expansions in Airy functions

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21: 18.24 Hahn Class: Asymptotic Approximations

… ►This expansion is in terms of the parabolic cylinder function and its derivative. … ►This expansion is in terms of confluent hypergeometric functions. … ►Both expansions are in terms of parabolic cylinder functions. … ►Dunster (2001b) provides various asymptotic expansions for $C_n(x;a)$ as $n\to \mathrm{\infty }$, in terms of elementary functions or in terms of Bessel functions. … ►This expansion is uniformly valid in any compact $x$-interval on the real line and is in terms of parabolic cylinder functions. …

22: 10.20 Uniform Asymptotic Expansions for Large Order

§10.20 Uniform Asymptotic Expansions for Large Order

… ►uniformly for $z$ $\in (0,\mathrm{\infty })$ in all cases, where $\mathrm{Ai}$ and $\mathrm{Bi}$ are the Airy functions (§9.2). … ► ►

§10.20(iii) Double Asymptotic Properties

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

23: Bibliography V

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Airy Functions and Applications to Physics. Second edition, Imperial College Press, London.

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External Links:

ISBN 978-1-84816-548-9; 1-84816-548-X, MathReview Entry, ZentralBlatt

Referenced by:

§1.17(ii), §9.10(ii), §9.10(ix), §9.11(v), §9.16, §9.16, §9.5(i).

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Expansion of vacuum magnetic fields in toroidal harmonics. Comput. Phys. Comm. 81 (1-2), pp. 74–90.

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External Links:

Document

Referenced by:

§14.31(i).

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Symbolic evaluation of coefficients in Airy-type asymptotic expansions. J. Math. Anal. Appl. 269 (1), pp. 317–331.

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External Links:

ISSN 0022-247X, Document, MathReview, ZentralBlatt

Referenced by:

§2.4(v).

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Expansions in products of Heine-Stieltjes polynomials. Constr. Approx. 15 (4), pp. 467–480.

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External Links:

ISSN 0176-4276, Document, MathReview (Luoqing Li), ZentralBlatt

Referenced by:

§31.15(iii).

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A note on the asymptotic expansion of generalized hypergeometric functions. Anal. Appl. (Singap.) 12 (1), pp. 107–115.

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External Links:

ISSN 0219-5305, Document, MathReview (Roberto S. Costas-Santos), ZentralBlatt

Referenced by:

§16.11(ii), §16.11(ii).

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24: 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

►Let ${𝒞}_{\nu }\left(x\right)$, ${\rho }_{\nu }(t)$, and ${\sigma }_{\nu }(t)$ be defined as in §10.21(ii) and $M\left(x\right)$, $\theta \left(x\right)$, $N\left(x\right)$, and $\varphi \left(x\right)$ denote the modulus and phase functions for the Airy functions and their derivatives as in §9.8. … ►Here $a_m$ and $a_m^{\prime }$ denote respectively the zeros of the Airy function $\mathrm{Ai}\left(z\right)$ and its derivative ${\mathrm{Ai}}^{\prime }\left(z\right)$; see §9.9. … ►Higher coefficients in the asymptotic expansions in this subsection can be obtained by expressing the cross-products in terms of the modulus and phase functions (§10.18), and then reverting the asymptotic expansion for the difference of the phase functions. …

25: 2.2 Transcendental Equations

… ►An important case is the reversion of asymptotic expansions for zeros of special functions. In place of (2.2.1) assume that …where $F_0=f_0$ and $s{F}_s$ ($s\ge 1$) is the coefficient of $x^{-1}$ in the asymptotic expansion of ${(f(x))}^s$ (Lagrange’s formula for the reversion of series). Conditions for the validity of the reversion process in $ℂ$ are derived in Olver (1997b, pp. 14–16). Applications to real and complex zeros of Airy functions are given in Fabijonas and Olver (1999). …

