Strictly speaking the graph of f is as shown in

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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 Square-wave 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 square-wave 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 square-wave 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 square-wave function We see that, as n increases, Sn x becomes a better approximation to the square-wave 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.

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