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1: 18.16 Zeros

… ►Let $j_{\alpha ,m}$ be the $m$th positive zero of the Bessel function $J_{\alpha }\left(x\right)$ (§10.21(i)). … ►Let ${\varphi }_m=j_{\alpha ,m}/\rho$. … ►For three additional terms in this expansion see Gatteschi (2002). … ►For further information on the zeros of the classical orthogonal polynomials, see Szegő (1975, Chapter VI), Erdélyi et al. (1953b, §§10.16 and 10.17), Gatteschi (1987, 2002), López and Temme (1999a), and Temme (1990a). ►

§18.16(vii) Discriminants

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2: 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).

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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).

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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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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).

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Four remarks on eigenvalues of Lamé’s equation. Anal. Appl. (Singap.) 2 (2), pp. 161–175.

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External Links:

ISSN 0219-5305, Document, MathReview, ZentralBlatt

Referenced by:

§29.3(ii), §29.3(vii), §29.5, §29.7(i), §29.7(i).

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3: 32.7 Bäcklund Transformations

… ►satisfies ${\text{P}}_{\text{V}}$ with … ►

§32.7(vii) Sixth Painlevé Equation

►Let $w_j(z_j)=w_j(z_j;{\alpha }_j,{\beta }_j,{\gamma }_j,{\delta }_j)$, $j=0,1,2,3$, be solutions of ${\text{P}}_{\text{VI}}$ with … ► ${\text{P}}_{\text{VI}}$ also has quadratic and quartic transformations. …Also, …

4: 25.21 Software

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§25.21(vii) Fermi–Dirac and Bose–Einstein Integrals

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5: 19.2 Definitions

… ►where $p_j$ is a polynomial in $t$ while $\rho$ and $\sigma$ are rational functions of $t$. … ►Here $a,b,p$ are real parameters, and $k_c$ and $x$ are real or complex variables, with $p\ne 0$, $k_c\ne 0$. … ►If , then $k_c$ is pure imaginary. … ►

§19.2(iv) A Related Function: $R_C(x,y)$

… ►For the special cases of $R_C(x,x)$ and $R_C(0,y)$ see (19.6.15). …

6: 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).

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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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On the $G$-function. VII, VIII. Nederl. Akad. Wetensch., Proc. 49, pp. 1063–1072, 1165–1175 = Indagationes Math. 8, 661–670, 713–723 (1946).

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§16.11(ii).

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Jacobian Elliptic Function Tables. Dover Publications Inc., New York.

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External Links:

MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§22.20(vii).

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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).

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7: 35.9 Applications

… ►In multivariate statistical analysis based on the multivariate normal distribution, the probability density functions of many random matrices are expressible in terms of generalized hypergeometric functions of matrix argument ${}_p{}^{}F_q^{}$, with $p\le 2$ and $q\le 1$. … ►For other statistical applications of ${}_p{}^{}F_q^{}$ functions of matrix argument see Perlman and Olkin (1980), Groeneboom and Truax (2000), Bhaumik and Sarkar (2002), Richards (2004) (monotonicity of power functions of multivariate statistical test criteria), Bingham et al. (1992) (Procrustes analysis), and Phillips (1986) (exact distributions of statistical test criteria). … ►For applications of the integral representation (35.5.3) see McFarland and Richards (2001, 2002) (statistical estimation of misclassification probabilities for discriminating between multivariate normal populations). The asymptotic approximations of §35.7(iv) are applied in numerous statistical contexts in Butler and Wood (2002). …

8: Bibliography W

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Prime Divisors of the Bernoulli and Euler Numbers. In Number Theory for the Millennium, III (Urbana, IL, 2000), pp. 357–374.

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External Links:

FullText, MathReview (Arnold M. Adelberg), ZentralBlatt

Referenced by:

§24.20.

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Uniform asymptotic expansion of $J_{\nu }(\nu a)$ via a difference equation. Numer. Math. 91 (1), pp. 147–193.

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External Links:

ISSN 0029-599X, Document, MathReview (J. P. Singhal), ZentralBlatt

Referenced by:

§10.20(ii).

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The Algebraic Eigenvalue Problem. Monographs on Numerical Analysis. Oxford Science Publications, The Clarendon Press, Oxford University Press, Oxford.

