Watson expansions

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21: 5.17 Barnes’ $G$-Function (Double Gamma Function)

… ►When $z\to \mathrm{\infty }$ in , ►

5.17.5 $$\mathrm{Ln}G\left(z+1\right)\sim \frac{1}{4}{z}^2+z\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)-\left(\frac{1}{2}z(z+1)+\frac{1}{12}\right)\ln z-\ln A+\sum _{k=1}^{\mathrm{\infty }}\frac{B_{2k+2}}{2k(2k+1)(2k+2)z^{2k}}.$$

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Symbols:

$G\left(z\right)$: Barnes’ $G$-function (or double gamma function), $B_n$: Bernoulli numbers, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer, $z$: complex variable and $A$: Glaisher’s constant

Referenced by:

§5.17, §5.17, Erratum (V1.0.7) for Equation (5.17.5), Erratum (V1.1.11) for Equation (5.17.5)

Permalink:

http://dlmf.nist.gov/5.17.E5

Encodings:

pMML, png, TeX

Errata (effective with 1.1.11):

For consistency we have replaced $\mathrm{Ln}z$ by $\ln z$.

Errata (effective with 1.0.7):

Originally the term $z\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)$ was incorrectly stated as $z\mathrm{\Gamma }\left(z+1\right)$. (This error was reported subsequently by Nick Jones on December 11, 2013.)

Reported 2013-08-01 by Gergő Nemes

See also:

Annotations for §5.17 and Ch.5

…

22: 10.31 Power Series

… ►When $\nu$ is not an integer the corresponding expansion for $K_{\nu }\left(z\right)$ is obtained from (10.25.2) and (10.27.4). …

23: 6.12 Asymptotic Expansions

§6.12 Asymptotic Expansions

►

§6.12(i) Exponential and Logarithmic Integrals

… ►For the function $\chi$ see §9.7(i). … ►

§6.12(ii) Sine and Cosine Integrals

… ► …

24: 7.12 Asymptotic Expansions

§7.12 Asymptotic Expansions

►

§7.12(i) Complementary Error Function

… ►

§7.12(ii) Fresnel Integrals

… ►For exponentially-improved expansions use (7.5.7), (7.5.10), and §7.12(i). ►

§7.12(iii) Goodwin–Staton Integral

…

25: 10.18 Modulus and Phase Functions

… ►

§10.18(iii) Asymptotic Expansions for Large Argument

… ►

10.18.19 $$N_{\nu }^2\left(x\right)\sim \frac{2}{\pi x}\left(1-\frac{1}{2}\frac{\mu -3}{{(2x)}^2}-\frac{1}{2\cdot 4}\frac{(\mu -1)(\mu -45)}{{(2x)}^4}-\mathrm{\cdots }\right),$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $N_{\nu }\left(x\right)$: modulus of derivatives of Bessel functions, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.2.30 (corrected)

Referenced by:

§10.18(iii), §10.40(i), §10.40(i)

Permalink:

http://dlmf.nist.gov/10.18.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §10.18(iii), §10.18 and Ch.10

►the general term in this expansion being … ►

10.18.21 $${\varphi }_{\nu }\left(x\right)\sim x-\left(\frac{1}{2}\nu -\frac{1}{4}\right)\pi +\frac{\mu +3}{2(4x)}+\frac{{\mu }^2+46\mu -63}{6{(4x)}^3}+\frac{{\mu }^3+185{\mu }^2-2053\mu +1899}{5{(4x)}^5}+\mathrm{\cdots }.$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, ${\varphi }_{\nu }\left(x\right)$: phase of derivatives of Bessel functions, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.2.31

Referenced by:

§10.18(iii), Erratum (V1.1.8) for Section 10.18(iii)

Permalink:

http://dlmf.nist.gov/10.18.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §10.18(iii), §10.18 and Ch.10

►In (10.18.17) and (10.18.18) the remainder after $n$ terms does not exceed the $(n+1)$th term in absolute value and is of the same sign, provided that $n>\nu -\frac{1}{2}$ for (10.18.17) and $-\frac{3}{2}\le \nu \le \frac{3}{2}$ for (10.18.18).

26: 2.11 Remainder Terms; Stokes Phenomenon

… ► … ►Application of Watson’s lemma (§2.4(i)) yields … ►

§2.11(iii) Exponentially-Improved Expansions

… ►In this way we arrive at hyperasymptotic expansions. … ► …

27: 10.8 Power Series

… ►When $\nu$ is not an integer the corresponding expansions for $Y_{\nu }\left(z\right)$, $H_{\nu }^{(1)}\left(z\right)$, and $H_{\nu }^{(2)}\left(z\right)$ are obtained by combining (10.2.2) with (10.2.3), (10.4.7), and (10.4.8). …

28: 10.43 Integrals

… ►For further properties of the Bickley function, including asymptotic expansions and generalizations, see Amos (1983c, 1989) and Luke (1962, Chapter 8). … ►For asymptotic expansions of the direct transform (10.43.30) see Wong (1981), and for asymptotic expansions of the inverse transform (10.43.31) see Naylor (1990, 1996). … ►For collections of integrals of the functions $I_{\nu }\left(z\right)$ and $K_{\nu }\left(z\right)$, including integrals with respect to the order, see Apelblat (1983, §12), Erdélyi et al. (1953b, §§7.7.1–7.7.7 and 7.14–7.14.2), Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2000, §§5.5, 6.5–6.7), Gröbner and Hofreiter (1950, pp. 197–203), Luke (1962), Magnus et al. (1966, §3.8), Marichev (1983, pp. 191–216), Oberhettinger (1972), Oberhettinger (1974, §§1.11 and 2.7), Oberhettinger (1990, §§1.17–1.20 and 2.17–2.20), Oberhettinger and Badii (1973, §§1.15 and 2.13), Okui (1974, 1975), Prudnikov et al. (1986b, §§1.11–1.12, 2.15–2.16, 3.2.8–3.2.10, and 3.4.1), Prudnikov et al. (1992a, §§3.15, 3.16), Prudnikov et al. (1992b, §§3.15, 3.16), Watson (1944, Chapter 13), and Wheelon (1968).

29: 20.2 Definitions and Periodic Properties

… ►Corresponding expansions for ${\theta }_j^{\prime }\left(z\right|\tau )$, $j=1,2,3,4$, can be found by differentiating (20.2.1)–(20.2.4) with respect to $z$. …

30: Bibliography W

… ►

The cubic transformation of the hypergeometric function. Quart. J. Pure and Applied Math. 41, pp. 70–79.

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External Links:

ZentralBlatt

Referenced by:

§15.8(v), §15.8(v).

Encodings:

BibTeX

►

Generating functions of class-numbers. Compositio Math. 1, pp. 39–68.

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External Links:

ISSN 0010-437X, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§20.7(v), §20.8.

Encodings:

BibTeX

►

The surface of an ellipsoid. Quart. J. Math., Oxford Ser. 6, pp. 280–287.

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External Links:

Document, ZentralBlatt

Referenced by:

§19.33(i).

Encodings:

BibTeX

►

Two tables of partitions. Proc. London Math. Soc. (2) 42, pp. 550–556.

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External Links:

Document, ZentralBlatt

Referenced by:

§27.21.

Encodings:

BibTeX

… ►

A table of Ramanujan’s function $\tau (n)$ . Proc. London Math. Soc. (2) 51, pp. 1–13.

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External Links:

Document, MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§27.21.

Encodings:

BibTeX

…