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11: Bibliography V

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Accurate calculation of prolate spheroidal radial functions of the first kind and their first derivatives. Quart. Appl. Math. 60 (3), pp. 589–599.

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Notes:

For software, see Van Buren (website)

External Links:

ISSN 0033-569X, MathReview (Alan M. Cohen), ZentralBlatt

Referenced by:

§30.16(iii).

Encodings:

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Mathieu and Spheroidal Wave Functions: Fortran Programs for their Accurate Calculation

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External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, A. L. Van Buren and J. E. Boisvert (2002), A. L. Van Buren and J. E. Boisvert (2004), A. L. Van Buren and J. E. Boisvert (2007).

Encodings:

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Rational approximations for exponential integrals $E_n(x)$ . Acad. Roy. Belg. Bull. Cl. Sci. (5) 56, pp. 1064–1072.

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External Links:

MathReview (A. Magnus), ZentralBlatt

Referenced by:

3rd item.

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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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Asymptotic expansion of the generalized hypergeometric function ${}_p{}^{}F_q^{}(z)$ as $z\to \mathrm{\infty }$ for . Anal. Appl. (Singap.) 21 (2), pp. 535–545.

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External Links:

ISSN 0219-5305, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§16.11(i), §16.11, Erratum (V1.1.9) for Section 16.11(i).

Encodings:

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12: 10.73 Physical Applications

… ►For this problem and its further generalizations, see Korenev (2002, Chapter 4, §37) and Gray et al. (1922, Chapter I, §1, Chapter XVI, §4). … ►See Jackson (1999, Chapter 3, §§3.7, 3.8, 3.11, 3.13), Lamb (1932, Chapter V, §§100–102; Chapter VIII, §§186, 191–193; Chapter X, §§303, 304), Happel and Brenner (1973, Chapter 3, §3.3; Chapter 7, §7.3), Korenev (2002, Chapter 4, §43), and Gray et al. (1922, Chapter XI). … ►Consequently, Bessel functions $J_n\left(x\right)$, and modified Bessel functions $I_n\left(x\right)$, are central to the analysis of microwave and optical transmission in waveguides, including coaxial and fiber. … ►See Korenev (2002). On separation of variables into cylindrical coordinates, the Bessel functions $J_n\left(x\right)$, and modified Bessel functions $I_n\left(x\right)$ and $K_n\left(x\right)$, all appear. …

13: 32.14 Combinatorics

… ►Let $S_N$ be the group of permutations $𝝅$ of the numbers $1,2,\mathrm{\dots },N$ (§26.2). With , $𝝅(m_1),𝝅(m_2),\mathrm{\dots },𝝅(m_n)$ is said to be an increasing subsequence of $𝝅$ of length $n$ when . Let ${\mathrm{\ell }}_N(𝝅)$ be the length of the longest increasing subsequence of $𝝅$. …and $w(x)$ satisfies ${\text{P}}_{\text{II}}$ with $\alpha =0$ and boundary conditions … ►See Forrester and Witte (2001, 2002) for other instances of Painlevé equations in random matrix theory.

14: 22.9 Cyclic Identities

… ►The following notation is a generalization of that of Khare and Sukhatme (2002). … ► … ►These identities are cyclic in the sense that each of the indices $m,n$ in the first product of, for example, the form $s_{m,p}^{(4)}{s}_{n,p}^{(4)}$ are simultaneously permuted in the cyclic order: $m\to m+1\to m+2\to \mathrm{\cdots }p\to 1\to 2\to \mathrm{\cdots }m-1$; $n\to n+1\to n+2\to \mathrm{\cdots }p\to 1\to 2\to \mathrm{\cdots }n-1$. … ►

22.9.23 $$s_{1,3}^{(4)}{d}_{1,3}^{(4)}{c}_{2,3}^{(4)}{c}_{3,3}^{(4)}+s_{2,3}^{(4)}{d}_{2,3}^{(4)}{c}_{3,3}^{(4)}{c}_{1,3}^{(4)}+s_{3,3}^{(4)}{d}_{3,3}^{(4)}{c}_{1,3}^{(4)}{c}_{2,3}^{(4)}=\frac{{\kappa }^2}{1-{\kappa }^2}\left(s_{1,3}^{(4)}{d}_{1,3}^{(4)}+s_{2,3}^{(4)}{d}_{2,3}^{(4)}+s_{2,3}^{(4)}{d}_{2,3}^{(4)}\right).$$

