uniform asymptotic approximations for large parameters

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

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§18.29.

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

Referenced by:

§18.15(i), §18.15(i).

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

Referenced by:

§18.15(i).

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An asymptotic expansion of $W_{k,m}(z)$ with large variable and parameters. Math. Comp. 27 (122), pp. 429–436.

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FullText, Document, MathReview (J. A. Donaldson), ZentralBlatt

Referenced by:

§13.19.

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

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Document, MathReview (L. Berg), ZentralBlatt

Referenced by:

§8.11(v).

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12: 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. …where $z$ is a real or complex variable and $u$ is a large real or complex parameter. … ►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. … ►Then for large $u$ asymptotic approximations of the solutions $w$ can be constructed in terms of Bessel functions, or modified Bessel functions, of variable order (in fact the order depends on $u$ and $\alpha$). …

13: Bibliography T

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Uniform asymptotic approximation of Fermi-Dirac integrals. J. Comput. Appl. Math. 31 (3), pp. 383–387.

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ISSN 0377-0427, Document, MathReview, ZentralBlatt

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§25.12(iii).

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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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FullText, MathReview (G. Meinardus), ZentralBlatt

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§8.25(iii).

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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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FullText, MathReview, ZentralBlatt

Referenced by:

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

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Uniform asymptotics for the incomplete gamma functions starting from negative values of the parameters. Methods Appl. Anal. 3 (3), pp. 335–344.

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ISSN 1073-2772, FullText, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§8.12, §8.2(i), §8.2(ii), §8.6(ii).

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

Referenced by:

§13.8(iv).

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14: 30.9 Asymptotic Approximations and Expansions

§30.9 Asymptotic Approximations and Expansions

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§30.9(i) Prolate Spheroidal Wave Functions

… ►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). … ►

15: 10.69 Uniform Asymptotic Expansions for Large Order

§10.69 Uniform Asymptotic Expansions for Large Order

… ►All fractional powers take their principal values. ►All four expansions also enjoy the same kind of double asymptotic property described in §10.41(iv). … ►

16: 10.21 Zeros

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§10.21(vi) McMahon’s Asymptotic Expansions for Large Zeros

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§10.21(vii) Asymptotic Expansions for Large Order

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§10.21(viii) Uniform Asymptotic Approximations for Large Order

… ►For derivations and further information, including extensions to uniform asymptotic expansions, see Olver (1954, 1960). The latter reference includes numerical tables of the first few coefficients in the uniform asymptotic expansions. …

17: Bibliography D

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Uniform asymptotic expansions for prolate spheroidal functions with large parameters. SIAM J. Math. Anal. 17 (6), pp. 1495–1524.

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ISSN 0036-1410, Document, MathReview (R. K. Gupta), ZentralBlatt

Referenced by:

§30.9(i).

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Uniform asymptotic expansions for oblate spheroidal functions I: Positive separation parameter $\lambda$ . Proc. Roy. Soc. Edinburgh Sect. A 121 (3-4), pp. 303–320.

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ISSN 0308-2105, MathReview (Archibald D. MacGillivray), ZentralBlatt

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§30.9(ii).

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Uniform asymptotic approximation of Mathieu functions. Methods Appl. Anal. 1 (2), pp. 143–168.

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ISSN 1073-2772, Pdf, MathReview (J. M. H. Peters), ZentralBlatt

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§28.8(iv).

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Uniform asymptotic expansions for associated Legendre functions of large order. Proc. Roy. Soc. Edinburgh Sect. A 133 (4), pp. 807–827.

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ISSN 0308-2105, Document, MathReview, ZentralBlatt

Referenced by:

§14.15(i), §14.15(i), §14.15(ii), §14.15(ii), §14.26.

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Uniform asymptotic approximations for incomplete Riemann zeta functions. J. Comput. Appl. Math. 190 (1-2), pp. 339–353.

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ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

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§8.22(ii).

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18: 2.4 Contour Integrals

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§2.4(i) Watson’s Lemma

… ►Then … ►For examples and extensions (including uniformity and loop integrals) see Olver (1997b, Chapter 4), Wong (1989, Chapter 1), and Temme (1985). … ►

§2.4(v) Coalescing Saddle Points: Chester, Friedman, and Ursell’s Method

… ►The problem of obtaining an asymptotic approximation to $I(\alpha ,z)$ that is uniform with respect to $\alpha$ in a region containing $\widehat{\alpha }$ is similar to the problem of a coalescing endpoint and saddle point outlined in §2.3(v). …

19: Bibliography L

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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

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§19.27(vi).

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Asymptotics of the first Appell function $F_1$ with large parameters II. Integral Transforms Spec. Funct. 24 (12), pp. 982–999.

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

Referenced by:

§16.13, §16.13.

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Asymptotics of the first Appell function $F_1$ with large parameters. Integral Transforms Spec. Funct. 24 (9), pp. 715–733.

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

Referenced by:

§16.13, §16.13.

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Asymptotic expansions of the Whittaker functions for large order parameter. Methods Appl. Anal. 6 (2), pp. 249–256.

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Notes:

Dedicated to Richard A. Askey on the occasion of his 65th birthday, Part II

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ISSN 1073-2772, Pdf, MathReview, ZentralBlatt

Referenced by:

§13.21(i), §13.24(ii).

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

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Document, MathReview (Colin Clark), ZentralBlatt

Referenced by:

§36.12(iii), §36.14(i).

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20: 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. …where $k$ is a large real parameter and $𝐲=\{y_1,y_2,\mathrm{\dots }\}$ is a set of additional (nonasymptotic) parameters. … ►The leading-order uniform asymptotic approximation is given by … ►For further information concerning integrals with several coalescing saddle points see Arnol’d et al. (1988), Berry and Howls (1993, 1994), Bleistein (1967), Duistermaat (1974), Ludwig (1966), Olde Daalhuis (2000), and Ursell (1972, 1980).