5 Gamma FunctionProperties5.1 Special Notation5.3 Graphics

§5.2 Definitions

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Referenced by:

§1.15(vi), §10.23(iii), §10.43(ii), §10.44(iii), §25.5(ii)

Permalink:

http://dlmf.nist.gov/5.2

See also:

Annotations for Ch.5

Contents

1. §5.2(i) Gamma and Psi Functions
2. §5.2(ii) Euler’s Constant
3. §5.2(iii) Pochhammer’s Symbol

§5.2(i) Gamma and Psi Functions

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Notes:

See Olver (1997b, pp. 31 and 39) or Temme (1996b, pp. 41 and 53).

Referenced by:

§10.1, §10.22(ii), §10.31, §10.8, §13.1, §14.1, §14.17(iv), §15.1, §19.12, §2.3(ii), §2.5(i), §25.1, §32.11(ii), §33.1, §33.6, §6.10(ii), §6.6, §8.1, §8.19(iv)

Permalink:

http://dlmf.nist.gov/5.2.i

See also:

Annotations for §5.2 and Ch.5

Euler’s Integral

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Keywords:

Euler’s integral, analytic properties, definition, gamma function, psi function, reciprocal, zeros

See also:

Annotations for §5.2(i), §5.2 and Ch.5

5.2.1		$$\mathrm{\Gamma }\left(z\right)={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{z-1}dt,$$
$\mathrm{\Re }z>0$.
ⓘ Defines: $\mathrm{\Gamma }\left(z\right)$: gamma function Symbols: $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable A&S Ref: 6.1.1 Referenced by: §10.43(iii), (25.11.27), (25.11.28), (25.5.6), (25.5.7), §5.9(i), §5.9(ii), §8.21(ii), (9.12.17) Permalink: http://dlmf.nist.gov/5.2.E1 Encodings: TeX, pMML, png See also: Annotations for §5.2(i), §5.2(i), §5.2 and Ch.5

When $\mathrm{\Re }z\le 0$, $\mathrm{\Gamma }\left(z\right)$ is defined by analytic continuation. It is a meromorphic function with no zeros, and with simple poles of residue ${(-1)}^n/n!$ at $z=-n$. $1/\mathrm{\Gamma }\left(z\right)$ is entire, with simple zeros at $z=-n$.

5.2.2		$$\psi \left(z\right)={\mathrm{\Gamma }}^{\prime }\left(z\right)/\mathrm{\Gamma }\left(z\right),$$
$z\ne 0,-1,-2,\mathrm{\dots }$.
ⓘ Defines: $\psi \left(z\right)$: psi (or digamma) function Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function and $z$: complex variable A&S Ref: 6.3.1 Permalink: http://dlmf.nist.gov/5.2.E2 Encodings: TeX, pMML, png See also: Annotations for §5.2(i), §5.2(i), §5.2 and Ch.5

$\psi \left(z\right)$ is meromorphic with simple poles of residue $-1$ at $z=-n$.

§5.2(ii) Euler’s Constant

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Keywords:

Euler’s constant

Notes:

See Olver (1997b, p. 34) or Temme (1996b, p. 10).

Referenced by:

§10.22(iii), §10.24, §10.45, §10.8, §10.9(i), §11.6(i), §13.1, §14.1, §14.8(i), §2.10(i), §2.5(iii), §2.6(iii), §25.1, §25.2(iv), §26.8(vii), §27.11, §3.12, §33.5(iii), §5.1, §6.1, §7.1, §9.12(viii)

Permalink:

http://dlmf.nist.gov/5.2.ii

See also:

Annotations for §5.2 and Ch.5

5.2.3		$$\gamma =\underset{n\to \mathrm{\infty }}{lim}\left(1+\frac{1}{2}+\frac{1}{3}+\mathrm{\cdots }+\frac{1}{n}-\ln n\right)=\mathrm{0.57721\hspace{0.33em}56649\hspace{0.33em}01532\hspace{0.33em}86060}\mathrm{\dots }.$$
ⓘ Defines: $\gamma$: Euler’s constant Symbols: $\ln z$: principal branch of logarithm function and $n$: nonnegative integer A&S Ref: 6.1.3 (where the 10D value is given, and Table 1.1 where the 24D value is given.) Notes: For more digits see OEIS Sequence A001620; see also Sloane (2003). Referenced by: §4.4(iii) Permalink: http://dlmf.nist.gov/5.2.E3 Encodings: TeX, pMML, png See also: Annotations for §5.2(ii), §5.2 and Ch.5

§5.2(iii) Pochhammer’s Symbol

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Defines:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial)

Keywords:

Lah numbers, Pochhammer’s symbol, definitions

Referenced by:

§3.9(v), Erratum (V1.0.17) for Subsection 5.2(iii)

Permalink:

http://dlmf.nist.gov/5.2.iii

Addition (effective with 1.2.0):

A paragraph under (5.2.8) about Lah numbers and (5.2.9) was added.

