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Use the integration theorem to show that Cos nx 효(72-3a2) =D Σ(-1)”-1! -T

Use the integration theorem to show that Cos nx 효(72-3a2) =D Σ(-1)”-1! -T

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Solve c and d, using the table might help

1. The Fourier series for the function f(x) = ;x, -T < x <T is sin nx E(-1)" n-1 n n=1 a) Sketch the graph of the function f, which is the pointwise sum of the series on R. b) Use Parseval's formula to show that En=1 72 1 c) Use the integration theorem to show that COS nx 2(r² – 3x²) = E(-1)"-1 -T < x < T n2 n=1 and sin nx ba(n² – a²) = E(-1)*-1 n3 -T < x < T n=1 d) Discuss whether the sum of each Fourier series in c) is piecewise continuous, continuous, piecewise C' or C on R. Sketch the graphs of the sum functions.
Transcribed Image Text:1. The Fourier series for the function f(x) = ;x, -T < x <T is sin nx E(-1)" n-1 n n=1 a) Sketch the graph of the function f, which is the pointwise sum of the series on R. b) Use Parseval's formula to show that En=1 72 1 c) Use the integration theorem to show that COS nx 2(r² – 3x²) = E(-1)"-1 -T < x < T n2 n=1 and sin nx ba(n² – a²) = E(-1)*-1 n3 -T < x < T n=1 d) Discuss whether the sum of each Fourier series in c) is piecewise continuous, continuous, piecewise C' or C on R. Sketch the graphs of the sum functions.
Table 5.1: Examples of Fourier series f(x) zao +2(am cos nx + b, sin nx) n=1 i) x, 0<x < T (-1)n+1 sin nx n n=1 sin nx ii) (T – x), 0< x < T n=1 sin(2n – 1)x 2n – 1 ii) 7, 0< x < T n=1 Σ (-1)ª-1 Cos(2n – 1)x 2n – 1 iv) T, n < x < T uf > * > 0 n=1 COS nx v) (7? – 32), 0 < x < + E(-1)"-1. n2 n=1 COs nx vi) (T – 2)? – RT², 0 < x < T n2 23D1 cos (2n – 1)x (2n – 1)2 vii) 7(T – 2), 0< x < ™ n=1 (-1)n-1 Sin(2n – 1)x (2n – 1)2 0 < x < }T TI, liT(T – x), T < x < T viii) n=1
Transcribed Image Text:Table 5.1: Examples of Fourier series f(x) zao +2(am cos nx + b, sin nx) n=1 i) x, 0<x < T (-1)n+1 sin nx n n=1 sin nx ii) (T – x), 0< x < T n=1 sin(2n – 1)x 2n – 1 ii) 7, 0< x < T n=1 Σ (-1)ª-1 Cos(2n – 1)x 2n – 1 iv) T, n < x < T uf > * > 0 n=1 COS nx v) (7? – 32), 0 < x < + E(-1)"-1. n2 n=1 COs nx vi) (T – 2)? – RT², 0 < x < T n2 23D1 cos (2n – 1)x (2n – 1)2 vii) 7(T – 2), 0< x < ™ n=1 (-1)n-1 Sin(2n – 1)x (2n – 1)2 0 < x < }T TI, liT(T – x), T < x < T viii) n=1
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