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11: 8.20 Asymptotic Expansions of $E_p\left(z\right)$

… ►Where the sectors of validity of (8.20.2) and (8.20.3) overlap the contribution of the first term on the right-hand side of (8.20.3) is exponentially small compared to the other contribution; compare §2.11(ii). ►For an exponentially-improved asymptotic expansion of $E_p\left(z\right)$ see §2.11(iii). …

12: 13.19 Asymptotic Expansions for Large Argument

… ►Error bounds and exponentially-improved expansions are derivable by combining §§13.7(ii) and 13.7(iii) with (13.14.2) and (13.14.3). … ►

13: 2.11 Remainder Terms; Stokes Phenomenon

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§2.11(iii) Exponentially-Improved Expansions

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§2.11(v) Exponentially-Improved Expansions (continued)

… ►For another approach see Paris (2001a, b). ►

§2.11(vi) Direct Numerical Transformations

… ►Further improvements in accuracy can be realized by making a second application of the Euler transformation; see Olver (1997b, pp. 540–543). …

14: Bille C. Carlson

… ►Symmetric integrals and their degenerate cases allow greatly shortened integral tables and improved algorithms for numerical computation. …

15: 7.12 Asymptotic Expansions

… ►For these and other error bounds see Olver (1997b, pp. 109–112), with $\alpha =\frac{1}{2}$ and $z$ replaced by $z^2$; compare (7.11.2). ►For re-expansions of the remainder terms leading to larger sectors of validity, exponential improvement, and a smooth interpretation of the Stokes phenomenon, see §§2.11(ii)–2.11(iv) and use (7.11.3). … ►They are bounded by $|\csc \left(4\mathrm{ph}z\right)|$ times the first neglected terms when . … ►For exponentially-improved expansions use (7.5.7), (7.5.10), and §7.12(i). …

16: Bibliography N

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Error bounds and exponential improvement for Hermite’s asymptotic expansion for the gamma function. Appl. Anal. Discrete Math. 7 (1), pp. 161–179.

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External Links:

ISSN 1452-8630, Document, FullText, MathReview (Junesang Choi), ZentralBlatt

Referenced by:

§5.11(ii), §5.11(ii).

Encodings:

BibTeX

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Error bounds and exponential improvement for the asymptotic expansion of the Barnes $G$-function. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470 (2172), pp. 20140534, 14.

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External Links:

ISSN 1364-5021, MathReview (Mehdi Hassani), ZentralBlatt

Referenced by:

§5.17, §5.17.

Encodings:

BibTeX

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Error bounds and exponential improvements for the asymptotic expansions of the gamma function and its reciprocal. Proc. Roy. Soc. Edinburgh Sect. A 145 (3), pp. 571–596.

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External Links:

ISSN 0308-2105, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§5.11(i), §5.11(i).

Encodings:

BibTeX

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17: 5.11 Asymptotic Expansions

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§5.11(ii) Error Bounds and Exponential Improvement

… ►For error bounds for (5.11.8) and an exponentially-improved extension, see Nemes (2013b). … ►For re-expansions of the remainder terms in (5.11.1) and (5.11.3) in series of incomplete gamma functions with exponential improvement (§2.11(iii)) in the asymptotic expansions, see Berry (1991), Boyd (1994), and Paris and Kaminski (2001, §6.4). …

18: 8.11 Asymptotic Approximations and Expansions

… ►where $\delta$ denotes an arbitrary small positive constant. … ►For an exponentially-improved asymptotic expansion (§2.11(iii)) see Olver (1991a). … ►Sharp error bounds and an exponentially-improved extension for (8.11.7) can be found in Nemes (2016). … ►For error bounds and an exponentially-improved extension for this later expansion, see Nemes (2015c). … ►For sharp error bounds and an exponentially-improved extension, see Nemes (2016). …

19: 12.9 Asymptotic Expansions for Large Variable

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§12.9(ii) Bounds and Re-Expansions for the Remainder Terms

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20: 7.20 Mathematical Applications

… ►The complementary error function also plays a ubiquitous role in constructing exponentially-improved asymptotic expansions and providing a smooth interpretation of the Stokes phenomenon; see §§2.11(iii) and 2.11(iv). …