scaled%20gamma%20function

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1: 5.11 Asymptotic Expansions

§5.11 Asymptotic Expansions

… ►The scaled gamma function ${\mathrm{\Gamma }}^{\ast }\left(z\right)$ is defined in (5.11.3) and its main property is ${\mathrm{\Gamma }}^{\ast }\left(z\right)\sim 1$ as $z\to \mathrm{\infty }$ in the sector $|\mathrm{ph}z|\le \pi -\delta$. … ►

§5.11(ii) Error Bounds and Exponential Improvement

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§5.11(iii) Ratios

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2: Errata

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In previous versions of the DLMF, in §8.18(ii), the notation $\tilde{\mathrm{\Gamma }}(z)$ was used for the scaled gamma function ${\mathrm{\Gamma }}^{\ast }\left(z\right)$. Now in §8.18(ii), we adopt the notation which was introduced in Version 1.1.7 (October 15, 2022) and correspondingly, Equation (8.18.13) has been removed. In place of Equation (8.18.13), it is now mentioned to see (5.11.3).

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Equations: (5.9.2_5), (5.9.10_1), (5.9.10_2), (5.9.11_1), (5.9.11_2), the definition of the scaled gamma function ${\mathrm{\Gamma }}^{\ast }\left(z\right)$ was inserted after the first equals sign in (5.11.3), post equality added in (7.17.2) which gives “$={\sum }_{m=0}^{\mathrm{\infty }}{a}_m{t}^{2m+1}$”, (7.17.2_5), (31.11.3_1), (31.11.3_2) with some explanatory text.

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The unnumbered equation

$$\underset{c\to -n}{lim}\frac{F(a,b;c;z)}{\mathrm{\Gamma }\left(c\right)}=𝐅(a,b;-n;z)=\frac{{\left(a\right)}_{n+1}{\left(b\right)}_{n+1}}{(n+1)!}{z}^{n+1}F(a+n+1,b+n+1;n+2;z),$$ $n=0,1,2,\mathrm{\dots }$

was added in the second paragraph. An equation number will be assigned in an expanded numbering scheme that is under current development. Additionally, the discussion following (15.2.6) was expanded.

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Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

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14.19.2 $$P_{\nu -\frac{1}{2}}^{\mu }\left(\cosh \xi \right)=\frac{\mathrm{\Gamma }\left(\frac{1}{2}-\mu \right)}{{\pi }^{1/2}{\left(1-{\mathrm{e}}^{-2\xi }\right)}^{\mu }{\mathrm{e}}^{(\nu +(1/2))\xi }}𝐅(\frac{1}{2}-\mu ,\frac{1}{2}+\nu -\mu ;1-2\mu ;1-{\mathrm{e}}^{-2\xi }),$$ $\mu \ne \frac{1}{2},\frac{3}{2},\frac{5}{2},\mathrm{\dots }$

Originally the argument to $𝐅$ in this equation was incorrect (${\mathrm{e}}^{-2\xi }$, rather than $1-{\mathrm{e}}^{-2\xi }$), and the condition on $\mu$ was too weak ($\mu \ne \frac{1}{2}$, rather than $\mu \ne \frac{1}{2},\frac{3}{2},\frac{5}{2},\mathrm{\dots }$). Also, the factor multiplying $𝐅$ was rewritten to clarify the poles; originally it was $\frac{\mathrm{\Gamma }\left(1-2\mu \right)2^{2\mu }}{\mathrm{\Gamma }\left(1-\mu \right){\left(1-{\mathrm{e}}^{-2\xi }\right)}^{\mu }{\mathrm{e}}^{(\nu +(1/2))\xi }}$.

Reported 2010-11-02 by Alvaro Valenzuela.

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3: 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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Incomplete gamma functions for large values of their variables. Adv. in Appl. Math. 34 (3), pp. 467–485.

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External Links:

ISSN 0196-8858, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§8.12.

Encodings:

BibTeX

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An asymptotic expansion of the double gamma function. J. Approx. Theory 111 (2), pp. 298–314.

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External Links:

ISSN 0021-9045, Document, MathReview (C. M. Joshi), ZentralBlatt

Referenced by:

§5.17.

Encodings:

BibTeX

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Algorithm 309. Gamma function with arbitrary precision. Comm. ACM 10 (8), pp. 511–512.

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External Links:

ISSN 0001-0782, Document

Referenced by:

§5.24(ii).

Encodings:

BibTeX

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Application of the $\tau$-function theory of Painlevé equations to random matrices: ${\mathrm{P}}_{\mathrm{VI}}$, the JUE, CyUE, cJUE and scaled limits. Nagoya Math. J. 174, pp. 29–114.

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External Links:

ISSN 0027-7630, MathReview, ZentralBlatt

Referenced by:

§32.10(vi), §32.6(v).

Encodings:

BibTeX

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4: Bibliography G

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Algorithm 542: Incomplete gamma functions. ACM Trans. Math. Software 5 (4), pp. 482–489.

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Notes:

Single-precision Fortran.

External Links:

ISSN 0098-3500, Document, GAMS (module 542; package TOMS), ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

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Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. Phys. 38 (1), pp. 77–81.

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External Links:

MathReview (H. O. Pollak), ZentralBlatt

Referenced by:

§5.6(i), §6.8, §7.8, §8.10.

Encodings:

BibTeX

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A harmonic mean inequality for the gamma function. SIAM J. Math. Anal. 5 (2), pp. 278–281.

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External Links:

Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§5.6(i).

Encodings:

BibTeX

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A computational procedure for incomplete gamma functions. ACM Trans. Math. Software 5 (4), pp. 466–481.

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External Links:

ISSN 0098-3500, Document, ZentralBlatt

Referenced by:

§8.25(iv), §8.25(v).

Encodings:

BibTeX

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §10.77(ix).

Encodings:

BibTeX

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