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21: Bibliography V

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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).

Encodings:

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Expansion of vacuum magnetic fields in toroidal harmonics. Comput. Phys. Comm. 81 (1-2), pp. 74–90.

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External Links:

Document

Referenced by:

§14.31(i).

Encodings:

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Determinants and Their Applications in Mathematical Physics. Applied Mathematical Sciences, Vol. 134, Springer-Verlag, New York.

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External Links:

ISBN 0-387-98558-1, MathReview (Jorma Kaarlo Merikoski), ZentralBlatt

Referenced by:

§1.3(ii).

Encodings:

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Symbolic evaluation of coefficients in Airy-type asymptotic expansions. J. Math. Anal. Appl. 269 (1), pp. 317–331.

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External Links:

ISSN 0022-247X, Document, MathReview, ZentralBlatt

Referenced by:

§2.4(v).

Encodings:

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Expansions in products of Heine-Stieltjes polynomials. Constr. Approx. 15 (4), pp. 467–480.

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External Links:

ISSN 0176-4276, Document, MathReview (Luoqing Li), ZentralBlatt

Referenced by:

§31.15(iii).

Encodings:

BibTeX

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22: Bibliography U

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Mathematics of Computation Unpublished Mathematical Tables Collection.

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Notes:

Archives of American Mathematics, Center for American History, The University of Texas at Austin. Covers 1944–1994. An online guide is arranged chronologically by date of issuance within Mathematics of Computation

External Links:

Guide

Referenced by:

§27.21.

Encodings:

BibTeX

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23: Bibliography D

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Asymptotic Methods in Analysis. 2nd edition, Bibliotheca Mathematica, Vol. IV, North-Holland Publishing Co., Amsterdam.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§2.2, Ch.2, §26.7(iv), §26.7(iv), §4.13.

Encodings:

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Atomic motions in a rigid sphere gas as a problem in neutron transport. Nucl. Sci. Eng. 24 (2), pp. 142–152.

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Referenced by:

§18.39(iii).

Encodings:

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A fast new public code for computing photon orbits in a Kerr spacetime. The Astrophysical Journal 696, pp. 1616–1629.

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External Links:

Document

Referenced by:

§19.35(ii).

Encodings:

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Bernoulli Numbers. Bibliography (1713–1990). Queen’s Papers in Pure and Applied Mathematics, Vol. 87, Queen’s University, Kingston, ON.

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Notes:

A frequently updated version, with links to reviews, exists online

External Links:

Online version, MathReview, ZentralBlatt

Referenced by:

Ch.24.

Encodings:

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Über die Bestimmung der mittleren Werthe in der Zahlentheorie. Abhandlungen der Königlich Preussischen Akademie der Wissenschaften von 1849, pp. 69–83 (German).

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Notes:

Reprinted in G. Lejeune Dirichlet’s Werke, Band II, pp. 49–66, Reimer, Berlin, 1897 and with corrections in G. Lejeune Dirichlet’s Werke, AMS Chelsea Publishing Series, Providence, RI, 1969.

External Links:

ISBN 0-8284-0225-6, Link, MathReview

Referenced by:

§27.11.

Encodings:

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24: Bibliography O

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On the asymptotics for late coefficients in uniform asymptotic expansions of integrals with coalescing saddles. Methods Appl. Anal. 7 (4), pp. 727–745.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Raimundas Vidunas), ZentralBlatt

Referenced by:

§36.12(iii).

Encodings:

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Triorthogonal systems in spaces of constant curvature in which the equation ${\mathrm{\Delta }}_{2}u+\lambda u=0$ allows a complete separation of variables. Mat. Sbornik N.S. 27(69) (3), pp. 379–426 (Russian).

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Notes:

In Russian.

External Links:

MathReview (H. Busemann), ZentralBlatt

Referenced by:

§31.17(i).

Encodings:

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Computing $\pi (x)$: The combinatorial method. Revista do DETUA 4 (6), pp. 759–768.

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Notes:

Online fulltext contains postpublication annotations.

External Links:

FullText

Referenced by:

§27.18.

Encodings:

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A paradox in asymptotics. SIAM J. Math. Anal. 1 (4), pp. 533–534.

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External Links:

Document, MathReview (L. Hoischen), ZentralBlatt

Referenced by:

§2.10(iv).

Encodings:

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Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York.

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Notes:

Reprinted in 2000 by the Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pennsylvania

External Links:

MathReview (J. F. Traub), ZentralBlatt (Nelli Dimitrova (Sofia))

Referenced by:

§3.8(vii).

Encodings:

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25: 19.9 Inequalities

… ►Throughout this subsection , except in (19.9.4). …The left-hand inequalities in (19.9.2) and (19.9.3) are equivalent, but the right-hand inequality of (19.9.3) is sharper than that of (19.9.2) when . …The lower bound in (19.9.4) is sharper than $2/\pi$ when $0\le k^2\le 0.9960$. … ►Further inequalities for $K\left(k\right)$ and $E\left(k\right)$ can be found in Alzer and Qiu (2004), Anderson et al. (1992a, b, 1997), and Qiu and Vamanamurthy (1996). … ►Inequalities for both $F(\varphi ,k)$ and $E(\varphi ,k)$ involving inverse circular or inverse hyperbolic functions are given in Carlson (1961b, §4). …

