§ 5.2. Definitions

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

§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

Notes:: See Olver (1997b, pp. 31 and 39) or Temme (1996, pp. 41 and 53).
Referenced by:: §2.3(ii), §2.5(i)
Permalink:: http://dlmf.nist.gov/5.2.SS1

¶ Euler's Integral

5.2.1 $\Gamma\!\left(z\right)=\int _{0}^{\infty}e^{{-t}}t^{{z-1}}dt,$ $\realpart{z}>0$ .

Defines:: $\Gamma\!\left(z\right)$ : Gamma function
Symbols:: $z$ : complex variable
A&S Ref:: 6.1.1
Referenced by:: §5.9(i), §5.9(ii), §9.12(vi)
Permalink:: http://dlmf.nist.gov/5.2.E1
Encodings:: TeX, pMathML, png

When $\realpart{z}\le 0$ , $\Gamma\!\left(z\right)$ 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 $\psi\!\left(z\right)={{\Gamma}^{{\prime}}}\!\left(z\right)/\Gamma\!\left(z\right),$ $z\neq 0,-1,-2,\dots$ .

Defines:: $\psi\!\left(z\right)$ : Psi or digamma function
Symbols:: $\Gamma\!\left(z\right)$ : Gamma function and $z$ : complex variable
A&S Ref:: 6.3.1
Permalink:: http://dlmf.nist.gov/5.2.E2
Encodings:: TeX, pMathML, png

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

§ 5.2(ii). Euler's Constant

Notes:: See Olver (1997b, p. 34) or Temme (1996, p. 10).
Referenced by:: §2.10(i), §2.5(iii), §2.6(iii), §27.11, Tab.5.1.1, §9.12(viii)
Permalink:: http://dlmf.nist.gov/5.2.SS2

5.2.3 $\EulerConstant=\lim _{{n\to\infty}}\left(1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n}-\ln n\right)=0.57721\; 56649\; 0 1532\; 86060\;\dots.$

Notes:: For more digits see OEIS Sequence A001620; see also Sloane (2003).
Defines:: $\EulerConstant$ : Euler's constant
Symbols:: $n$ : nonnegative integer
A&S Ref:: 6.1.3 (where the 10D value is given, and Table 1.1 where the 24D value is given.)
Permalink:: http://dlmf.nist.gov/5.2.E3
Encodings:: TeX, pMathML, png