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Symbols:

$s_{m,p}^{(r)}$: cyclic quantity, $c_{m,p}^{(r)}$: cyclic quantity, $d_{m,p}^{(r)}$: cyclic quantity and $\kappa$: change of variable

Permalink:

http://dlmf.nist.gov/22.9.E23

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See also:

Annotations for §22.9(iv), §22.9(iv), §22.9 and Ch.22

… ►For extensions of the identities given in §§22.9(ii)–22.9(iv), and also to related elliptic functions, see Khare and Sukhatme (2002), Khare et al. (2003). …

15: 32.8 Rational Solutions

… ►Special rational solutions of ${\text{P}}_{\text{III}}$ are … ►Then ${\text{P}}_{\text{III}}$ has rational solutions iff … ►Special rational solutions of ${\text{P}}_{\text{IV}}$ are … ►Special rational solutions of ${\text{P}}_{\text{V}}$ are … ►For determinantal representations see Masuda et al. (2002). …

16: 4.44 Other Applications

… ►For an application of the Lambert $W$-function to generalized Gaussian noise see Chapeau-Blondeau and Monir (2002). …

17: Ronald F. Boisvert

… ►Boisvert was awarded the Outstanding Contribution to ACM Award in 2000, the Keene State College Alumni Achievement Award in 2002, the U. …

18: Bibliography K

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Quantum Calculus. Universitext, Springer-Verlag, New York.

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External Links:

ISBN 0-387-95341-8, MathReview (P. D. F. Ion), ZentralBlatt

Referenced by:

Ch.17.

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Replica field theories, Painlevé transcendents, and exact correlation functions. Phys. Rev. Lett. 89 (25), pp. (250201–1)–(250201–4).

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External Links:

ISSN 0031-9007, Document, MathReview, ZentralBlatt

Referenced by:

§32.16.

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Cyclic identities involving Jacobi elliptic functions. J. Math. Phys. 43 (7), pp. 3798–3806.

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External Links:

ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§22.9(i), §22.9(ii), §22.9(iii), §22.9(iv), §22.9(v).

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Jacobi Functions and Analysis on Noncompact Semisimple Lie Groups. In Special Functions: Group Theoretical Aspects and Applications, pp. 1–85.

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Notes:

Reprinted by Kluwer Academic Publishers, Boston, 2002.

External Links:

ISBN 90-277-1822-9; 1-4020-0319-6, MathReview (Jacques Faraut), ZentralBlatt

Referenced by:

§14.31(ii), §15.17(iii), §15.9(ii).

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Bessel Functions and their Applications. Analytical Methods and Special Functions, Vol. 8, Taylor & Francis Ltd., London-New York.

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Notes:

Translated from the Russian by E. V. Pankratiev

External Links:

ISBN 0-415-28130-X, MathReview (L. Debnath), ZentralBlatt

Referenced by:

§10.73(i), §10.73(i), §10.73(i).

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19: Bibliography M

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A determinant formula for a class of rational solutions of Painlevé V equation. Nagoya Math. J. 168, pp. 1–25.

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External Links:

ISSN 0027-7630, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.8(v).

Encodings:

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Exact misclassification probabilities for plug-in normal quadratic discriminant functions. II. The heterogeneous case. J. Multivariate Anal. 82 (2), pp. 299–330.

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External Links:

ISSN 0047-259X, Document, MathReview (Peter A. Lachenbruch), ZentralBlatt

Referenced by:

§35.9.

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Phenomenological equations of state for the quark-gluon plasma. Phys. Rev. D 65 (3), pp. (034009–1)–(034009–10).

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External Links:

Document

Referenced by:

§24.18.

Encodings:

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Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.

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External Links:

Document, ZentralBlatt

Referenced by:

§27.13(iv).

Encodings:

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CODATA recommended values of the fundamental physical constants: 2002. Rev. Mod.Phys. 77, pp. 1–107.

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Referenced by:

§18.39(ii).

Encodings:

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20: 32.16 Physical Applications

… ►See also McCoy et al. (1977), Jimbo et al. (1980), Essler et al. (1996), and Kanzieper (2002). …