exponentially-improved expansions

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21: 13.29 Methods of Computation

… ►However, this accuracy can be increased considerably by use of the exponentially-improved forms of expansion supplied by the combination of (13.7.10) and (13.7.11), or by use of the hyperasymptotic expansions given in Olde Daalhuis and Olver (1995a). …

22: 11.11 Asymptotic Expansions of Anger–Weber Functions

§11.11 Asymptotic Expansions of Anger–Weber Functions

… ►For sharp error bounds and exponentially-improved extensions, see Nemes (2018). … ►where … ►The later references also contain exponentially-improved extensions of (11.11.8) and (11.11.10). …

23: 11.9 Lommel Functions

… ►and … ►

§11.9(ii) Expansions in Series of Bessel Functions

… ►

§11.9(iii) Asymptotic Expansion

►For fixed $\mu$ and $\nu$, … ►For an error bound for (11.9.9) and an exponentially-improved extension see Nemes (2015b). …

24: Bibliography D

… ►

Plancherel-Rotach asymptotic expansion for some polynomials from indeterminate moment problems. Constr. Approx. 40 (1), pp. 61–104.

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External Links:

ISSN 0176-4276, Document, Link, MathReview (Francisco Marcellán), ZentralBlatt

Referenced by:

§2.9(iii), §2.9(iii).

Encodings:

BibTeX

… ►

Asymptotic Expansions: Their Derivation and Interpretation. Academic Press, London-New York.

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External Links:

MathReview, ZentralBlatt

Referenced by:

(11.11.11), §11.6(i), §11.6(iii), §2.11(v), Ch.2, §8.1, Notations, Notations.

Encodings:

BibTeX

… ►

On Vandermonde’s theorem, and some more general expansions. Proc. Edinburgh Math. Soc. 25, pp. 114–132.

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External Links:

ZentralBlatt

Referenced by:

§15.4(ii).

Encodings:

BibTeX

… ►

Error bounds for exponentially improved asymptotic solutions of ordinary differential equations having irregular singularities of rank one. Methods Appl. Anal. 3 (1), pp. 109–134.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Roderick Wong), ZentralBlatt

Referenced by:

§2.11(v).

Encodings:

BibTeX

… ►

Uniform asymptotic expansions for Charlier polynomials. J. Approx. Theory 112 (1), pp. 93–133.

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External Links:

ISSN 0021-9045, Document, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§18.24.

Encodings:

BibTeX

…

25: 5.17 Barnes’ $G$-Function (Double Gamma Function)

… ►When $z\to \mathrm{\infty }$ in , ►

5.17.5 $$\mathrm{Ln}G\left(z+1\right)\sim \frac{1}{4}{z}^2+z\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)-\left(\frac{1}{2}z(z+1)+\frac{1}{12}\right)\ln z-\ln A+\sum _{k=1}^{\mathrm{\infty }}\frac{B_{2k+2}}{2k(2k+1)(2k+2)z^{2k}}.$$

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Symbols:

$G\left(z\right)$: Barnes’ $G$-function (or double gamma function), $B_n$: Bernoulli numbers, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer, $z$: complex variable and $A$: Glaisher’s constant

Referenced by:

§5.17, §5.17, Erratum (V1.0.7) for Equation (5.17.5), Erratum (V1.1.11) for Equation (5.17.5)

Permalink:

http://dlmf.nist.gov/5.17.E5

Encodings:

pMML, png, TeX

Errata (effective with 1.1.11):

For consistency we have replaced $\mathrm{Ln}z$ by $\ln z$.

Errata (effective with 1.0.7):

Originally the term $z\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)$ was incorrectly stated as $z\mathrm{\Gamma }\left(z+1\right)$. (This error was reported subsequently by Nick Jones on December 11, 2013.)

Reported 2013-08-01 by Gergő Nemes

See also:

Annotations for §5.17 and Ch.5

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

26: 25.11 Hurwitz Zeta Function

… ►The function $\zeta (s,a)$ was introduced in Hurwitz (1882) and defined by the series expansion … ►For other series expansions similar to (25.11.10) see Coffey (2008). … ►

§25.11(xii) $a$-Asymptotic Behavior

… ►As $a\to \mathrm{\infty }$ in the sector , with $s(\ne 1)$ and $\delta$ fixed, we have the asymptotic expansion … ►For an exponentially-improved form of (25.11.43) see Paris (2005b).