scaled gamma function

SearchAdvancedHelp

(0.009 seconds)

1—10 of 40 matching pages

1: 8.18 Asymptotic Expansions of $I_x(a,b)$

… ►

General Case

►For the scaled gamma function ${\mathrm{\Gamma }}^{\ast }\left(z\right)$ see (5.11.3). … ►

8.18.14 $$I_x(a,b)\sim Q(b,a\zeta )-\frac{{(2\pi b)}^{-1/2}}{{\mathrm{\Gamma }}^{\ast }\left(b\right)}{\left(\frac{x}{x_0}\right)}^a{\left(\frac{1-x}{1-x_0}\right)}^b\sum _{k=0}^{\mathrm{\infty }}\frac{h_k(\zeta ,\mu )}{a^k},$$

ⓘ

Symbols:

$I_x(a,b)$: incomplete beta function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $Q(a,z)$: normalized incomplete gamma function, ${\mathrm{\Gamma }}^{\ast }\left(z\right)$: scaled gamma function, $k$: nonnegative integer, $a$: parameter, $b$: parameter, $x$: variable, $\mu$ , $\zeta$: variable and $h_k(\zeta ,\mu )$: coefficients

Permalink:

http://dlmf.nist.gov/8.18.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8

…

2: 5.11 Asymptotic Expansions

… ►

5.11.3 $$\mathrm{\Gamma }\left(z\right)={\mathrm{e}}^{-z}{z}^z{\left(\frac{2\pi }{z}\right)}^{1/2}{\mathrm{\Gamma }}^{\ast }\left(z\right)\sim {\mathrm{e}}^{-z}{z}^z{\left(\frac{2\pi }{z}\right)}^{1/2}\sum _{k=0}^{\mathrm{\infty }}\frac{g_k}{z^k},$$

ⓘ

Defines:

${\mathrm{\Gamma }}^{\ast }\left(z\right)$: scaled gamma function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer, $z$: complex variable and $g_k$: coefficients

A&S Ref:

6.1.37

Referenced by:

§10.19(i), §13.8(iv), §5.11(i), §5.11(i), §5.11(i), §5.11(ii), §5.11(ii), §5.21, §5.9(i), §8.11(ii), §8.12, (8.18.13), (8.18.13), §8.18(ii), §8.18(ii), Erratum (V1.1.7) for Additions, Erratum (V1.1.8) for Rearrangement

Permalink:

http://dlmf.nist.gov/5.11.E3

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

The definition of the scaledgammafunction ${\mathrm{\Gamma }}^{\ast }\left(z\right)$ was inserted after the first equals sign.

Changes (effective with 1.0.11):

An unnecessary bracket around the sum was removed.

See also:

Annotations for §5.11(i), §5.11 and Ch.5

… ►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 $|\mathrm{ph}z|\le \pi -\delta$. …

3: 5.9 Integral Representations

… ►

5.9.11_1 $${\mathrm{\Gamma }}^{\ast }\left(z\right)=1-\frac{1}{2\pi \mathrm{i}}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-2\pi t}{\mathrm{\Gamma }}^{\ast }\left(t{\mathrm{e}}^{\mathrm{i}\pi /2}\right)}{t+\mathrm{i}z}dt+\frac{1}{2\pi \mathrm{i}}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-2\pi t}{\mathrm{\Gamma }}^{\ast }\left(t{\mathrm{e}}^{-\mathrm{i}\pi /2}\right)}{t-\mathrm{i}z}dt,$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{\Gamma }}^{\ast }\left(z\right)$: scaled gamma function and $z$: complex variable

Source:

Boyd (1994, (2.16), p. 617)

Referenced by:

§5.9(i), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/5.9.E11_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

►

5.9.11_2 $$\frac{1}{{\mathrm{\Gamma }}^{\ast }\left(z\right)}=1-\frac{1}{2\pi \mathrm{i}}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-2\pi t}{\mathrm{\Gamma }}^{\ast }\left(t{\mathrm{e}}^{\mathrm{i}\pi /2}\right)}{t-\mathrm{i}z}dt+\frac{1}{2\pi \mathrm{i}}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-2\pi t}{\mathrm{\Gamma }}^{\ast }\left(t{\mathrm{e}}^{-\mathrm{i}\pi /2}\right)}{t+\mathrm{i}z}dt,$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{\Gamma }}^{\ast }\left(z\right)$: scaled gamma function and $z$: complex variable

Source:

Boyd (1994, (4.2), p. 628)

Referenced by:

§5.9(i), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/5.9.E11_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

►where , and the scaled gamma function ${\mathrm{\Gamma }}^{\ast }\left(z\right)$ is defined in (5.11.3). …

