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13.30. In each part of Problem 13.29, tell where the discontinuities of f x are located and to what value the series converges at the discontunities. Ans. (a) x 0; 2; 4; . . . ; 0 b no discontinuities (c) x 0; 10; 20; . . . ; 20 (d) x 3; 9; 15; . . . ; 3
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& 13.31. Expand f x Ans: 2 x x 6
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FOURIER SERIES 0<x<4 in a Fourier series of period 8. 4<x<8 & ' 16 x 1 3x 1 5x 2 cos 2 cos cos 2 4 4 4  3 5
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13.32. (a) Expand f x cos x; 0 < x < , in a Fourier sine series. (b) How should f x be de ned at x 0 and x  so that the series will converge to f x for 0 @ x @  Ans: a
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13.33. (a) Expand in a Fourier series f x cos x; 0 < x <  if the period is ; and (b) compare with the result of Problem 13.32, explaining the similarities and di erences if any. Ans. Answer is the same as in Problem 13.32. 13.34. Expand f x Ans: 0<x<4 in a series of (a) sines, (b) cosines. 4<x<8  1 1  32 X 1 n nx 16 X 2 cos n=2 cos n 1 nx sin b 2 sin cos a 2 2 8 8  n 1 n2  n 1 n2 x 8 x &
13.35. Prove that for 0 @ x @ ,   2 cos 2x cos 4x cos 6x a x  x 6 12 22 32 b x  x   8 sin x sin 3x sin 5 3 3  13 3 5
13.36. Use the preceding problem to show that a
1 X1 2 ; 2 6 n n 1
1 X 1 n 1 n 1
2 ; 12
1 X 1 n 1 3 : 3 32 n 1 2n 1
p 1 1 1 1 1 1 32 2 13.37. Show that 3 3 3 3 3 3 . 16 1 3 5 7 9 11 DIFFERENTIATION AND INTEGRATION OF FOURIER SERIES 13.38. (a) Show that for  < x < ,   sin x sin 2x sin 3x x 2 1 2 3
(b) By integrating the result of (a), show that for  @ x @ ,   2 cos x cos 2x cos 3x x2 4 2 2 2 3 1 2 3 (c) By integrating the result of (b), show that for  @ x @ ,   sin x sin 2x sin 3x 3 3 x  x  x 12 13 2 3 13.39. (a) Show that for  < x < ,   1 2 3 4 x cos x sin x 2 sin 2x sin 3x sin 4x 2 1 3 2 4 3 5   1 cos 2x cos 3x cos 4x x sin x 1 cos x 2 2 1 3 2 4 3 5
(b) Use (a) to show that for  @ x @ ,
CHAP. 13]
FOURIER SERIES
13.40. By di erentiating the result of Problem 13.35(b), prove that for 0 @ x @ ,    4 cos x cos 3x cos 5x x 2  12 32 52 PARSEVAL S IDENTITY 13.41. By using Problem 13.35 and Parseval s identity, show that a
1 X1 4 n4 90 n 1 1 X1 6 n6 945 n 1
13.42. Show that
1 1 1 2 8 . 2 2 2 2 2 16 3 3 5 5 7
[Hint: Use Problem 13.11.]
13.43. Show that
1 4 ; 4 96 n 1 2n 1
1 6 . 6 960 n 1 2n 1
13.44. Show that
1 1 1 42 39 2 2 2 2 . 2 2 2 2 16 1 2 3 2 3 4 3 4 5
BOUNDARY-VALUE PROBLEMS 13.45. (a) Solve @U @2 U 2 2 subject to the conditions U 0; t 0; U 4; t 0; U x; 0 3 sin x 2 sin 5x, where @t @x 0 < x < 4; t > 0. a U x; t 3e 2 t sin x 2e 50 t sin 5x.
(b) Give a possible physical interpretation of the problem and solution. Ans: 13.46. Solve &
@U @2 U 2 subject to the conditions U 0; t 0; U 6; t 0; U x; 0 @t @x physically. 1 X 1 cos m=3 ! m2 2 t=36 mx e 2 sin Ans: U x; t m 6 m 1
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