asymptotic approximations for large parameters

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11: 34.8 Approximations for Large Parameters

§34.8 Approximations for Large Parameters

►For large values of the parameters in the $\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ symbols, different asymptotic forms are obtained depending on which parameters are large. … ►Semiclassical (WKBJ) approximations in terms of trigonometric or exponential functions are given in Varshalovich et al. (1988, §§8.9, 9.9, 10.7). Uniform approximations in terms of Airy functions for the $\mathit{3}j$ and $\mathit{6}j$ symbols are given in Schulten and Gordon (1975b). For approximations for the $\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ symbols with error bounds see Flude (1998), Chen et al. (1999), and Watson (1999): these references also cite earlier work.

12: 25.11 Hurwitz Zeta Function

… ►

§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 … ►Similarly, as $a\to \mathrm{\infty }$ in the sector , ►

25.11.44 $${\zeta }^{\prime }(-1,a)-\frac{1}{12}+\frac{1}{4}{a}^2-\left(\frac{1}{12}-\frac{1}{2}a+\frac{1}{2}{a}^2\right)\ln a\sim -\sum _{k=1}^{\mathrm{\infty }}\frac{B_{2k+2}}{(2k+2)(2k+1)2k}{a}^{-2k},$$

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Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $\sim$: Poincaré asymptotic expansion, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $a$: real or complex parameter

Keywords:

asymptoticapproximation

Source:

Elizalde (1986, (18), p. 349)

Permalink:

http://dlmf.nist.gov/25.11.E44

Encodings:

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See also:

Annotations for §25.11(xii), §25.11 and Ch.25

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25.11.45 $${\zeta }^{\prime }(-2,a)-\frac{1}{12}a+\frac{1}{9}{a}^3-\left(\frac{1}{6}a-\frac{1}{2}{a}^2+\frac{1}{3}{a}^3\right)\ln a\sim \sum _{k=1}^{\mathrm{\infty }}\frac{2B_{2k+2}}{(2k+2)(2k+1)2k(2k-1)}{a}^{-(2k-1)}.$$

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Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $\sim$: Poincaré asymptotic expansion, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $a$: real or complex parameter

Keywords:

asymptoticapproximation

Source:

Elizalde (1986, (19), p. 350)

Permalink:

http://dlmf.nist.gov/25.11.E45

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(xii), §25.11 and Ch.25

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13: 12.10 Uniform Asymptotic Expansions for Large Parameter

§12.10 Uniform Asymptotic Expansions for Large Parameter

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§12.10(vi) Modifications of Expansions in Elementary Functions

… ► … ►

Modified Expansions

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14: 14.26 Uniform Asymptotic Expansions

§14.26 Uniform Asymptotic Expansions

►The uniform asymptotic approximations given in §14.15 for $P_{\nu }^{-\mu }\left(x\right)$ and ${𝑸}_{\nu }^{\mu }\left(x\right)$ for are extended to domains in the complex plane in the following references: §§14.15(i) and 14.15(ii), Dunster (2003b); §14.15(iii), Olver (1997b, Chapter 12); §14.15(iv), Boyd and Dunster (1986). For an extension of §14.15(iv) to complex argument and imaginary parameters, see Dunster (1990b). ►See also Frenzen (1990), Gil et al. (2000), Shivakumar and Wong (1988), Ursell (1984), and Wong (1989) for uniform asymptotic approximations obtained from integral representations.

15: 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). … ►

12.11.4 $$u_{a,s}\sim 2^{\frac{1}{2}}\mu \left(p_0(\alpha )+\frac{p_1(\alpha )}{{\mu }^4}+\frac{p_2(\alpha )}{{\mu }^8}+\mathrm{\cdots }\right),$$

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Defines:

$u_{a,s}$: zeros (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $s$: nonnegative integer, $a$: real or complex parameter, $\alpha$ and $p_n(\zeta )$: coefficients

Referenced by:

§12.11(iii)

Permalink:

http://dlmf.nist.gov/12.11.E4

Encodings:

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See also:

Annotations for §12.11(iii), §12.11 and Ch.12

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16: 15.12 Asymptotic Approximations

§15.12 Asymptotic Approximations

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§15.12(i) Large Variable

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§15.12(ii) Large $c$

… ►As $\lambda \to \mathrm{\infty }$, … ►For other extensions, see Wagner (1986), Temme (2003) and Temme (2015, Chapters 12 and 28).

17: 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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Asymptotic approximations and error bounds. SIAM Rev. 22 (2), pp. 188–203.

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External Links:

ISSN 0036-1445, Document, MathReview (L. Berg), ZentralBlatt

Referenced by:

§2.7(iii).

Encodings:

BibTeX

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Whittaker functions with both parameters large: Uniform approximations in terms of parabolic cylinder functions. Proc. Roy. Soc. Edinburgh Sect. A 86 (3-4), pp. 213–234.

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External Links:

ISSN 0308-2105, MathReview (P. F. Hsieh), ZentralBlatt

Referenced by:

§13.20(iii), §13.20(iv).

Encodings:

BibTeX

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18: Bibliography T

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Laguerre polynomials: Asymptotics for large degree. Technical report Technical Report AM-R8610, CWI, Amsterdam, The Netherlands.

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External Links:

FullText

Referenced by:

§2.4(vi).

Encodings:

BibTeX

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On the computation of the incomplete gamma functions for large values of the parameters. In Algorithms for approximation (Shrivenham, 1985), Inst. Math. Appl. Conf. Ser. New Ser., Vol. 10, pp. 479–489.

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External Links:

FullText, MathReview (G. Meinardus), ZentralBlatt

Referenced by:

§8.25(iii).

Encodings:

BibTeX

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Computational aspects of incomplete gamma functions with large complex parameters. In Approximation and Computation. A Festschrift in Honor of Walter Gautschi, R. V. M. Zahar (Ed.), International Series of Numerical Mathematics, Vol. 119, pp. 551–562.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§8.11(iii), §8.20(ii), §8.25(i), §8.25(iii), 2nd item, 3rd item.

Encodings:

BibTeX

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Large parameter cases of the Gauss hypergeometric function. J. Comput. Appl. Math. 153 (1-2), pp. 441–462.

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External Links:

ISSN 0377-0427, Document, MathReview (J. P. Singhal), ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

BibTeX

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Asymptotic expansions of Kummer hypergeometric functions for large values of the parameters. Integral Transforms Spec. Funct. 33 (1), pp. 16–31.

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External Links:

ISSN 1065-2469, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§13.8(iv).

Encodings:

BibTeX

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19: 18.15 Asymptotic Approximations

§18.15 Asymptotic Approximations

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§18.15(i) Jacobi

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§18.15(ii) Ultraspherical

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§18.15(iii) Legendre

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§18.15(iv) Laguerre

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20: 18.26 Wilson Class: Continued

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§18.26(v) Asymptotic Approximations

►For asymptotic expansions of Wilson polynomials of large degree see Wilson (1991), and for asymptotic approximations to their largest zeros see Chen and Ismail (1998). ►Koornwinder (2009) rescales and reparametrizes Racah polynomials and Wilson polynomials in such a way that they are continuous in their four parameters, provided that these parameters are nonnegative. Moreover, if one or more of the new parameters becomes zero, then the polynomial descends to a lower family in the Askey scheme.