The product nm is used in the definition of Γ(nz ) to show that the equation
(12.146) simplifies after a lot of careful algebra to
φ(z) = limm→∞nnz−^1 [(m−1)!]nmn−^21 mmn
(nm −1)! (nm )nz = limm→∞[(m−1)!]nmn−^21 mmn−^1
(nm −1)! (12.147)The equation (12.147) shows that φ(z) is independent of z and is a constant. To
find the value of the constant, select a value of z where equation (12.143) can be
evaluated. Selecting the value z= 1 /n one finds after simplification the product
formula first derived by Gauss^5 and Legendre^6
nnzΓ(z)Γ(
z+1
n)
Γ(
z+2
n)
···Γ(
z+n− 1
n)
=n^1 /^2 (2 π)n− 21
Γ(nz ) (12.148)Using equation (12.148) one can produce the special cases
n= 2 Γ(z)Γ(
z+1
2)
=Γ(2z) (2π)^1 /^221 /^2 −^2 zn= 3 Γ(z)Γ(
z+^1
3)
Γ(
z+^2
3)
=Γ(3z)(2π) 3^1 /^2 −^3 z(12.149)Example 12-8. (Summation)
The function Ψ(x) defined by Ψ(x) =
d
dx ln Γ(x) =Γ′(x)Γ(x) occurs in representing
the summation of many finite and infinite convergent series. The Gamma function
satisfies
Γ(x+ 1) = xΓ(x)so that
lnΓ(x+ 1) = ln x+ lnΓ(x) (12.150)Differentiate equation (12.150) and show
Ψ(x+ 1) =^1
x+ Ψ(x) or Ψ(x+ 1) −Ψ(x) =^1
x(12.151)This demonstrates that Ψ(x)satisfies the difference equation
∆Ψ(x) =^1 x or ∆Ψ(a+n) = a+^1 n, a is constant (12.152)
(^5) Carl Friedrich Gauss (1777-1855) A famous German mathematician.
(^6) Adrien-Marie Legendre (1752-1833) A famous French mathematician.