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21: 33.23 Methods of Computation

… ►Cancellation errors increase with increases in $\rho$ and $|r|$, and may be estimated by comparing the final sum of the series with the largest partial sum. Use of extended-precision arithmetic increases the radial range that yields accurate results, but eventually other methods must be employed, for example, the asymptotic expansions of §§33.11 and 33.21. … ►On the other hand, the irregular solutions of §§33.2(iii) and 33.14(iii) need to be integrated in the direction of decreasing radii beginning, for example, with values obtained from asymptotic expansions (§§33.11 and 33.21). … ►Thompson and Barnett (1985, 1986) and Thompson (2004) use combinations of series, continued fractions, and Padé-accelerated asymptotic expansions (§3.11(iv)) for the analytic continuations of Coulomb functions. … ►Seaton (1984) estimates the accuracies of these approximations. …

22: 18.15 Asymptotic Approximations

§18.15 Asymptotic Approximations

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§18.15(i) Jacobi

… ►For higher coefficients see Baratella and Gatteschi (1988), and for another estimate of the error term in a related expansion see Wong and Zhao (2003). … ►

§18.15(ii) Ultraspherical

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§18.15(iii) Legendre

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23: 2.8 Differential Equations with a Parameter

§2.8 Differential Equations with a Parameter

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§2.8(i) Classification of Cases

… ►For error bounds, more delicate error estimates, extensions to complex $\xi$ and $u$, zeros, connection formulas, extensions to inhomogeneous equations, and examples, see Olver (1997b, Chapters 11, 13), Olver (1964b), Reid (1974a, b), Boyd (1987), and Baldwin (1991). … ►For error bounds, more delicate error estimates, extensions to complex $\xi$, $\nu$, and $u$, zeros, and examples see Olver (1997b, Chapter 12), Boyd (1990a), and Dunster (1990a). … ►

24: Bibliography V

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Newton polyhedra and estimates of oscillatory integrals. Funkcional. Anal. i Priložen. 10 (3), pp. 13–38 (Russian).

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Notes:

English translation in Functional Anal. Appl. 18 (1976), pp. 175-196.

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ISSN 0374-1990, Document, MathReview (Helmut Hamm), ZentralBlatt

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§36.6.

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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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ISSN 0022-247X, Document, MathReview, ZentralBlatt

Referenced by:

§2.4(v).

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A new estimate of the function $\zeta (1+it)$ . Izv. Akad. Nauk SSSR. Ser. Mat. 22, pp. 161–164 (Russian).

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External Links:

MathReview (H. Davenport), ZentralBlatt

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§27.12.

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

Referenced by:

§16.11(ii), §16.11(ii).

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Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.

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ISSN 0176-4276, Document, MathReview, ZentralBlatt

Referenced by:

§30.16(i), §30.16(ii), §30.16(ii).

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25: Bibliography P

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An asymptotic formula for the incomplete gamma function. Ž. Vyčisl. Mat. i Mat. Fiz. 5, pp. 118–121 (Russian).

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 5(1), pp. 162–166

External Links:

Document, MathReview (E. Lukacs), ZentralBlatt

Referenced by:

§8.11(iv).

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On two-sided estimates, uniform with respect to the real argument and index, for modified Bessel functions. Mat. Zametki 65 (5), pp. 681–692 (Russian).

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Notes:

English translation in Math. Notes 65 (1999) pp. 571-581.

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ISSN 0025-567X, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.37.

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The asymptotic behaviour of Pearcey’s integral for complex variables. Proc. Roy. Soc. London Ser. A 432 (1886), pp. 391–426.

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ISSN 0962-8444, Document, Link, MathReview (J. S. Joel), ZentralBlatt

Referenced by:

§36.12(i).

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Error bounds for the uniform asymptotic expansion of the incomplete gamma function. J. Comput. Appl. Math. 147 (1), pp. 215–231.

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ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§8.12.

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Exponential asymptotics of the Mittag-Leffler function. Proc. Roy. Soc. London Ser. A 458, pp. 3041–3052.

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ISSN 1364-5021, Document, MathReview (Roderick Wong), ZentralBlatt

Referenced by:

§10.46.

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26: 28.8 Asymptotic Expansions for Large $q$

§28.8 Asymptotic Expansions for Large $q$

… ►For error estimates see Kurz (1979), and for graphical interpretation see Figure 28.2.1. … ►

§28.8(ii) Sips’ Expansions

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Barrett’s Expansions

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27: 2.10 Sums and Sequences

§2.10 Sums and Sequences

… ► … ►As a first estimate for large $n$ … ►

§2.10(iii) Asymptotic Expansions of Entire Functions

… ►See also Flajolet and Odlyzko (1990).

28: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►

§33.22(v) Asymptotic Solutions

… ►The WKBJ approximations of §33.23(vii) may also be used to estimate the penetrability. …

29: Bibliography B

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Asymptotic methods in enumeration. SIAM Rev. 16 (4), pp. 485–515.

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Document, MathReview (Bernard Harris), ZentralBlatt

Referenced by:

Ch.2.

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Uniform asymptotic smoothing of Stokes’s discontinuities. Proc. Roy. Soc. London Ser. A 422, pp. 7–21.

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ISSN 0080-4630, FullText, MathReview (André Hautot), ZentralBlatt

Referenced by:

§2.11(iv).

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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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Error estimates for the solution of linear systems. SIAM J. Sci. Comput. 21 (2), pp. 764–781.

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ISSN 1095-7197, Document, MathReview (Dimitrios Noutsos), ZentralBlatt

Referenced by:

§3.2(iii).

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On the asymptotic expansion of some integrals. Arch. Math. (Basel) 42 (3), pp. 253–259.

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ISSN 0003-889X, Document, MathReview (E. Riekstiņš), ZentralBlatt

Referenced by:

§2.5(ii).

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30: Bibliography K

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Asymptotic behavior of the solutions of the Painlevé equation of the first kind. Differ. Uravn. 24 (10), pp. 1684–1695 (Russian).

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Notes:

In Russian. English translation: Differential Equations 24(1988), no. 10, pp. 1107–1115 (1989)

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ISSN 0374-0641, MathReview, ZentralBlatt

Referenced by:

§32.11(i).

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Asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity. J. Comput. Appl. Math. 205 (1), pp. 186–206.

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ISSN 0377-0427, Document, FullText, MathReview (José Luis López), ZentralBlatt

Referenced by:

§19.12, §19.12, §19.36(iv), §19.36(iv).

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Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator. Adv. Math. 333, pp. 796–821.

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ISSN 0001-8708, Document, Link, MathReview (Martin Axel Hallnäs)

Referenced by:

§18.14(i).

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Estimates of trigonometric sums and their applications. Uspehi Mat. Nauk 13 (4 (82)), pp. 185–192 (Russian).

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External Links:

MathReview (R. A. Rankin), ZentralBlatt

Referenced by:

§27.12.

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Asymptotic solution of Maxwell’s equations near caustics. Izv. Vuz. Radiofiz. 7, pp. 1049–1056.

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Referenced by:

§36.14(i).

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