double asymptotic property

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1: 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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2: 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). …

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)

Referenced by:

(11.11.10), (11.11.8), §11.11(iii).

Encodings:

BibTeX

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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

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:

BibTeX

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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:

BibTeX

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10: Bibliography F

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Algorithm 838: Airy functions. ACM Trans. Math. Software 30 (4), pp. 491–501.

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Notes:

Single- and double-precision Fortran, maximum accuracy 32S.

External Links:

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

Referenced by:

3rd item, 2nd item, 1st item.

Encodings:

BibTeX

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Notes:

The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.

External Links:

GAMS (package FDLIBM)

Referenced by:

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

Encodings:

BibTeX

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An asymptotic expansion of the double gamma function. J. Approx. Theory 111 (2), pp. 298–314.

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External Links:

ISSN 0021-9045, Document, MathReview (C. M. Joshi), ZentralBlatt

Referenced by:

§5.17.

Encodings:

BibTeX

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The transformation properties of the sixth Painlevé equation and one-parameter families of solutions. Lett. Nuovo Cimento (2) 30 (17), pp. 539–544.

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External Links:

ISSN 0024-1318, MathReview (D. A. Lutz)

Referenced by:

§32.10(vi).

Encodings:

BibTeX

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Computing the hypergeometric function. J. Comput. Phys. 137 (1), pp. 79–100.

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Notes:

Double-precision Fortran

External Links:

ISSN 0021-9991, Document, Code, MathReview (J. P. Singhal), ZentralBlatt

Referenced by:

§15.19(i), 1st item, 2nd item.

Encodings:

BibTeX

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