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21: Bibliography F

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FDLIBM (free C library)

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Notes:: The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.
External Links:: GAMS (package FDLIBM)
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

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S. Fempl (1960) Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

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External Links:: MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §6.15.
Encodings:: BibTeX

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H. E. Fettis (1970) On the reciprocal modulus relation for elliptic integrals. SIAM J. Math. Anal. 1 (4), pp. 524–526.

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External Links:: Document, MathReview (B. C. Carlson), ZentralBlatt
Referenced by:: §19.7(i).
Encodings:: BibTeX

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F. Feuillebois (1991) Numerical calculation of singular integrals related to Hankel transform. Comput. Math. Appl. 21 (2-3), pp. 87–94.

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Notes:: Fortran, minimum precision 6D.
External Links:: ISSN 0898-1221, Document, MathReview, ZentralBlatt
Referenced by:: §10.77(ix).
Encodings:: BibTeX

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G. Freud (1969) On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

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External Links:: ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)
Referenced by:: §18.32.
Encodings:: BibTeX

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22: 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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O. Vallée and M. Soares (2010) 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).
Encodings:: BibTeX

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A. J. van der Poorten (1980) Some Wonderful Formulas

\ldots

an Introduction to Polylogarithms. In Proceedings of the Queen’s Number Theory Conference, 1979 (Kingston, Ont., 1979), R. Ribenboim (Ed.), Queen’s Papers in Pure and Appl. Math., Vol. 54, Kingston, Ont., pp. 269–286.

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External Links:: MathReview (F. Beukers), ZentralBlatt
Referenced by:: (25.6.10), (25.6.8), (25.6.9), §25.6(i), §25.6.
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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23: 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.

24: 20.11 Generalizations and Analogs

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20.11.1

G(m,n)=\sum\limits_{k=0}^{n-1}e^{-\pi ik^{2}m/n};

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Defines:: $G(m,n)$ : Gauss sum (locally)
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $m$ : integer and $n$ : integer
Permalink:: http://dlmf.nist.gov/20.11.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §20.11(i), §20.11 and Ch.20

… ►However, in this case

q

is no longer regarded as an independent complex variable within the unit circle, because

k

is related to the variable

\tau=\tau(k)

of the theta functions via (20.9.2). … ►For applications to rapidly convergent expansions for

\pi

see Chudnovsky and Chudnovsky (1988), and for applications in the construction of elliptic-hypergeometric series see Rosengren (2004). … ►For specialization to the one-dimensional theta functions treated in the present chapter, see Rauch and Lebowitz (1973) and §21.7(iii). … ►Such sets of twelve equations include derivatives, differential equations, bisection relations, duplication relations, addition formulas (including new ones for theta functions), and pseudo-addition formulas. …

25: 25.11 Hurwitz Zeta Function

… ►The Riemann zeta function is a special case: … ►When

a=\frac{1}{2}

, (25.11.10) reduces to (25.8.3); compare (25.11.11). ►For other series expansions similar to (25.11.10) see Coffey (2008). … ►When

a=1

, (25.11.35) reduces to (25.2.3). … ►uniformly with respect to bounded nonnegative values of

\alpha

. …

26: Bibliography C

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F. Calogero (1978) Asymptotic behaviour of the zeros of the (generalized) Laguerre polynomial

{L}_{n}^{\alpha}(x)

as the index

\alpha\rightarrow\infty

and limiting formula relating Laguerre polynomials of large index and large argument to Hermite polynomials. Lett. Nuovo Cimento (2) 23 (3), pp. 101–102.

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External Links:: ISSN 0024-1318, Document, MathReview (R. A. Askey)
Referenced by:: §18.7(iii).
Encodings:: BibTeX

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R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

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External Links:: ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt
Referenced by:: §26.8(vii).
Encodings:: BibTeX

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W. J. Cody (1991) Performance evaluation of programs related to the real gamma function. ACM Trans. Math. Software 17 (1), pp. 46–54.

