bessel functions with an additional exponential or rational factor. Comput. Phys. Comm. 178 (8), pp. 578–590.

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Notes:

Associated computer program available online

External Links:

ISSN 0010-4655, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§10.74(vii), §10.74(vii).

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

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External Links:

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

Referenced by:

§14.29.

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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.

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26: Software Index

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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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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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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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Software associated with papers published in the journal Computer Physics Communications.

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A cross index of mathematical software in use at NIST.

27: Bibliography

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Some series of the zeta and related functions. Analysis (Munich) 18 (2), pp. 131–144.

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External Links:

ISSN 0174-4747, MathReview (Jack Quine), ZentralBlatt

Referenced by:

(25.11.37), (25.11.38), (25.11.39), (25.11.40), §25.11, §25.11(xi), (25.8.1), (25.8.4), §25.8.

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SPHEREPACK 2.0: A Model Development Facility. NCAR Technical Note Technical Report TN-436-STR, National Center for Atmospheric Research.

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Notes:

SPHEREPACK 2.0 is a collection of Fortran programs that facilitates computer modeling of geophysical processes. Accurate solutions are obtained via the spectral method that uses both scalar and vector spherical harmonic transforms. The package also contains utility programs for computing the associated Legendre functions. SPHEREPACK 3.1 was released in August 2003.

External Links:

FullText, SPHEREPACK 3.1

Referenced by:

1st item.

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Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.

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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.

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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.

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Associated Laguerre and Hermite polynomials. Proc. Roy. Soc. Edinburgh 96A, pp. 15–37.

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External Links:

MathReview Entry

Referenced by:

(18.30.16), (18.30.17), (18.30.18), (18.30.19), (18.30.20), §18.30(iii), §18.30(iv), §18.30, §18.30(v), (18.35.10).

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28: Bibliography O

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

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ISSN 0377-0427, Document, MathReview (Boris Petrovich Osilenker), ZentralBlatt

Referenced by:

§25.8.

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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.

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Bessel Functions. Part III: Zeros and Associated Values. Royal Society Mathematical Tables, Volume 7, Cambridge University Press, Cambridge-New York.

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External Links:

ISBN 0-521-06153-9, MathReview (L. Fox), ZentralBlatt

Referenced by:

§10.21(vi), §10.21(viii), §10.58, §10.74(vi), 1st item, 2nd item.

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On the asymptotic solution of second-order differential equations having an irregular singularity of rank one, with an application to Whittaker functions. J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (2), pp. 225–243.

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External Links:

Document, MathReview (T. E. Hull), ZentralBlatt

Referenced by:

§13.19, §13.7(ii).

Encodings:

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

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External Links:

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

Referenced by:

§14.15(v), §14.15(v).

Encodings:

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