Theory of Statistics - George Mason University

C.2 General Mathematical Functions and Operators 857
Γ(α) The complete gamma function:
Γ(α) =
∞
t
0
α−1 e −t dt. (C.2)
(This is called Euler’s integral.) Integration by parts immediately
gives the replication formula
Γ(α + 1) = αΓ(α),
and so if α is a positive integer, Γ(α + 1) = α!. More
generally, Γ(α + 1) can be taken as the definition of α!.
This does not exist for negative integers, but does for
all other real α.
Direct evaluation of the integral yields Γ(1/2) = √ π.
Using this and the replication formula, with some manipulation
we get for the positive integer j
Γ(j + 1/2) =
1 · 2 · · ·(2j − 1)
2j √
π.
The notation Γd(α) denotes the multivariate gamma
function, where α is a d-vector. (In other literature this
notation denotes the incomplete univariate gamma function,
for which I use γ(α, d); see below.)
Associated with the gamma function are some other useful functions:
ψ(α) The psi function or the digamma function:
ψ ′ (α) The trigamma function,
ψ(α) = d log(Γ(α))/dα. (C.3)
ψ ′ (α) = dψ(α)/dα. (C.4)
More general are the polygamma functions, for n =
1, 2, . . ., ψ (n) (α) = d (n) ψ(α)/(dα) (n) , which for a fixed
n, is called the (n + 2)-gamma function.
γ(α, x) The incomplete gamma function,
γ(α, x) =
This is also often denoted as Γx(α).
Theory of Statistics c○2000–2013 James E. Gentle
x
t
0
α−1 e −t dt. (C.5)