FourierSeries-1

A n a n 0 figure 5 energy spectra 2 4 6 8 0 n 10 2 4 a

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Unformatted text preview: eate a richer sound through emphasizing certain harmonics by assigning larger Fourier coefﬁcients (and therefore higher corresponding energies). Exercises 7–11 Click here for solutions. S Find the Fourier series of the function. 7. f x 1 0 if x if 1 8. f x 0 1 0 if 2 x 0 if 0 x 1 if 1 x 2 1–6 A function f is given on the interval and f is , periodic with period 2 . (a) Find the Fourier coefﬁcients of f . (b) Find the Fourier series of f . For what values of x is f x equal to its Fourier series? ; (c) Graph f and the partial sums S2, S4, and S6 of the Fourier series. 1 1. f x if if 0 1 x x 9. f x 0 3. f x fx fx 4 fx 0 fx 8 fx x 10. f x 0 x 1 11. f t x sin 3 t , ■ x, 1 x 1 1 t fx 2 fx 1 x2 0 cos x 5. f x 1 6. f x ■ 4 x 4. f x if if 0 fx 2 if 4 x 0 if 0 x 4 ■ 0 x 2. f x 1 x ■ 1 0 ■ ■ if if 0 if if if 0 ■ x x ■ ■ ■ ■ 0 if E sin t 0 ■ ■ ■ ■ ■ ■ ■ t, where t represents time, is passed through a so-called half-wave rectiﬁer that clips the negative part of the wave. Find the Fourier series of the resulting periodic function 2 x ■ 12. A voltage E sin 0 x 2 x ■ if 0 t ■ ■ ■ ■ ■ ■ 0 ft ft t 2 ft 10 ■ FOURIER SERIES 18. Use the result of Example 2 to show that 13–16 Sketch the graph of the sum of the Fourier series of f without actually calculating the Fourier series. 13. f x 3 1 if 4 x 0 if 0 x 4 14. f x x 1 15. f x 3 x, 1 x e x, 2 x 2 ■ ■ ■ if 1 x 0 x if 0 x 1 ■ ■ 2 1 72 8 1 1 3 1 5 1 7 4 20. Use the given graph of f and Simpson’s Rule with n ■ ■ ■ ■ ■ ■ ■ ■ 8 to estimate the Fourier coefﬁcients a 0, a1, a 2, b1, and b2. Then use them to graph the second partial sum of the Fourier series and compare with the graph of f . y 17. (a) Show that, if x2 1 52 19. Use the result of Example 1 to show that 1 16. f x 1 32 1 1 1 3 x 1, then 1 n1 n 4 n2 2 cos n x 1 (b) By substituting a speciﬁc value of x, show that n1 1 n2 2 6 0.25 x FOURIER SERIES ■ 11 SOLUTIONS 1. (a) a0 = an = 1 2π π −π 1 π π −π 1 π f (x) dx = π −π π...
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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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