asymptotic approximations to zeros

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31: 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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An application of the finite element approximation method to find the complex zeros of the modified Bessel function $K_n(z)$ . Math. Comp. 33 (148), pp. 1299–1306.

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ISSN 0025-5718, FullText, MathReview (R. P. Boas, Jr.), ZentralBlatt

Referenced by:

§10.74(vi), 2nd item.

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On the asymptotics of the Meixner-Pollaczek polynomials and their zeros. Constr. Approx. 17 (1), pp. 59–90.

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ISSN 0176-4276, Document, MathReview (C. M. Joshi), ZentralBlatt

Referenced by:

§18.24.

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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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Monotonic sequences related to zeros of Bessel functions. Numer. Algorithms 49 (1-4), pp. 221–233.

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ISSN 1017-1398, Document, FullText, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§10.21(iv), §10.21(iv).

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32: 10.70 Zeros

§10.70 Zeros

►Asymptotic approximations for large zeros are as follows. … ►

$$\text{<span class="ltx\_text hilite">zeros</span> of }{\mathrm{ber}}_{\nu }x\sim \sqrt{2}(t-f(t)),$$ $t=(m-\frac{1}{2}\nu -\frac{3}{8})\pi$,

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$$\text{<span class="ltx\_text hilite">zeros</span> of }{\mathrm{bei}}_{\nu }x\sim \sqrt{2}(t-f(t)),$$ $t=(m-\frac{1}{2}\nu +\frac{1}{8})\pi$,

… ►In the case $\nu =0$, numerical tabulations (Abramowitz and Stegun (1964, Table 9.12)) indicate that each of (10.70.2) corresponds to the $m$th zero of the function on the left-hand side. …

33: 36.7 Zeros

§36.7 Zeros

… ► … ►Close to the $y$-axis the approximate location of these zeros is given by … ►The zeros are approximated by solutions of the equation …Outside the bifurcation set (36.4.10), each rib is flanked by a series of zero lines in the form of curly “antelope horns” related to the “outside” zeros (36.7.2) of the cusp canonical integral. …

34: 25.10 Zeros

§25.10 Zeros

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§25.10(i) Distribution

… ►These are called the trivial zeros. … ►Calculations relating to the zeros on the critical line make use of the real-valued function …

35: Bibliography V

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Rational approximations for exponential integrals $E_n(x)$ . Acad. Roy. Belg. Bull. Cl. Sci. (5) 56, pp. 1064–1072.

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MathReview (A. Magnus), ZentralBlatt

Referenced by:

3rd item.

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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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Approximation of eigenvalues of some differential equations by zeros of orthogonal polynomials. J. Comput. Appl. Math. 213 (2), pp. 488–500.

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ISSN 0377-0427, Document, FullText, MathReview (Javier Segura), ZentralBlatt

Referenced by:

item (f), §28.34(ii), §29.20(i).

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Asymptotic expansion of the generalized hypergeometric function ${}_p{}^{}F_q^{}(z)$ as $z\to \mathrm{\infty }$ for . Anal. Appl. (Singap.) 21 (2), pp. 535–545.

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ISSN 0219-5305, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§16.11(i), §16.11, Erratum (V1.1.9) for Section 16.11(i).

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On the localization and computation of zeros of Bessel functions. Z. Angew. Math. Mech. 77 (6), pp. 467–475.

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ISSN 0044-2267, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vi).

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36: 16.4 Argument Unity

… ►When $k=1$ the function is said to be balanced or Saalschützian. … ►The last condition is equivalent to the sum of the top parameters plus $2$ equals the sum of the bottom parameters, that is, the series is 2-balanced. … ►The function ${}_3{}^{}F_2^{}(a,b,c;d,e;1)$ is analytic in the parameters $a,b,c,d,e$ when its series expansion converges and the bottom parameters are not negative integers or zero. …A detailed treatment of analytic continuation in (16.4.11) and asymptotic approximations as the variables $a,b,c,d,e$ approach infinity is given by Aomoto (1987). … ► …

37: 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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Introduction to Asymptotics: A Treatment Using Nonstandard Analysis. World Scientific Publishing Co. Inc., River Edge, NJ.

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ISBN 981-02-2915-1, MathReview (W. A. J. Luxemburg), ZentralBlatt

Referenced by:

Ch.2.

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Asymptotics of the hypergeometric function. Math. Methods Appl. Sci. 24 (6), pp. 369–389.

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ISSN 0170-4214, Document, MathReview, ZentralBlatt

Referenced by:

§14.15(iii), §15.12(iii).

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38: Bibliography H

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Computer Approximations. SIAM Ser. in Appl. Math., John Wiley & Sons Inc., New York.

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Notes:

Reprinted by Krieger Publishing Company, 1978.

External Links:

ZentralBlatt

Referenced by:

§3.11(i), §3.11(iii), §4.47(ii), §5.23(i), §5.23(i).

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Approximations for Digital Computers. Princeton University Press, Princeton, N.J..

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Notes:

Assisted by Jeanne T. Hayward and James P. Wong, Jr.

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MathReview (P. W. Ketchum), ZentralBlatt

Referenced by:

1st item, 1st item.

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On the distribution of the zeros of generalized Airy functions. Math. Comp. 29 (131), pp. 863–877.

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FullText, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§9.13(i).

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Applied and Computational Complex Analysis. Vol. 1: Power Series—Integration—Conformal Mapping—Location of Zeros. Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York.

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Notes:

Reprinted in 1988.

External Links:

ISBN 0-471-37244-7, MathReview (M. Marden), ZentralBlatt

Referenced by:

§1.10(vii), §1.9(vi), Ch.3.

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Error bounds for asymptotic approximations of zeros of Hankel functions occurring in diffraction problems. J. Mathematical Phys. 11 (8), pp. 2501–2504.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.21(xiv).

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39: 2.3 Integrals of a Real Variable

… ►For the Fourier integral … ►(In other words, differentiation of (2.3.8) with respect to the parameter $\lambda$ (or $\mu$) is legitimate.) … ►

§2.3(iv) Method of Stationary Phase

… ►For extensions to oscillatory integrals with more general $t$-powers and logarithmic singularities see Wong and Lin (1978) and Sidi (2010). … ►

§2.3(vi) Asymptotics of Mellin Transforms

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40: 14.20 Conical (or Mehler) Functions

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

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

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

… ►For extensions to complex arguments (including the range ), asymptotic expansions, and explicit error bounds, see Dunster (1991). … ►

§14.20(x) Zeros and Integrals

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