Environmental Engineering Reference
In-Depth Information
Solution.
By the properties of the
δ
-function,
+
∞
e
−
st
t
=
0
=
e
−
st
d
t
L
[
δ
(
t
)] =
δ
(
t
)
=
1
.
0
In applications, we can find corresponding relations between
f
(
s
)
and
f
(
t
)
from the
tables of Laplace transformations.
Similar to the Fourier transformation,
f
(
t
)
must satisfy certain conditions for the
existence of
f
(
s
)
. It can be shown that the Laplace transformation
+
∞
f
e
−
st
d
t
(
s
)=
L
[
f
(
t
)] =
f
(
t
)
0
exists in
Re
(
s
)
>
k
and is an analytical function if: (1)
f
(
t
)
≡
0for
t
<
0, (2) for
t
is continuous in any finite region except a finite number of discontinuous
points of the first kind, (3) the increasing speed of
f
≥
0,
f
(
t
)
(
t
)
is less than an exponential
→
+
∞
function as
t
so that there exist constants
c
and
k
such that
c
e
kt
|
(
)
|≤
,
<
<
+
∞
.
f
t
0
t
≥
(
)
Here
k
0iscalledthe
increasing exponent
of
f
t
.
The integral
f
(
s
)
is, in fact, absolutely and uniformly convergent in
Re
(
s
)=
β
>
k
,i.e.
+
∞
+
∞
f
e
−
st
d
t
c
β
−
c
e
−
(
β
−
k
)
t
d
t
(
t
)
≤
=
k
.
0
0
Also
+
∞
0
−
f
)
=
e
−
st
d
t
(
s
tf
(
t
)
exists in
Re
(
s
)=
β
>
k
because
+
∞
c
+
∞
0
−
e
−
st
d
t
c
(
β
−
t
e
−
(
β
−
k
)
t
d
t
tf
(
t
)
≤
=
2
.
k
)
0
f
Therefore,
(
s
)
not only exists in
Re
(
s
)=
β
>
k
but also is an analytical function.
B.2.2 Properties of Laplace Transformation
The Laplace transformation and its inverse transformation are defined by integrals.
By the properties of integration, we can easily obtain the following properties of
Laplace transformations.
Linearity
Let
L
f
1
(
f
2
(
[
f
1
(
t
)] =
s
)
,
L
[
f
2
(
t
)] =
s
)
. For any two constants
α
and
β
,wehave
L
[
α
f
1
(
t
)+
β
f
2
(
t
)] =
α
L
[
f
1
(
t
)]+
β
L
[
f
2
(
t
)]
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