Environmental Engineering Reference
In-Depth Information
Appendix B
Integral Transformations
B.1 Fourier Integral Transformation
B.1.1 Fourier Integral
A periodic function can be expanded into a Fourier series. We show here that a non-
periodic function can be expressed by using a Fourier integral.
Fourier Series of Exponential Form
of period
T
satisfies the Dirichlet condition, it can be
expanded into a Fourier series. Therefore, at any continuous point of
g
If a periodic function
g
(
t
)
(
t
)
we have
⎧
⎨
a
0
2
+
+
∞
n
=
1
(
g
(
t
)=
a
n
cos
n
ω
t
+
b
n
sin
n
ω
t
)
T
2
2
T
a
n
=
g
(
t
)
cos
n
ω
t
d
t
,
n
=
0
,
1
,
2
, ··· ,
⎩
2
−
T
2
2
T
b
n
=
g
(
t
)
sin
n
ω
t
d
t
,
n
=
1
,
2
, ··· ,
2
−
2
T
. By applying the Euler formula e
±
i
n
ω
t
where
ω
=
=
cos
n
ω
t
±
isin
n
ω
t
,wehave
a
n
−
e
−
i
n
ω
t
+
∞
n
=
1
a
0
2
+
i
b
n
a
n
+
i
b
n
e
i
n
ω
t
g
(
t
)=
+
.
2
2
Let
a
0
2
,
a
n
−
i
b
n
a
n
+
i
b
n
c
0
=
c
n
=
,
c
−
n
=
,
n
=
1
,
2
, ··· .
2
2
Search WWH ::
Custom Search