Schläfli–Sommerfeld integrals

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11: Bibliography R

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Composite approximations to the solutions of the Orr-Sommerfeld equation. Studies in Appl. Math. 51, pp. 341–368.

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MathReview (E. A. Boyd), ZentralBlatt

Referenced by:

(9.13.25), (9.13.26), (9.13.28), (9.13.29), (9.13.32), (9.13.33), (9.13.34), §9.13(ii), §9.13(ii).

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Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow. Studies in Appl. Math. 53, pp. 91–110.

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MathReview (T. W. Kao), ZentralBlatt

Referenced by:

§2.8(iii).

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Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. II. The general theory. Studies in Appl. Math. 53, pp. 217–224.

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MathReview (T. W. Kao), ZentralBlatt

Referenced by:

§2.8(iii).

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Computation of Hankel (Bessel) functions of complex index and argument by numerical integration of a Schläfli contour integral. Ž. Vyčisl. Mat. i Mat. Fiz. 13, pp. 1415–1424, 1636.

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English translation in U.S.S.R. Computational Math. and Math. Phys. 13(6), pp. 58–67

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ISSN 0044-4669, Document, MathReview (H. Zegeling), ZentralBlatt

Referenced by:

§10.74(iii).

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Dawson’s integral and the sampling theorem. Computers in Physics 3 (2), pp. 85–87.

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Referenced by:

§7.25(vi).

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12: 10.9 Integral Representations

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Schläfli’s and Related Integrals

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§10.9(ii) Contour Integrals

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Schläfli–Sommerfeld Integrals

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Schläfli’s Integral

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13: 9.16 Physical Applications

… ►The function $\mathrm{Ai}\left(x\right)$ first appears as an integral in two articles by G. … ►A quite different application is made in the study of the diffraction of sound pulses by a circular cylinder (Friedlander (1958)). … ►In the study of the stability of a two-dimensional viscous fluid, the flow is governed by the Orr–Sommerfeld equation (a fourth-order differential equation). …

14: 28.18 Integrals and Integral Equations

§28.18 Integrals and Integral Equations

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15: 9.13 Generalized Airy Functions

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§9.13(ii) Generalizations from Integral Representations

►Reid (1972) and Drazin and Reid (1981, Appendix) introduce the following contour integrals in constructing approximate solutions to the Orr–Sommerfeld equation for fluid flow: … ►and the difference equation … ►Connection formulas for the solutions of (9.13.31) include … ►

16: Bibliography S

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Gaussian-transform method for molecular integrals. I. Formulation for energy integrals. J. Chem. Phys. 43 (2), pp. 398–414.

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Document, MathReview (R. K. Nesbet)

Referenced by:

§8.24(i).

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Algorithm AS 239. Chi-squared and incomplete gamma integral. Appl. Statist. 37 (3), pp. 466–473.

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Notes:

Double-precision Fortran.

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FullText, Code

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5th item.

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Non-linear integral equations for Heun functions. Proc. Edinburgh Math. Soc. (2) 16, pp. 281–289.

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Document, MathReview (J. D. DePree), ZentralBlatt

Referenced by:

§31.10(ii).

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The Use of Integral Transforms. McGraw-Hill, New York.

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ISBN 00-705-9436-8, ZentralBlatt

Referenced by:

§1.14(v), §10.43(v), §14.31(ii).

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Atombau und Spektrallinien. Vieweg, Braunschweig.

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Notes:

English translation: Wave Mechanics published by Methuen, London, 1930.

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ZentralBlatt

Referenced by:

§33.1.

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17: 33.1 Special Notation

… ►The main functions treated in this chapter are first the Coulomb radial functions $F_{\mathrm{\ell }}(\eta ,\rho )$, $G_{\mathrm{\ell }}(\eta ,\rho )$, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ (Sommerfeld (1928)), which are used in the case of repulsive Coulomb interactions, and secondly the functions $f(ϵ,\mathrm{\ell };r)$, $h(ϵ,\mathrm{\ell };r)$, $s(ϵ,\mathrm{\ell };r)$, $c(ϵ,\mathrm{\ell };r)$ (Seaton (1982, 2002a)), which are used in the case of attractive Coulomb interactions. …

18: 19.35 Other Applications

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§19.35(i) Mathematical

►Generalizations of elliptic integrals appear in analysis of modular theorems of Ramanujan (Anderson et al. (2000)); analysis of Selberg integrals (Van Diejen and Spiridonov (2001)); use of Legendre’s relation (19.7.1) to compute $\pi$ to high precision (Borwein and Borwein (1987, p. 26)). ►

§19.35(ii) Physical

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19: 6.1 Special Notation

… ►Unless otherwise noted, primes indicate derivatives with respect to the argument. ►The main functions treated in this chapter are the exponential integrals $\mathrm{Ei}\left(x\right)$, $E_1\left(z\right)$, and $\mathrm{Ein}\left(z\right)$; the logarithmic integral $\mathrm{li}\left(x\right)$; the sine integrals $\mathrm{Si}\left(z\right)$ and $\mathrm{si}\left(z\right)$; the cosine integrals $\mathrm{Ci}\left(z\right)$ and $\mathrm{Cin}\left(z\right)$.

20: 25.7 Integrals

§25.7 Integrals

►For definite integrals of the Riemann zeta function see Prudnikov et al. (1986b, §2.4), Prudnikov et al. (1992a, §3.2), and Prudnikov et al. (1992b, §3.2).