752 CHAPTER 37. SOME FUNDAMENTAL FUNCTIONS AND TRANSFORMS

Proof: By the monotone convergence theorem, for n ∈ N

Γ(α) = limn→∞

∫ n

1/ne−ttα−1 ≤ lim sup

n→∞

(∫ 1

1/ntα−1dt +

∫ n

1Ce−t/2

)≤ 1

α+ lim

n→∞

(−2Ce−

12 n +2Ce−

12

)< ∞

The explanation for the constant is as follows. For t ≥ 1 and m a positive integer larger thanα,

tα−1

et/2 <tn−1

et/2

which converges to 0 as t → ∞ which is easily shown by an appeal to L’Hospital’s rule.Hence

tα−1e−t ≤Cet/2e−t =Ce−t/2.■

Proposition 37.1.3 For n a positive integer, n!=Γ(n+1). In general, Γ(1)= 1,Γ(α +1)=αΓ(α)

Proof: First of all, Γ(1) = limδ→0∫

δ−1

δe−tdt = limδ→0

(e−δ − e−(δ

−1))= 1. Next,

for α > 0,

Γ(α +1) = limδ→0

∫δ−1

δ

e−ttα dt = limδ→0

[−e−ttα |δ

−1

δ+α

∫δ−1

δ

e−ttα−1dt

]

= limδ→0

(e−δ

δα − e−(δ

−1)δ−α +α

∫δ−1

δ

e−ttα−1dt

)= αΓ(α)

Now it is defined that 0! = 1 and so Γ(1) = 0!. Suppose that Γ(n+1) = n!, what ofΓ(n+2)? Is it (n+1)!? if so, then by induction, the proposition is established. Fromwhat was just shown,

Γ(n+2) = Γ(n+1)(n+1) = n!(n+1) = (n+1)!

and so this proves the proposition. ■

37.2 Laplace TransformEverything holds for a much more general set of assumptions if you have a more modernversion of the integral. This is why I am using notation which corresponds to this moregeneral situation. All of the functions considered here are assumed piecewise continuouswith finitely many jumps in every finite interval. Then such a function f is said to be inL1 ([0,∞)) if ∫

∞

0| f (t)|dt < ∞

Similar usages of this symbol are defined synonomously. Sometimes I will just write L1

to indicate that the absolute value of the function is integrable. Here is the definition of aLaplace transform.