asymptotic expansion for large argument

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21: Bibliography L

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

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

Referenced by:

§19.27(vi).

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

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Document, MathReview (Colin Clark), ZentralBlatt

Referenced by:

§36.12(iii), §36.14(i).

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22: 11.13 Methods of Computation

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§11.13(ii) Series Expansions

►Although the power-series expansions (11.2.1) and (11.2.2), and the Bessel-function expansions of §11.4(iv) 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 $|z|$ and/or $|\nu |$ the asymptotic expansions given in §11.6 should be used instead. … ►Then from the limiting forms for small argument (§§11.2(i), 10.7(i), 10.30(i)), limiting forms for large argument (§§11.6(i), 10.7(ii), 10.30(ii)), and the connection formulas (11.2.5) and (11.2.6), it is seen that ${𝐇}_{\nu }\left(x\right)$ and ${𝐋}_{\nu }\left(x\right)$ can be computed in a stable manner by integrating forwards, that is, from the origin toward infinity. The solution ${𝐊}_{\nu }\left(x\right)$ needs to be integrated backwards for small $x$, and either forwards or backwards for large $x$ depending whether or not $\nu$ exceeds $\frac{1}{2}$. …

23: Bibliography D

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Plancherel-Rotach asymptotic expansion for some polynomials from indeterminate moment problems. Constr. Approx. 40 (1), pp. 61–104.

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ISSN 0176-4276, Document, Link, MathReview (Francisco Marcellán), ZentralBlatt

Referenced by:

§2.9(iii), §2.9(iii).

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Asymptotic Expansions: Their Derivation and Interpretation. Academic Press, London-New York.

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External Links:

MathReview, ZentralBlatt

Referenced by:

(11.11.11), §11.6(i), §11.6(iii), §2.11(v), Ch.2, §8.1, Notations, Notations.

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Uniform asymptotic expansions for prolate spheroidal functions with large parameters. SIAM J. Math. Anal. 17 (6), pp. 1495–1524.

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ISSN 0036-1410, Document, MathReview (R. K. Gupta), ZentralBlatt

Referenced by:

§30.9(i).

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Uniform asymptotic expansions for Charlier polynomials. J. Approx. Theory 112 (1), pp. 93–133.

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ISSN 0021-9045, Document, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§18.24.

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Uniform asymptotic expansions for associated Legendre functions of large order. Proc. Roy. Soc. Edinburgh Sect. A 133 (4), pp. 807–827.

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ISSN 0308-2105, Document, MathReview, ZentralBlatt

Referenced by:

§14.15(i), §14.15(i), §14.15(ii), §14.15(ii), §14.26.

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24: Bibliography W

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Asymptotic expansions of some matrix argument hypergeometric functions, with applications to macromolecules. Ann. Inst. Statist. Math. 45 (3), pp. 467–475.

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ISSN 0020-3157, Document, MathReview (N. K. Thakare), ZentralBlatt

Referenced by:

§35.9.

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Wave functions for large arguments by the amplitude-phase method. Phys. Rev. 52, pp. 1123–1127.

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Document, ZentralBlatt

Referenced by:

§33.5(ii).

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An asymptotic expansion of $W_{k,m}(z)$ with large variable and parameters. Math. Comp. 27 (122), pp. 429–436.

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FullText, Document, MathReview (J. A. Donaldson), ZentralBlatt

Referenced by:

§13.19.

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On uniform asymptotic expansion of definite integrals. J. Approximation Theory 7 (1), pp. 76–86.

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Document, MathReview (L. Berg), ZentralBlatt

Referenced by:

§8.11(v).

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Asymptotic expansions of the Kontorovich-Lebedev transform. Appl. Anal. 12 (3), pp. 161–172.

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ISSN 0003-6811, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

§10.43(v).

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25: 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

Referenced by:

§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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Uniform asymptotic expansions for hypergeometric functions with large parameters. III. Analysis and Applications (Singapore) 8 (2), pp. 199–210.

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ISSN 0219-5305, Document, MathReview (Javier Segura), 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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Some new asymptotic expansions for Bessel functions of large orders. Proc. Cambridge Philos. Soc. 48 (3), pp. 414–427.

