Fourier series over any range 633
=
2
3
⎡
⎢
⎢
⎢
⎣
tsin
(
2 πnt
3
)
(
2 πn
3
) +
cos
(
2 πnt
3
)
(
2 πn
3
) 2
⎤
⎥
⎥
⎥
⎦
3
0
by parts
=
2
3
⎡
⎢
⎢
⎢
⎣
⎧
⎪⎪
⎪⎨
⎪⎪⎪
⎩
3sin2πn
(
2 πn
3
)+
cos2πn
(
2 πn
3
) 2
⎫
⎪⎪
⎪⎬
⎪⎪⎪
⎭
−
⎧
⎪⎪
⎪⎨
⎪⎪
⎪⎩
0 +
cos0
(
2 πn
3
) 2
⎫
⎪⎪
⎪⎬
⎪⎪
⎪⎭
⎤
⎥
⎥
⎥
⎦
= 0
bn=
2
L
∫ L 2
−L
2
f(t)sin
(
2 πnt
L
)
dt
=
2
L
∫L
0
tsin
(
2 πnt
L
)
dt
=
2
3
∫ 3
0
tsin
(
2 πnt
3
)
dt
=
2
3
⎡
⎢
⎢
⎢
⎣
−tcos
(
2 πnt
3
)
(
2 πn
3
) +
sin
(
2 πnt
3
)
(
2 πn
3
) 2
⎤
⎥
⎥
⎥
⎦
3
0
by parts
=
2
3
⎡
⎢
⎢
⎢
⎣
⎧
⎪⎪
⎪⎨
⎪⎪
⎪⎩
−3cos2πn
(
2 πn
3
) +
sin2πn
(
2 πn
3
) 2
⎫
⎪⎪
⎪⎬
⎪⎪
⎪⎭
−
⎧
⎪⎪
⎪⎨
⎪⎪
⎪⎩
0 +
sin0
(
2 πn
3
) 2
⎫
⎪⎪
⎪⎬
⎪⎪
⎪⎭
⎤ ⎥ ⎥ ⎥ ⎦ =
2
3
⎡
⎢
⎢
⎣
−3cos2πn
(
2 πn
3
)
⎤
⎥
⎥
⎦=
− 3
πn
cos2πn=
− 3
πn
Henceb 1 =
− 3
π
,b 2 =
− 3
2 π
,b 3 =
− 3
3 π
andsoon.
Thus the Fourier series for the functionf(t)intherange
0to3isgivenby:
f(t)=
3
2
−
3
π
[
sin
(
2 πt
3
)
+
1
2
sin
(
4 πt
3
)
+
1
3
sin
(
6 πt
3
)
+ ···
]
Now try the following exercise
Exercise 232 Further problemson Fourier
series over any rangeL
- 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
(
3 πt
10
)
+
1
5
sin
(
5 πt
10
)
+···
]
- Find the Fourier series for f(x)=x in the
rangex=0tox=5.
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
f(x) =
5
2
−
5
π
[
sin
(
2 πx
5
)
+
1
2
sin
(
4 πx
5
)
+
1
3
sin
(
6 πx
5
)
+···
]
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
- 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
(πx
2
)
+
1
3
sin
(
3 πx
2
)
+
1
5
sin
(
5 πx
2
)
+···
)
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