Lecture Notes in Differential Equations - Bruce E. Shapiro

338 LESSON 32. THE LAPLACE TRANSFORM
The Gamma Function
Definition 32.9. The Gamma Function is defined by
Γ(x) =
∫ ∞
for x ∈ R but x not equal to a negative integer.
0
u x−1 e −u du (32.45)
.
Figure 32.2: The Gamma function Γ(t).
10
5
4 3 2 1 1 2 3 4
5
10
If we make a change of variable u = st for some constant s and new variable
t in (32.57), then du = sdt and
Thus
or
Γ(x) =
∫ ∞
0
∫ ∞
(st) x−1 e −st sdt = s x t x−1 e −st dt (32.46)
0
∫ ∞
Γ(x + 1) = s x+1 t x e −st dt = L[t x ] (32.47)
L[t x ] =
0
Γ(x + 1)
s x+1 (32.48)
The gamma function has an interesting factorial like property. Decreasing
the argument in (32.49) by 1,
Γ(x) = L [ t x−1] (32.49)
But since
t x−1 = 1 d
x dt tx (32.50)