double asymptotic property

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1: 10.69 Uniform Asymptotic Expansions for Large Order

… ►All fractional powers take their principal values. ►All four expansions also enjoy the same kind of double asymptotic property described in §10.41(iv). …

2: 10.41 Asymptotic Expansions for Large Order

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§10.41(iv) Double Asymptotic Properties

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§10.41(v) Double Asymptotic Properties (Continued)

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3: 10.20 Uniform Asymptotic Expansions for Large Order

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§10.20(iii) Double Asymptotic Properties

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4: 10.74 Methods of Computation

… ►Moreover, because of their double asymptotic properties (§10.41(v)) these expansions can also be used for large $x$ or $|z|$, whether or not $\nu$ is large. …

5: 12.10 Uniform Asymptotic Expansions for Large Parameter

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$${𝖠}_2(\tau )=\frac{1}{288}{\tau }^2(6160{\tau }^4+18480{\tau }^3+19404{\tau }^2+8028\tau +945).$$

… ►In addition, it enjoys a double asymptotic property: it holds if either or both $\mu$ and $t$ tend to infinity. …The proof of the double asymptotic property then follows with the aid of error bounds; compare §10.41(iv). …

6: Mathematical Introduction

… ►In addition, there is a comprehensive account of the great variety of analytical methods that are used for deriving and applying the extremely important asymptotic properties of the special functions, including double asymptotic properties (Chapter 2 and §§10.41(iv), 10.41(v)). …

7: 13.21 Uniform Asymptotic Approximations for Large $\kappa$

… ►For asymptotic expansions having double asymptotic properties see Skovgaard (1966). …

8: 2.1 Definitions and Elementary Properties

§2.1 Definitions and Elementary Properties

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§2.1(iii) Asymptotic Expansions

… ►If the set $𝐗$ in §2.1(iii) is a closed sector $\alpha \le \mathrm{ph}x\le \beta$, then by definition the asymptotic property (2.1.13) holds uniformly with respect to $\mathrm{ph}x\in [\alpha ,\beta ]$ as $|x|\to \mathrm{\infty }$. The asymptotic property may also hold uniformly with respect to parameters. … ►As in §2.1(iv), generalized asymptotic expansions can also have uniformity properties with respect to parameters. …

9: Bibliography N

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The resurgence properties of the large order asymptotics of the Anger-Weber function I. J. Class. Anal. 4 (1), pp. 1–39.

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External Links:

ISSN 1848-5979, Document, Link, MathReview (José Luis López), ZentralBlatt

Referenced by:

(11.11.10), (11.11.8), §11.11(iii).

Encodings:

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The resurgence properties of the large order asymptotics of the Anger-Weber function II. J. Class. Anal. 4 (2), pp. 121–147.

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External Links:

ISSN 1848-5979, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

(11.11.10), (11.11.8), §11.11(iii).

Encodings:

BibTeX

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Notes:

Double precision. In Japanese, with some English.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index.

Encodings:

BibTeX

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Notes:

Single and double precision. See Kahaner et al. (1989).

External Links:

GAMS (package NMS)

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

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Evaluation of negative energy Coulomb (Whittaker) functions. Comput. Phys. Comm. 159 (1), pp. 55–62.

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Notes:

Double-precision Fortran-90 code.

External Links:

Document, Code

Referenced by:

§33.23(vi), 8th item.

Encodings:

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10: Bibliography S

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Some properties of polynomial sets of type zero. Duke Math. J. 5, pp. 590–622.

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External Links:

ISSN 0012-7094, Link, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§18.2(xii).

Encodings:

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On the double points of a Mathieu equation. J. Comput. Appl. Math. 107 (1), pp. 111–125.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§28.7.

Encodings:

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Staudt and arithmetical properties of Bernoulli numbers. Historia Sci. (2) 5 (1), pp. 69–74.

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External Links:

ISSN 0285-4821, MathReview (Peter M. Neumann), ZentralBlatt

Referenced by:

§24.10(ii).

Encodings:

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Some combinatorial properties of Jack symmetric functions. Adv. Math. 77 (1), pp. 76–115.

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External Links:

ISSN 0001-8708, Document, MathReview (Dennis White), ZentralBlatt

Referenced by:

§18.37(iii).

Encodings:

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The relation between asymptotic properties of the second Painlevé equation in different directions towards infinity. Differ. Uravn. 23 (5), pp. 834–842 (Russian).

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Notes:

In Russian. English translation: Differential Equations 23(1987), no. 5, pp. 569–576.

External Links:

ISSN 0374-0641, MathReview (T. Chanturiya), ZentralBlatt

Referenced by:

§32.11(ii).

Encodings:

BibTeX

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