asymptotic approximations for large zeros

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11: 28.34 Methods of Computation

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Use of asymptotic expansions and approximations for large $q$ (§§28.8(i), 28.16). See also Zhang and Jin (1996, pp. 482–485).

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Solution of the continued-fraction equations (28.6.16)–(28.6.19) and (28.15.2) by successive approximation. See Blanch (1966), Shirts (1993a), and Meixner and Schäfke (1954, §2.87).

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Asymptotic approximations by zeros of orthogonal polynomials of increasing degree. See Volkmer (2008). This method also applies to eigenvalues of the Whittaker–Hill equation (§28.31(i)) and eigenvalues of Lamé functions (§29.3(i)).

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Use of asymptotic expansions and approximations for large $q$ (§§28.8(ii)–28.8(iv)).

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Use of asymptotic expansions for large $z$ or large $q$. See §§28.25 and 28.26.

12: 8.13 Zeros

§8.13 Zeros

… ►For asymptotic approximations for $x_{+}(a)$ and $x_{-}(a)$ as $a\to -\mathrm{\infty }$ see Tricomi (1950b), with corrections by Kölbig (1972b). For more accurate asymptotic approximations see Thompson (2012). … ►For information on the distribution and computation of zeros of $\gamma (a,\lambda a)$ and $\mathrm{\Gamma }(a,\lambda a)$ in the complex $\lambda$-plane for large values of the positive real parameter $a$ see Temme (1995a). … ►Approximations to $a_n^{\ast }$, $x_n^{\ast }$ for large $n$ can be found in Kölbig (1970). …

13: 6.18 Methods of Computation

… ►For large $x$ or $|z|$ these series suffer from slow convergence or cancellation (or both). … ►For large $x$ and $\left|z\right|$, expansions in inverse factorial series (§6.10(i)) or asymptotic expansions (§6.12) are available. The attainable accuracy of the asymptotic expansions can be increased considerably by exponential improvement. … ►

§6.18(iii) Zeros

►Zeros of $\mathrm{Ci}\left(x\right)$ and $\mathrm{si}\left(x\right)$ can be computed to high precision by Newton’s rule (§3.8(ii)), using values supplied by the asymptotic expansion (6.13.2) as initial approximations. …

14: 2.4 Contour Integrals

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§2.4(i) Watson’s Lemma

… ►For examples see Olver (1997b, pp. 315–320). ►

§2.4(iii) Laplace’s Method

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§2.4(v) Coalescing Saddle Points: Chester, Friedman, and Ursell’s Method

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§2.4(vi) Other Coalescing Critical Points

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15: Bibliography T

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Uniform asymptotic approximation of Fermi-Dirac integrals. J. Comput. Appl. Math. 31 (3), pp. 383–387.

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ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§25.12(iii).

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Laguerre polynomials: Asymptotics for large degree. Technical report Technical Report AM-R8610, CWI, Amsterdam, The Netherlands.

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Referenced by:

§2.4(vi).

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On the computation of the incomplete gamma functions for large values of the parameters. In Algorithms for approximation (Shrivenham, 1985), Inst. Math. Appl. Conf. Ser. New Ser., Vol. 10, pp. 479–489.

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FullText, MathReview (G. Meinardus), ZentralBlatt

Referenced by:

§8.25(iii).

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Computational aspects of incomplete gamma functions with large complex parameters. In Approximation and Computation. A Festschrift in Honor of Walter Gautschi, R. V. M. Zahar (Ed.), International Series of Numerical Mathematics, Vol. 119, pp. 551–562.

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FullText, MathReview, ZentralBlatt

Referenced by:

§8.11(iii), §8.20(ii), §8.25(i), §8.25(iii), 2nd item, 3rd item.

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Asymptotic expansions of Kummer hypergeometric functions for large values of the parameters. Integral Transforms Spec. Funct. 33 (1), pp. 16–31.

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ISSN 1065-2469, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§13.8(iv).

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16: Bibliography N

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The resurgence properties of the large order asymptotics of the Anger-Weber function I. J. Class. Anal. 4 (1), pp. 1–39.

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ISSN 1848-5979, Document, Link, MathReview (José Luis López), ZentralBlatt

Referenced by:

(11.11.10), (11.11.8), §11.11(iii).

