gamma function

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21: 5.6 Inequalities

§5.6 Inequalities

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5.6.2 $$\frac{1}{\mathrm{\Gamma }\left(x\right)}+\frac{1}{\mathrm{\Gamma }\left(1/x\right)}\le 2,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function and $x$: real variable

Referenced by:

§5.6(i)

Permalink:

http://dlmf.nist.gov/5.6.E2

Encodings:

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Annotations for §5.6(i), §5.6 and Ch.5

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5.6.3 $$\frac{1}{{(\mathrm{\Gamma }\left(x\right))}^2}+\frac{1}{{(\mathrm{\Gamma }\left(1/x\right))}^2}\le 2,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function and $x$: real variable

Referenced by:

§5.6(i)

Permalink:

http://dlmf.nist.gov/5.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.6(i), §5.6 and Ch.5

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5.6.6 $$|\mathrm{\Gamma }\left(x+\mathrm{i}y\right)|\le |\mathrm{\Gamma }\left(x\right)|,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{i}$: imaginary unit, $x$: real variable and $y$: real variable

A&S Ref:

6.1.26

Referenced by:

§5.6(ii)

Permalink:

http://dlmf.nist.gov/5.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §5.6(ii), §5.6 and Ch.5

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5.6.8 $$\left|\frac{\mathrm{\Gamma }\left(z+a\right)}{\mathrm{\Gamma }\left(z+b\right)}\right|\le \frac{1}{{|z|}^{b-a}}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $z$: complex variable, $a$: real or complex variable and $b$: real or complex variable

Referenced by:

§5.6(ii)

Permalink:

http://dlmf.nist.gov/5.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §5.6(ii), §5.6 and Ch.5

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22: 5.23 Approximations

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§5.23(i) Rational Approximations

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§5.23(ii) Expansions in Chebyshev Series

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§5.23(iii) Approximations in the Complex Plane

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

23: 8.9 Continued Fractions

§8.9 Continued Fractions

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8.9.1 $$\mathrm{\Gamma }\left(a+1\right){\mathrm{e}}^z{\gamma }^{\ast }(a,z)=\frac{1}{1-}\frac{z}{a+1+}\frac{z}{a+2-}\frac{(a+1)z}{a+3+}\frac{2z}{a+4-}\frac{(a+2)z}{a+5+}\frac{3z}{a+6-}\mathrm{\cdots },$$ $a\ne -1,-2,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.31 (in modified form.)

Permalink:

http://dlmf.nist.gov/8.9.E1

Encodings:

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Annotations for §8.9 and Ch.8

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8.9.2 $$z^{-a}{\mathrm{e}}^z\mathrm{\Gamma }(a,z)=\frac{z^{-1}}{1+}\frac{(1-a)z^{-1}}{1+}\frac{z^{-1}}{1+}\frac{(2-a)z^{-1}}{1+}\frac{2z^{-1}}{1+}\frac{(3-a)z^{-1}}{1+}\frac{3z^{-1}}{1+}\mathrm{\cdots },$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\mathrm{ph}$: phase, $z$: complex variable and $a$: parameter

Referenced by:

§8.19(vii)

Permalink:

http://dlmf.nist.gov/8.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.9 and Ch.8

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24: 8.6 Integral Representations

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§8.6(i) Integrals Along the Real Line

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§8.6(ii) Contour Integrals

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Mellin–Barnes Integrals

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§8.6(iii) Compendia

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25: 8.27 Approximations

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§8.27(i) Incomplete Gamma Functions

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DiDonato (1978) gives a simple approximation for the function $F(p,x)=x^{-p}{\mathrm{e}}^{x^2/2}{\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{-t^2/2}{t}^{p}dt$ (which is related to the incomplete gamma function by a change of variables) for real $p$ and large positive $x$. This takes the form $F(p,x)=4x/h(p,x)$, approximately, where $h(p,x)=3(x^2-p)+\sqrt{{(x^2-p)}^2+8(x^2+p)}$ and is shown to produce an absolute error $O\left(x^{-7}\right)$ as $x\to \mathrm{\infty }$.

