# §5.2 Definitions

## §5.2(i) Gamma and Psi Functions

### Euler’s Integral

 5.2.1 $\mathop{\Gamma\/}\nolimits\!\left(z\right)=\int_{0}^{\infty}e^{-t}t^{z-1}% \mathrm{d}t,$ $\Re{z}>0$. Defines: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{e}$: base of exponential function, $\int$: integral, $\Re{}$: real part and $z$: complex variable A&S Ref: 6.1.1 Referenced by: §10.43(iii), §5.9(i), §5.9(ii), §8.21(ii), §9.12(vi) Permalink: http://dlmf.nist.gov/5.2.E1 Encodings: TeX, pMML, png See also: Annotations for 5.2(i)

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

 5.2.2 $\mathop{\psi\/}\nolimits\!\left(z\right)=\mathop{\Gamma\/}\nolimits'\!\left(z% \right)/\mathop{\Gamma\/}\nolimits\!\left(z\right),$ $z\neq 0,-1,-2,\dots$. Defines: $\mathop{\psi\/}\nolimits\!\left(\NVar{z}\right)$: psi (or digamma) function Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: 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)

$\mathop{\psi\/}\nolimits\!\left(z\right)$ is meromorphic with simple poles of residue $-1$ at $z=-n$.

## §5.2(ii) Euler’s Constant

 5.2.3 $\gamma=\lim_{n\to\infty}\left(1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n}-% \mathop{\ln\/}\nolimits n\right)=0.57721\;56649\;01532\;86060\;\dots.$ Defines: $\gamma$: Euler’s constant Symbols: $\mathop{\ln\/}\nolimits\NVar{z}$: 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(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 Permalink: http://dlmf.nist.gov/5.2.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 5.2(iii) 5.2.5 $\displaystyle{\left(a\right)_{n}}$ $\displaystyle=\mathop{\Gamma\/}\nolimits\!\left(a+n\right)/\mathop{\Gamma\/}% \nolimits\!\left(a\right),$ $a\neq 0,-1,-2,\dots$.