central in imaginary direction

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11: Staff

… ►Frank W. J. Olver [December 15, 1924-April 23, 2013] served as Editor-in-Chief and Mathematics Editor for the DLMF project from its inception until his death on April 23, 2013. … ►

Ian J. Thompson, Lawrence Livermore National Laboratory, Chap. 33

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Mourad E. H. Ismail, University of Central Florida

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Ian J. Thompson, Lawrence Livermore National Laboratory, for Chap. 33

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12: 18.39 Applications in the Physical Sciences

§18.39 Applications in the Physical Sciences

… ►The spectrum is entirely discrete as in §1.18(v). … ►In what follows the radial and spherical radial eigenfunctions corresponding to (18.39.27) are found in four different notations, with identical eigenvalues, all of which appear in the current and past mathematical and theoretical physics and chemistry literatures, regarding this central problem. … ►noting that the ${\psi }_{p,l}(r)$ are real, follows from the fact that the Schrödinger operator of (18.39.28) is self-adjoint, or from the direct derivation of Dunkl (2003). … ►

§18.39(iii) Non Classical Weight Functions of Utility in DVR Method in the Physical Sciences

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13: 10.74 Methods of Computation

… ►A comprehensive and powerful approach is to integrate the differential equations (10.2.1) and (10.25.1) by direct numerical methods. … ►In the interval , $J_{\nu }\left(x\right)$ needs to be integrated in the forward direction and $Y_{\nu }\left(x\right)$ in the backward direction, with initial values for the former obtained from the power-series expansion (10.2.2) and for the latter from asymptotic expansions (§§10.17(i) and 10.20(i)). In the interval either direction of integration can be used for both functions. ►Similarly, to maintain stability in the interval the integration direction has to be forwards in the case of $I_{\nu }\left(x\right)$ and backwards in the case of $K_{\nu }\left(x\right)$, with initial values obtained in an analogous manner to those for $J_{\nu }\left(x\right)$ and $Y_{\nu }\left(x\right)$. … ►

§10.74(viii) Functions of Imaginary Order

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14: 15.19 Methods of Computation

… ►For $z\in ℂ$ it is possible to use the linear transformations in such a way that the new arguments lie within the unit circle, except when $z={\mathrm{e}}^{\pm \pi \mathrm{i}/3}$. …However, by appropriate choice of the constant $z_0$ in (15.15.1) we can obtain an infinite series that converges on a disk containing $z={\mathrm{e}}^{\pm \pi \mathrm{i}/3}$. … ►A comprehensive and powerful approach is to integrate the hypergeometric differential equation (15.10.1) by direct numerical methods. … ►Gauss quadrature approximations are discussed in Gautschi (2002b). … ►For example, in the half-plane $\mathrm{\Re }z\le \frac{1}{2}$ we can use (15.12.2) or (15.12.3) to compute $F(a,b;c+N+1;z)$ and $F(a,b;c+N;z)$, where $N$ is a large positive integer, and then apply (15.5.18) in the backward direction. …

15: 18.2 General Orthogonal Polynomials

… ►In the former case we also require … ►If the orthogonality interval is $(-\mathrm{\infty },\mathrm{\infty })$ or $(0,\mathrm{\infty })$, then the role of $d/dx$ can be played by ${\delta }_x$, the central-difference operator in the imaginary direction (§18.1(i)). … ►For usage of the zeros of an OP in Gauss quadrature see §3.5(v). … ►Alternatives for numerical calculation of the recursion coefficients in terms of the moments are discussed in these references, and in §18.40(ii). … ►for $x,y$ in the support of the orthogonality measure and $z$ such that the series in (18.2.41) converges absolutely for all these $x,y$. …

16: Bibliography C

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Conduction of Heat in Solids. 2nd edition, Clarendon Press, Oxford.

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External Links:

ISBN 0198533039

Referenced by:

§7.21, Ch.7.

Encodings:

BibTeX

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Caustics in cylindrical ducts. Proc. Roy. Soc. London Ser. A 455, pp. 2529–2548.

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External Links:

ISSN 1364-5021, Document, MathReview, ZentralBlatt

Referenced by:

§36.14(iv).

Encodings:

BibTeX

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Algorithm AS 310. Computing the non-central beta distribution function. Appl. Statist. 46 (1), pp. 146–156.

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Notes:

Single-precision Fortran.

External Links:

FullText, Code, ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

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$\gamma$ rays from an extranuclear direct capture process. Nuclear Physics 24 (1), pp. 89–101.

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External Links:

Document

Referenced by:

§33.22(v).

Encodings:

BibTeX

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Some non-central distribution problems in multivariate analysis. Ann. Math. Statist. 34 (4), pp. 1270–1285.

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External Links:

FullText, MathReview (I. Olkin), ZentralBlatt

Referenced by:

§35.4(i), §35.4(ii).

Encodings:

BibTeX

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17: 16.25 Methods of Computation

… ►They are similar to those described for confluent hypergeometric functions, and hypergeometric functions in §§13.29 and 15.19. There is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations). This occurs when the wanted solution is intermediate in asymptotic growth compared with other solutions. In these cases integration, or recurrence, in either a forward or a backward direction is unstable. …

18: Sidebar 9.SB1: Supernumerary Rainbows

… ►The faint line below the main colored arc is a ‘supernumerary rainbow’, produced by the interference of different sun-rays traversing a raindrop and emerging in the same direction. …Airy invented his function in 1838 precisely to describe this phenomenon more accurately than Young had done in 1800 when pointing out that supernumerary rainbows require the wave theory of light and are impossible to explain with Newton’s picture of light as a stream of independent corpuscles. The house in the picture is Newton’s birthplace. …

19: Brian D. Sleeman

… ► 1939 in London, U. … ►Sleeman published numerous papers in applied analysis, multiparameter spectral theory, direct and inverse scattering theory, and mathematical medicine. He is author of the book Multiparameter spectral theory in Hilbert space, published by Pitman in 1978, and coauthor (with D. … Plank) of Differential equations and mathematical biology, published by CRC Press in 2003, with a second edition in 2010. ►Sleeman was elected a Fellow of the Royal Society of Edinburgh in 1976 and is the founding editor of the journal Computational and Mathematical Methods in Medicine. …

20: Peter A. Clarkson

… ► 1957 in Westow, U. … ► Kruskal, he developed the “direct method” for determining symmetry solutions of partial differential equations in New similarity reductions of the Boussinesq equation (with M. … Ablowitz) was published by Cambridge University Press in 1991. … Tuszyński), published by Oxford University Press in 1997. … ►Institute of Physics in 1999, and of the U. …