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The resurgence properties of the large order asymptotics of the Anger-Weber function II. J. Class. Anal. 4 (2), pp. 121–147.

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ISSN 1848-5979, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

(11.11.10), (11.11.8), §11.11(iii).

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On the large argument asymptotics of the Lommel function via Stieltjes transforms. Asymptot. Anal. 91 (3-4), pp. 265–281.

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ISSN 0921-7134, Document, MathReview Entry, ZentralBlatt

Referenced by:

§11.6(iii), §11.6(iii), §11.9(iii), §11.9(iii).

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Error Bounds for the Large-Argument Asymptotic Expansions of the Hankel and Bessel Functions. Acta Appl. Math. 150, pp. 141–177.

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ISSN 0167-8019, Document, MathReview Entry, ZentralBlatt

Referenced by:

(9.7.16), (9.7.17), §9.7(iii), §9.7(iv).

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Error bounds for the large-argument asymptotic expansions of the Lommel and allied functions. Stud. Appl. Math. 140 (4), pp. 508–541.

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ISSN 0022-2526, Document, Link, MathReview (Pedro J. Pagola), ZentralBlatt

Referenced by:

§11.11(i), §11.11(i).

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17: 2.3 Integrals of a Real Variable

… ►Then … ►For the Fourier integral … ►

§2.3(iv) Method of Stationary Phase

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§2.3(v) Coalescing Peak and Endpoint: Bleistein’s Method

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§2.3(vi) Asymptotics of Mellin Transforms

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18: 34.8 Approximations for Large Parameters

§34.8 Approximations for Large Parameters

►For large values of the parameters in the $\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ symbols, different asymptotic forms are obtained depending on which parameters are large. … ►and the symbol $o\left(1\right)$ denotes a quantity that tends to zero as the parameters tend to infinity, as in §2.1(i). ►Semiclassical (WKBJ) approximations in terms of trigonometric or exponential functions are given in Varshalovich et al. (1988, §§8.9, 9.9, 10.7). Uniform approximations in terms of Airy functions for the $\mathit{3}j$ and $\mathit{6}j$ symbols are given in Schulten and Gordon (1975b). …

19: Bibliography C

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Asymptotic behaviour of the zeros of the (generalized) Laguerre polynomial $L_n^{\alpha }(x)$ as the index $\alpha \to \mathrm{\infty }$ and limiting formula relating Laguerre polynomials of large index and large argument to Hermite polynomials. Lett. Nuovo Cimento (2) 23 (3), pp. 101–102.

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ISSN 0024-1318, Document, MathReview (R. A. Askey)

Referenced by:

§18.7(iii).

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Asymptotic approximations for symmetric elliptic integrals. SIAM J. Math. Anal. 25 (2), pp. 288–303.

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ISSN 0036-1410, Document, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§19.12, §19.27(ii), §19.27(iii), §19.27(iv), §19.27(v), §19.27(vi).

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Asymptotics of the largest zeros of some orthogonal polynomials. J. Phys. A 31 (25), pp. 5525–5544.

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ISSN 0305-4470, Document, MathReview (Tim H. Baker), ZentralBlatt

Referenced by:

§18.26(v), §18.29.

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Further formulas for calculating approximate values of the zeros of certain combinations of Bessel functions. IEEE Trans. Microwave Theory Tech. 11 (6), pp. 546–547.

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ISSN 0018-9480, FullText

Referenced by:

§10.21(x).

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The asymptotic nature of zeros of cross-product Bessel functions. Quart. J. Mech. Appl. Math. 19 (4), pp. 511–522.

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ISSN 0033-5614, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.21(x).

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20: 2.8 Differential Equations with a Parameter

… ►Zeros of $f(z)$ are also called turning points. … ►In both cases uniform asymptotic approximations are obtained in terms of Bessel functions of order $1/(\lambda +2)$. … ►For further examples of uniform asymptotic approximations in terms of parabolic cylinder functions see §§13.20(iii), 13.20(iv), 14.15(v), 15.12(iii), 18.24. ►For further examples of uniform asymptotic approximations in terms of Bessel functions or modified Bessel functions of variable order see §§13.21(ii), 14.15(ii), 14.15(iv), 14.20(viii), 30.9(i), 30.9(ii). ►For examples of uniform asymptotic approximations in terms of Whittaker functions with fixed second parameter see §18.15(i) and §28.8(iv). …