Fourier series for a non-periodic function over range 2π 621
Whenθ=π, f(θ )=π^2
Hence π^2 =4 π^2
3+ 4(
cosπ+1
4cos2π+1
9cos3π+1
16cos4π+···)− 4 π(
sinπ+1
2sin2π+1
3sin3π+···)i.e. π^2 −4 π^2
3= 4(
− 1 +1
4−1
9+1
16−···)
− 4 π( 0 )−π^2
3= 4(
− 1 +1
4−1
9+1
16−···)π^2
3= 4(
1 −1
4+1
9−1
16+···)Henceπ^2
12= 1 −1
4+1
9−1
16+···or
π^2
12= 1 −1
22+1
32−1
42+···Now try the following exercise
Exercise 229 Further problemson Fourier
series of non-periodic functions over a range
of 2π- Show that the Fourier series for the function
f(x)=x over the rangex=0tox= 2 π is
given by:
f(x)=π− 2
(
sinx+^12 sin2x+^13 sin3x+^14 sin4x+···)- Determine the Fourier series for the function
defined by:
f(t)={
1 −t, when−π<t< 0
1 +t, when 0<t<πDraw a graph of the function within and
outside of the given range.⎡
⎢
⎢
⎣f(t)=π
2+ 1 −4
π(
cost+cos3t
32
+
cos5t
52+···)⎤
⎥
⎥
⎦- Find the Fourier series for the function
f(x)=x⎡+πwithin the range−π<x<π.
⎢
⎢
⎣f(x)=π+ 2(
sinx−1
2sin2x+1
3sin3x−···)⎤
⎥
⎥
⎦- Determine the Fourier series up to and
including the third harmonic for the
function defined by:
f(x)={
x, when 0<x<π
2 π−x, whenπ<x< 2 πSketch a graph of the function within and
outsideofthegivenrange,assumingtheperiod
is 2π.
⎡
⎢
⎢
⎣f(x)=π
2−4
π(
cosx+cos3x
32+cos5x
52+···)⎤
⎥
⎥
⎦- Expand the function f(θ )=θ^2 in a Fourier
series in the range−π<θ<π.
Sketch the function within and outside of the
given range.
⎡
⎢
⎢
⎢
⎣
f(θ )=π^2
3− 4(
cosθ−1
22cos2θ+1
32cos3θ−···)⎤
⎥
⎥
⎥
⎦- For the Fourier series obtained in Problem 5,
letθ=πand deduce the series for
∑∞
n= 11
n^2
[
1 +1
22+1
32+1
42+1
52+···=π^2
6]- Show that the Fourier series for the triangular
waveform shown in Fig. 67.5 is given by:
y=8
π^2(
sinθ−1
32sin3θ+1
52sin5θ−1
72sin7θ+···)