Fourier Series
When the French mathematician Joseph Fourier (1768–1830) was trying to solve a prob
lem in heat conduction, he needed to express a function
as an inﬁnite series of sine and
cosine functions:
Earlier, Daniel Bernoulli and Leonard Euler had used such series while investigating prob
lems concerning vibrating strings and astronomy.
The series in Equation 1 is called a
trigonometric series
or
Fourier series
and it turns
out that expressing a function as a Fourier series is sometimes more advantageous than
expanding it as a power series. In particular, astronomical phenomena are usually periodic,
as are heartbeats, tides, and vibrating strings, so it makes sense to express them in terms
of periodic functions.
We start by assuming that the trigonometric series converges and has a continuous func
tion
as its sum on the interval
, that is,
Our aim is to ﬁnd formulas for the coefﬁcients
and
in terms of
. Recall that for a
power series
we found a formula for the coefﬁcients in terms of deriv
atives:
. Here we use integrals.
If we integrate both sides of Equation 2 and assume that it’s permissible to integrate the
series termbyterm, we get
But
because
is an integer. Similarly,
. So
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sin
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sin
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y
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sin
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a
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µ
f
±
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²
³
b
1
sin
x
³
b
2
sin 2
x
³
b
3
sin 3
x
³¶¶¶
±
a
0
³
a
1
cos
x
³
a
2
cos 2
x
³
a
3
cos 3
x
f
±
x
²
±
a
0
³
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n
cos
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1
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View Full Documentand solving for
gives
To determine
for
we multiply both sides of Equation 2 by
(where
is
an integer and
) and integrate termbyterm from
to
:
We’ve seen that the ﬁrst integral is 0. With the help of Formulas 81, 80, and 64 in the Table
of Integrals, it’s not hard to show that
for all
and
So the only nonzero term in (4) is
and we get
Solving for
, and then replacing
by , we have
Similarly, if we multiply both sides of Equation 2 by
and integrate from
to
,
we get
We have derived Formulas 3, 5, and 6 assuming
is a continuous function such that
Equation 2 holds and for which the termbyterm integration is legitimate. But we can still
consider the Fourier series of a wider class of functions: A
piecewise continuous function
on
is continuous except perhaps for a ﬁnite number of removable or jump disconti
nuities. (In other words, the function has no inﬁnite discontinuities. See Section 2.4 for a
discussion of the different types of discontinuities.)
±
a
,
b
²
f
n
±
1, 2, 3, .
. .
b
n
±
1
±
y
²
f
³
x
´
sin
nx
dx
6
²
sin
mx
n
±
1, 2, 3, .
. .
a
n
±
1
y
²
f
³
x
´
cos
5
n
m
a
m
y
²
f
³
x
´
cos
±
a
m
a
m
y
²
cos
cos
±
µ
0
for
n
²
m
for
n
±
m
m
n
y
²
sin
cos
±
0
±
a
0
y
²
cos
³
¶
´
n
±
1
a
n
y
²
cos
cos
³
¶
´
n
±
1
b
n
y
²
sin
cos
4
y
²
f
³
x
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 Spring '08
 Bray,C
 Math, Fourier Series, Cos, Continuous function, Joseph Fourier

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