FOURIER SERIES FOR A NON-PERIODIC FUNCTION OVER RANGE 2π 667L
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+···)Hence
π^2
12= 1 −1
4+1
9−1
16+···orπ^2
12= 1 −1
22+1
32−1
42+···Now try the following exercise.
Exercise 241 Further problems on Fourier
series of non-periodic functions over a range
of 2π- Show that the Fourier series for the func-
tionf(x)=xover the rangex=0tox= 2 π
is given by:
f(x)=π− 2(
sinx+^12 sin 2x+^13 sin 3x+^14 sin 4x+···)- Determine the Fourier series for the function
defined by:
f(t)={
1 −t, when−π<t< 01 +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+cos 3t
32+cos 5t
52+···)⎤⎥
⎥
⎦- Find the Fourier series for the function
f(x)=x+⎡πwithin the range−π<x<π.
⎢
⎢
⎣f(x)=π+ 2(
sinx−1
2sin 2x+1
3sin 3x−···)⎤⎥
⎥
⎦- Determine the Fourier series up to and
including the third harmonic for the
function defined by:
f(x)={
x, when 0<x<π2 π−x, whenπ<x< 2 πSketch a graph of the function within and
outside of the given range, assuming the
period is 2π.
⎡⎢
⎢
⎣f(x)=π
2−4
π(
cosx+cos 3x
32+cos 5x
52+···)⎤⎥
⎥
⎦- Expand the functionf(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
22cos 2θ+1
32cos 3θ−···)⎤⎥
⎥
⎥
⎦- 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. 70.5 is given by:
y=8
π^2(
sinθ−1
32sin 3θ+1
52sin 5θ−1
72sin 7θ+···)in the range 0 to 2π.(^0) π 2 π
1
− 1
y
θ
Figure 70.5