asymptotic expansions for large zeros

SearchAdvancedHelp

(0.006 seconds)

21—30 of 40 matching pages

21: 2.3 Integrals of a Real Variable

… ►Then … ►For the Fourier integral … ►

§2.3(ii) Watson’s Lemma

… ►Then the series obtained by substituting (2.3.7) into (2.3.1) and integrating formally term by term yields an asymptotic expansion: … ►Then …

22: Bibliography T

… ►

The asymptotic expansion of the incomplete gamma functions. SIAM J. Math. Anal. 10 (4), pp. 757–766.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (E. Riekstiņš), ZentralBlatt

Referenced by:

§8.12.

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

FullText

Referenced by:

§2.4(vi).

Encodings:

BibTeX

… ►

Asymptotics of zeros of incomplete gamma functions. Ann. Numer. Math. 2 (1-4), pp. 415–423.

ⓘ

Notes:

Special functions (Torino, 1993)

External Links:

ISSN 1021-2655, FullText, MathReview (E. R. Love), ZentralBlatt

Referenced by:

§8.13(ii).

Encodings:

BibTeX

… ►

Asymptotic expansions of Kummer hypergeometric functions with three asymptotic parameters a, b and z. Indagationes Mathematicae.

ⓘ

External Links:

ISSN 0019-3577, Document, Link

Referenced by:

§13.8(iv).

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

§13.8(iv).

Encodings:

BibTeX

…

23: Bibliography S

… ►

Uniform asymptotic expansions of modified Mathieu functions. J. Reine Angew. Math. 247, pp. 1–17.

ⓘ

External Links:

MathReview (J. Meixner), ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

… ►

Asymptotic expansions of Mellin transforms and analogues of Watson’s lemma. SIAM J. Math. Anal. 16 (4), pp. 896–906.

ⓘ

External Links:

ISSN 0036-1410, Document, Link, MathReview (John S. Lew), ZentralBlatt

Referenced by:

§2.3(vi).

Encodings:

BibTeX

… ►

A simple approach to asymptotic expansions for Fourier integrals of singular functions. Appl. Math. Comput. 216 (11), pp. 3378–3385.

ⓘ

External Links:

ISSN 0096-3003, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§2.3(iv).

Encodings:

BibTeX

►

Asymptotic expansion of Mellin transforms in the complex plane. Int. J. Pure Appl. Math. 71 (3), pp. 465–480.

ⓘ

External Links:

ISSN 1311-8080, MathReview Entry, ZentralBlatt

Referenced by:

§2.3(vi).

Encodings:

BibTeX

… ►

Uniform asymptotic expansions of Hermite polynomials. M. Phil. thesis, City University of Hong Kong.

ⓘ

External Links:

Abstract

Referenced by:

§18.16(v).

Encodings:

BibTeX

…

24: Bibliography L

… ►

Uniform asymptotic expansions of symmetric elliptic integrals. Constr. Approx. 17 (4), pp. 535–559.

ⓘ

External Links:

ISSN 0176-4276, Document, MathReview (Valdir A. Menegatto), ZentralBlatt

Referenced by:

§19.27(vi).

Encodings:

BibTeX

… ►

Asymptotics of the first Appell function $F_1$ with large parameters II. Integral Transforms Spec. Funct. 24 (12), pp. 982–999.

ⓘ

External Links:

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

Referenced by:

§16.13, §16.13.

Encodings:

BibTeX

… ►

Asymptotic expansions of the Whittaker functions for large order parameter. Methods Appl. Anal. 6 (2), pp. 249–256.

ⓘ

Notes:

Dedicated to Richard A. Askey on the occasion of his 65th birthday, Part II

External Links:

ISSN 1073-2772, Pdf, MathReview, ZentralBlatt

Referenced by:

§13.21(i), §13.24(ii).

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

§24.16(i), §24.16(i).

Encodings:

BibTeX

… ►

Uniform asymptotic expansions at a caustic. Comm. Pure Appl. Math. 19, pp. 215–250.

ⓘ

External Links:

Document, MathReview (Colin Clark), ZentralBlatt

Referenced by:

§36.12(iii), §36.14(i).

Encodings:

BibTeX

…

25: 9.12 Scorer Functions

… ►

§9.12(viii) Asymptotic Expansions

… ►For other phase ranges combine these results with the connection formulas (9.12.11)–(9.12.14) and the asymptotic expansions given in §9.7. … ►

Integrals

… ►

§9.12(ix) Zeros

… ►For the above properties and further results, including the distribution of complex zeros, asymptotic approximations for the numerically large real or complex zeros, and numerical tables see Gil et al. (2003c). …

26: Bibliography C

… ►

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.

