Fourier series over any range 633
=
2
3
⎡
⎢
⎢
⎢
⎣
t sin
2πnt
3
2πn
3
+
cos
2πnt
3
2πn
3
2
⎤
⎥
⎥
⎥
⎦
3
0
by parts
=
2
3
⎡
⎢
⎢
⎢
⎣
⎧
⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎩
3 sin2πn
2πn
3
+
cos 2πn
2πn
3
2
⎫
⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎭
−
⎧
⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎩
0 +
cos 0
2πn
3
2
⎫
⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎭
⎤
⎥
⎥
⎥
⎦
= 0
b n =
2
L
L
2
−L
2
f (t ) sin
2πnt
L
dt
=
2
L
L
0
t sin
2πnt
L
dt
=
2
3
3
0
t sin
2πnt
3
dt
=
2
3
⎡
⎢
⎢
⎢
⎣
−t cos
2πnt
3
2πn
3
+
sin
2πnt
3
2πn
3
2
⎤
⎥
⎥
⎥
⎦
3
0
by parts
=
2
3
⎡
⎢
⎢
⎢
⎣
⎧
⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎩
−3 cos 2πn
2πn
3
+
sin 2πn
2πn
3
2
⎫
⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎭
−
⎧
⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎩
0 +
sin 0
2πn
3
2
⎫
⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎭
⎤
⎥
⎥
⎥
⎦
=
2
3
⎡
⎢
⎢
⎣
−3 cos 2πn
2πn
3
⎤
⎥
⎥
⎦ =
−3
πn
cos 2πn =
−3
πn
Hence b 1 =
−3
π
, b 2 =
−3
2π
, b 3 =
−3
3π
and so on.
Thus the Fourier series for the function f (t ) in the range
0 to 3 is given by:
f (t) =
3
2
−
3
π
sin
2πt
3
+
1
2
sin
4πt
3
+
1
3
sin
6πt
3
+ · · ·
Now try the following exercise
Exercise 232 Further problems on Fourier
series over any range L
1. The voltage from a square wave generator is
of the form:
v(t ) =
0, −10 < t < 0
5,
0 < t < 10
and is periodic of period 20. Show that the
Fourier series for the function is given by:
v(t ) =
5
2
+
10
π
sin
πt
10
+
1
3
sin
3πt
10
+
1
5
sin
5πt
10
+ · · ·
2. Find the Fourier series for f (x) = x in the
range x = 0 to x = 5.
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
f (x) =
5
2
−
5
π
sin
2π x
5
+
1
2
sin
4π x
5
+
1
3
sin
6π x
5
+ · · ·
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
3. A periodic function of period 4 is defined by:
f (x) =
−3, −2 < x < 0
+3,
0 < x < 2
Sketch the function and obtain the Fourier
series for the function.
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
f (x) =
12
π
sin
π x
2
+
1
3
sin
3π x
2
+
1
5
sin
5π x
2
+ · · ·
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