4: 14.3 Definitions and Hypergeometric Representations

… ►

14.3.4

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $m$: nonnegative integer, $\nu$: general degree and : real variable

Referenced by:

§14.3(i)

Permalink:

http://dlmf.nist.gov/14.3.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(i), §14.3(i), §14.3 and Ch.14

… ►

14.3.5

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $m$: nonnegative integer, $\nu$: general degree and : real variable

Referenced by:

§14.3(i)

Permalink:

http://dlmf.nist.gov/14.3.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(i), §14.3(i), §14.3 and Ch.14

… ►

14.3.13 $$w_1(\nu ,\mu ,x)=\frac{2^{\mu }\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +\frac{1}{2}\right)}{\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +1\right)}{\left(1-x^2\right)}^{-\mu /2}𝐅(-\frac{1}{2}\nu -\frac{1}{2}\mu ,\frac{1}{2}\nu -\frac{1}{2}\mu +\frac{1}{2};\frac{1}{2};x^2),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $x$: real variable, $\mu$: general order, $\nu$: general degree and $w_2$: function $w_1$,

Referenced by:

§10.59

Permalink:

http://dlmf.nist.gov/14.3.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(iii), §14.3 and Ch.14

►

14.3.14 $$w_2(\nu ,\mu ,x)=\frac{2^{\mu }\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +1\right)}{\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +\frac{1}{2}\right)}x{\left(1-x^2\right)}^{-\mu /2}𝐅(\frac{1}{2}-\frac{1}{2}\nu -\frac{1}{2}\mu ,\frac{1}{2}\nu -\frac{1}{2}\mu +1;\frac{3}{2};x^2).$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $x$: real variable, $\mu$: general order, $\nu$: general degree and $w_2$: function $w_1$,

Referenced by:

§10.59, §14.3(i), §14.5(i)

Permalink:

http://dlmf.nist.gov/14.3.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(iii), §14.3 and Ch.14

… ►

14.3.19

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $\mu$: general order, $\nu$: general degree and : real variable

Permalink:

http://dlmf.nist.gov/14.3.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(iii), §14.3 and Ch.14

…

5: Errata

… ►

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).

… ►

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.

…

6: 13.8 Asymptotic Approximations for Large Parameters

… ►where ${\mathrm{\Gamma }}^{\ast }\left(a\right)$ is the scaled gamma function defined in (5.11.3). …

7: 5.23 Approximations

… ►See Schmelzer and Trefethen (2007) for a survey of rational approximations to various scaled versions of $\mathrm{\Gamma }\left(z\right)$. …

8: 15.14 Integrals

… ►

15.14.1 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}𝐅(\genfrac{}{}{0pt}{}{a,b}{c};-x)dx=\frac{\mathrm{\Gamma }\left(s\right)\mathrm{\Gamma }\left(a-s\right)\mathrm{\Gamma }\left(b-s\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-s\right)},$$ $\min (\mathrm{\Re }a,\mathrm{\Re }b)>\mathrm{\Re }s>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $x$: real variable, $s$: nonnegative integer, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Keywords:

Mellin transform

Referenced by:

§15.14

Permalink:

http://dlmf.nist.gov/15.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.14 and Ch.15

…

9: 16.2 Definition and Analytic Properties

… ►

16.2.5 $${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)={}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)/\left(\mathrm{\Gamma }\left(b_1\right)\mathrm{\cdots }\mathrm{\Gamma }\left(b_q\right)\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a_1\right)}_k\mathrm{\cdots }{\left(a_p\right)}_k}{\mathrm{\Gamma }\left(b_1+k\right)\mathrm{\cdots }\mathrm{\Gamma }\left(b_q+k\right)}\frac{z^k}{k!};$$

ⓘ

Defines:

${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

§16.2(v)

Permalink:

http://dlmf.nist.gov/16.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §16.2(v), §16.2 and Ch.16

…

10: 14.23 Values on the Cut

… ►

14.23.3 $${𝑸}_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)=\frac{{\mathrm{e}}^{\mp \nu \pi \mathrm{i}/2}{\pi }^{3/2}{\left(1-x^2\right)}^{\mu /2}}{2^{\nu +1}}\left(\frac{x𝐅(\frac{1}{2}\mu -\frac{1}{2}\nu +\frac{1}{2},\frac{1}{2}\nu +\frac{1}{2}\mu +1;\frac{3}{2};x^2)}{\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +\frac{1}{2}\right)}\mp \mathrm{i}\frac{𝐅(\frac{1}{2}\mu -\frac{1}{2}\nu ,\frac{1}{2}\nu +\frac{1}{2}\mu +\frac{1}{2};\frac{1}{2};x^2)}{\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +1\right)\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +1\right)}\right).$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.23

Permalink:

http://dlmf.nist.gov/14.23.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.23 and Ch.14

…