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21: 10.40 Asymptotic Expansions for Large Argument

§10.40 Asymptotic Expansions for Large Argument

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Products

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$\nu$-Derivative

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§10.40(iv) Exponentially-Improved Expansions

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22: 11.13 Methods of Computation

… ►Although the power-series expansions (11.2.1) and (11.2.2), and the Bessel-function expansions of §11.4(iv) converge for all finite values of $z$, they are cumbersome to use when $|z|$ is large owing to slowness of convergence and cancellation. For large $|z|$ and/or $|\nu |$ the asymptotic expansions given in §11.6 should be used instead. … ►Then from the limiting forms for small argument (§§11.2(i), 10.7(i), 10.30(i)), limiting forms for large argument (§§11.6(i), 10.7(ii), 10.30(ii)), and the connection formulas (11.2.5) and (11.2.6), it is seen that ${𝐇}_{\nu }\left(x\right)$ and ${𝐋}_{\nu }\left(x\right)$ can be computed in a stable manner by integrating forwards, that is, from the origin toward infinity. The solution ${𝐊}_{\nu }\left(x\right)$ needs to be integrated backwards for small $x$, and either forwards or backwards for large $x$ depending whether or not $\nu$ exceeds $\frac{1}{2}$. …

23: 28.34 Methods of Computation

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Use of asymptotic expansions and approximations for large $q$ (§§28.8(i), 28.16). See also Zhang and Jin (1996, pp. 482–485).

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Use of asymptotic expansions and approximations for large $q$ (§§28.8(ii)–28.8(iv)).

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Use of asymptotic expansions for large $z$ or large $q$. See §§28.25 and 28.26.

24: 29.16 Asymptotic Expansions

… ►The approximations for Lamé polynomials hold uniformly on the rectangle $0\le \mathrm{\Re }z\le K$, $0\le \mathrm{\Im }z\le K^{\prime }$, when $nk$ and $n{k}^{\prime }$ assume large real values. …

25: 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta \right|$

§33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta \right|$

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§33.10(i) Large $\rho$

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§33.10(ii) Large Positive $\eta$

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§33.10(iii) Large Negative $\eta$

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26: Ronald F. Boisvert

… ►He is also Editor at Large for the DLMF Project. …

27: 28.26 Asymptotic Approximations for Large $q$

§28.26 Asymptotic Approximations for Large $q$

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§28.26(ii) Uniform Approximations

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28: 12.11 Zeros

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§12.11(ii) Asymptotic Expansions of Large Zeros

… ►When $a>-\frac{1}{2}$ the zeros are asymptotically given by $z_{a,s}$ and $\overline{z_{a,s}}$, where $s$ is a large positive integer and … ►

§12.11(iii) Asymptotic Expansions for Large Parameter

►For large negative values of $a$ the real zeros of $U(a,x)$, $U^{\prime }(a,x)$, $V(a,x)$, and $V^{\prime }(a,x)$ can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). …

29: 28.8 Asymptotic Expansions for Large $q$

§28.8 Asymptotic Expansions for Large $q$

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§28.8(ii) Sips’ Expansions

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§28.8(iii) Goldstein’s Expansions

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Barrett’s Expansions

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30: Bibliography O

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Uniform asymptotic expansions for hypergeometric functions with large parameters. I. Analysis and Applications (Singapore) 1 (1), pp. 111–120.

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External Links:

ISSN 0219-5305, Document, MathReview, ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

BibTeX

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Uniform asymptotic expansions for hypergeometric functions with large parameters. II. Analysis and Applications (Singapore) 1 (1), pp. 121–128.

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External Links:

ISSN 0219-5305, Document, MathReview, ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

BibTeX

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Uniform asymptotic expansions for hypergeometric functions with large parameters. III. Analysis and Applications (Singapore) 8 (2), pp. 199–210.

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External Links:

ISSN 0219-5305, Document, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

BibTeX

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Some new asymptotic expansions for Bessel functions of large orders. Proc. Cambridge Philos. Soc. 48 (3), pp. 414–427.

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External Links:

Document, MathReview (F. Oberhettinger), ZentralBlatt

Referenced by:

§10.19(iii), §10.21(vii).

Encodings:

BibTeX

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Tables for Bessel Functions of Moderate or Large Orders. National Physical Laboratory Mathematical Tables, Vol. 6. Department of Scientific and Industrial Research, Her Majesty’s Stationery Office, London.

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External Links:

MathReview (I. A. Stegun), ZentralBlatt

Referenced by:

§10.20(i), §10.41(ii), 3rd item, 3rd item.

Encodings:

BibTeX

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