Fourier series over any range 633
=2
3⎡
⎢
⎢
⎢
⎣tsin(
2 πnt
3)(
2 πn
3) +cos(
2 πnt
3)(
2 πn
3) 2⎤
⎥
⎥
⎥
⎦30
by parts=2
3⎡
⎢
⎢
⎢
⎣⎧
⎪⎪
⎪⎨⎪⎪⎪
⎩3sin2πn
(
2 πn
3)+cos2πn
(
2 πn
3) 2⎫
⎪⎪
⎪⎬⎪⎪⎪
⎭−⎧
⎪⎪
⎪⎨⎪⎪
⎪⎩0 +cos0
(
2 πn
3) 2⎫
⎪⎪
⎪⎬⎪⎪
⎪⎭⎤
⎥
⎥
⎥
⎦= 0bn=2
L∫ L 2−L
2f(t)sin(
2 πnt
L)
dt=2
L∫L0tsin(
2 πnt
L)
dt=2
3∫ 30tsin(
2 πnt
3)
dt=2
3⎡
⎢
⎢
⎢
⎣−tcos(
2 πnt
3)(
2 πn
3) +sin(
2 πnt
3)(
2 πn
3) 2⎤
⎥
⎥
⎥
⎦30
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
πncos2πn=− 3
πnHenceb 1 =
− 3
π,b 2 =− 3
2 π,b 3 =− 3
3 πandsoon.Thus the Fourier series for the functionf(t)intherange
0to3isgivenby:f(t)=3
2−3
π[
sin(
2 πt
3)
+1
2sin(
4 πt
3)+1
3sin(
6 πt
3)
+ ···]Now try the following exerciseExercise 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< 10and 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
3sin(
3 πt
10)+1
5sin(
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
2sin(
4 πx
5)+1
3sin(
6 πx
5)
+···]⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦- A periodic function of period 4 is defined by:
f(x)={
− 3 , − 2 <x< 0
+ 3 , 0 <x< 2Sketch the function and obtain the Fourier
series for the function.
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
f(x)=12
π(
sin(πx
2)+1
3sin(
3 πx
2)+1
5sin(
5 πx
2)
+···)⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