§5.2 Definitions

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 $z=-n$.