1.2 Gamma Function
3
1.2
Gamma Function
1.2.1 Fundamental Equations
The gamma function Γ is a generalization of the factorial function. The following
equations can be found in [1] and [3]. For positive integers n
Γ (n) = (n − 1)! , and
(1.1a)
n!! =
2
1
2 n Γ (
1
2 n + 1)
: n even
π
−
1
2 2
1
2 (n+1) Γ (
1
2 n + 1)
: n odd
.
(1.1b)
The gamma function can be defined by
Γ (z) =
∞
0
z
z−1 exp(−t)dt , and
(1.2)
holds the recurrence relation
Γ (z + 1) = z Γ (z) .
(1.3)
Euler’s reflection formula
Γ (1 − z)Γ (z) =
π
sin(πz)
z /
∈ Z.
(1.4)
Plays an important rôle for evaluating Γ (z). Figure 1.1 shows the absolute value of
the gamma function in the complex plane. Under complex conjugation the gamma
function becomes
Γ (¯ z) = Γ (z) and ln(Γ (¯ z)) = ln(Γ (z)).
(1.5)
For z = −n, n ∈ N Γ (z) has a simple pole, Fig. 1.1, with residuum
Res(Γ, −n) =
−n
n!
.
Some special values are
Γ (ix) =
π
x sinh(πx)
with x real,
(1.6a)
Γ (
1
2
) = π
1
2 , and
(1.6b)
|Γ (x + iy)| ≤ |Γ (x)|.
(1.6c)