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21: Bibliography N

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An extension of Laplace’s method. Constr. Approx. 51 (2), pp. 247–272.

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External Links:

ISSN 0176-4276, Document, Link, MathReview (José Luis López), ZentralBlatt

Referenced by:

(11.11.16), (11.11.17), §11.11(iii).

Encodings:

BibTeX

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An extension of Turán’s inequality. Math. Inequal. Appl. 18 (1), pp. 321–335.

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External Links:

ISSN 1331-4343, Document, Link, MathReview (Gradimir V. Milovanovic), ZentralBlatt

Referenced by:

§18.14(ii).

Encodings:

BibTeX

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22: Frank W. J. Olver

… ►He is particularly known for his extensive work in the study of the asymptotic solution of differential equations, i. …

23: 1.12 Continued Fractions

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§1.12(iv) Contraction and Extension

… ►Conversely, $C$ is called an extension of $C^{\prime }$. …

24: 19.14 Reduction of General Elliptic Integrals

… ►It then improves the classical method by first applying Hermite reduction to (19.2.3) to arrive at integrands without multiple poles and uses implicit full partial-fraction decomposition and implicit root finding to minimize computing with algebraic extensions. …A similar remark applies to the transformations given in Erdélyi et al. (1953b, §13.5) and to the choice among explicit reductions in the extensive table of Byrd and Friedman (1971), in which one limit of integration is assumed to be a branch point of the integrand at which the integral converges. …

25: 15.12 Asymptotic Approximations

… ►For this result and an extension to an asymptotic expansion with error bounds see Jones (2001). … ►For $U(a,z)$ see §12.2, and for an extension to an asymptotic expansion see Olde Daalhuis (2003a). … ►For $\mathrm{Ai}\left(z\right)$ see §9.2, and for further information and an extension to an asymptotic expansion see Olde Daalhuis (2003b). … ►For other extensions, see Wagner (1986), Temme (2003) and Temme (2015, Chapters 12 and 28).

26: 36.6 Scaling Relations

… ►For the results in this section and more extensive lists of exponents see Berry (1977) and Varčenko (1976).

27: 11.11 Asymptotic Expansions of Anger–Weber Functions

… ►For sharp error bounds and exponentially-improved extensions, see Nemes (2018). … ►The later references also contain exponentially-improved extensions of (11.11.8) and (11.11.10). For an extension of (11.11.17) (and (11.11.16)) into a complete asymptotic expansion, see Nemes (2020). …

28: 2.10 Sums and Sequences

… ►In both expansions the remainder term is bounded in absolute value by the first neglected term in the sum, and has the same sign, provided that in the case of (2.10.7), truncation takes place at $s=2m-1$, where $m$ is any positive integer satisfying $m\ge \frac{1}{2}(\alpha +1)$. ►For extensions of the Euler–Maclaurin formula to functions $f(x)$ with singularities at $x=a$ or $x=n$ (or both) see Sidi (2004, 2012b, 2012a). … ►For an extension to integrals with Cauchy principal values see Elliott (1998). … ►For extensions to $\alpha \le 0$, higher terms, and other examples, see Olver (1997b, Chapter 8). … ►For other examples and extensions see Olver (1997b, Chapter 8), Olver (1970), Wong (1989, Chapter 2), and Wong and Wyman (1974). …

29: 4.10 Integrals

… ►Extensive compendia of indefinite and definite integrals of logarithms and exponentials include Apelblat (1983, pp. 16–47), Bierens de Haan (1939), Gröbner and Hofreiter (1949, pp. 107–116), Gröbner and Hofreiter (1950, pp. 52–90), Gradshteyn and Ryzhik (2000, Chapters 2–4), and Prudnikov et al. (1986a, §§1.3, 1.6, 2.3, 2.6).

30: 5.17 Barnes’ $G$-Function (Double Gamma Function)

… ►For error bounds and an exponentially-improved extension, see Nemes (2014a). …