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21: 10.41 Asymptotic Expansions for Large Order

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§10.41(ii) Uniform Expansions for Real Variable

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§10.41(iii) Uniform Expansions for Complex Variable

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§10.41(iv) Double Asymptotic Properties

… ►Similar analysis can be developed for the uniform asymptotic expansions in terms of Airy functions given in §10.20. …

22: Bibliography L

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A uniform asymptotic expansion for Krawtchouk polynomials. J. Approx. Theory 106 (1), pp. 155–184.

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ISSN 0021-9045, Document, MathReview (Ahmed I. Zayed), ZentralBlatt

Referenced by:

§18.24.

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Uniform asymptotic expansions of symmetric elliptic integrals. Constr. Approx. 17 (4), pp. 535–559.

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ISSN 0176-4276, Document, MathReview (Valdir A. Menegatto), ZentralBlatt

Referenced by:

§19.27(vi).

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Approximation of orthogonal polynomials in terms of Hermite polynomials. Methods Appl. Anal. 6 (2), pp. 131–146.

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ISSN 1073-2772, Pdf, MathReview, ZentralBlatt

Referenced by:

§18.15(vi), §18.16(vi).

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Uniform approximations of Bernoulli and Euler polynomials in terms of hyperbolic functions. Stud. Appl. Math. 103 (3), pp. 241–258.

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ISSN 0022-2526, Document, Link, MathReview, ZentralBlatt

Referenced by:

§24.11, §24.11.

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Uniform asymptotic expansions at a caustic. Comm. Pure Appl. Math. 19, pp. 215–250.

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External Links:

Document, MathReview (Colin Clark), ZentralBlatt

Referenced by:

§36.12(iii), §36.14(i).

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23: 10.72 Mathematical Applications

… ►Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter. … ►These expansions are uniform with respect to $z$, including the turning point $z_0$ and its neighborhood, and the region of validity often includes cut neighborhoods (§1.10(vi)) of other singularities of the differential equation, especially irregular singularities. … ►If $f(z)$ has a double zero $z_0$, or more generally $z_0$ is a zero of order $m$, $m=2,3,4,\mathrm{\dots }$, then uniform asymptotic approximations (but not expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order $1/(m+2)$. … ►These asymptotic expansions are uniform with respect to $z$, including cut neighborhoods of $z_0$, and again the region of uniformity often includes cut neighborhoods of other singularities of the differential equation. … ►These approximations are uniform with respect to both $z$ and $\alpha$, including $z=z_0(a)$, the cut neighborhood of $z=0$, and $\alpha =a$. …

24: 34.8 Approximations for Large Parameters

§34.8 Approximations for Large Parameters

… ►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.

25: 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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26: 24.18 Physical Applications

§24.18 Physical Applications

►Bernoulli polynomials appear in statistical physics (Ordóñez and Driebe (1996)), in discussions of Casimir forces (Li et al. (1991)), and in a study of quark-gluon plasma (Meisinger et al. (2002)). ►Euler polynomials also appear in statistical physics as well as in semi-classical approximations to quantum probability distributions (Ballentine and McRae (1998)).

27: 28.26 Asymptotic Approximations for Large $q$

§28.26 Asymptotic Approximations for Large $q$

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§28.26(ii) Uniform Approximations

… ►For asymptotic approximations for ${\mathrm{M}}_{\nu }^{(3,4)}(z,h)$ see also Naylor (1984, 1987, 1989).

28: 9.16 Physical Applications

… ►The Airy functions constitute uniform approximations whose region of validity includes the turning point and its neighborhood. …The use of Airy function and related uniform asymptotic techniques to calculate amplitudes of polarized rainbows can be found in Nussenzveig (1992) and Adam (2002). … ►Again, the quest for asymptotic approximations that are uniformly valid solutions to this equation in the neighborhoods of critical points leads (after choosing solvable equations with similar asymptotic properties) to Airy functions. … ►This reference provides several examples of applications to problems in quantum mechanics in which Airy functions give uniform asymptotic approximations, valid in the neighborhood of a turning point. … ►An application of the Scorer functions is to the problem of the uniform loading of infinite plates (Rothman (1954b, a)).

29: Bibliography R

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Rational Chebyshev approximation by Remes’ algorithms. Numer. Math. 7 (4), pp. 322–330.

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ISSN 0029-599X, Document, MathReview (C. W. Clenshaw), ZentralBlatt

Referenced by:

§3.11(i).

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Remark on Algorithm 498: Airy functions using Chebyshev series approximations. ACM Trans. Math. Software 7 (3), pp. 404–405.

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ISSN 0098-3500, Document

Referenced by:

1st item.

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Composite approximations to the solutions of the Orr-Sommerfeld equation. Studies in Appl. Math. 51, pp. 341–368.

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MathReview (E. A. Boyd), ZentralBlatt

Referenced by:

(9.13.25), (9.13.26), (9.13.28), (9.13.29), (9.13.32), (9.13.33), (9.13.34), §9.13(ii), §9.13(ii).

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Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow. Studies in Appl. Math. 53, pp. 91–110.

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MathReview (T. W. Kao), ZentralBlatt

Referenced by:

§2.8(iii).

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Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. II. The general theory. Studies in Appl. Math. 53, pp. 217–224.

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External Links:

MathReview (T. W. Kao), ZentralBlatt

Referenced by:

§2.8(iii).

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30: Bibliography B

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Uniform approximation for potential scattering involving a rainbow. Proc. Phys. Soc. 89 (3), pp. 479–490.

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External Links:

Document, ZentralBlatt

Referenced by:

§36.14(iii), §9.16.

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Uniform approximation: A new concept in wave theory. Science Progress (Oxford) 57, pp. 43–64.

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Referenced by:

§36.14(i), §9.16.

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Uniform asymptotic expansion of Charlier polynomials. Methods Appl. Anal. 1 (3), pp. 294–313.

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ISSN 1073-2772, Pdf, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§18.24.

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A uniform asymptotic formula for orthogonal polynomials associated with $\exp (-x^4)$ . J. Approx. Theory 98, pp. 146–166.

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ISSN 0021-9045, Document, MathReview (V. L. Deshpande), ZentralBlatt

Referenced by:

§18.32.

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Padé-type Approximation and General Orthogonal Polynomials. International Series of Numerical Mathematics, Vol. 50, Birkhäuser Verlag, Basel.

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ISBN 3-7643-1100-2, MathReview (A. Magnus), ZentralBlatt

Referenced by:

§3.11(iv).

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