Lecture Notes in Differential Equations - Bruce E. Shapiro

346 LESSON 32. THE LAPLACE TRANSFORM
The Laplace Transform of the step function is easily calculated,
L[U(t − a)] =
=
∫ ∞
0
∫ ∞
a
= e−as
s
U(t − a)e −st dt (32.113)
e −st dt (32.114)
(32.115)
We will also find the Laplace Transforms of shifted functions to be useful:
L[f(t)U(t − a)] =
=
Here we make a change of variable
∫ ∞
0
∫ ∞
a
f(t)U(t − a)e −st dt (32.116)
f(t)e −st dt (32.117)
x = t − a (32.118)
so that
L[f(t)U(t − a)] =
∫ ∞
0
f(x + a)e −s(x+a) dx (32.119)
∫ ∞
= e −sa f(x + a)e −sx dx (32.120)
0
= e −sa L[f(t + a)] (32.121)
A similar result is obtained for
L[f(t − a)U(t − a)] =
=
Now define
so that
∫ ∞
0
∫ ∞
L[f(t − a)U(t − a)] =
a
f(t − a)U(t − a)e −st dt (32.122)
f(t − a)e −st dt (32.123)
x = t − a (32.124)
∫ ∞
0
f(t)e −s(x+a) dx (32.125)
∫ ∞
= e −sa f(t)e −sx dx (32.126)
0
= e −as F (s) (32.127)