
30.5. Incomplete Gamma and Beta Functions 1519
30.5 Incomplete Gamma and Beta Functions
30.5.1 In com plete Gamma Function
◮ Integral representations. Recurrence formulas.
Definitions:
γ(α, x) =
Z
x
0
e
−t
t
α−1
dt, Re α > 0,
Γ(α, x) =
Z
∞
x
e
−t
t
α−1
dt = Γ(α) − γ(α, x).
Recurrence formulas:
γ(α + 1, x) = αγ(α, x) − x
α
e
−x
,
γ(α + 1, x) = (x + α)γ(α, x) + (1 − α)xγ(α − 1, x),
Γ(α + 1, x) = αΓ(α, x) + x
α
e
−x
.
Special cases:
γ(n + 1, x) = n!
1 −e
−x
n
X
k=0
x
k
k!
, n = 0, 1, . . . ;
Γ(n + 1, x) = n! e
−x
n
X
k=0
x
k
k!
, n = 0, 1, . . . ;
Γ(−n, x) =
(−1)
n
n!
Γ(0, x) − e
−x
n−1
X
k=0
(−1)
k
k!
x
k+1
, n = 1, 2, . . .
◮ Expansions as x → 0 and x → ∞. Relation to other functions.
Asymptotic expansions ...