5 Gamma Function Properties 5.1 Special Notation 5.3 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

§5.2(i) Gamma and Psi Functions
§5.2(ii) Euler’s Constant
§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		$\Gamma\left(z\right)=\int_{0}^{\infty}e^{-t}t^{z-1}\,\mathrm{d}t,$
$\Re z>0$ .
ⓘ Defines: $\Gamma\left(\NVar{z}\right)$ : gamma function Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : 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 $\Re z\leq 0$ , $\Gamma\left(z\right)$ is defined by analytic continuation. It is a meromorphic function with no zeros, and with simple poles of residue $(-1)^{n}/n!$ at $z=-n$ . $1/\Gamma\left(z\right)$ is entire, with simple zeros at $z=-n$ .

5.2.2		$\psi\left(z\right)=\Gamma'\left(z\right)/\Gamma\left(z\right),$
$z\neq 0,-1,-2,\dots$ .
ⓘ Defines: $\psi\left(\NVar{z}\right)$ : psi (or digamma) function Symbols: $\Gamma\left(\NVar{z}\right)$ : 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		$\gamma=\lim_{n\to\infty}\left(1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n}-\ln n% \right)=0.57721\;56649\;01532\;86060\;\dots.$
ⓘ Defines: $\gamma$ : Euler’s constant Symbols: $\ln\NVar{z}$ : 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:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : 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	$\displaystyle{\left(a\right)_{0}}$	$\displaystyle=1,$
5.2.4	$\displaystyle{\left(a\right)_{n}}$	$\displaystyle=a(a+1)(a+2)\cdots(a+n-1),$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : 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	$\displaystyle{\left(a\right)_{n}}$	$\displaystyle=\Gamma\left(a+n\right)/\Gamma\left(a\right),$
$a\neq 0,-1,-2,\dots$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : 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: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : 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		${\left(-m\right)_{n}}=\begin{cases}\frac{(-1)^{n}m!}{(m-n)!},&0\leq n\leq m,\\ 0,&n>m,\end{cases}$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : 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	$\displaystyle{\left(a\right)_{2n}}$	$\displaystyle=2^{2n}{\left(\frac{a}{2}\right)_{n}}{\left(\frac{a+1}{2}\right)_% {n}},$
5.2.8	$\displaystyle{\left(a\right)_{2n+1}}$	$\displaystyle=2^{2n+1}{\left(\frac{a}{2}\right)_{n+1}}{\left(\frac{a+1}{2}% \right)_{n}}.$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : 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) ${\left(x\right)_{n}}=x(x+1)\cdots(x+n-1)$ and falling factorials $(-1)^{n}{\left(-x\right)_{n}}=x(x-1)\cdots(x-n+1)$ can be expressed in terms of each other via

5.2.9	$\displaystyle{\left(x\right)_{n}}$	$\displaystyle=\sum_{k=0}^{n}L(n,k)x(x-1)\cdots(x-k+1),$
5.2.9	$\displaystyle x(x-1)\cdots(x-n+1)$	$\displaystyle=\sum_{k=0}^{n}(-1)^{n-k}L(n,k){\left(x\right)_{k}},$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : 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 $L(n,k)=\genfrac{(}{)}{0.0pt}{}{n-1}{k-1}\frac{n!}{k!}$ is the Lah number.

5.1 Special Notation 5.3 Graphics

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§5.2 Definitions

Contents

§5.2(i) Gamma and Psi Functions

Euler’s Integral

§5.2(ii) Euler’s Constant

§5.2(iii) Pochhammer’s Symbol