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41: 18.32 OP’s with Respect to Freud Weights

… ►For a uniform asymptotic expansion in terms of Airy functions (§9.2) for the OP’s in the case $Q(x)=x^4$ see Bo and Wong (1999). …

42: 36.12 Uniform Approximation of Integrals

§36.12 Uniform Approximation of Integrals

… ►The canonical integrals (36.2.4) provide a basis for uniform asymptotic approximations of oscillatory integrals. … ►The leading-order uniform asymptotic approximation is given by …

43: Bibliography J

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Uniform asymptotic expansions for Meixner polynomials. Constr. Approx. 14 (1), pp. 113–150.

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External Links:

ISSN 0176-4276, Document, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§18.24, §2.4(vi).

Encodings:

BibTeX

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Derivation of Green-type, transitional and uniform asymptotic expansions from differential equations. V. Angular oblate spheroidal wavefunctions ${\overline{ps}}_n^r(\eta ,h)$ and ${\overline{qs}}_n^r(\eta ,h)$ for large $h$ . Proc. Roy. Soc. London Ser. A 321, pp. 545–555.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§30.9(ii).

Encodings:

BibTeX

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44: Bibliography N

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Uniform asymptotic expansion for the incomplete beta function. SIGMA Symmetry Integrability Geom. Methods Appl. 12, pp. 101, 5 pages.

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External Links:

ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§8.18(ii), §8.18(ii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v).

Encodings:

BibTeX

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Uniform Asymptotic Approximations of Solutions of Second-order Linear Differential Equations, with a Coalescing Simple Turning Point and Simple Pole. Ph.D. Thesis, University of Maryland, College Park, MD.

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Referenced by:

§2.8(vi).

Encodings:

BibTeX

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45: 2.1 Definitions and Elementary Properties

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§2.1(iv) Uniform Asymptotic Expansions

… ►As in §2.1(iv), generalized asymptotic expansions can also have uniformity properties with respect to parameters. …

46: 13.20 Uniform Asymptotic Approximations for Large $\mu$

§13.20 Uniform Asymptotic Approximations for Large $\mu$

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§13.20(i) Large $\mu$, Fixed $\kappa$

… ►For uniform approximations valid when $\mu$ is large, $x/\mathrm{i}\in (0,\mathrm{\infty })$, and $\kappa /\mathrm{i}\in [0,\mu /\delta ]$, see Olver (1997b, pp. 401–403). … ►For uniform approximations of $M_{\kappa ,\mathrm{i}\mu }\left(z\right)$ and $W_{\kappa ,\mathrm{i}\mu }\left(z\right)$, $\kappa$ and $\mu$ real, one or both large, see Dunster (2003a). …

47: 19.33 Triaxial Ellipsoids

… ►Let a homogeneous magnetic ellipsoid with semiaxes $a,b,c$, volume $V=4\pi abc/3$, and susceptibility $\chi$ be placed in a previously uniform magnetic field $H$ parallel to the principal axis with semiaxis $c$. The external field and the induced magnetization together produce a uniform field inside the ellipsoid with strength $H/(1+L_c\chi )$, where $L_c$ is the demagnetizing factor, given in cgs units by …

48: 30.9 Asymptotic Approximations and Expansions

… ►For uniform asymptotic expansions in terms of Airy or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1986). … ►For uniform asymptotic expansions in terms of elementary, Airy, or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1992, 1995). …

49: 8.22 Mathematical Applications

… ►For further information on ${\zeta }_x(s)$, including zeros and uniform asymptotic approximations, see Kölbig (1970, 1972a) and Dunster (2006). …

50: 18.24 Hahn Class: Asymptotic Approximations

… ►When the parameters $\alpha$ and $\beta$ are fixed and the ratio $n/N=c$ is a constant in the interval (0,1), uniform asymptotic formulas (as $n\to \mathrm{\infty }$ ) of the Hahn polynomials $Q_n(z;\alpha ,\beta ,N)$ can be found in Lin and Wong (2013) for $z$ in three overlapping regions, which together cover the entire complex plane. …