About the Project
5 Gamma FunctionProperties

§5.2 Definitions

  1. §5.2(i) Gamma and Psi Functions
  2. §5.2(ii) Euler’s Constant
  3. §5.2(iii) Pochhammer’s Symbol

§5.2(i) Gamma and Psi Functions

Euler’s Integral

5.2.1 Γ(z)=0ettz1dt,

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),

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++1nlnn)=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+n1),
5.2.5 (a)n =Γ(a+n)/Γ(a),
5.2.6 (a)n=(1)n(an+1)n,
5.2.7 (m)n={(1)nm!(mn)!,0nm,0,n>m,
5.2.8 (a)2n =22n(a2)n(a+12)n,
(a)2n+1 =22n+1(a2)n+1(a+12)n.

Pochhammer symbols (rising factorials) (x)n=x(x+1)(x+n1) and falling factorials (1)n(x)n=x(x1)(xn+1) can be expressed in terms of each other via

5.2.9 (x)n =k=0nL(n,k)x(x1)(xk+1),
x(x1)(xn+1) =k=0n(1)nkL(n,k)(x)k,

in which L(n,k)=(n1k1)n!k! is the Lah number.