ⓘ

External Links:

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

Referenced by:

§18.7(iii).

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

§18.26(v), §18.29.

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

§10.21(x).

Encodings:

BibTeX

… ►

On the asymptotic expansion of Airy’s integral. Proc. Glasgow Math. Assoc. 6, pp. 113–115.

ⓘ

External Links:

Document, MathReview (L. Berg), ZentralBlatt

Referenced by:

(9.5.7), §9.5(ii).

Encodings:

BibTeX

►

Asymptotic Expansions. Cambridge Tracts in Mathematics and Mathematical Physics, Cambridge University Press, New York.

ⓘ

External Links:

MathReview (A. E. Danese), ZentralBlatt

Referenced by:

Ch.2.

Encodings:

BibTeX

…

27: Bibliography P

… ►

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

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§8.12.

Encodings:

BibTeX

►

A uniform asymptotic expansion for the incomplete gamma function. J. Comput. Appl. Math. 148 (2), pp. 323–339.

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview (Dumitru Acu), ZentralBlatt

Referenced by:

(8.11.6), §8.11(iii), (8.12.18), §8.12, §8.12, §8.12, Erratum (V1.0.16) for Equation (8.12.18).

Encodings:

BibTeX

►

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

ⓘ

External Links:

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

Referenced by:

§10.46.

Encodings:

BibTeX

►

The asymptotic expansion of a generalised incomplete gamma function. J. Comput. Appl. Math. 151 (2), pp. 297–306.

ⓘ

External Links:

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

Referenced by:

§8.16.

Encodings:

BibTeX

►

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.

ⓘ

External Links:

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

Referenced by:

§10.20(ii).

Encodings:

BibTeX

…

28: 10.75 Tables

… ►

Olver (1962) provides tables for the uniform asymptotic expansions given in §10.20(i), including $\zeta$ and ${(4\zeta /(1-x^2))}^{\frac{1}{4}}$ as functions of $x$ $(=z)$ and the coefficients $A_k(\zeta )$, $B_k(\zeta )$, $C_k(\zeta )$, $D_k(\zeta )$ as functions of $\zeta$. These enable $J_{\nu }\left(\nu x\right)$, $Y_{\nu }\left(\nu x\right)$, $J_{\nu }^{\prime }\left(\nu x\right)$, $Y_{\nu }^{\prime }\left(\nu x\right)$ to be computed to 10S when $\nu \ge 15$, except in the neighborhoods of zeros.

… ►

Real Zeros

… ►

Olver (1960) tabulates $j_{n,m}$, $J_n^{\prime }\left(j_{n,m}\right)$, $j_{n,m}^{\prime }$, $J_n\left(j_{n,m}^{\prime }\right)$, $y_{n,m}$, $Y_n^{\prime }\left(y_{n,m}\right)$, $y_{n,m}^{\prime }$, $Y_n\left(y_{n,m}^{\prime }\right)$, $n=0(\frac{1}{2})20⁤\frac{1}{2}$, $m=1(1)50$, 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as $n\to \mathrm{\infty }$; see §10.21(viii), and more fully Olver (1954).

… ►

Complex Zeros

… ►

Olver (1960) tabulates $a_{n,m}^{\prime }$, ${𝗃}_n\left(a_{n,m}^{\prime }\right)$, $b_{n,m}^{\prime }$, ${𝗒}_n\left(b_{n,m}^{\prime }\right)$, $n=1(1)20$, $m=1(1)50$, 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as $n\to \mathrm{\infty }$.

…

29: 9.17 Methods of Computation

… ►

§9.17(i) Maclaurin Expansions

►Although the Maclaurin-series expansions of §§9.4 and 9.12(vi) converge for all finite values of $z$, they are cumbersome to use when $|z|$ is large owing to slowness of convergence and cancellation. 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. …

30: 2.4 Contour Integrals

… ►

§2.4(i) Watson’s Lemma

… ►Then … ►For large $t$, the asymptotic expansion of $q(t)$ may be obtained from (2.4.3) by Haar’s method. This depends on the availability of a comparison function $F(z)$ for $Q(z)$ that has an inverse transform …For examples see Olver (1997b, pp. 315–320). ►

§2.4(iii) Laplace’s Method

…