asymptotic expansion for large argument

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11: 11.9 Lommel Functions

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§11.9(iii) Asymptotic Expansion

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12: 2.11 Remainder Terms; Stokes Phenomenon

… ►For large $|z|$, with $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$ (), the Whittaker function of the second kind has the asymptotic expansion (§13.19) …

13: 35.10 Methods of Computation

§35.10 Methods of Computation

►For small values of $\Vert 𝐓\Vert$ the zonal polynomial expansion given by (35.8.1) can be summed numerically. For large $\Vert 𝐓\Vert$ the asymptotic approximations referred to in §35.7(iv) are available. … ►See Yan (1992) for the ${}_1{}^{}F_1^{}$ and ${}_2{}^{}F_1^{}$ functions of matrix argument in the case $m=2$, and Bingham et al. (1992) for Monte Carlo simulation on $𝐎(m)$ applied to a generalization of the integral (35.5.8). … ►These algorithms are extremely efficient, converge rapidly even for large values of $m$, and have complexity linear in $m$.

14: 33.11 Asymptotic Expansions for Large $\rho$

§33.11 Asymptotic Expansions for Large $\rho$

►For large $\rho$, with $\mathrm{\ell }$ and $\eta$ fixed, … ►

$f(\eta ,\rho )\sim {\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{f}_k,$

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$g(\eta ,\rho )\sim {\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{g}_k,$

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$\widehat{f}(\eta ,\rho )\sim {\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{\widehat{f}}_k,$

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15: 10.1 Special Notation

… ►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).

16: 14.32 Methods of Computation

… ►In particular, for small or moderate values of the parameters $\mu$ and $\nu$ the power-series expansions of the various hypergeometric function representations given in §§14.3(i)–14.3(iii), 14.19(ii), and 14.20(i) can be selected in such a way that convergence is stable, and reasonably rapid, especially when the argument of the functions is real. … ►

Application of the uniform asymptotic expansions for large values of the parameters given in §§14.15 and 14.20(vii)–14.20(ix).

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17: 14.26 Uniform Asymptotic Expansions

§14.26 Uniform Asymptotic Expansions

►The uniform asymptotic approximations given in §14.15 for $P_{\nu }^{-\mu }\left(x\right)$ and ${𝑸}_{\nu }^{\mu }\left(x\right)$ for are extended to domains in the complex plane in the following references: §§14.15(i) and 14.15(ii), Dunster (2003b); §14.15(iii), Olver (1997b, Chapter 12); §14.15(iv), Boyd and Dunster (1986). For an extension of §14.15(iv) to complex argument and imaginary parameters, see Dunster (1990b). ►See also Frenzen (1990), Gil et al. (2000), Shivakumar and Wong (1988), Ursell (1984), and Wong (1989) for uniform asymptotic approximations obtained from integral representations.

18: Bibliography P

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On two-sided estimates, uniform with respect to the real argument and index, for modified Bessel functions. Mat. Zametki 65 (5), pp. 681–692 (Russian).

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Notes:

English translation in Math. Notes 65 (1999) pp. 571-581.

External Links:

ISSN 0025-567X, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.37.

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

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§8.12.

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

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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).

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

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External Links:

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

Referenced by:

§8.16.

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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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External Links:

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

Referenced by:

§10.20(ii).

Encodings:

BibTeX

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19: 10.74 Methods of Computation

… ►The power-series expansions given in §§10.2 and 10.8, together with the connection formulas of §10.4, can be used to compute the Bessel and Hankel functions when the argument $x$ or $z$ is sufficiently small in absolute value. … ►If $x$ or $|z|$ is large compared with ${|\nu |}^2$, then the asymptotic expansions of §§10.17(i)–10.17(iv) are available. … ►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. … ►In the case of the spherical Bessel functions the explicit formulas given in §§10.49(i) and 10.49(ii) are terminating cases of the asymptotic expansions given in §§10.17(i) and 10.40(i) for the Bessel functions and modified Bessel functions. …

20: 2.10 Sums and Sequences

… ►for large $n$. … ►This identity can be used to find asymptotic approximations for large $n$ when the factor $v_j$ changes slowly with $j$, and $u_j$ is oscillatory; compare the approximation of Fourier integrals by integration by parts in §2.3(i). … ►

§2.10(iii) Asymptotic Expansions of Entire Functions

… ►Hence … ►

Example

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