A Fourier series is an in nite series of the form
a +
X1
n=1
bn cos(n!x) +
X1
n=1
cn sin(n!x):
Virtually any periodic function that arises in applications can be represented as the
sum of a Fourier series. For example, consider the three functions whose graph are
shown below:
These are known, respectively, as the triangle wave (x), the sawtooth wave N(x),
and the square wave (x). Each of these functions can be expressed as the sum of
a Fourier series:
(x) = cos x +
cos 3x
32 +
cos 5x
52 +
cos 7x
72 +
cos 9x
92 +
N(x) = sin x +
sin 2x
2
+
sin 3x
3
+
sin 4x
4
+
sin 5x
5
+
(x) = sin x +
sin 3x
3
+
sin 5x
5
+
sin 7x
7
+
sin 9x
9
+
Fourier series are critically important to the study of di erential equations, and they
have many applications throughout the sciences. In addition, Fourier series played
an important role in the development of analysis, and the desire to prove theorems
about their convergence was a large part of the motivation for the development of
Lebesgue integration.
These notes develop Fourier series on the level of calculus. We will not be worrying
about convergence, and we will not be not be proving that any given function is
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