Environmental Engineering Reference
In-Depth Information
When
f
(
t
)
is an odd or even function, in particular, we have
⎧
⎨
⎧
⎨
+
∞
+
∞
f
(
t
)=
b
(
ω
)
sin
ω
t
d
ω
,
f
(
t
)=
a
(
ω
)
cos
ω
t
d
ω
,
0
0
and
+
∞
+
∞
⎩
⎩
2
π
2
π
b
(
ω
)=
f
(
τ
)
sin
ωτ
d
τ
,
a
(
ω
)=
f
(
τ
)
cos
ωτ
d
τ
,
0
0
respectively.
B.1.2 Fourier Transformation
Definition
If the function
f
(
t
)
satisfies certain conditions, we may express it by using a Fourier
integral. Let
+
∞
f
e
−
iω
t
d
t
(
ω
)=
f
(
t
)
.
(B.4)
−
∞
By Eq. (B.3) we have
+
∞
1
2
f
e
iω
t
d
f
(
t
)=
(
ω
)
ω
.
(B.5)
π
−
∞
Thereforewe may express
f
f
. Equations (B.4)
and (B.5) are called the
Fourier transformation
and the
inverse Fourier transforma-
tion
, respectively. The
f
(
ω
)(
f
(
t
))
by the integral of
f
(
t
)(
(
ω
))
(
ω
)
is called the
image function
of
f
(
t
)
.The
f
(
t
)
is called
the
inverse image function
of
f
(
ω
)
. For convenience in applications, Eqs. (B.4)
and (B.5) are written as
f
F
−
1
f
(
ω
)
.
(
ω
)=
F
[
f
(
t
)]
,
f
(
t
)=
Example 1.
Find the Fourier transformation and the integral expression of the ex-
ponentially decaying function
0
,
t
<
0
,
f
(
t
)=
e
−
β
t
,
0
≤
t
,
β
>
0
.
Solution.
By Eq. (B.4), we obtain the image function
+
∞
1
β
+
ω
=
β
−
i
ω
f
e
−
(
β
+
iω
)
t
d
t
(
ω
)=
F
[
f
(
t
)] =
=
2
,
i
β
2
+
ω
0
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