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Document, MathReview (F. Oberhettinger), ZentralBlatt

Referenced by:

§10.19(iii), §10.21(vii).

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26: Bibliography F

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Uniform asymptotic expansions for hypergeometric functions with large parameters IV. Anal. Appl. (Singap.) 12 (6), pp. 667–710.

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ISSN 0219-5305, Document, Link, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§15.12(iii), §15.12(iii).

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Uniform asymptotic expansions of certain classes of Meijer $G$-functions for a large parameter. SIAM J. Math. Anal. 4 (3), pp. 482–507.

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Document, MathReview (C. M. Joshi), ZentralBlatt

Referenced by:

§16.22.

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Uniform asymptotic expansions of a class of Meijer $G$-functions for a large parameter. SIAM J. Math. Anal. 14 (6), pp. 1204–1253.

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ISSN 0036-1410, Document, MathReview (Roop N. Kesarwani), ZentralBlatt

Referenced by:

§16.22.

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Tables of Weber Parabolic Cylinder Functions and Other Functions for Large Arguments. National Physical Laboratory Mathematical Tables, Vol. 4. Department of Scientific and Industrial Research, Her Majesty’s Stationery Office, London.

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MathReview (L. J. Slater), ZentralBlatt

Referenced by:

3rd item, 2nd item.

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Error bounds for the asymptotic expansion of the ratio of two gamma functions with complex argument. SIAM J. Math. Anal. 23 (2), pp. 505–511.

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ISSN 0036-1410, Document, FullText, MathReview (A. Reinfelds), ZentralBlatt

Referenced by:

§5.11(iii), §5.11(iii).

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27: 9.7 Asymptotic Expansions

§9.7 Asymptotic Expansions

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§9.7(iii) Error Bounds for Real Variables

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§9.7(iv) Error Bounds for Complex Variables

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§9.7(v) Exponentially-Improved Expansions

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28: Bibliography T

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

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External Links:

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

Referenced by:

§8.12.

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

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FullText

Referenced by:

§2.4(vi).

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

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ISSN 0019-3577, Document, Link

Referenced by:

§13.8(iv).

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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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External Links:

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

Referenced by:

§13.8(iv).

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Evaluation of the exponential integral for large complex arguments. J. Research Nat. Bur. Standards 52, pp. 313–317.

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External Links:

MathReview (H. Bückner), ZentralBlatt

Referenced by:

§6.18(i).

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29: 15.19 Methods of Computation

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§15.19(i) Maclaurin Expansions

… ►For $z\in ℝ$ it is always possible to apply one of the linear transformations in §15.8(i) in such a way that the hypergeometric function is expressed in terms of hypergeometric functions with an argument in the interval $[0,\frac{1}{2}]$. ►For $z\in ℂ$ it is possible to use the linear transformations in such a way that the new arguments lie within the unit circle, except when $z={\mathrm{e}}^{\pm \pi \mathrm{i}/3}$. … ►Large values of $|a|$ or $|b|$, for example, delay convergence of the Gauss series, and may also lead to severe cancellation. … ►Initial values for moderate values of $|a|$ and $|b|$ can be obtained by the methods of §15.19(i), and for large values of $|a|$, $|b|$, or $|c|$ via the asymptotic expansions of §§15.12(ii) and 15.12(iii). …

30: 10.75 Tables

… ►Also, for additional listings of tables pertaining to complex arguments see Babushkina et al. (1997). … ►

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.

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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).

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Olver (1962) provides tables for the uniform asymptotic expansions given in §10.41(ii), including $\eta$ and the coefficients $U_k(p)$, $V_k(p)$ as functions of $p={(1+x^2)}^{-\frac{1}{2}}$. These enable $I_{\nu }\left(\nu x\right)$, $K_{\nu }\left(\nu x\right)$, $I_{\nu }^{\prime }\left(\nu x\right)$, $K_{\nu }^{\prime }\left(\nu x\right)$ to be computed to 10S when $\nu \ge 16$.

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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 $n\to \mathrm{\infty }$.

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