asymptotic expansions for large order

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21: 16.11 Asymptotic Expansions

§16.11 Asymptotic Expansions

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§16.11(i) Formal Series

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§16.11(ii) Expansions for Large Variable

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§16.11(iii) Expansions for Large Parameters

… ►Asymptotic expansions for the polynomials ${}_{p+2}{}^{}F_q^{}(-r,r+a_0,𝐚;𝐛;z)$ as $r\to \mathrm{\infty }$ through integer values are given in Fields and Luke (1963b, a) and Fields (1965).

22: 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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External Links:

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

Referenced by:

(11.11.10), (11.11.8), §11.11(iii).

Encodings:

BibTeX

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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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External Links:

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

Referenced by:

(11.11.10), (11.11.8), §11.11(iii).

Encodings:

BibTeX

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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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External Links:

ISSN 0921-7134, Document, MathReview Entry, ZentralBlatt

Referenced by:

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

Encodings:

BibTeX

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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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External Links:

ISSN 0167-8019, Document, MathReview Entry, ZentralBlatt

Referenced by:

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

Encodings:

BibTeX

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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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External Links:

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

Referenced by:

§11.11(i), §11.11(i).

Encodings:

BibTeX

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23: 25.11 Hurwitz Zeta Function

… ►The function $\zeta (s,a)$ was introduced in Hurwitz (1882) and defined by the series expansion … ►For other series expansions similar to (25.11.10) see Coffey (2008). … ►

§25.11(xii) $a$-Asymptotic Behavior

… ►As $a\to \mathrm{\infty }$ in the sector , with $s(\ne 1)$ and $\delta$ fixed, we have the asymptotic expansion … ►Similarly, as $a\to \mathrm{\infty }$ in the sector , …

24: 10.68 Modulus and Phase Functions

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§10.68(iii) Asymptotic Expansions for Large Argument

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10.68.16 $$M_{\nu }\left(x\right)=\frac{{\mathrm{e}}^{x/\sqrt{2}}}{{(2\pi x)}^{\frac{1}{2}}}\left(1-\frac{\mu -1}{8\sqrt{2}}\frac{1}{x}+\frac{{(\mu -1)}^2}{256}\frac{1}{x^2}-\frac{(\mu -1)({\mu }^2+14\mu -399)}{6144\sqrt{2}}\frac{1}{x^3}+O\left(\frac{1}{x^4}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $M_{\nu }\left(x\right)$: modulus of Bessel functions, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.10.21

Permalink:

http://dlmf.nist.gov/10.68.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §10.68(iii), §10.68 and Ch.10

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10.68.17 $$\ln M_{\nu }\left(x\right)=\frac{x}{\sqrt{2}}-\frac{1}{2}\ln \left(2\pi x\right)-\frac{\mu -1}{8\sqrt{2}}\frac{1}{x}-\frac{(\mu -1)(\mu -25)}{384\sqrt{2}}\frac{1}{x^3}-\frac{(\mu -1)(\mu -13)}{128}\frac{1}{x^4}+O\left(\frac{1}{x^5}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $M_{\nu }\left(x\right)$: modulus of Bessel functions, $\ln z$: principal branch of logarithm function, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.10.22

Permalink:

http://dlmf.nist.gov/10.68.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §10.68(iii), §10.68 and Ch.10

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10.68.18 $${\theta }_{\nu }\left(x\right)=\frac{x}{\sqrt{2}}+\left(\frac{1}{2}\nu -\frac{1}{8}\right)\pi +\frac{\mu -1}{8\sqrt{2}}\frac{1}{x}+\frac{\mu -1}{16}\frac{1}{x^2}-\frac{(\mu -1)(\mu -25)}{384\sqrt{2}}\frac{1}{x^3}+O\left(\frac{1}{x^5}\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, ${\theta }_{\nu }\left(x\right)$: phase of Bessel functions, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.10.23

Referenced by:

§10.68(i), §10.70

Permalink:

http://dlmf.nist.gov/10.68.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §10.68(iii), §10.68 and Ch.10

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10.68.20 $$\ln N_{\nu }\left(x\right)=-\frac{x}{\sqrt{2}}+\frac{1}{2}\ln \left(\frac{\pi }{2x}\right)+\frac{\mu -1}{8\sqrt{2}}\frac{1}{x}+\frac{(\mu -1)(\mu -25)}{384\sqrt{2}}\frac{1}{x^3}-\frac{(\mu -1)(\mu -13)}{128}\frac{1}{x^4}+O\left(\frac{1}{x^5}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $N_{\nu }\left(x\right)$: modulus of derivatives of Bessel functions, $\ln z$: principal branch of logarithm function, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.10.25

Permalink:

http://dlmf.nist.gov/10.68.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §10.68(iii), §10.68 and Ch.10

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25: 13.9 Zeros

… ► … ►where $n$ is a large positive integer, and the logarithm takes its principal value (§4.2(i)). … ►where $n$ is a large positive integer. … ►where $n$ is a large positive integer. …

26: Bibliography P

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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.

Encodings:

BibTeX

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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).

Encodings:

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

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External Links:

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

Referenced by:

§10.46.

Encodings:

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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.

Encodings:

BibTeX

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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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27: 2.11 Remainder Terms; Stokes Phenomenon

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§2.11(i) Numerical Use of Asymptotic Expansions

… ► ►In order to guard against this kind of error remaining undetected, the wanted function may need to be computed by another method (preferably nonasymptotic) for the smallest value of the (large) asymptotic variable $x$ that is intended to be used. … ►For second-order differential equations, see Olde Daalhuis and Olver (1995a), Olde Daalhuis (1995, 1996), and Murphy and Wood (1997). … ► …

28: 12.11 Zeros

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§12.11(ii) Asymptotic Expansions of Large Zeros

… ►When $a>-\frac{1}{2}$ the zeros are asymptotically given by $z_{a,s}$ and $\overline{z_{a,s}}$, where $s$ is a large positive integer and … ►

§12.11(iii) Asymptotic Expansions for Large Parameter

►For large negative values of $a$ the real zeros of $U(a,x)$, $U^{\prime }(a,x)$, $V(a,x)$, and $V^{\prime }(a,x)$ can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). For example, let the $s$th real zeros of $U(a,x)$ and $U^{\prime }(a,x)$, counted in descending order away from the point $z=2\sqrt{-a}$, be denoted by $u_{a,s}$ and $u_{a,s}^{\prime }$, respectively. …

29: 15.12 Asymptotic Approximations

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§15.12(i) Large Variable

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§15.12(ii) Large $c$

… ►As $\lambda \to \mathrm{\infty }$, … ►For this result and an extension to an asymptotic expansion with error bounds see Jones (2001). … ►For other extensions, see Wagner (1986), Temme (2003) and Temme (2015, Chapters 12 and 28).

30: 11.9 Lommel Functions

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§11.9(ii) Expansions in Series of Bessel Functions

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§11.9(iii) Asymptotic Expansion

►For fixed $\mu$ and $\nu$, … ►For uniform asymptotic expansions, for large $\nu$ and fixed $\mu =-1,0,1,2,\mathrm{\dots }$, of solutions of the inhomogeneous modified Bessel differential equation that corresponds to (11.9.1) see Olver (1997b, pp. 388–390). …