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4
Chapter 2
Fourier Series
Solutions to Exercises 2.1
1.
(a) cos
x
has period 2
π
.
(b) cos
πx
has period
T
=
2
π
π
= 2.
(c) cos
2
3
x
has
period
T
=
2
π
2
/
3
=3
π
.
(d) cos
x
has period 2
π
, cos 2
x
has period
π
,2
π
,3
π
,
˙
.. A
common period of cos
x
and cos 2
x
is 2
π
. So cos
x
+ cos 2
x
has period 2
π
.
5.
This is the special case
p
=
π
of Exercise 6(b).
9.
(a) Suppose that
f
and
g
are
T
periodic. Then
f
(
x
+
T
)
·
g
(
x
+
T
)=
f
(
x
)
·
g
(
x
),
and so
f
·
g
is
T
periodic. Similarly,
f
(
x
+
T
)
g
(
x
+
T
)
=
f
(
x
)
g
(
x
)
,
and so
f/g
is
T
periodic.
(b) Suppose that
f
is
T
periodic and let
h
(
x
f
(
x/a
). Then
h
(
x
+
aT
f
±
x
+
aT
a
²
=
f
³
x
a
+
T
´
=
f
³
x
a
´
(because
f
is
T
periodic)
=
h
(
x
)
.
Thus
h
has period
aT
. Replacing
a
by 1
/a
, we Fnd that the function
f
(
ax
) has
period
T/a
.
(c) Suppose that
f
is
T
periodic. Then
g
(
f
(
x
+
T
)) =
g
(
f
(
x
)), and so
g
(
f
(
x
)) is
also
T
periodic.
13.
µ
π/
2

π/
2
f
(
x
)
dx
=
µ
π/
2
0
1
dx
=
π/
2
.
17.
By Exercise 16,
F
is 2 periodic, because
¶
2
0
f
(
t
)
dt
= 0 (this is clear from
the graph of
f
). So it is enough to describe
F
on any interval of length 2. ±or
0
<x<
2, we have
F
(
x
µ
x
0
(1

t
)
dt
=
t

t
2
2
·
·
·
x
0
=
x

x
2
2
.
±or all other
x
,
F
(
x
+2)=
F
(
x
). (b) The graph of
F
over the interval [0
,
2] consists
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This note was uploaded on 12/22/2011 for the course MAP 3305 taught by Professor Stuartchalk during the Fall '11 term at UNF.
 Fall '11
 StuartChalk

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