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Properties of Laplace Transform

Since many of the functions we will be dealing with will be solutions to differential

equations and hence continuous, the above assumptions are completely justified.
A useful fact is that the inverse Laplace transform has the linearity property.

Theorem 2.12È

Let F (s) and F (s) be functions whose inverse Laplace transforms exist.
1
2
Then for any scalars α, β

−1
−1
−1
L [αF (s) + βF (s)] = αL [F (s)] + βL [F (s)].
2
1
2
1
.
. . . . . . .
Example 2.22,
( )
1 1 1 1
−1 t −t
L + = e + e = cosh t, t ≥ 0.
2(s − 1) 2(s + 1) 2 2
. . . . .
One of the practical features of the Laplace transform is that it can be applied
to discontinuous functions f. In these instances, it must be borne in mind that

when the inverse transform is invoked, there are other functions with the same
−1
L F(s).
An important function occurring in electrical systems is the (delayed) unit step
function
{
1, t ≥ a,
u (t) =
a
0, t < a,

for a ≥ 0. This function delays its output until t = a and then assumes a constant
value of one unit. In the literature, the unit step function is also commonly defined
as
{
1, t > a,
u (t) =
a
0, t < a,

for a ≥ 0, and is known as the Heaviside (step) function. Both definitions of u (t)
a
have the same Laplace transform and so from that point of view are
indistinguishable. When a = 0, we will write u (t) = u(t).
a
Another common notation for the unit step function u (t) is u(t−a) or h(t−a).
a
Compute the Laplace transform,

∫ ∫ ∞
∞ ∞ e −st e −as
L(u (t)) = e −st u (t)dt = e −st dt = = (Re(s) > 0).
a
a
0 a −s 0 s
It is appropriate to write with either interpretation of u (t)
a

( −as )
e
−1
L = u (t),
a
s
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