uniform asymptotic approximation

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31: 2.4 Contour Integrals

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

32: 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(iv) Double Asymptotic Properties

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§10.41(v) Double Asymptotic Properties (Continued)

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

33: 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. …

34: 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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35: 18.32 OP’s with Respect to Freud Weights

… ►However, for asymptotic approximations in terms of elementary functions for the OP’s, and also for their largest zeros, see Levin and Lubinsky (2001) and Nevai (1986). 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). ►For asymptotic approximations to OP’s that correspond to Freud weights with more general functions $Q(x)$ see Deift et al. (1999a, b), Bleher and Its (1999), and Kriecherbauer and McLaughlin (1999). …

36: 18.24 Hahn Class: Asymptotic Approximations

§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. … ►Asymptotic approximations are also provided for the zeros of $K_n(x;p,N)$ in various cases depending on the values of $p$ and $\mu$. … ►For asymptotic approximations for the zeros of $M_n(nx;\beta ,c)$ in terms of zeros of $\mathrm{Ai}\left(x\right)$ (§9.9(i)), see Jin and Wong (1999) and Khwaja and Olde Daalhuis (2012). … ►Similar approximations are included for Jacobi, Krawtchouk, and Meixner polynomials.

37: 18.35 Pollaczek Polynomials

… ►See Bo and Wong (1996) for an asymptotic expansion of $P_n^{(\frac{1}{2})}(\cos \left(n^{-\frac{1}{2}}\theta \right);a,b)$ as $n\to \mathrm{\infty }$, with $a$ and $b$ fixed. This expansion is in terms of the Airy function $\mathrm{Ai}\left(x\right)$ and its derivative (§9.2), and is uniform in any compact $\theta$-interval in $(0,\mathrm{\infty })$. Also included is an asymptotic approximation for the zeros of $P_n^{(\frac{1}{2})}(\cos \left(n^{-\frac{1}{2}}\theta \right);a,b)$. …

38: Bibliography O

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On the resurgence properties of the uniform asymptotic expansion of the incomplete gamma function. Methods Appl. Anal. 5 (4), pp. 425–438.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Éric Delabaere), ZentralBlatt

Referenced by:

§8.12.

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On the asymptotics for late coefficients in uniform asymptotic expansions of integrals with coalescing saddles. Methods Appl. Anal. 7 (4), pp. 727–745.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Raimundas Vidunas), ZentralBlatt

Referenced by:

§36.12(iii).

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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).

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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).

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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:

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39: 2.9 Difference Equations

§2.9 Difference Equations

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§2.9(iii) Other Approximations

►For asymptotic approximations to solutions of second-order difference equations analogous to the Liouville–Green (WKBJ) approximation for differential equations (§2.7(iii)) see Spigler and Vianello (1992, 1997) and Spigler et al. (1999). … ►For discussions of turning points, transition points, and uniform asymptotic expansions for solutions of linear difference equations of the second order see Wang and Wong (2003, 2005). … ►

40: 11.11 Asymptotic Expansions of Anger–Weber Functions

§11.11 Asymptotic Expansions of Anger–Weber Functions

… ► … ►For an extension of (11.11.17) (and (11.11.16)) into a complete asymptotic expansion, see Nemes (2020). ►When $\nu$ is real and positive, all of (11.11.10)–(11.11.17) can be regarded as special cases of two asymptotic expansions given in Olver (1997b, pp. 352–360) for ${𝐀}_{-\nu }\left(\lambda \nu \right)$ as $\nu \to +\mathrm{\infty }$, one being uniform for , and the other being uniform for $\lambda \ge 1$. … ►Lastly, corresponding asymptotic approximations and expansions for ${𝐉}_{\nu }\left(\lambda \nu \right)$ and ${𝐄}_{\nu }\left(\lambda \nu \right)$, with or $\lambda >1$, follow from (11.10.15) and (11.10.16) and the corresponding asymptotic expansions for the Bessel functions $J_{\nu }\left(z\right)$ and $Y_{\nu }\left(z\right)$; see §10.19(ii). …