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21: 33.11 Asymptotic Expansions for Large $\rho$

§33.11 Asymptotic Expansions for Large $\rho$

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

33.11.4 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )={\mathrm{e}}^{\pm \mathrm{i}{\theta }_{\mathrm{\ell }}}(f(\eta ,\rho )\pm \mathrm{i}g(\eta ,\rho )),$$

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Symbols:

${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $f(\eta ,\rho )$: function and $g(\eta ,\rho )$: function

Permalink:

http://dlmf.nist.gov/33.11.E4

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.22):

The arguments of some of the functions in these equations were included to improve clarity of the presentation.

See also:

Annotations for §33.11 and Ch.33

… ►

33.11.7 $$g(\eta ,\rho )\widehat{f}(\eta ,\rho )-f(\eta ,\rho )\widehat{g}(\eta ,\rho )=1.$$

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Symbols:

$\rho$: nonnegative real variable, $\eta$: real parameter, $f(\eta ,\rho )$: function, $g(\eta ,\rho )$: function, $\widehat{f}(\eta ,\rho )$: function and $\widehat{g}(\eta ,\rho )$: function

A&S Ref:

14.5.8

Referenced by:

§33.11, Erratum (V1.0.22) for Equations (33.11.2)–(33.11.7), Erratum (V1.0.22) for Equations (33.11.2)–(33.11.7)

Permalink:

http://dlmf.nist.gov/33.11.E7

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.22):

The arguments of some of the functions in this equation were included to improve clarity of the presentation.

See also:

Annotations for §33.11 and Ch.33

…

22: Bibliography P

… ►

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

…

23: 35.10 Methods of Computation

§35.10 Methods of Computation

… ►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 $m$.

24: 9.7 Asymptotic Expansions

… ►

9.7.3 $$\chi (x)\equiv {\pi }^{1/2}\mathrm{\Gamma }\left(\frac{1}{2}x+1\right)/\mathrm{\Gamma }\left(\frac{1}{2}x+\frac{1}{2}\right).$$

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Defines:

$\chi (x)$: function (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter and $\equiv$: equals by definition

Source:

Olver (1997b, (13.02), p. 225)

Referenced by:

Erratum (V1.0.17) for Equations (9.7.3), (9.7.4)

Permalink:

http://dlmf.nist.gov/9.7.E3

Encodings:

pMML, png, TeX

Modification (effective with 1.0.17):

Originally the function $\chi$ was presented with argument given by a positive integer $n$. It has now been clarified to be valid for argument given by a positive real number $x$.

See also:

Annotations for §9.7(i), §9.7 and Ch.9

►For large $x$, ►

9.7.4 $$\chi (x)\sim {(\frac{1}{2}\pi x)}^{1/2}.$$

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter and $\chi (x)$: function

Source:

Olver (1997b, p. 225)

Referenced by:

Erratum (V1.0.17) for Equations (9.7.3), (9.7.4)

Permalink:

http://dlmf.nist.gov/9.7.E4

Encodings:

pMML, png, TeX

Modification (effective with 1.0.17):

Originally the formula was presented with argument given by a positive integer $n$. It has now been clarified to be valid for argument given by a positive real number $x$.

See also:

Annotations for §9.7(i), §9.7 and Ch.9

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25: 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).

26: 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).

…

27: 14.26 Uniform Asymptotic Expansions

§14.26 Uniform Asymptotic Expansions

… ►For an extension of §14.15(iv) to complex argument and imaginary parameters, see Dunster (1990b). …

28: Bibliography Z

… ►

Chebyshev series approximations for the Bessel function $Y_n(z)$ of complex argument. Appl. Math. Comput. 88 (2-3), pp. 275–286.

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External Links:

ISSN 0096-3003, Document, MathReview (Sven Ehrich), ZentralBlatt

Referenced by:

§10.76(ii).

Encodings:

BibTeX

►

Computation of Special Functions. John Wiley & Sons Inc., New York.

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Notes:

Includes diskette containing a large collection of mathematical function software written in Fortran. Implementation in double precision. Maximum accuracy 16S.

External Links:

ISBN 0-471-11963-6, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

6th item, 3rd item, 1st item, 11st item, 7th item, 2nd item, 6th item, 3rd item, 2nd item, 5th item, 8th item, 4th item, 2nd item, §19.37(ii), §19.37(iii), §19.37(iii), §22.21, §24.19(i), §28.1, item (b), item (d), 8th item, 3rd item, 6th item, §5.22(ii), §5.22(iii), 2nd item, 2nd item, 4th item, 2nd item, 2nd item, §8.17(v), 4th item, 2nd item, 5th item, 3rd item, 3rd item, 3rd item, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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29: 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. … ►And since there are no error terms they could, in theory, be used for all values of $z$; however, there may be severe cancellation when $|z|$ is not large compared with $n^2$. …

30: Bibliography L

… ►

The solutions of the Mathieu equation with a complex variable and at least one parameter large. Trans. Amer. Math. Soc. 36 (3), pp. 637–695.

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External Links:

ISSN 0002-9947, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

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Recurrence relations for hypergeometric functions of unit argument. Math. Comp. 45 (172), pp. 521–535.

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External Links:

ISSN 0025-5718, FullText, MathReview (S. Conde), ZentralBlatt

Referenced by:

§16.4(iii).

Encodings:

BibTeX

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Corrigenda: “Recurrence relations for hypergeometric functions of unit argument” [Math. Comp. 45 (1985), no. 172, 521–535; MR 86m:33004]. Math. Comp. 48 (178), pp. 853.

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External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

§16.4(iii).

Encodings:

BibTeX

… ►

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

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.

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External Links:

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

Referenced by:

§24.16(i), §24.16(i).

Encodings:

BibTeX

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