asymptotic approximations for large zeros

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21: Bibliography O

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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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ISSN 0219-5305, Document, MathReview, ZentralBlatt

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§15.12(iii).

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Uniform asymptotic expansions for hypergeometric functions with large parameters. II. Analysis and Applications (Singapore) 1 (1), pp. 121–128.

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ISSN 0219-5305, Document, MathReview, ZentralBlatt

Referenced by:

§15.12(iii).

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A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order. Proc. Cambridge Philos. Soc. 47, pp. 699–712.

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Document, MathReview (J. Todd), ZentralBlatt

Referenced by:

§10.21(vii).

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Asymptotic approximations and error bounds. SIAM Rev. 22 (2), pp. 188–203.

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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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ISSN 0308-2105, MathReview (P. F. Hsieh), ZentralBlatt

Referenced by:

§13.20(iii), §13.20(iv).

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22: 3.8 Nonlinear Equations

… ►For real functions $f(x)$ the sequence of approximations to a real zero $\xi$ will always converge (and converge quadratically) if either: … ►Let $z^2-sz-t$ be an approximation to the real quadratic factor of $p(z)$ that corresponds to a pair of conjugate complex zeros or to a pair of real zeros. … ►Initial approximations to the zeros can often be found from asymptotic or other approximations to $f(z)$, or by application of the phase principle or Rouché’s theorem; see §1.10(iv). … ►For describing the distribution of complex zeros of solutions of linear homogeneous second-order differential equations by methods based on the Liouville–Green (WKB) approximation, see Segura (2013). … ►For moderate or large values of $n$ it is not uncommon for the magnitude of the right-hand side of (3.8.14) to be very large compared with unity, signifying that the computation of zeros of polynomials is often an ill-posed problem. …

23: Bibliography Z

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Algorithm 985: Simple, Efficient, and Relatively Accurate Approximation for the Evaluation of the Faddeyeva Function. ACM Trans. Math. Softw. 44 (2), pp. 22:1–22:9.

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ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

4th item, §7.25(iii).

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Distribution of zeros of Gauss and Kummer hypergeometric functions. A semiclassical approach. Ann. Numer. Math. 2 (1-4), pp. 457–472.

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ISSN 1021-2655, MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§13.9(i), §15.13.

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A note on the $\tau$-method approximations for the Bessel functions $Y_0(z)$ and $Y_1(z)$ . Comput. Math. Appl. 31 (9), pp. 63–70.

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ISSN 0898-1221, Document, MathReview, ZentralBlatt

Referenced by:

§10.76(ii).

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Chebyshev series approximations for the Bessel function $Y_n(z)$ of complex argument. Appl. Math. Comput. 88 (2-3), pp. 275–286.

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ISSN 0096-3003, Document, MathReview (Sven Ehrich), ZentralBlatt

Referenced by:

§10.76(ii).

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Approximations to -, di- and tri-logarithms. J. Comput. Appl. Math. 202 (2), pp. 450–459.

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ISSN 0377-0427, Document, FullText, MathReview (F. Beukers), ZentralBlatt

Referenced by:

§25.18(i), §25.18(i).

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24: Bibliography P

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Exactification of the method of steepest descents: The Bessel functions of large order and argument. Proc. Roy. Soc. London Ser. A 460, pp. 2737–2759.

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ISSN 1364-5021, Document, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§10.20(ii).

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Chebyshev series approximations for the zeros of the Bessel functions. J. Comput. Phys. 53 (1), pp. 188–192.

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ISSN 0021-9991, Document, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§10.76(ii).

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Approximation for the turning points of Bessel functions. J. Comput. Phys. 64 (1), pp. 253–257.

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ISSN 0021-9991, Document, MathReview (Evangelos Ifantis), ZentralBlatt

Referenced by:

§10.76(ii).

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On the maximum errors of polynomial approximations defined by interpolation and by least squares criteria. Comput. J. 9 (4), pp. 404–407.

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Document, MathReview, ZentralBlatt

Referenced by:

§3.11(ii).

