6.1: THE LAPLACE TRANSFORM
KIAM HEONG KWA
A tool of frequent occurrence in modern mathematics on studying
an unknown function
f
is to transform the function
f
into another
function by means of a
linear integral transformation
T
:
(1)
f
(
t
)
7→ T
f
(
s
) =
T
[
f
(
t
)](
s
) =
Z
b
a
K
(
s, t
)
f
(
t
)
dt.
Equation (1) gives the value at the point
s
of the transformed function
T
f
, called the
transform
of
f
, in terms of the values of
f
at points in
the interval (
a, b
). Either
a
or
b
or both may be infinite. The function
K
of two variables is referred to as the
kernel
of the transformation
T
.
Such a linear transformation
T
satisfies the
linearity property
(2)
T
(
αf
+
βg
) =
α
T
f
+
β
T
g
for all functions
f
and
g
under consideration and for all scalars
α
and
β
.
The transformation
T
will be particularly useful if, from the knowl
edge of
T
f
, we are able to determine
f
uniquely. The process of recov
ering the unknown function
f
from its transform
T
f
is known as the
inversion problem
.
In this case, it is customary to denote the inverse
transformation by
T

1
, so that
T

1
(
T
f
) =
f.
The usefulness of an integral transformation arises from the fact that
manipulating the transformed functions is often equivalent to handling
the unknown functions under consideration and may be simpler. In this
way, an entire problem may be converted to a simpler one and solved
and then the original problem is solved by inverting the transformation.
We shall consider a particular type of linear integral transformation,
called the onesided
Laplace transform
:
(3)
f
(
t
)
7→ L
f
(
s
) =
L
[
f
(
t
)](
s
) =
Z
∞
0
e

st
f
(
t
)
dt
= lim
τ
→∞
Z
τ
0
e

st
f
(
t
)
dt.
Date
: February 21, 2011.
1
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KIAM HEONG KWA
The Laplace transform can also be defined for complexvalued func
tions with the parameter
s
being complex also[1, 2].
As indicated in the following example, not every function possesses
a Laplace transform.
Example 1.
The Laplace transform
L
[
e
t
2
](
s
)
is not welldefined for
any
s
. This is so since for every
s
, if
t
*
>
max
{
1
 
s

,
1 +

s
}
, then
Z
τ
t
*
e

st
e
t
2
dt
=
Z
τ
t
*
e
t
(
t

s
)
dt
≥
Z
τ
t
*
e
t
dt
=
e
τ

e
t
*
for all
τ
≥
t
*
, from which it follows that
Z
∞
t
*
e

st
e
t
2
dt
≥
lim
τ
→∞
(
e
τ

e
t
*
) =
∞
.
This implies that the improper integral defining
L
[
e
t
2
](
s
)
also diverges.
In order to show that there is a large class of functions that admits
Laplace transforms, we first make a few definitions in the following.
Definition 1.
A function
f
is said to have a
jump discontinuity
at a
point
τ
if both the onesided limits
f
(
τ

) = lim
t
→
τ

f
(
t
)
and
f
(
τ
+) = lim
t
→
τ
+
f
(
t
)
exist as finite numbers and
f
(
τ

)
6
=
f
(
τ
+)
.
Definition 2.
A function
f
is said to be
piecewise continuous
on the
interval
[0
,
∞
)
if the following conditions hold:
(1)
f
(0+) = lim
t
→
0+
f
(
t
)
exists as a finite number,
(2)
f
is continuous on every finite interval
(0
, b
)
except possibly at
a finite number of points
t
1
, t
2
,
· · ·
, t
n
in
(0
, b
)
at which
f
has
jump discontinuities.
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 Winter '11
 Kwa
 Math, Differential Equations, Equations, Continuous function, dt, KIAM HEONG KWA

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