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21: 10.20 Uniform Asymptotic Expansions for Large Order

§10.20 Uniform Asymptotic Expansions for Large Order

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10.20.5 $$Y_{\nu }\left(\nu z\right)\sim -{\left(\frac{4\zeta }{1-z^2}\right)}^{\frac{1}{4}}\left(\frac{\mathrm{Bi}\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{1}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{A_k(\zeta )}{{\nu }^{2k}}+\frac{{\mathrm{Bi}}^{\prime }\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{5}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{B_k(\zeta )}{{\nu }^{2k}}\right),$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\sim$: Poincaré asymptotic expansion, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\zeta (z)$: solution, $A_k(\zeta )$: coefficients and $B_k(\zeta )$: coefficients

A&S Ref:

9.3.36

Permalink:

http://dlmf.nist.gov/10.20.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.20(i), §10.20 and Ch.10

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10.20.9 $$\begin{array}{c}H_{\nu }^{(1)}{}_{}{}^{\prime }\left(\nu z\right)\\ H_{\nu }^{(2)}{}_{}{}^{\prime }\left(\nu z\right)\end{array}\}\sim \frac{4{\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}}{z}{\left(\frac{1-z^2}{4\zeta }\right)}^{\frac{1}{4}}\left(\frac{{\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}\mathrm{Ai}\left({\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}{\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{4}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{C_k(\zeta )}{{\nu }^{2k}}+\frac{{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}{\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{2}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{D_k(\zeta )}{{\nu }^{2k}}\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\zeta (z)$: solution, $C_k(\zeta )$: coefficients and $D_k(\zeta )$: coefficients

A&S Ref:

9.3.45

Referenced by:

§10.20(ii)

Permalink:

http://dlmf.nist.gov/10.20.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.20(i), §10.20 and Ch.10

… ►For asymptotic properties of the expansions (10.20.4)–(10.20.6) with respect to large values of $z$ see §10.41(v).

22: 33.20 Expansions for Small $|ϵ|$

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§33.20(iv) Uniform Asymptotic Expansions

►For a comprehensive collection of asymptotic expansions that cover $f(ϵ,\mathrm{\ell };r)$ and $h(ϵ,\mathrm{\ell };r)$ as $ϵ\to 0\pm$ and are uniform in $r$, including unbounded values, see Curtis (1964a, §7). …

23: 8.12 Uniform Asymptotic Expansions for Large Parameter

§8.12 Uniform Asymptotic Expansions for Large Parameter

… ►A different type of uniform expansion with coefficients that do not possess a removable singularity at $z=a$ is given by … ►Lastly, a uniform approximation for $\mathrm{\Gamma }(a,ax)$ for large $a$, with error bounds, can be found in Dunster (1996a). ►For other uniform asymptotic approximations of the incomplete gamma functions in terms of the function $\mathrm{erfc}$ see Paris (2002b) and Dunster (1996a). ►

Inverse Function

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24: Bibliography F

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Uniform asymptotic expansions for hypergeometric functions with large parameters IV. Anal. Appl. (Singap.) 12 (6), pp. 667–710.

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External Links:

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

Referenced by:

§15.12(iii), §15.12(iii).

Encodings:

BibTeX

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Uniform asymptotic expansions of certain classes of Meijer $G$-functions for a large parameter. SIAM J. Math. Anal. 4 (3), pp. 482–507.

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External Links:

Document, MathReview (C. M. Joshi), ZentralBlatt

Referenced by:

§16.22.

Encodings:

BibTeX

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Uniform asymptotic expansions of a class of Meijer $G$-functions for a large parameter. SIAM J. Math. Anal. 14 (6), pp. 1204–1253.

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External Links:

ISSN 0036-1410, Document, MathReview (Roop N. Kesarwani), ZentralBlatt

Referenced by:

§16.22.

Encodings:

BibTeX

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A uniform asymptotic expansion of the Jacobi polynomials with error bounds. Canad. J. Math. 37 (5), pp. 979–1007.

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External Links:

ISSN 0008-414X, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.15(i), §18.16(ii).

Encodings:

BibTeX

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Uniform asymptotic expansions of Laguerre polynomials. SIAM J. Math. Anal. 19 (5), pp. 1232–1248.

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External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§18.15(iv).

Encodings:

BibTeX

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25: Bibliography W

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Uniform asymptotics of the Stieltjes-Wigert polynomials via the Riemann-Hilbert approach. J. Math. Pures Appl. (9) 85 (5), pp. 698–718.

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External Links:

ISSN 0021-7824, Document, MathReview (Rabah Khaldi), ZentralBlatt

Referenced by:

§18.29.

Encodings:

BibTeX

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Estimates for the error term in a uniform asymptotic expansion of the Jacobi polynomials. Anal. Appl. (Singap.) 1 (2), pp. 213–241.

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External Links:

ISSN 0219-5305, Document, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§18.15(i), §18.15(i).

Encodings:

BibTeX

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Uniform asymptotic expansion of the Jacobi polynomials in a complex domain. Proc. Roy. Soc. London Ser. A 460, pp. 2569–2586.

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External Links:

ISSN 1364-5021, Document, MathReview (Pet”r Rusev), ZentralBlatt

Referenced by:

§18.15(i).

Encodings:

BibTeX

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On a uniform treatment of Darboux’s method. Constr. Approx. 21 (2), pp. 225–255.

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External Links:

ISSN 0176-4276, Document, MathReview Entry, ZentralBlatt

Referenced by:

§2.10(iv).

Encodings:

BibTeX

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On uniform asymptotic expansion of definite integrals. J. Approximation Theory 7 (1), pp. 76–86.

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External Links:

Document, MathReview (L. Berg), ZentralBlatt

Referenced by:

§8.11(v).

Encodings:

BibTeX

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26: 8.25 Methods of Computation

… ►DiDonato and Morris (1986) describes an algorithm for computing $P(a,x)$ and $Q(a,x)$ for $a\ge 0$, $x\ge 0$, and $a+x\ne 0$ from the uniform expansions in §8.12. …

27: 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

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Modified Expansions

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28: 13.22 Zeros

… ►Asymptotic approximations to the zeros when the parameters $\kappa$ and/or $\mu$ are large can be found by reversion of the uniform approximations provided in §§13.20 and 13.21. …

29: 31.16 Mathematical Applications

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§31.16(i) Uniformization Problem for Heun’s Equation

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30: 34.8 Approximations for Large Parameters

… ►Uniform approximations in terms of Airy functions for the $\mathit{3}j$ and $\mathit{6}j$ symbols are given in Schulten and Gordon (1975b). …