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Luke (1969b, pp. 25, 40–41) gives Chebyshev-series expansions for $\mathrm{\Gamma }(a,\omega z)$ (by specifying parameters) with , and $\gamma (a,\omega z)$ with $0\le \omega \le 1$; see also Temme (1994b, §3).

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26: 5.8 Infinite Products

§5.8 Infinite Products

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5.8.1 $$\mathrm{\Gamma }\left(z\right)=\underset{k\to \mathrm{\infty }}{lim}\frac{k!k^z}{z(z+1)\mathrm{\cdots }(z+k)},$$ $z\ne 0,-1,-2,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $!$: factorial (as in $n!$), $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.1.2

Referenced by:

§5.8

Permalink:

http://dlmf.nist.gov/5.8.E1

Encodings:

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Annotations for §5.8 and Ch.5

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5.8.2 $$\frac{1}{\mathrm{\Gamma }\left(z\right)}=z{\mathrm{e}}^{\gamma z}\prod _{k=1}^{\mathrm{\infty }}\left(1+\frac{z}{k}\right){\mathrm{e}}^{-z/k},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.1.3

Permalink:

http://dlmf.nist.gov/5.8.E2

Encodings:

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See also:

Annotations for §5.8 and Ch.5

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5.8.3 $${\left|\frac{\mathrm{\Gamma }\left(x\right)}{\mathrm{\Gamma }\left(x+\mathrm{i}y\right)}\right|}^2=\prod _{k=0}^{\mathrm{\infty }}\left(1+\frac{y^2}{{(x+k)}^2}\right),$$ $x\ne 0,-1,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $x$: real variable and $y$: real variable

A&S Ref:

6.1.25 (where the formula for the reciprocal is given.)

Referenced by:

§5.8

Permalink:

http://dlmf.nist.gov/5.8.E3

Encodings:

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Annotations for §5.8 and Ch.5

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5.8.5 $$\prod _{k=0}^{\mathrm{\infty }}\frac{(a_1+k)(a_2+k)\mathrm{\cdots }(a_m+k)}{(b_1+k)(b_2+k)\mathrm{\cdots }(b_m+k)}=\frac{\mathrm{\Gamma }\left(b_1\right)\mathrm{\Gamma }\left(b_2\right)\mathrm{\cdots }\mathrm{\Gamma }\left(b_m\right)}{\mathrm{\Gamma }\left(a_1\right)\mathrm{\Gamma }\left(a_2\right)\mathrm{\cdots }\mathrm{\Gamma }\left(a_m\right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $m$: nonnegative integer, $k$: nonnegative integer, $a_k$: coefficient and $b_k$: coefficient

Referenced by:

§5.8

Permalink:

http://dlmf.nist.gov/5.8.E5

Encodings:

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See also:

Annotations for §5.8 and Ch.5

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27: 5.18 $q$-Gamma and $q$-Beta Functions

§5.18 $q$-Gamma and $q$-Beta Functions

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§5.18(ii) $q$-Gamma Function

… ►Also, $\ln {\mathrm{\Gamma }}_q\left(x\right)$ is convex for $x>0$, and the analog of the Bohr–Mollerup theorem (§5.5(iv)) holds. … ►

5.18.10 $$\underset{q\to 1-}{lim}{\mathrm{\Gamma }}_q\left(z\right)=\mathrm{\Gamma }\left(z\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, $q$: real or complex variable and $z$: complex variable

Permalink:

http://dlmf.nist.gov/5.18.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §5.18(ii), §5.18 and Ch.5

… ►For the $q$-digamma or $q$-psi function ${\psi }_q(z)={\mathrm{\Gamma }}_q^{\prime }\left(z\right)/{\mathrm{\Gamma }}_q\left(z\right)$ see Salem (2013). …

