# §5.2 Definitions

## §5.2(i) Gamma and Psi Functions

### Euler’s Integral

 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, $\mathrm{e}$: 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

 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

 5.2.4 $\displaystyle{\left(a\right)_{0}}$ $\displaystyle=1,$ $\displaystyle{\left(a\right)_{n}}$ $\displaystyle=a(a+1)(a+2)\cdots(a+n-1),$ ⓘ 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 $\displaystyle{\left(a\right)_{2n}}$ $\displaystyle=2^{2n}{\left(\frac{a}{2}\right)_{n}}{\left(\frac{a+1}{2}\right)_% {n}},$ $\displaystyle{\left(a\right)_{2n+1}}$ $\displaystyle=2^{2n+1}{\left(\frac{a}{2}\right)_{n+1}}{\left(\frac{a+1}{2}% \right)_{n}}.$ ⓘ 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