Suggested 2024-02-26 by Eric Shirley

Addition (effective with 1.0.17):

Equations (5.2.6)–(5.2.8) were added.

See also:

Annotations for §5.2 and Ch.5

5.2.4		${\left(a\right)}_0$	$=1,$
		${\left(a\right)}_n$	$=a(a+1)(a+2)\mathrm{\cdots }(a+n-1),$
ⓘ Symbols: ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer and $a$: real or complex variable Referenced by: (25.11.10), (25.8.2) Permalink: http://dlmf.nist.gov/5.2.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §5.2(iii), §5.2 and Ch.5
5.2.5		${\left(a\right)}_n$	$=\mathrm{\Gamma }\left(a+n\right)/\mathrm{\Gamma }\left(a\right),$
$a\ne 0,-1,-2,\mathrm{\dots }$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer and $a$: real or complex variable A&S Ref: 6.1.22 Referenced by: (25.11.28), (25.5.7), (25.8.2) Permalink: http://dlmf.nist.gov/5.2.E5 Encodings: TeX, pMML, png See also: Annotations for §5.2(iii), §5.2 and Ch.5

5.2.6		$${\left(-a\right)}_n={(-1)}^n{\left(a-n+1\right)}_n,$$
ⓘ Symbols: ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer and $a$: real or complex variable Referenced by: §5.2(iii), Erratum (V1.0.17) for Subsection 5.2(iii) Permalink: http://dlmf.nist.gov/5.2.E6 Encodings: TeX, pMML, png Addition (effective with 1.0.17): This equation was added. See also: Annotations for §5.2(iii), §5.2 and Ch.5

5.2.7
ⓘ Symbols: ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $m$: nonnegative integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/5.2.E7 Encodings: TeX, pMML, png Addition (effective with 1.0.17): This equation was added. See also: Annotations for §5.2(iii), §5.2 and Ch.5

5.2.8		${\left(a\right)}_{2n}$	$=2^{2n}{\left({\displaystyle \frac{a}{2}}\right)}_n{\left({\displaystyle \frac{a+1}{2}}\right)}_n,$
		${\left(a\right)}_{2n+1}$	$=2^{2n+1}{\left({\displaystyle \frac{a}{2}}\right)}_{n+1}{\left({\displaystyle \frac{a+1}{2}}\right)}_n.$
ⓘ Symbols: ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer and $a$: real or complex variable Referenced by: §5.2(iii), Erratum (V1.0.17) for Subsection 5.2(iii) Permalink: http://dlmf.nist.gov/5.2.E8 Encodings: TeX, TeX, pMML, pMML, png, png Addition (effective with 1.0.17): This equation was added. Suggested by Tom Koornwinder See also: Annotations for §5.2(iii), §5.2 and Ch.5

Pochhammer symbols (rising factorials) ${\left(x\right)}_n=x(x+1)\mathrm{\cdots }(x+n-1)$ and falling factorials ${(-1)}^n{\left(-x\right)}_n=x(x-1)\mathrm{\cdots }(x-n+1)$ can be expressed in terms of each other via

5.2.9		${\left(x\right)}_n$	$={\displaystyle \sum _{k=0}^n}L(n,k)x(x-1)\mathrm{\cdots }(x-k+1),$
		$x(x-1)\mathrm{\cdots }(x-n+1)$	$={\displaystyle \sum _{k=0}^n}{(-1)}^{n-k}L(n,k){\left(x\right)}_k,$
ⓘ Symbols: ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer, $k$: nonnegative integer, $x$: real variable and $L(n,k)$: Lah number Referenced by: §5.2(iii), Erratum (V1.2.0) for Chapter 5 Addition Permalink: http://dlmf.nist.gov/5.2.E9 Encodings: TeX, TeX, pMML, pMML, png, png Addition (effective with 1.2.0): This equation was added. Suggested by Eric Shirley See also: Annotations for §5.2(iii), §5.2 and Ch.5

in which $L(n,k)=\left(\genfrac{}{}{0.0pt}{}{n-1}{k-1}\right)\frac{n!}{k!}$ is the Lah number.

5.1 Special Notation5.3 Graphics

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