Exercise
4.2.6
:
Suppose f
(
t
)
is defined on
(

⇡
,
⇡
]
as
f
(
t
)
=
8
>
>
<
>
>
:

1
if

⇡
<
t
0
,
1
if
0
<
t
⇡
.
Extend periodically and compute the Fourier series of f
(
t
)
.
Exercise
4.2.7
:
Suppose
f
(
t
)
is defined on
(

⇡
,
⇡
]
as
t
3
. Extend periodically and compute the
Fourier series of f
(
t
)
.
Exercise
4.2.8
:
Suppose
f
(
t
)
is defined on
[

⇡
,
⇡
]
as
t
2
. Extend periodically and compute the
Fourier series of f
(
t
)
.
There is another form of the Fourier series using complex exponentials that is sometimes easier
to work with.
Exercise
4.2.9
:
Let
f
(
t
)
=
a
0
2
+
1
X
n
=
1
a
n
cos(
nt
)
+
b
n
sin(
nt
)
.
Use Euler’s formula e
i
✓
=
cos(
✓
)
+
i
sin(
✓
)
to show that there exist complex numbers c
m
such that
f
(
t
)
=
1
X
m
=
1
c
m
e
imt
.
Note that the sum now ranges over all the integers including negative ones. Do not worry about
convergence in this calculation. Hint: It may be better to start from the complex exponential form
and write the series as
c
0
+
1
X
m
=
1
c
m
e
imt
+
c

m
e

imt
.
166
CHAPTER 4. FOURIER SERIES AND PDES
Exercise
4.2.101
:
Suppose
f
(
t
)
is defined on
[

⇡
,
⇡
]
as
f
(
t
)
=
sin
(
t
)
. Extend periodically and
compute the Fourier series.
Exercise
4.2.102
:
Suppose
f
(
t
)
is defined on
(

⇡
,
⇡
]
as
f
(
t
)
=
sin
(
⇡
t
)
. Extend periodically and
compute the Fourier series.
Exercise
4.2.103
:
Suppose
f
(
t
)
is defined on
(

⇡
,
⇡
]
as
f
(
t
)
=
sin
2
(
t
)
. Extend periodically and
compute the Fourier series.
Exercise
4.2.104
:
Suppose
f
(
t
)
is defined on
(

⇡
,
⇡
]
as
f
(
t
)
=
t
4
. Extend periodically and compute
the Fourier series.
4.3. MORE ON THE FOURIER SERIES
167
4.3
More on the Fourier series
Note: 2 lectures, §9.2 – §9.3 in [EP], §10.3 in [BD]
Before reading the lecture, it may be good to first try Project IV (Fourier series) from the
IODE website:
. After reading the lecture it may be good to
continue with Project V (Fourier series again).
4.3.1
2
L
periodic functions
We have computed the Fourier series for a 2
⇡
periodic function, but what about functions of di
↵
erent
periods. Well, fear not, the computation is a simple case of change of variables. We can just rescale
the independent axis. Suppose that we have a 2
L
periodic function
f
(
t
) (
L
is called the
half period
).
Let
s
=
⇡
L
t
. Then the function
g
(
s
)
=
f
✓
L
⇡
s
◆
is 2
⇡
periodic. We want to also rescale all our sines and cosines. We want to write
f
(
t
)
=
a
0
2
+
1
X
n
=
1
a
n
cos
✓
n
⇡
L
t
◆
+
b
n
sin
✓
n
⇡
L
t
◆
.
If we change variables to
s
we see that
g
(
s
)
=
a
0
2
+
1
X
n
=
1
a
n
cos(
ns
)
+
b
n
sin(
ns
)
.
We compute
a
n
and
b
n
as before. After we write down the integrals we change variables from
s
back to
t
.
a
0
=
1
⇡
Z
⇡

⇡
g
(
s
)
ds
=
1
L
Z
L

L
f
(
t
)
dt
,
a
n
=
1
⇡
Z
⇡

⇡
g
(
s
) cos(
ns
)
ds
=
1
L
Z
L

L
f
(
t
) cos
✓
n
⇡
L
t
◆
dt
,
b
n
=
1
⇡
Z
⇡

⇡
g
(
s
) sin(
ns
)
ds
=
1
L
Z
L

L
f
(
t
) sin
✓
n
⇡
L
t
◆
dt
.
The two most common half periods that show up in examples are
⇡
and 1 because of the
simplicity.
We should stress that we have done no new mathematics, we have only changed
variables. If you understand the Fourier series for 2
⇡
periodic functions, you understand it for 2
L

periodic functions. All that we are doing is moving some constants around, but all the mathematics
is the same.
168
CHAPTER 4. FOURIER SERIES AND PDES
Example 4.3.1:
Let
f
(
t
)
=

t

for

1
<
t
1,
extended periodically. The plot of the periodic extension is given in Figure 4.8. Compute the Fourier
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 Fall '14
 Equations, The Land, Elementary algebra, Partial differential equation