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1: 5.11 Asymptotic Expansions

… ►Wrench (1968) gives exact values of $g_k$ up to $g_{20}$. …For explicit formulas for $g_k$ in terms of Stirling numbers see Nemes (2013a), and for asymptotic expansions of $g_k$ as $k\to \mathrm{\infty }$ see Boyd (1994) and Nemes (2015a). ►

Terminology

►The expansion (5.11.1) is called Stirling’s series (Whittaker and Watson (1927, §12.33)), whereas the expansion (5.11.3), or sometimes just its leading term, is known as Stirling’s formula (Abramowitz and Stegun (1964, §6.1), Olver (1997b, p. 88)). …

2: Bibliography N

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On simplified asymptotic formulas for a class of Mathieu functions. SIAM J. Math. Anal. 15 (6), pp. 1205–1213.

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External Links:

ISSN 0036-1410, Document, MathReview (Harold Exton), ZentralBlatt

Referenced by:

§28.26(ii).

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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).

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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).

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An explicit formula for the coefficients in Laplace’s method. Constr. Approx. 38 (3), pp. 471–487.

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External Links:

ISSN 0176-4276, Document, Link, MathReview (Pedro J. Pagola), ZentralBlatt

Referenced by:

§5.11(i), §5.11(i).

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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).

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3: Bibliography G

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Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.

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ISSN 1023-6198, Document, FullText, MathReview (Thomas Britz), ZentralBlatt

Referenced by:

§17.7(iii).

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Construction of Gauss-Christoffel quadrature formulas. Math. Comp. 22, pp. 251–270.

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External Links:

ISSN 0025-5718, Document, Link, MathReview (R. E. Barnhill)

Referenced by:

§18.39(iii).

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §10.77(ix).

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Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.

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External Links:

FullText, MathReview (B. M. Stewart), ZentralBlatt

Referenced by:

§24.15(iii), §24.6.

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Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:

ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt

Referenced by:

§18.38(iii).

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4: 18.39 Applications in the Physical Sciences

… ►Here are three examples of solutions for (18.39.8) for explicit choices of $V(x)$ and with the ${\psi }_n(x)$ corresponding to the discrete spectrum. … ►Explicit normalization is given for the second, third, and fourth of these, paragraphs c) and d), below. … ►thus recapitulating, for $Z=1$, line 11 of Table 18.8.1, now shown with explicit normalization for the measure $dr$. … ►see Bethe and Salpeter (1957, p. 13), Pauling and Wilson (1985, pp. 130, 131); and noting that this differs from the Rodrigues formula of (18.5.5) for the Laguerre OP’s, in the omission of an $n!$ in the denominator. … ►Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry. …