FourierSeries-1

75 05 025 0 5 25 0 25 5 x f ourier series 13 x a0

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Unformatted text preview: Use f (x) = if 0 ≤ x ≤ 4 f (x) dx = f (x) cos 0 1 8 0 −4 , L = 4. −x dx = 1 nπx dx = L 1 4 0 −4 4 nπx dx = (cos (nπ ) − 1) = 4 (nπ )2 −x cos if n is even 2 −8/(nπ) bn = 1 L L −L if n is odd f (x) sin nπx dx = L 1 4 0 −4 −x sin 4/nπ if n is even −4/nπ nπx 4 dx = cos (nπ) = 4 nπ if n is odd Fourier Series: 1+ ∞ k=1 − 4 π π 8 4 π cos sin (2k − 1)x − (2k − 1) x + sin (2k)x (2k − 1)π 4 (2k − 1)2 π2 4 (2k)π 4 y 4 3 2 1 0 -5 -2.5 0 2.5 5 x 11. Use f (x) = {sin(3πt) if − 1 ≤ t ≤ 1 , L = 1. Note: This can be done instantly if one observes that the period of sin(3πt) is 2 , and the period of f (x) = 2 which 3 is an integer multiple of 2 . Therefore f (x) is the same as sin(3πt) for all t, and its Fourier series is therefore 3 sin(3πt). We can get this result using the standard coefficient formulas: a0 = 1 2L L −L f (x) dx = 1 2 1 −1 sin(3πx) dx = 0 11 nπx 1 f (x) cos dx = −1 sin(3πx) cos(nπx) dx L −1 L = 0 [applying change of variables to a formula in the section] an = 1L nπx 1 dx = −1 sin(3πx) sin(nπx) dx f (x) sin L −L L sin nπ 6 if n = 3 π (−9 + n2 ) = [using integral table and addition formula = 1 if n = 3 bn = 0 if n = 3 1 if n = 3 14 ■ FOURIER SERIES Fourier Series: sin(3πx) y 1 0.5 0 -5 -2.5 0 2.5 5 x -0.5 -1 13. 3 −1 3 −1 if −5 ≤ x < −4 y 3 if −4 ≤ x < 0 if 0 ≤ x < 4 2 if 4 ≤ x < 5 1 0 -5 -2.5 0 2.5 5 x -1 15. y 1 0.5 0 -1.25 0 1.25 2.5 3.75 x -0.5 -1 17. (a) We find the Fourier series for f (x) = {x2 a0 = 1 2L L −L f (x) dx = 1 2 1 −1 x2 dx = if −1 ≤ x ≤ 1, L = 1 1 3 4 (nπ)2 11 nπx 4 1 an = f (x) cos cos nπ = dx = −1 x2 cos(nπx) dx = 4 L −1 L (nπ )2 − (nπ)2 bn = 1 L L −L f (x) sin So we have x2 = 1 3 + nπx dx = L ∞ n=1 (−1)n 1 −1 x2 sin(nπx) dx = 0 because x2 sin(nπx) is odd. 4 cos(nπx) for −1 ≤ x ≤ 1. (nπ)2 (b) We let x = 1 in the above to obtain 1= 1 3 + ∞ n=1 (−1)n 4 cos(nπ) (nπ)2 2 3 = 4 n2 π2 n=1 ∞ ∞1 π2 = 2 6 n=1 n if n even if n...
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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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