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11: 20 Theta Functions

Chapter 20 Theta Functions

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12: 36 Integrals with Coalescing Saddles

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13: 28 Mathieu Functions and Hill’s Equation

Chapter 28 Mathieu Functions and Hill’s Equation

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14: 10.73 Physical Applications

… ►The Helmholtz equation, $({\nabla }^2+k^2)\psi =0$, follows from the wave equation …This equation governs problems in acoustic and electromagnetic wave propagation. …See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25). … ►

§10.73(ii) Spherical Bessel Functions

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15: 30.9 Asymptotic Approximations and Expansions

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§30.9(i) Prolate Spheroidal Wave Functions

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§30.9(ii) Oblate Spheroidal Wave Functions

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§30.9(iii) Other Approximations and Expansions

… ►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). …

16: 30.10 Series and Integrals

… ►Integrals and integral equations for ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ are given in Arscott (1964b, §8.6), Erdélyi et al. (1955, §16.13), Flammer (1957, Chapter 5), and Meixner (1951). …

17: 8 Incomplete Gamma and Related
Functions

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18: Bibliography N

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On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

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External Links:

ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§28.26(ii).

Encodings:

BibTeX

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Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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External Links:

FullText, MathReview (I. A. Stegun), ZentralBlatt

Referenced by:

§19.37(iv).

Encodings:

BibTeX

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Scattering theory of waves and particles. Dover Publications, Inc., Mineola, NY.

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Notes:

Reprint of the 1982 second edition [Springer, New York; MR0666397 (84f:81001)], with list of errata prepared for this edition by the author

External Links:

ISBN 0-486-42535-5, MathReview Entry

Referenced by:

§1.18(vi), §1.18(viii).

Encodings:

BibTeX

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A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.

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Notes:

See for additions and corrections same journal, Sect. B 75, 149–163 (1975)

External Links:

MathReview, ZentralBlatt

Referenced by:

§7.14(iii), §7.21, §7.7(iii).

Encodings:

BibTeX

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Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations. Institute of Physics Publishing, Bristol.

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External Links:

ISBN 0-7503-0610-6, MathReview (Peter Giblin), ZentralBlatt

Referenced by:

§36.14(ii).

Encodings:

BibTeX

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19: 8.26 Tables

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Khamis (1965) tabulates $P(a,x)$ for $a=0.05(.05)10(.1)20(.25)70$, $0.0001\le x\le 250$ to 10D.

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Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_n\left(x\right)$ for $n=2,3,4,10,20$, $x=0(.01)2$ to 7D; also $(x+n){\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2,3,4,10,20$, $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

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Pagurova (1961) tabulates $E_n\left(x\right)$ for $n=0(1)20$, $x=0(.01)2(.1)10$ to 4-9S; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ for $p=0(.1)1$, $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

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Zhang and Jin (1996, Table 19.1) tabulates $E_n\left(x\right)$ for $n=1,2,3,5,10,15,20$, $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

20: 23 Weierstrass Elliptic and Modular
Functions

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