Page 369 - Schaum's Outline of Theory and Problems of Advanced Calculus

360 FOURIER SERIES [CHAP. 13

2 x 0 < x < 4
in a Fourier series of period 8.
13.31. Expand f ðxÞ¼
x 6 4 < x < 8
16 x 1 3 x 1 5 x
Ans: cos þ cos þ cos þ
2 4 3 2 4 5 2 4
13.32. (a) Expand f ðxÞ¼ cos x; 0 < x < ,ina Fourier sine series.
(b) How should f ðxÞ be defined at x ¼ 0 and x ¼ so that the series will converge to f ðxÞ for 0 @ x @ ?
8 X n sin 2nx
1
Ans: ðaÞ ðbÞ f ð0Þ¼ f ð Þ¼ 0
4n 1
2
n¼1
13.33. (a) Expand in a Fourier series f ðxÞ¼ cos x; 0 < x < if the period is ; and (b)compare with the result of
Problem 13.32, explaining the similarities and differences if any.
Ans. Answer is the same as in Problem 13.32.

x 0 < x < 4
8 x 4 < x < 8
13.34. Expand f ðxÞ¼ in a series of (a) sines, (b)cosines.
32 X 1 n n x 16 X 2cos n =2 cos n 1 n x
1
1
Ans: ðaÞ sin sin ðbÞ cos
2 n 2 2 8 2 n 2 8
n¼1 n¼1
13.35. Prove that for 0 @ x @ ,
2 cos 2x cos 4x cos 6x
ðaÞ xð xÞ¼ 2 þ 2 þ 2 þ
6 1 2 3
8 sin x sin 3x sin 5

1 3 5
ðbÞ xð xÞ¼ 3 þ 3 þ 3 þ
13.36. Use the preceding problem to show that
1 1 2 1 n 1 2 1 n 1 3
X X ð 1Þ X ð 1Þ
; ; :
ðaÞ 2 ¼ ðbÞ 2 ¼ ðcÞ 3 ¼
n 6 n 12 32
n¼1 n¼1 n¼1 ð2n 1Þ
1 1 1 1 1 1 3 2 p ffiffiffi 2
13.37. Show that þ þ þ ¼ .
1 3 3 3 5 3 7 3 9 3 11 3 16
DIFFERENTIATION AND INTEGRATION OF FOURIER SERIES
13.38. (a) Show that for < x < ,

sin x sin 2x sin 3x
x ¼ 2 þ
1 2 3
(b)Byintegrating the result of (a), show that for @ x @ ,
2 cos x cos 2x cos 3x
2
x ¼ 4 2 2 þ 2
3 1 2 3
(c)Byintegrating the result of (b), show that for @ x @ ,

sin x sin 2x sin 3x
xð xÞð þ xÞ¼ 12 þ
1 3 2 3 3 3
13.39. (a) Show that for < x < ,
1 2 3 4
x cos x ¼ sin x þ 2 sin 2x sin 3x þ sin 4x
2 1 3 2 4 3 5
(b)Use (a)to show that for @ x @ ,

1 cos 2x cos 3x cos 4x
x sin x ¼ 1 cos x 2 þ
2 1 3 2 4 3 5