where
g
(
x
) is an odd function,
g
(
x
) =
(
3
/
2
,
x >
0
,

3
/
2
,
x <
0
.
2. Calculate Fourier Series for the function
f
(
x
), defined on [

5
,
5], where
f
(
x
) = 3
H
(
x

2)
.
By a similar method,
f
(
x
) =
9
5
+
∞
X
n
=1
•

3
πn
sin
2
πn
5
cos
πnx
5
+
3
πn
cos
2
πn
5

(

1)
n
¶
sin
πnx
5
‚
.
3. Calculate Fourier Series for the function,
f
(
x
), defined as follows:
(a)
x
∈
[

4
,
4], and
f
(
x
) = 5
.
Comparing
f
(
x
) with the general Fourier Series expression with
L
= 4,
g
(
x
) =
a
0
2
+
∞
X
n
=1
a
n
cos
πnx
4
+
b
n
sin
πnx
4
¶
,
we can see that
a
0
= 10,
a
n
=
b
n
= 0 for
n >
0 will give
f
(
x
) =
g
(
x
).
(b)
x
∈
[

π, π
], and
f
(
x
) = 21 + 2 sin 5
x
+ 8 cos 2
x.
Again, for
L
=
π
, we have
g
(
x
) =
a
0
2
+
∞
X
n
=1
(
a
n
cos
nx
+
b
n
sin
nx
)
,
and setting
a
0
= 42,
a
2
= 8,
b
5
= 2 and the rest of the coefficients zero,
we obtain
f
(
x
) =
g
(
x
).
(c)
x
∈
[

π, π
], and
f
(
x
) =
8
X
n
=1
c
n
sin
nx,
with
c
n
= 1
/n.
2