Page 776 - Handbook Of Integral Equations

10.4. Gamma Function. Beta Function

Definition. Integral representations
The gamma function, Γ(z), is an analytic function of the complex argument z everywhere,
except for the points z =0, –1, –2, ...
For Re z >0,
∞

e dt.
Γ(z)= t z–1 –t
0
For –(n +1) < Re z < –n, where n = 0,1,2, ... ,
n
m
∞ (–1)
–t
Γ(z)= e – t z–1 dt.
0 m!
m=0
Euler formula
n! n z
Γ(z) = lim (z ≠ 0, –1, –2, ... ).
n→∞ z(z +1) ... (z + n)

Simplest properties

Γ(z +1) = zΓ(z), Γ(n +1) = n!, Γ(1) = Γ(2)=1.

Symmetry formulas

π π
Γ(z)Γ(–z)= – , Γ(z)Γ(1 – z)= ,
z sin(πz) sin(πz)
1 1 π

Γ + z Γ – z = .
2 2 cos(πz)
Multiple argument formulas

2 2z–1 1
Γ(2z)= √ Γ(z)Γ z + ,
π 2
3 3z–1/2 1 2
Γ(3z)= Γ(z)Γ z + Γ z + ,
2π 3 3
n–1
! k
(1–n)/2 nz–1/2
Γ(nz)=(2π) n Γ z + .
n
k=0
Fractional values of the argument
√
1
√ 1 π
Γ = π, Γ n + = (2n – 1)!!,
2 2 2 n
1 √ 1 n 2 π
n √
Γ – = –2 π, Γ – n =(–1) .
2 2 (2n – 1)!!
Asymptotic expansion (Stirling formula)
√
–3
–z z–1/2
Γ(z)= 2πe z 1+ 1 z –1 + 1 z –2 + O(z ) (|arg |z < π).
12 288

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 761