What's New
About the Project
NIST
5 Gamma FunctionProperties

§5.2 Definitions

Contents

§5.2(i) Gamma and Psi Functions

Euler’s Integral

5.2.1 Γ(z)=0e-ttz-1dt,
z>0.

When z0, Γ(z) 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/Γ(z) is entire, with simple zeros at z=-n.

5.2.2 ψ(z)=Γ(z)/Γ(z),
z0,-1,-2,.

ψ(z) is meromorphic with simple poles of residue -1 at z=-n.

§5.2(ii) Euler’s Constant

5.2.3 γ=limn(1+12+13++1n-lnn)=0.57721 56649 01532 86060.

§5.2(iii) Pochhammer’s Symbol

5.2.4 (a)0 =1,
(a)n =a(a+1)(a+2)(a+n-1),
5.2.5 (a)n =Γ(a+n)/Γ(a),
a0,-1,-2,.