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Algorithm 498: Airy functions using Chebyshev series approximations. ACM Trans. Math. Software 1 (4), pp. 372–379.

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ISSN 0098-3500, Document, GAMS (module 498; package TOMS), ZentralBlatt

Referenced by:

1st item, 4th item.

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25: 2.7 Differential Equations

… ►For large $s$, … ►

§2.7(iii) Liouville–Green (WKBJ) Approximation

►For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows: … ►By approximating … ►The first of these references includes extensions to complex variables and reversions for zeros. …

26: Bibliography L

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Inequalities and approximations for zeros of Bessel functions of small order. SIAM J. Math. Anal. 14 (2), pp. 383–388.

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ISSN 0036-1410, Document, MathReview (S. Ahmed), ZentralBlatt

Referenced by:

§10.21(iv).

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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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Large degree asymptotics of generalized Bernoulli and Euler polynomials. J. Math. Anal. Appl. 363 (1), pp. 197–208.

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ISSN 0022-247X, Document, Link, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§24.16(i), §24.16(i).

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27: Bibliography J

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Asymptotic formulas for the zeros of the Meixner polynomials. J. Approx. Theory 96 (2), pp. 281–300.

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ISSN 0021-9045, Document, MathReview (N. Hayek Calil), ZentralBlatt

Referenced by:

§18.24.

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REMES2 — a Fortran program to calculate rational minimax approximations to a given function. Technical Report Technical Report AECL-4210, Atomic Energy of Canada Limited. Chalk River Nuclear Laboratories, Chalk River, Ontario.

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Referenced by:

§3.11(iii).

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Asymptotic behavior of integrals. SIAM Rev. 14 (2), pp. 286–317.

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Document, MathReview (A. Erdelyi), ZentralBlatt

Referenced by:

Ch.2.

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Parabolic cylinder functions of large order. J. Comput. Appl. Math. 190 (1-2), pp. 453–469.

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ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§12.10(vi).

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Derivation of Green-type, transitional and uniform asymptotic expansions from differential equations. V. Angular oblate spheroidal wavefunctions ${\overline{ps}}_n^r(\eta ,h)$ and ${\overline{qs}}_n^r(\eta ,h)$ for large $h$ . Proc. Roy. Soc. London Ser. A 321, pp. 545–555.

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FullText, MathReview, ZentralBlatt

Referenced by:

§30.9(ii).

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28: 9.17 Methods of Computation

… ►For large $|z|$ the asymptotic expansions of §§9.7 and 9.12(viii) should be used instead. … ►

§9.17(v) Zeros

►Zeros of the Airy functions, and their derivatives, can be computed to high precision via Newton’s rule (§3.8(ii)) or Halley’s rule (§3.8(v)), using values supplied by the asymptotic expansions of §9.9(iv) as initial approximations. …See also Fabijonas et al. (2004). ►For the computation of the zeros of the Scorer functions and their derivatives see Gil et al. (2003c).

29: 14.20 Conical (or Mehler) Functions

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§14.20(vii) Asymptotic Approximations: Large $\tau$, Fixed $\mu$

… ►For asymptotic expansions and explicit error bounds, see Olver (1997b, pp. 473–474). … ►

§14.20(viii) Asymptotic Approximations: Large $\tau$, $0\le \mu \le A\tau$

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§14.20(ix) Asymptotic Approximations: Large $\mu$, $0\le \tau \le A\mu$

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§14.20(x) Zeros and Integrals

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30: 10.74 Methods of Computation

… ►For large positive real values of $\nu$ the uniform asymptotic expansions of §§10.20(i) and 10.20(ii) can be used. Moreover, because of their double asymptotic properties (§10.41(v)) these expansions can also be used for large $x$ or $|z|$, whether or not $\nu$ is large. … ►Methods for obtaining initial approximations to the zeros include asymptotic expansions (§§10.21(vi)-10.21(ix)), graphical intersection of $2D$ graphs in $ℝ$ (e. … ►

Real Zeros

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Complex Zeros

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