Environmental Engineering Reference
In-Depth Information
Also, we have the convolution theorem
F
−
1
=
f
1
f
2
=
f
1
∗
f
2
.
B.2 Laplace Transformation
The Laplace transformation draws from an extension of the Fourier transforma-
tion. When the conditions for the Fourier transformation cannot be satisfied, we can
sometimes apply a Laplace transformation.
B.2.1 Laplace Transformation
Functions must be absolutely integrable in
to have a Fourier transforma-
tion under the classical definition. Many functions, some of which are important,
cannot however satisfy this condition. Since the Fourier transformation can only be
applied to functions defined in
(
−
∞
,
+
∞
)
, its application is limited. In order to relax
these limits, we need to introduce a new integral transformation.
Consider the function
(
−
∞
,
+
∞
)
ϕ
(
t
)
,
t
∈
(
−
∞
,
+
∞
)
,
e
−
β
t
g
(
t
)=
ϕ
(
t
)
I
(
t
)
,
β
>
0
,
where
I
(
t
)
is a unit function. Through multiplying
ϕ
(
t
)
by
I
(
t
)
, we reduce the do-
by e
−
β
t
, we increase the
main
(
−
∞
,
+
∞
)
to
(
0
,
+
∞
)
. Through multiplying
ϕ
(
t
)
speed of tending to zero of
ϕ
(
t
)
such that the absolutely integrable condition can be
satisfied.
Consider the Fourier transformation of
g
(
t
)
,
+
∞
+
∞
+
∞
e
−
β
t
e
−
iω
t
d
t
e
−
(
β
+
iω
)
t
d
t
e
−
st
d
t
g
(
ω
)=
ϕ
(
t
)
I
(
t
)
=
f
(
t
)
=
f
(
t
)
,
−
∞
0
0
g
s
−
β
i
f
where
f
(
t
)=
ϕ
(
t
)
I
(
t
)
,
s
=
β
+
i
ω
.Let
g
(
ω
)=
=
(
s
)
. The complex-
valued function
f
(
s
)
thus comes from the integral
transformation of
f
(
t
)
,
+
∞
e
−
st
d
t
. This can be used to introduce the Laplace transformation.
Suppose that
f
f
(
t
)
0
(
t
)
is defined for
t
≥
0. When
t
>
0,
f
(
t
)
≡
0. Assume that the
+
∞
e
−
st
d
t
is convergent in a certain region of complex variables. Let
integral
f
(
t
)
0
+
∞
f
e
−
st
d
t
(
s
)=
f
(
t
)
,
(B.23)
0
This is called the
Laplace transformation
of
f
(
t
)
and denoted by
f
(
s
)=
L
[
f
(
t
)]
.
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