26: 19.5 Maclaurin and Related Expansions

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19.5.1 $$K\left(k\right)=\frac{\pi }{2}\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}\right)}_m{\left(\frac{1}{2}\right)}_m}{m!m!}{k}^{2m}=\frac{\pi }{2}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\frac{1}{2},\frac{1}{2}}{1};k^2),$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $!$: factorial (as in $n!$), $m$: nonnegative integer and $k$: real or complex modulus

Source:

Carlson (1961b, (2.4))

Referenced by:

§19.36(iv), §19.5

Permalink:

http://dlmf.nist.gov/19.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.5 and Ch.19

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19.5.2 $$E\left(k\right)=\frac{\pi }{2}\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(-\frac{1}{2}\right)}_m{\left(\frac{1}{2}\right)}_m}{m!m!}{k}^{2m}=\frac{\pi }{2}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-\frac{1}{2},\frac{1}{2}}{1};k^2),$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $!$: factorial (as in $n!$), $m$: nonnegative integer and $k$: real or complex modulus

Source:

Carlson (1961b, (2.5))

Referenced by:

§19.36(iv)

Permalink:

http://dlmf.nist.gov/19.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.5 and Ch.19

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19.5.3 $$D\left(k\right)=\frac{\pi }{4}\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(\frac{3}{2}\right)}_m{\left(\frac{1}{2}\right)}_m}{(m+1)!m!}{k}^{2m}=\frac{\pi }{4}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\frac{3}{2},\frac{1}{2}}{2};k^2),$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $D\left(k\right)$: complete elliptic integral of Legendre’s type, $!$: factorial (as in $n!$), $m$: nonnegative integer and $k$: real or complex modulus

Permalink:

http://dlmf.nist.gov/19.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.5 and Ch.19

… ►Coefficients of terms up to ${\lambda }^{49}$ are given in Lee (1990), along with tables of fractional errors in $K\left(k\right)$ and $E\left(k\right)$, $0.1\le k^2\le 0.9999$, obtained by using 12 different truncations of (19.5.6) in (19.5.8) and (19.5.9). … ►Series expansions of $F(\varphi ,k)$ and $E(\varphi ,k)$ are surveyed and improved in Van de Vel (1969), and the case of $F(\varphi ,k)$ is summarized in Gautschi (1975, §1.3.2). …

27: Bibliography W

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Surface Waves. In Handbuch der Physik, Vol. 9, Part 3, pp. 446–778.

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External Links:

Online edition, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§11.12.

Encodings:

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Lectures on Quantum Mechanics. Cambridge University Press, Cambridge, UK.

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External Links:

ISBN 978-1-107-02872-2

Referenced by:

§18.39(ii).

Encodings:

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Harmonic Analysis. In Studies in Real and Complex Analysis, I. I. Hirschman (Ed.), Studies in Mathematics, Vol. 3, pp. 124–178.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§1.15(iii), §1.15(iv), §1.15(v).

Encodings:

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On the functions associated with the parabolic cylinder in harmonic analysis. Proc. London Math. Soc. 35, pp. 417–427.

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External Links:

Document, ZentralBlatt

Referenced by:

§12.1, §12.5(i), §12.5(ii), §12.9(i).

Encodings:

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Applications of some recent results in asymptotic expansions. Congr. Numer. 37, pp. 145–182.

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External Links:

ISSN 0384-9864, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§19.27(vi).

Encodings:

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28: 28.33 Physical Applications

… ►Mathieu functions occur in practical applications in two main categories: … ►Physical problems involving Mathieu functions include vibrational problems in elliptical coordinates; see (28.32.1). We shall derive solutions to the uniform, homogeneous, loss-free, and stretched elliptical ring membrane with mass $\rho$ per unit area, and radial tension $\tau$ per unit arc length. …In elliptical coordinates (28.32.2) becomes (28.32.3). … ►More complete bibliographies will be found in McLachlan (1947) and Meixner and Schäfke (1954). …

29: 3.11 Approximation Techniques

… ►This is because in the notation of §3.11(i) … ►with coefficients given in Table 3.11.1. … ►The error curve is shown in Figure 3.11.1. … ►In consequence we can solve the system … ►is of fundamental importance in the FFT algorithm. …

30: Bibliography M

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Hypergeometric Functions.

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Notes:

Unpublished Lecture Notes, University of London. An online version exists.

External Links:

FullText

Referenced by:

§35.7(ii), §35.8(iii).

Encodings:

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On the Representation of Meijer’s $G$-Function in the Vicinity of Singular Unity. In Complex Analysis and Applications ’81 (Varna, 1981), pp. 383–398.

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External Links:

MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§16.21.

Encodings:

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Geometrical and Catastrophe Optics Methods in Scattering. In Physical Acoustics, A. D. Pierce and R. N. Thurston (Eds.), Vol. 21, pp. 1–234.

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Referenced by:

§36.14(ii).

Encodings:

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A $q$-analog of hypergeometric series well-poised in $\mathrm{𝑆𝑈}(n)$ and invariant $G$-functions. Adv. in Math. 58 (1), pp. 1–60.

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External Links:

ISSN 0001-8708, Document, MathReview (Robert A. Gustafson), ZentralBlatt

Referenced by:

§17.15.

Encodings:

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A $q$-analog of the Gauss summation theorem for hypergeometric series in $U(n)$ . Adv. in Math. 72 (1), pp. 59–131.

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External Links:

ISSN 0001-8708, Document, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.15.

Encodings:

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