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Fall 2003 Math 308/501–502
7 Laplace Transforms
7.2 Deﬁnition of the Laplace Transform
Wed, 08/Oct
c
±
2003, Art Belmonte
Summary
Deﬁnitions
•
Given a function
f
of
t
,the
Laplace transform
of
f
is a
function
L
{
f
}
(
s
)
=
F
(
s
)
deﬁned by
L
{
f
}
(
s
)
=
±
∞
0
f
(
t
)
e

st
dt
=
lim
T
→∞
±
T
0
f
(
t
)
e

,
when this limit exists. Here
s
>
0 and may be further
restricted depending on
f
. The transform is an
operator
which acts on a function to produce yet another function.
•
A function
f
is
piecewise continuous
on
(
0
,
∞
)
if it has
only ﬁnitely many jump discontinuities on any ﬁnite
subinterval of
(
0
,
∞
)
.
•
In a similar manner,
f
is
piecewise differentiable
on
(
0
,
∞
)
if it is continuous and its derivative is piecewise continuous.
•
A function
f
is of
exponential order
if

f
(
t
)
 ≤
Ce
at
for
t
>
0, where
C
and
a
are constants.
Existence Theorem for Laplace Transforms
If f is a function deﬁned on
[0
,
∞
)
that is piecewise continuous
and of exponential order (say

f
(
t
)
 ≤
), then the Laplace
transform
L
{
f
}
(
s
)
exists for s
>
a (at least).
Linearity of the Laplace Transform
Let the Laplace transforms of
f
,
f
1
,and
f
2
exist for
s
>α
and let
c
be any constant. Then we have
L
{
f
1
+
f
2
}
=
L
{
f
1
} +
L
{
f
2
}
L
{
cf
}
=
c
L
{
f
}
Notes
When computing Laplace transforms strictly with a pencil, we use
integration by parts, L’Hospital’s rule, etc. On the other hand, the
MATLAB Symbolic Math Toolbox (SMT) command
laplace
computes Laplace transforms at one fell swoop. In between these
two extremes is a middle ground, where one mimicks hand work
semiautomatically with the SMT commands
int
,
subs
limit
.
This is how
Hand Examples
below were computed. They were
veriﬁed via
laplace
.
Small Table of Laplace Transforms
f
(
t
)
F
(
s
)
=
L
{
f
}
(
s
)
Restrictions
1
1
s
s
>
0
e
1
s

a
s
>
a
t
n
n
!
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 Math

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