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The Laplace Transform
Definition and properties of Laplace Transform, piecewise continuous
functions, the Laplace Transform method of solving initial value problems
The method of Laplace transforms is a system that relies on algebra (rather
than calculusbased methods) to solve linear differential equations.
While it
might seem to be a somewhat cumbersome method at times, it is a very
powerful tool that enables us to readily deal with linear differential equations
with discontinuous forcing functions.
Definition
:
Let
f
(
t
) be defined for
t
≥ 0.
The Laplace transform of
f
(
t
),
denoted by
F
(
s
) or
L
{
f
(
t
)}, is given by
L
{
f
(
t
)} =
∫
∞

=
0
)
(
)
(
dt
t
f
e
s
F
st
.
Provided that this (improper) integral exists, i.e. that the integral is
convergent.
For functions continuous on [0,
∞
), the above transformation is
onetoone
.
That is, different continuous functions will have different transforms.
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View Full Document Example
:
Let
f
(
t
) = 1, then
s
s
F
1
)
(
=
,
s
> 0
.
L
{
f
(
t
)} =
∞

∞

∞


=
=
∫
∫
0
0
0
1
)
(
st
st
st
e
s
dt
e
dt
t
f
e
The integral is divergent whenever
s
≤ 0.
However, when
s
> 0, it
converges to
( )
)
(
1
)
1
(
1
0
1
0
s
F
s
s
e
s
=
=


=


.
Example
:
Let
f
(
t
) =
t
, then
2
1
)
(
s
s
F
=
,
s
> 0
.
[This is left to you as an exercise.]
Example
:
Let
f
(
t
) =
e
at
, then
a
s
s
F

=
1
)
(
,
s
>
a
.
L
{
f
(
t
)} =
∞

∞

∞


=
=
∫
∫
0
)
(
0
)
(
0
1
t
s
a
t
s
a
at
st
e
s
a
dt
e
dt
e
e
The integral is divergent whenever
s
≤
a
.
However, when
s
>
a
, it
converges to
( )
)
(
1
)
1
(
1
0
1
0
s
F
a
s
s
a
e
s
a
=

=


=


.
Definition
:
A function
f
(
t
) is called
piecewise continuous
if it only has
finitely many (or none whatsoever – a continuous function is considered to
be “piecewise continuous”!) discontinuities on any interval [
a
,
b
], and that
both onesided limits exist as
t
approaches each of those discontinuity from
within
the interval.
The last part of the definition means that
f
could have
removable and/or jump discontinuities only; it cannot have any infinity
discontinuity.
Theorem
:
Suppose that
1.
f
is piecewise continuous on the interval 0 ≤
t
≤
A
for any
A
> 0.
2.
│
f
(
t
)│ ≤
K
e
at
when
t
≥
M
, for any real constant
a
, and any positive
constants
K
and
M
.
(This means that
f
is “of exponential order”, i.e.
its rate of growth is no faster than that of exponential functions.)
Then the Laplace transform,
F
(
s
) =
L
{
f
(
t
)}, exists for
s
>
a
.
Note
:
The above theorem gives a sufficient
condition for the existence of
Laplace transforms.
It is not
a necessary
condition.
A function does not
need to satisfy the two conditions in order to have a Laplace transform.
Examples of such functions that nevertheless have Laplace transforms are
logarithmic functions and the unit impulse function.
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View Full DocumentSome properties of the Laplace Transform
1.
L
{0} = 0
2.
L
{
f
(
t
)
±
g
(
t
)} =
L
{
f
(
t
)}
±
L
{
g
(
t
)}
3.
L
{
c
f
(
t
)} =
c
L
{
f
(
t
)}
, for any constant
c
.
Properties 2 and 3 together means that the Laplace transform is
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This note was uploaded on 07/16/2011 for the course MATH 251 taught by Professor Chezhongyuan during the Spring '08 term at Pennsylvania State University, University Park.
 Spring '08
 CHEZHONGYUAN
 Differential Equations, Calculus, Algebra, Equations, Partial Differential Equations

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