24
The Laplace Transform (Intro)
The Laplace transform is a mathematical tool based on integration that has a number of appli
cations. It particular, it can simplify the solving of many differential equations. We will find
it particularly useful when dealing with nonhomogeneous equations in which the forcing func
tions are not continuous. This makes it a valuable tool for engineers and scientists dealing with
“realworld” applications.
By the way, the Laplace transform is just one of many “integral transforms” in general use.
Conceptually and computationally, it is probably the simplest. If you understand the Laplace
transform, then you will find it much easier to pick up the other transforms as needed.
24.1
Basic Definition and Examples
Definition, Notation and Other Basics
Let
f
be a ‘suitable’ function (more on that later). The
Laplace transform of
f
, denoted by
either
F
or
L
[
f
] , is the function given by
F
(
s
)
=
L
[
f
]

s
=
integraldisplay
∞
0
f
(
t
)
e
−
st
dt
.
(24.1)
!
◮
Example 24.1:
For our first example, let us use
f
(
t
)
=
braceleftBigg
1
if
t
≤
2
0
if
2
<
t
.
This is the relatively simple discontinuous function graphed in figure 24.1a. To compute the
Laplace transform of this function, we need to break the integral into two parts:
F
(
s
)
=
L
[
f
]

s
=
integraldisplay
∞
0
f
(
t
)
e
−
st
dt
=
integraldisplay
2
0
f
(
t
)
bracehtipupleft
bracehtipdownrightbracehtipdownleft
bracehtipupright
1
e
−
st
dt
+
integraldisplay
∞
2
f
(
t
)
bracehtipupleft
bracehtipdownrightbracehtipdownleft
bracehtipupright
0
e
−
st
dt
=
integraldisplay
2
0
e
−
st
dt
+
integraldisplay
∞
2
0
dt
=
integraldisplay
2
0
e
−
st
dt
.
471
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472
The Laplace Transform
(a)
(b)
T
S
1
2
2
1
1
2
0
Figure 24.1:
The graph of
(a)
the discontinuous function
f
(
t
)
from example 24.1 and
(b)
its Laplace transform
F
(
s
)
.
So, if
s
negationslash=
0
,
F
(
s
)
=
integraldisplay
2
0
e
−
st
dt
=
e
−
st
−
s
vextendsingle
vextendsingle
vextendsingle
vextendsingle
2
t
=
0
= −
1
s
bracketleftbig
e
−
s
·
2
−
e
−
s
·
0
bracketrightbig
=
1
s
bracketleftbig
1
−
e
−
2
s
bracketrightbig
.
And if
s
=
0
,
F
(
s
)
=
F
(
0
)
=
integraldisplay
2
0
e
−
0
·
t
dt
=
integraldisplay
2
0
1
dt
=
2
.
This is the function sketched in figure 24.1b. (Using L’Hôpital’s rule, you can easily show that
F
(
s
)
→
F
(
0
)
as
s
→
0
. So, despite our need to compute
F
(
s
)
separately when
s
=
0
,
F
is a continuous function.)
As the example just illustrated, we really are ‘transforming’ the function
f
(
t
)
into another
function
F
(
s
)
. This process of transforming
f
(
t
)
to
F
(
s
)
is also called the
Laplace transform
and, unsurprisingly, is denoted by
L
. Thus, when we say “the Laplace transform”, we can be
referring to either the transformed function
F
(
s
)
or to the process of computing
F
(
s
)
from
f
(
t
)
.
Some other quick notes:
1.
There are standard notational conventions that simplify bookkeeping. The functions ‘to
be transformed’ are (almost) always denoted by lower case Roman letters —
f
,
g
,
h
,
etc. — and
t
is (almost) always used as the variable in the formulas for these functions
(because, in applications, these are typically functions of time).
The corresponding
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 Summer '09
 Differential Equations, Equations, Derivative, Continuous function, e−st dt

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