§5.2 Definitions

Referenced by:: §1.15(vi), ¶ ‣ §10.23(iii), §10.43(ii), §10.44(iii), §25.5(ii)
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 (1996b, pp. 41 and 53).
Referenced by:: §10.1, §10.22(ii), §10.31, §10.8, §13.1, §14.1, §14.17(iv), §15.1, §19.12, §2.3(ii), ¶ ‣ §2.5(i), §25.1, §32.11(ii), §33.1, §33.6, §6.10(ii), §6.6, §8.1, §8.19(iv)
Permalink:: http://dlmf.nist.gov/5.2.i

¶ Euler’s Integral

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

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

Defines:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function
Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : 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

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

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

$\mathop{\psi\/}\nolimits\!\left(z\right)$ 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 (1996b, p. 10).
Keywords:: Euler’s constant
Referenced by:: §10.22(iii), §10.24, §10.45, §10.8, ¶ ‣ §10.9(i), §11.6(i), §13.1, §14.1, §14.8(i), ¶ ‣ §2.10(i), ¶ ‣ §2.5(iii), §2.6(iii), §25.1, §25.2(iv), §26.8(vii), §27.11, §3.12, §33.5(iii), §5.1, §6.1, §7.1, ¶ ‣ §9.12(viii)
Permalink:: http://dlmf.nist.gov/5.2.ii

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

Notes:: For more digits see OEIS Sequence A001620; see also Sloane (2003).
Defines:: $\EulerConstant$ : Euler’s constant
Symbols:: $\mathop{\ln\/}\nolimits z$ : 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.)
Referenced by:: §4.4(iii)
Permalink:: http://dlmf.nist.gov/5.2.E3
Encodings:: TeX, pMML, png