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Notes:

Paperback reprint of original edition, published in 1965 by Clarendon Press, Oxford.

External Links:

ISBN 0-19-853418-3, MathReview, ZentralBlatt

Referenced by:

§3.2(v), §3.2(vi), §3.2(vii).

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Computation with Recurrence Relations. Pitman, Boston, MA.

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External Links:

ISBN 0-273-08508-5, MathReview (S. Conde), ZentralBlatt

Referenced by:

§13.29(iv), §16.25, §2.9(iii), §3.10(iii), §3.6(iii), §3.6(v), §3.6(vii).

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Quadrature formulas for oscillatory integral transforms. Numer. Math. 39 (3), pp. 351–360.

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External Links:

ISSN 0029-599X, Document, MathReview (A.J. Rodrigues), ZentralBlatt

Referenced by:

§10.74(vii), §3.5(vii).

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9: Bibliography G

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Asymptotics and bounds for the zeros of Laguerre polynomials: A survey. J. Comput. Appl. Math. 144 (1-2), pp. 7–27.

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External Links:

ISSN 0377-0427, Document, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§18.16(iv), §18.16(iv), §18.16(vi).

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The triplets of helium. Philos. Trans. Roy. Soc. London Ser. A 228, pp. 151–196.

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External Links:

Document, ZentralBlatt (Möglich, Dr. F.)

Referenced by:

§34.3(vii).

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Matrix Computations. 3rd edition, Johns Hopkins University Press, Baltimore, MD.

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External Links:

ISBN 0-8018-5413-X; 0-8018-5414-8, MathReview, ZentralBlatt

Referenced by:

§3.2(i), §3.2(i), §3.2(ii), §3.2(vii).

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Constructing wavefunctions for nonlocal potentials. J. Chem. Phys. 52, pp. 6211–6217.

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External Links:

Document, MathReview (D. B. Fairlie)

Referenced by:

(9.12.22), (9.12.23), §9.12(vii), §9.17(iii).

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Painlevé Differential Equations in the Complex Plane. Studies in Mathematics, Vol. 28, Walter de Gruyter & Co., Berlin-New York.

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External Links:

ISBN 3-11-017379-4, MathReview, ZentralBlatt

Referenced by:

§32.10(i), §32.10(iii), §32.10(iv), §32.10(v), §32.10(vi), §32.2(iv), §32.7(iii), §32.7(iv), §32.7(v), §32.7(vi), §32.7(vii), §32.8(ii), §32.8(iii), §32.8(iv), §32.8(v), §32.9(i), §32.9(ii), §32.9(iii).

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10: Bibliography H

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Fast Hankel transform algorithm. IEEE Trans. Acoust. Speech Signal Process. 32 (3), pp. 666–671.

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External Links:

ISSN 0096-3518, FullText, ZentralBlatt

Referenced by:

§10.74(vii).

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A Course of Pure Mathematics. 10th edition, Cambridge University Press.

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Notes:

Numerous reprintings exist, including the Centenary Edition with Foreword by T. W. Körner (Cambridge Univerity Press, 2008).

External Links:

MathReview, ZentralBlatt

Referenced by:

§1.4(ii), §1.4(iii), §1.4(iv), §1.4(v), §1.4(vi), §1.4(vii), §4.4(iii), §4.6(i), §4.6(ii).

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Analytic evaluation of two-center STO electron repulsion integrals via ellipsoidal expansion. Internat. J. Quantum Chem. 88 (6), pp. 701–734.

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External Links:

Document

Referenced by:

§8.24(iii).

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Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. 2nd edition, Prentice Hall Inc., Upper Saddle River, NJ.

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External Links:

ISBN 0-13-041408-5; 978-0-13-041408-3, MathReview (W. P. Ziemer), ZentralBlatt

Referenced by:

§1.6(i).

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Coulomb Wave Functions. In Handbuch der Physik, Bd. 41/1, S. Flügge (Ed.), pp. 408–465.

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External Links:

MathReview (G. Källén)

Referenced by:

§33.2(iii), §33.23(vii), §33.23(vii), §33.24, §33.5(ii), §33.7, Ch.33.

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