26: 9.13 Generalized Airy Functions

§9.13 Generalized Airy Functions

… ►The distribution in $ℂ$ and asymptotic properties of the zeros of $A_n\left(z\right)$, $A_n^{\prime }\left(z\right)$, $B_n\left(z\right)$, and $B_n^{\prime }\left(z\right)$ are investigated in Swanson and Headley (1967) and Headley and Barwell (1975). … ► … ►Reid (1972) and Drazin and Reid (1981, Appendix) introduce the following contour integrals in constructing approximate solutions to the Orr–Sommerfeld equation for fluid flow: … ►

27: 10.19 Asymptotic Expansions for Large Order

§10.19 Asymptotic Expansions for Large Order

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§10.19(i) Asymptotic Forms

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§10.19(ii) Debye’s Expansions

… ►Here $\mathrm{Ai}$ and $\mathrm{Bi}$ are the Airy functions (§9.2), and … ►See also §10.20(i).

28: 12.11 Zeros

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§12.11(ii) Asymptotic Expansions of Large Zeros

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§12.11(iii) Asymptotic Expansions for Large Parameter

►For large negative values of $a$ the real zeros of $U(a,x)$, $U^{\prime }(a,x)$, $V(a,x)$, and $V^{\prime }(a,x)$ can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). …Here $\alpha ={\mu }^{-\frac{4}{3}}{a}_s$, $a_s$ denoting the $s$th negative zero of the function $\mathrm{Ai}$ (see §9.9(i)). … ►where $\beta ={\mu }^{-\frac{4}{3}}{a}_s^{\prime }$, $a_s^{\prime }$ denoting the $s$th negative zero of the function ${\mathrm{Ai}}^{\prime }$ and …

29: 9.9 Zeros

… ►They are denoted by $a_k$, $a_k^{\prime }$, $b_k$, $b_k^{\prime }$, respectively, arranged in ascending order of absolute value for $k=1,2,\mathrm{\dots }.$ … ►

§9.9(ii) Relation to Modulus and Phase

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§9.9(iv) Asymptotic Expansions

… ►For error bounds for the asymptotic expansions of $a_k$, $b_k$, $a_k^{\prime }$, and $b_k^{\prime }$ see Pittaluga and Sacripante (1991), and a conjecture given in Fabijonas and Olver (1999). ►

§9.9(v) Tables

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30: Bibliography J

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Further results on the computation of incomplete gamma functions. In Analytic Theory of Continued Fractions, II (Pitlochry/Aviemore, 1985), W. J. Thron (Ed.), Lecture Notes in Math. 1199, pp. 67–89.

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External Links:

MathReview (Marietta J. Tretter), ZentralBlatt

Referenced by:

§8.25(iv).

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Tables of Functions with Formulae and Curves. 4th edition, Dover Publications, New York.

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Notes:

Contains corrections and enlargements of earlier editions. Originally published by B. G. Teubner, Leipzig, in 1933. In German and English.

External Links:

MathReview, ZentralBlatt

Referenced by:

2nd item, §20.15, §5.1, Notations P, E. Jahnke, F. Emde, and F. Lösch (1966).

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Numerical calculation of Bessel, Hankel and Airy functions. Computer Physics Communications 183 (3), pp. 506–519.

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External Links:

Document, FullText, ZentralBlatt

Referenced by:

§10.19(iii), §10.19(iii).

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On the computation of incomplete gamma functions in the complex domain. J. Comput. Appl. Math. 12/13, pp. 401–417.

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External Links:

ISSN 0377-0427, Document, MathReview (M. Marsden), ZentralBlatt

Referenced by:

§8.25(iv), §8.9.

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Asymptotic behavior of the continued fraction coefficients of a class of Stieltjes transforms including the Binet function. In Orthogonal functions, moment theory, and continued fractions (Campinas, 1996), Lecture Notes in Pure and Appl. Math., Vol. 199, pp. 257–274.

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External Links:

MathReview (Olav Njåstad), ZentralBlatt

Referenced by:

§5.10, §5.10.

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