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External Links:: ISSN 0098-3500, Document, MathReview, ZentralBlatt
Referenced by:: §5.24(ii), §5.24(iii).
Encodings:: BibTeX

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M. Colman, A. Cuyt, and J. Van Deun (2011) Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

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External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §13.29(v), §15.19(v).
Encodings:: BibTeX

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M. D. Cooper, R. H. Jeppesen, and M. B. Johnson (1979) Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.

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External Links:: Document
Referenced by:: 5th item.
Encodings:: BibTeX

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27: 36.5 Stokes Sets

… ►The Stokes set is itself a cusped curve, connected to the cusp of the bifurcation set: … ►They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§36.4). … ►The first sheet corresponds to

x<0

and is generated as a solution of Equations (36.5.6)–(36.5.9). … ►This consists of three separate cusp-edged sheets connected to the cusp-edged sheets of the bifurcation set, and related by rotation about the

z

-axis by

2\pi/3

. … ►Red and blue numbers in each region correspond, respectively, to the numbers of real and complex critical points that contribute to the asymptotics of the canonical integral away from the bifurcation sets. …

28: 26.13 Permutations: Cycle Notation

… ►

\sigma\in\mathfrak{S}_{n}

is a one-to-one and onto mapping from

\{1,2,\ldots,n\}

to itself. … ►They are related to Stirling numbers of the first kind by …See §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations. … ►A permutation that consists of a single cycle of length

k

can be written as the composition of

k-1

two-cycles (read from right to left): … ►A permutation is even or odd according to the parity of the number of transpositions. …

29: Bibliography K

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R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:: ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:: Document, MathReview (Matti Jutila), ZentralBlatt
Referenced by:: §25.18(i).
Encodings:: BibTeX

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A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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External Links:: ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt
Referenced by:: §32.11(iv).
Encodings:: BibTeX

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S. Koizumi (1976) Theta relations and projective normality of Abelian varieties. Amer. J. Math. 98 (4), pp. 865–889.

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External Links:: FullText, MathReview (T. Oda), ZentralBlatt
Referenced by:: §21.6(ii).
Encodings:: BibTeX

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T. H. Koornwinder (2007b) The structure relation for Askey-Wilson polynomials. J. Comput. Appl. Math. 207 (2), pp. 214–226.

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External Links:: ISSN 0377-0427, Document, Link, MathReview Entry
Referenced by:: (18.9.18_5).
Encodings:: BibTeX

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30: Bibliography I

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K. Inkeri (1959) The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

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External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.12(i).
Encodings:: BibTeX

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M. E. H. Ismail, D. R. Masson, and M. Rahman (Eds.) (1997) Special Functions,

q

-Series and Related Topics. Fields Institute Communications, Vol. 14, American Mathematical Society, Providence, RI.

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External Links:: ISBN 0-8218-0524-X, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: Profile Mourad E.H. Ismail.
Encodings:: BibTeX

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M. E. H. Ismail and D. R. Masson (1991) Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math. 21 (1), pp. 359–375.

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External Links:: ISSN 0035-7596, MathReview (Joaquin Bustoz), ZentralBlatt
Referenced by:: §18.30(i).
Encodings:: BibTeX

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M. E. H. Ismail and M. E. Muldoon (1995) Bounds for the small real and purely imaginary zeros of Bessel and related functions. Methods Appl. Anal. 2 (1), pp. 1–21.

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External Links:: ISSN 1073-2772, FullText, MathReview (Andrea Laforgia), ZentralBlatt
Referenced by:: §10.21(v).
Encodings:: BibTeX

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A. R. Its and A. A. Kapaev (1987) The method of isomonodromic deformations and relation formulas for the second Painlevé transcendent. Izv. Akad. Nauk SSSR Ser. Mat. 51 (4), pp. 878–892, 912 (Russian).

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Notes:: In Russian. English translation: Math. USSR-Izv. 31(1988), no. 1, pp. 193–207
External Links:: ISSN 0373-2436, Document, MathReview (Alexander A. Pankov), ZentralBlatt
Referenced by:: §32.11(iii).
Encodings:: BibTeX

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