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
4
cos2π
+
1
9
cos3π+
1
16
cos4π+···
)
− 4 π
(
sinπ+
1
2
sin2π
+
1
3
sin3π+···
)
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
+···
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<π
Draw 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
2
sin2x
+
1
3
sin3x−···
)
⎤
⎥
⎥
⎦
- 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 π
Sketch 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
22
cos2θ
+
1
32
cos3θ−···
)
⎤
⎥
⎥
⎥
⎦
- For the Fourier series obtained in Problem 5,
letθ=πand deduce the series for
∑∞
n= 1
1
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
32
sin3θ+
1
52
sin5θ
−
1
72
sin7θ+···
)