28: 5.19 Mathematical Applications

§5.19 Mathematical Applications

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5.19.3 $$S=\psi \left(\frac{1}{2}\right)-2\psi \left(\frac{2}{3}\right)-\gamma =3\ln 3-2\ln 2-\frac{1}{3}\pi \sqrt{3}.$$

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Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function and $\ln z$: principal branch of logarithm function

Permalink:

http://dlmf.nist.gov/5.19.E3

Encodings:

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Annotations for §5.19(i), §5.19(i), §5.19 and Ch.5

… ►Many special functions $f(z)$ can be represented as a Mellin–Barnes integral, that is, an integral of a product of gamma functions, reciprocals of gamma functions, and a power of $z$, the integration contour being doubly-infinite and eventually parallel to the imaginary axis at both ends. … ►

§5.19(iii) $n$-Dimensional Sphere

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29: 8.4 Special Values

§8.4 Special Values

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8.4.2 $${\gamma }^{\ast }(a,0)=\frac{1}{\mathrm{\Gamma }\left(a+1\right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\gamma }^{\ast }(a,z)$: incomplete gamma function and $a$: parameter

Permalink:

http://dlmf.nist.gov/8.4.E2

Encodings:

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Annotations for §8.4 and Ch.8

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8.4.5 $$\mathrm{\Gamma }(1,z)={\mathrm{e}}^{-z},$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/8.4.E5

Encodings:

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Annotations for §8.4 and Ch.8

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8.4.9 $$P(n+1,z)=1-{\mathrm{e}}^{-z}{e}_n(z),$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $P(a,z)$: normalized incomplete gamma function, $z$: complex variable, $n$: nonnegative integer and $e_n(z)$: functions

A&S Ref:

6.5.13

Permalink:

http://dlmf.nist.gov/8.4.E9

Encodings:

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Annotations for §8.4 and Ch.8

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8.4.12 $${\gamma }^{\ast }(-n,z)=z^n,$$

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Symbols:

${\gamma }^{\ast }(a,z)$: incomplete gamma function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

6.5.14

Referenced by:

§8.13(i)

Permalink:

http://dlmf.nist.gov/8.4.E12

Encodings:

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Annotations for §8.4 and Ch.8

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30: 8.5 Confluent Hypergeometric Representations

§8.5 Confluent Hypergeometric Representations

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8.5.1 $$\gamma (a,z)=a^{-1}{z}^a{\mathrm{e}}^{-z}M(1,1+a,z)=a^{-1}{z}^{a}M(a,1+a,-z),$$ $a\ne 0,-1,-2,\mathrm{\dots }$.

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.12

Referenced by:

§8.11(ii), §8.14

Permalink:

http://dlmf.nist.gov/8.5.E1

Encodings:

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Annotations for §8.5 and Ch.8

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8.5.2 $${\gamma }^{\ast }(a,z)={\mathrm{e}}^{-z}𝐌(1,1+a,z)=𝐌(a,1+a,-z).$$

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Symbols:

$𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.7

Permalink:

http://dlmf.nist.gov/8.5.E2

Encodings:

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Annotations for §8.5 and Ch.8

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8.5.3 $$\mathrm{\Gamma }(a,z)={\mathrm{e}}^{-z}U(1-a,1-a,z)=z^a{\mathrm{e}}^{-z}U(1,1+a,z).$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.14, §8.19(vi)

Permalink:

http://dlmf.nist.gov/8.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.5 and Ch.8

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8.5.4 $$\gamma (a,z)=a^{-1}{z}^{\frac{1}{2}a-\frac{1}{2}}{\mathrm{e}}^{-\frac{1}{2}z}{M}_{\frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right).$$

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Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.5

Permalink:

http://dlmf.nist.gov/8.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §8.5 and Ch.8

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