Meijer%20G-function

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11: 15.12 Asymptotic Approximations

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15.12.13 $$G_0(\pm \beta )={\left(2+{\mathrm{e}}^{\pm \zeta }\right)}^{c-b-(1/2)}{\left(1+{\mathrm{e}}^{\pm \zeta }\right)}^{a-c+(1/2)}{\left(z-1-{\mathrm{e}}^{\pm \zeta }\right)}^{-a+(1/2)}\sqrt{\frac{\beta }{{\mathrm{e}}^{\zeta }-{\mathrm{e}}^{-\zeta }}}.$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $\zeta$: change of variable, $\beta$ and $G_0(\pm \zeta )$: function

Permalink:

http://dlmf.nist.gov/15.12.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(iii), §15.12 and Ch.15

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12: Bibliography F

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Notes:

The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.

External Links:

GAMS (package FDLIBM)

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

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External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§6.15.

Encodings:

BibTeX

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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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External Links:

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

Referenced by:

§16.22.

Encodings:

BibTeX

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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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External Links:

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

Referenced by:

§16.22.

Encodings:

BibTeX

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On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

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External Links:

ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)

Referenced by:

§18.32.

Encodings:

BibTeX

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13: Bibliography M

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On the Representation of Meijer’s $G$-Function in the Vicinity of Singular Unity. In Complex Analysis and Applications ’81 (Varna, 1981), pp. 383–398.

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External Links:

MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§16.21.

Encodings:

BibTeX

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Über die asymptotische Entwicklung von ${\displaystyle {\int }_0^{-\mathrm{\infty }-i(\arg w-\mu )}}{e}^{\nu z-w\sinh z}𝑑z$, für große Werte von $|w|$ und $|\nu |$. I, II. Proc. Akad. Wet. Amsterdam 35, pp. 1170–1180, 1291–1303 (German).

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External Links:

ISSN 0370-0348, ZentralBlatt

Referenced by:

(11.11.10), (11.11.8), §11.11(iii).

Encodings:

BibTeX

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On the $G$-function. VII, VIII. Nederl. Akad. Wetensch., Proc. 49, pp. 1063–1072, 1165–1175 = Indagationes Math. 8, 661–670, 713–723 (1946).

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§16.11(ii).

Encodings:

BibTeX

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The exponential integral distribution. Statist. Probab. Lett. 5 (3), pp. 209–211.

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External Links:

ISSN 0167-7152, Document, MathReview Entry, ZentralBlatt

Referenced by:

§8.19(xi).

Encodings:

BibTeX

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The $q$-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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External Links:

ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.27(v).

Encodings:

BibTeX

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14: 20 Theta Functions

Chapter 20 Theta Functions

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15: Software Index

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	Open Source													With Book						Commercial
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16 Generalized Hypergeometric Functions & Meijer G-Function
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20 Theta Functions
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16: Richard A. Askey

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16 Generalized Hypergeometric Functions & Meijer G-Function

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17: 10.17 Asymptotic Expansions for Large Argument

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10.17.17 $$R_{\mathrm{\ell }}^{\pm }(\nu ,z)={(-1)}^{\mathrm{\ell }}2\cos \left(\nu \pi \right)\left(\sum _{k=0}^{m-1}{(\pm \mathrm{i})}^k\frac{a_k(\nu )}{z^k}{G}_{\mathrm{\ell }-k}\left(\mp 2\mathrm{i}z\right)+R_{m,\mathrm{\ell }}^{\pm }(\nu ,z)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{i}$: imaginary unit, $G_p\left(z\right)$: rescaled terminant function, $m$: integer, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $a_k(\nu )$: polynomial coefficient and $R_{\mathrm{\ell }}^{\pm }(\nu ,z)$: remainder

Permalink:

http://dlmf.nist.gov/10.17.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §10.17(v), §10.17 and Ch.10

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18: 8 Incomplete Gamma and Related
Functions

Chapter 8 Incomplete Gamma and Related Functions

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19: 28 Mathieu Functions and Hill’s Equation

Chapter 28 Mathieu Functions and Hill’s Equation

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20: 8.26 Tables

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Khamis (1965) tabulates $P(a,x)$ for $a=0.05(.05)10(.1)20(.25)70$, $0.0001\le x\le 250$ to 10D.

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Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_n\left(x\right)$ for $n=2,3,4,10,20$, $x=0(.01)2$ to 7D; also $(x+n){\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2,3,4,10,20$, $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

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Pagurova (1961) tabulates $E_n\left(x\right)$ for $n=0(1)20$, $x=0(.01)2(.1)10$ to 4-9S; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ for $p=0(.1)1$, $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

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Zhang and Jin (1996, Table 19.1) tabulates $E_n\left(x\right)$ for $n=1,2,3,5,10,15,20$, $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.