Fourier series over any range 635
When n is even, a n = 0
a 1 =
−8
π 2 , a 3 =
−8
π 2 3 2 , a 5 =
−8
π 2 5 2 and so on.
Hence the half-range Fourier cosine series for f (x) in
the range 0 to 2 is given by:
f (x) = 1 −
8
π 2
cos
πx
2
+
1
3 2 cos
3πx
2
+
1
5 2 cos
5πx
2
+ · · ·
Problem 5. Find the half-range Fourier sine
series for the function f (x) = x in the range
0 ≤ x ≤ 2. Sketch the function within and outside of
the given range.
A half-range Fourier sine series indicates an odd function. Thus the graph of f (x) = x in the range 0 to 2 is
shown in Fig. 69.5 and is extended outside of this range
so as to be symmetrical about the origin, as shown by
the broken lines.
f (x )
f (x ) 5 x
x
22
24
2
4
2
0
6
22
Figure 69.5
From para. (c), for a half-range sine series:
f (x) =
∞
n=1
b n sin
nπ x
L
b n =
2
L
L
0
f (x) sin
nπ x
L
dx
=
2
2
2
0
x sin
nπ x
L
dx
=
⎡
⎢
⎣
−x cos
nπ x
2
nπ
2
+
sin
nπ x
2
nπ
2
2
⎤
⎥
⎦
2
0
=
⎡
⎢
⎣
⎛
⎜
⎝
−2 cos nπ
nπ
2
+
sin nπ
nπ
2
2
⎞
⎟
⎠ −
⎛
⎜
⎝0 +
sin 0
nπ
2
2
⎞
⎟
⎠
⎤
⎥
⎦
=
−2 cos nπ
nπ
2
=
−4
nπ
cos nπ
Hence b 1 =
−4
π
(−1) =
4
π
b 2 =
−4
2π
(1) =
−4
2π
b 3 =
−4
3π
(−1) =
4
3π
and so on.
Thus the half-range Fourier sine series in the range 0 to
2 is given by:
f (x)=
4
π
sin
πx
2
−
1
2
sin
2πx
2
+
1
3
sin
3πx
2
−
1
4
sin
4πx
2
+· · ·
Now try the following exercise
Exercise 233 Further problems on
half-range Fourier series over range L
1. Determine the half-range Fourier cosine series
for the function f (x) = x in the range
0 ≤ x ≤ 3. Sketch the function within and outside of the given range.
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
f (x) =
3
2
−
12
π 2
cos
π x
3
+
1
3 2 cos
3π x
3
+
1
5 2 cos
5π x
3
+ · · ·
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
2. Find the half-range Fourier sine series
for the function f (x) = x in the range
0 ≤ x ≤ 3. Sketch the function within and outside of the given range.
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
f (x) =
6
π
sin
π x
3
−
1
2
sin
2π x
3
+
1
3
sin
3π x
3
−
1
4
sin
4π x
3
+ · · ·
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