Fourier Series
Fourier series started life as a method to solve problems about the flow of heat through ordinary
materials. It has grown so far that if you search our library’s catalog for the keyword “Fourier” you will
find 618 entries as of this date. It is a tool in abstract analysis and electromagnetism and statistics
and radio communication and
. . .
. People have even tried to use it to analyze the stock market. (It
didn’t help.) The representation of musical sounds as sums of waves of various frequencies is an audible
example. It provides an indispensible tool in solving partial differential equations, and a later chapter
will show some of these tools at work.
5.1 Examples
The power series or Taylor series is based on the idea that you can write a general function as an infinite
series of powers. The idea of Fourier series is that you can write a function as an infinite series of sines
and cosines. You can also use functions other than trigonometric ones, but I’ll leave that generalization
aside for now, except to say that Legendre polynomials are an important example of functions used for
such more general expansions.
An example: On the interval
0
< x < L
the function
x
2
varies from 0 to
L
2
. It can be written
as the series of cosines
x
2
=
L
2
3
+
4
L
2
π
2
∞
X
1
(

1)
n
n
2
cos
nπx
L
=
L
2
3

4
L
2
π
2
cos
πx
L

1
4
cos
2
πx
L
+
1
9
cos
3
πx
L
 · · ·
(5
.
1)
To see if this is even plausible, examine successive partial sums of the series, taking one term, then two
terms, etc. Sketch the graphs of these partial sums to see if they start to look like the function they
are supposed to represent (left graph). The graphs of the series, using terms up to
n
= 5
do pretty
well at representing the parabola.
0
1
3
5
x
2
1
3
5
x
2
The same function can be written in terms of sines with another series:
x
2
=
2
L
2
π
∞
X
1
(

1)
n
+1
n

2
π
2
n
3
(
1

(

1)
n
)
)
sin
nπx
L
(5
.
2)
and again you can see how the series behaves by taking one to several terms of the series. (right graph)
The graphs show the parabola
y
=
x
2
and partial sums of the two series with terms up to
n
=
1
,
3
,
5
.
James Nearing, University of Miami
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
5—Fourier Series
2
The second form doesn’t work as well as the first one, and there’s a reason for that. The sine
functions all go to zero at
x
=
L
and
x
2
doesn’t, making it hard for the sum of sines to approximate
the desired function. They can do it, but it takes a lot more terms in the series to get a satisfactory
result. The series Eq. (
5.1
) has terms that go to zero as
1
/n
2
, while the terms in the series Eq. (
5.2
)
go to zero only as
1
/n
.*
5.2 Computing Fourier Series
How do you determine the details of these series starting from the original function? For the Taylor
series, the trick was to assume a series to be an infinitely long polynomial and then to evaluate it (and
its successive derivatives) at a point. You require that all of these values match those of the desired
function at that one point. That method won’t work in this case. (Actually I’ve read that it can work
here too, but with a ridiculous amount of labor and some mathematically suspect procedures.)
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Nearing
 Fourier Series, Heat, Mathematical analysis

Click to edit the document details