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1: 30.9 Asymptotic Approximations and Expansions

§30.9 Asymptotic Approximations and Expansions

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§30.9(i) Prolate Spheroidal Wave Functions

… ►The asymptotic behavior of ${\lambda }_n^m\left({\gamma }^2\right)$ and $a_{n,k}^m({\gamma }^2)$ as $n\to \mathrm{\infty }$ in descending powers of $2n+1$ is derived in Meixner (1944). …The asymptotic behavior of ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ and ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$ as $x\to \pm 1$ is given in Erdélyi et al. (1955, p. 151). …

2: 20 Theta Functions

Chapter 20 Theta Functions

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3: Bibliography B

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Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

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External Links:

Document

Referenced by:

§33.22(iv).

Encodings:

BibTeX

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Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

1st item, 4th item.

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Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.73(iv).

Encodings:

BibTeX

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Asymptotic behavior of the Pollaczek polynomials and their zeros. Stud. Appl. Math. 96, pp. 307–338.

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External Links:

ISSN 0022-2526, MathReview (Harold Exton), ZentralBlatt

Referenced by:

§18.35(iii).

Encodings:

BibTeX

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The behavior at unit argument of the hypergeometric function ${}_3{}^{}F_2^{}$ . SIAM J. Math. Anal. 18 (5), pp. 1227–1234.

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External Links:

ISSN 0036-1410, Document, MathReview (K. M. Saksena), ZentralBlatt

Referenced by:

§16.8(ii).

Encodings:

BibTeX

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4: Wolter Groenevelt

… ► 1976 in Leidschendam, the Netherlands) is an Associate Professor at the Delft University of Technology in Delft, The Netherlands. … ► in mathematics at the Delft University of Technology in 2004. … ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …

5: Ingram Olkin

… ► 2016) was Professor Emeritus of Statistics and Education in the Department of Statistics at Stanford University, California. … ►Olkin’s research covered a broad range of areas, including multivariate analysis, reliability theory, matrix theory, statistical models in the social and behavioral sciences, life distributions, and meta-analysis. …

6: 14.21 Definitions and Basic Properties

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§14.21(iii) Properties

… ►This includes, for example, the Wronskian relations (14.2.7)–(14.2.11); hypergeometric representations (14.3.6)–(14.3.10) and (14.3.15)–(14.3.20); results for integer orders (14.6.3)–(14.6.5), (14.6.7), (14.6.8), (14.7.6), (14.7.7), and (14.7.11)–(14.7.16); behavior at singularities (14.8.7)–(14.8.16); connection formulas (14.9.11)–(14.9.16); recurrence relations (14.10.3)–(14.10.7). …

7: 14.8 Behavior at Singularities

§14.8 Behavior at Singularities

… ►The behavior of ${𝖯}_{\nu }^{\mu }\left(x\right)$ and ${𝖰}_{\nu }^{\mu }\left(x\right)$ as $x\to -1+$ follows from the above results and the connection formulas (14.9.8) and (14.9.10). … ►

14.8.16 $${𝑸}_{-n-(1/2)}^{\mu }\left(x\right)\sim \frac{{\pi }^{1/2}\mathrm{\Gamma }\left(\mu +n+\frac{1}{2}\right)}{n!\mathrm{\Gamma }\left(\mu -n+\frac{1}{2}\right){(2x)}^{n+(1/2)}},$$ $n=1,2,3,\mathrm{\dots }$, $\mu -n+\frac{1}{2}\ne 0,-1,-2,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $x$: real variable, $n$: nonnegative integer and $\mu$: general order

Referenced by:

§14.21(iii), §14.8(iii)

Permalink:

http://dlmf.nist.gov/14.8.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §14.8(iii), §14.8 and Ch.14

8: Gergő Nemes

… ► 1988 in Szeged, Hungary) is a Research Fellow at the Alfréd Rényi Institute of Mathematics in Budapest, Hungary. … ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …

9: Peter L. Walker

… ►was Professor of Mathematics at the American University of Sharjah, Sharjah, United Arab Emirates, in 1997–2005. He began his academic career in 1964 at the University of Lancaster, U. …Since 1984 he has also taught at other Persian Gulf universities, including Sultan Qaboos University, Oman. … ►

20 Theta Functions

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10: William P. Reinhardt

… ► 1942 in San Francisco, California) is Professor of Chemistry and Adjunct Professor of Physics at the University of Washington, Seattle, currently Emeritus. His undergraduate and graduate degrees are from the University of California at Berkeley and Harvard University, respectively. …Reinhardt is a frequent visitor to the NIST Physics Laboratory in Gaithersburg, and to the Joint Quantum Institute (JQI) and Institute for Physical Sciences and Technology (ISTP) at the University of Maryland. … ►

20 Theta Functions

… ►In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.