442 Chapter 15Orthogonal expansions. Fourier analysis
6.For the square system of five charges in
Figure 15.18, show that the leading term in
the expansion in powers of 12 Rof the
electrostatic potential at P (in the plane of
the square) isV 1 = 1 qr
2
24 πε
0
R
3
, independent
of orientation.
Section 15.4
7.Confirm the relations (i) (15.33), (ii)
(15.34) and (iii)(15.36).
8.A periodic function with period 2πis
defined by
(i)Draw the graph of the function in the interval− 3 π 1 ≤ 1 x 1 ≤ 13 π. (ii)Find the Fourier series
of the function [Hint:f(x)is an even function of x]. (iii) Use the series to show that
[Hint: substitute a suitable value for xin the series].
9.A function with period 2 πis defined by
f(x) 1 = 1 x, −π 1 < 1 x 1 < 1 π
(i)Draw the graph of the function in the interval− 3 π 1 ≤ 1 x 1 ≤ 13 π.(ii)Find the Fourier
series of the function. [Hint:f(x)is an odd function of x.] (iii) Draw the graphs of the
first four partial sums of the series.
10.A function with period 2πis defined by
f(x) 1 = 1 x
2
, −π 1 < 1 x 1 < 1 π
(i)Draw the graph of the function in the interval− 3 π 1 ≤ 1 x 1 ≤ 13 π. (ii) Find the Fourier series
of the function. (iii) Use the series to show that
11.Show that the Fourier series of the periodic function defined by (see Figure 15.4)
is
11
2
22
13
4
35
6
ππ 57
+−
⋅
⋅
⋅
sin
cos cos cos
t
ttt
ft
tt
t
()
sin
=
≤≤
−≤≤
if
if
0
00
π
π
π
2
1
1
2
12
1
1
1
4
1
9
1
16
=
−
=− + − +
=
+
∑
nnn
∞
()
π
2
1
2
6
1
1
1
4
1
9
1
16
==++++
=
∑
nn
∞
π
4
1
1
3
1
5
1
7
=− + − +
fx
x
x
()
||
=
−<<
<<
1
22
0
2
if
if
ππ
π
π
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.P
+q
+q
+q
+q
− 4 q
θ
r
Figure 15.18