Chapter 7
Fourier series
Syllabus section:
6. Fourier series: full, half and arbitrary range series. Parseval’s Theorem.
Fourier series can be obtained for any function defined on a finite range, as in the SL section above. In
practice they provide a way to do various calculations with, and to analyse the behaviour of, functions which
are periodic, i.e. repeat the same values in a regular pattern. Such a function with period 2
ℓ
will obey an
equation
f
(
x
+
2
ℓ
) =
f
(
x
)
.
To begin with we will assume
ℓ
=
π
.
We know cos
nx
and sin
nx
for any integer
n
have period 2
π
. So,
of course, do the other trigonometric functions such as tan
x
but these have the disadvantage of becoming
unbounded at certain values (for tan
x
, at
x
=
π
/
2, for example).
Since cos
nx
and sin
nx
are the eigenfunctions of an SL system studied in the previous chapter, we can
write
f
=
∞
∑
0
(
a
n
cos
nx
+
b
n
sin
nx
)
for periodic piecewise differentiable
f
(in fact, for any function defined on a range of length 2
π
). We will
modify this way of writing the series slightly soon.
Such a series splits
f
into pieces of different frequency. Examples where this technique (or its general
izations for Fourier transforms, wavelets, etc) is useful and makes sense include: resolution of sound waves
into their different harmonics, optics, telecommunications, astronomy, climate variation, water waves, peri
odic behaviour of financial measures, etc. Fourier’s original application (in his “ Theorie analytique de la
chaleur” in 1822) was related to solutions of the equations governing the propagation of heat in a solid; as an
example they can be used to show why cellars maintain an almost constant temperature all year round – see
the appendix to this chapter.
We know from SL system theory that sin
nx
and cos
nx
form a set of “orthogonal functions”: see (6.23)
and (6.25).
76
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7.1
Full range Fourier series
The idea is to write a function
f
(
x
)
defined for a range of values of
x
of length 2
π
, say

π
≤
x
≤
π
, as a
series of trigonometric functions
f
=
1
2
a
0
+
∞
∑
1
a
n
cos
nx
+
∞
∑
1
b
n
sin
nx
.
(7.1)
Here the
1
2
a
0
is really a cos0
x
=
1 term (the eigenfunction for
λ
=
0): the reason for the half is that
integraltext
π

π
u
2
0
d
x
=
integraltext
π

π
1d
x
=
2
π
which is double the value of
integraltext
π

π
u
2
λ
d
x
for the other values of
λ
. There is no point
in including a sin0
x
=
0 term. Strictly, the use of the equals sign depends on convergence properties which
we describe later.
From the SL system results above we have that for all
m
≥
0
a
m
=
1
π
integraldisplay
π

π
f
cos
mx
d
x
.
It was to provide this nice form that we included the
1
2
with the
a
0
term:
1
2
a
0
is in fact the average value of
f
over the range. Similarly,
b
m
=
1
π
integraldisplay
π

π
f
sin
mx
d
x
.
Example 7.1.
Find the Fourier series for
f
(
x
) =
braceleftBig
0
if

π
<
x
<
0
x
if 0
<
x
<
π
.
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 Spring '09
 Fourier Series, Magnetism, Periodic function, Partial differential equation

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