Unformatted text preview: in the Table
of Integrals, it’s not hard to show that y sin n x cos m x d x y cos n x cos m x d x 0 for all n and m
0 for n
for n m
m So the only nonzero term in (4) is am and we get y f x cos m x d x am Solving for am , and then replacing m by n, we have 5 an 1 y f x cos n x d x n 1, 2, 3, . . . Similarly, if we multiply both sides of Equation 2 by sin m x and integrate from
we get 6 bn 1 y f x sin n x d x n to , 1, 2, 3, . . . We have derived Formulas 3, 5, and 6 assuming f is a continuous function such that
Equation 2 holds and for which the term-by-term integration is legitimate. But we can still
consider the Fourier series of a wider class of functions: A piecewise continuous function
on a, b is continuous except perhaps for a ﬁnite number of removable or jump discontinuities. (In other words, the function has no inﬁnite discontinuities. See Section 2.4 for a
discussion of the different types of discontinuities.) FOURIER SERIES ■ 3 7 Definition Let f be a piecewise continuous function on
Fourier series of f is the series a0 a n cos n x , . Then the bn sin n x n1 where the coefﬁcients an and bn in this series are deﬁned by
2 a0 1 an y y f x cos n x d x f x dx 1 bn y f x sin n x d x and are called the Fourier coefﬁcients of f .
Notice in Deﬁnition 7 that we are not saying f x is equal to its Fourier series. Later we
will discuss conditions under which that is actually true. For now we are just saying that
associated with any piecewise continuous function f on
is a certain series called
a Fourier series.
EXAMPLE 1 Find the Fourier coefﬁcients and Fourier series of the square-wave function f deﬁned by
1 if 0 fx x 0 and x fx 2 fx So f is periodic with period 2 and its graph is shown in Figure 1.
y ■ ■ Engineers use the square-wave function in
describing forces acting on a mechanical system
and electromotive forces in an electric circuit
(when a switch is turned on and off repeatedly).
Strictly speaking, the graph of f is as shown
in Figure 1(a), but i...
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