Lecture Notes in Differential Equations - Bruce E. Shapiro

334 LESSON 32. THE LAPLACE TRANSFORM
The existence of the Laplace Transform therefore depends on the existence
of the second integral We must show that this integral converges. But
Thus the second integral is bounded by
∫ ∞
M
|e −st f(t)| ≤ Ke −st e at = Ke (a−s)t (32.14)
e −st f(t)dt ≤
≤
∫ ∞
M
∫ ∞
M
= lim
T →∞
= lim
T →∞
|e −st f(t)|dt
Ke (a−s)t dt
∫ T
M
Ke (a−s)t dt
∣
K ∣∣∣
T
a − s e(a−s)t
M
K
= lim
T →∞ a − s [e(a−s)T − e (a−s)M ]
{ 0 if a < s
=
∞ if a ≥ s
Thus when s > a, the second integral in (32.13) vanishes and the total
integral converges (is defined).
Theorem 32.7 (Linearity). The Laplace Transform is lineary, e.g., for any
two functions f(t) and g(t) with Laplace Transforms F (s) and G(s), and
any two constants A and B,
L[Af(t) + Ag(t)] = AL[f(t)] + BL[g(t)] = AF (s) + BG(s) (32.15)
Proof. This property follows immediately from the linearity of the integral,
as follows:
L[Af(t) + Ag(t)] =
= A
∫ ∞
0
∫ ∞
0
(Af(t) + Bg(t))e −st dt (32.16)
f(t)e −st dt + B
∫ ∞
0
g(t)e −st dt (32.17)
= AL[f(t)] + BL[g(t)] = AF (s) + BG(s) (32.18)