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Unformatted text preview: t’s often represented as in
Figure 1(b), where you can see why it’s called a
square wave. 1 _π π 2π x π 0 2π x (a)
y
1 _π 0 FIGURE 1
Squarewave function (b)
SOLUTION Using the formulas for the Fourier coefﬁcients in Deﬁnition 7, we have a0 1
2 y f x dx 1
2 y 0 0 dx 1
2 y 0 1 dx 0 1
2 1
2 4 ■ FOURIER SERIES and, for n 1,
1 an y 1 sin n x
n 0 1 bn y ■■ 0
2
n Note that cos n equals 1 if n is even
1 if n is odd. 1
n 0 1
n 0 y 0 y 0 1 0 dx sin n 1 f x sin n x d x 1 cos n x
n and 1 f x cos n x d x y 0 sin 0 0 dx cos n 1 cos n x d x 0 y 0 sin x d x cos 0 if n is even
if n is odd The Fourier series of f is therefore
a0 a1 cos x a2 cos 2 x b1 sin x
1
2 0 0
2 1
2 2 a3 cos 3x b2 sin 2 x b3 sin 3x 0 sin 2 x 2
sin 3x
3 0
sin x 2
sin 3x
3 sin x Since odd integers can be written as n
Fourier series in sigma notation as
1
2 2
sin 5x
5 2k 2
k1 2k 1 0 sin 4 x 2
sin 5x
5 2
sin 7x
7 1, where k is an integer, we can write the sin 2 k 1x In Example 1 we found the Fourier series of the squarewave function, but we don’t
know yet whether this function is equal to its Fourier series. Let’s investigate this question
graphically. Figure 2 shows the graphs of some of the partial sums
Sn x 1
2 2 sin x 2
sin 3x
3 2
sin n x
n when n is odd, together with the graph of the squarewave function. FOURIER SERIES ■ 5 y y 1 y 1 1
S£ S¡
_π π x _π x π y π 1 1
S¡∞ S¡¡ x x y S¶ π x π _π y 1 _π S∞ _π x π _π FIGURE 2 Partial sums of the Fourier series for the squarewave function We see that, as n increases, Sn x becomes a better approximation to the squarewave
function. It appears that the graph of Sn x is approaching the graph of f x , except where
x 0 or x is an integer multiple of . In other words, it looks as if f is equal to the sum
of its Fourier series except at the points where f is discontinuous.
The following theorem, which we state without proof, says that this is typical of the
Fourier series of piecewise continuous functions. Recall that a piecewise continuous function has only a ﬁnite number of jump discontinuities on
, . At a number a where f
has...
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This note was uploaded on 12/05/2011 for the course MATH 41 taught by Professor Bray,c during the Spring '08 term at Duke.
 Spring '08
 Bray,C
 Math, Fourier Series

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