DLMF: GA.2 Definitions

§GA.2. Definitions

Sources:

Olver(1997)(Chapter 2, §§1 and 2), Temme(1996)(Chapters 1 and 3).

§GA.2(i)Gamma and Psi Functions
§GA.2(ii)Euler's Constant
§GA.2(iii)Pochhammer's Symbol

§GA.2(i). Gamma and Psi Functions

Notes:

Olver(1997)(pp. 31 and 39)

Temme(1996)(pp. 41 and 53)

Euler's Integral

Keywords:: Euler's integral, gamma function, psi function

GA.2.1 $\Gamma\!\left(z\right)=\int _{0}^{\infty}e^{{-t}}t^{{z-1}}dt,$ $\realpart{z}>0$ .

Notations:: $\Gamma\!\left(z\right)$ : Gamma function and $z$ : complex variable
A&S Ref:: 6.1.1
Referenced by:: §GA.9(i), §GA.9
Encodings:: pMathML, png, TeX

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 $z=-n$ .

GA.2.2 $\psi\!\left(z\right)={{\Gamma}^{{\prime}}}\!\left(z\right)/\Gamma\!\left(z\right),$ $z\neq 0,-1,-2,\dots$ .

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $\psi\!\left(z\right)$ : Psi or digamma function and $z$ : complex variable
A&S Ref:: 6.3.1
Encodings:: pMathML, png, TeX

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

§GA.2(ii). Euler's Constant

Notes:

Olver(1997)(p. 34)

Temme(1996)(p. 10)

Keywords:

Euler's constant

GA.2.3 $\EulerConstant=\lim _{{n\to\infty}}\left(1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n}-\mathrm{ln}n\right)=0.57721\; 56649\; 0 1532\; 86060\;\dots.$