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EGM 4313 S08 Homework Solution #8 1/17 EGM 4313 S08 - Homework#8 1. Problem Set 11.1, p.485, #21 Showing the details of your work, find the Fourier series of the given f(x), which is assumed to have theperiod of 2π. Sketch or graph the partial sums up to that including cos5x and sin5x.2( ),f xxxππ=−<<Solution:From (6) on page 480, we calculate coefficients in Fourier series as3220112233xax dxπππππππ−−===∫()222221cos1sin2 sin1sin2 cos2cos41nnaxnxdxxnxxnxdxnnxnxxnxnxdxnnnnπππππππππππππππ−−−−−−=⎡⎤⎢⎥=−⎢⎥⎣⎦⎡⎤⎢⎥=+−⎢⎥⎣⎦=−∫∫∫21sin0nbxnxdxπππ−==∫(because x2is a even function, bnhas to be zero)Therefore,()2214( )1cos3nnf xnxnπ∞==+−∑The following plot is generated based on f(x) with items up to n=5.[email protected](x) x^2;[email protected](x) pi^2/3‐4*cos(x)+cos(2*x)‐4/9*cos(3*x)+1/4*cos(4*x)‐4/25*cos(5*x);figure; hold;fplot(f1,[‐pi,pi])fplot(f2,[‐pi,pi])
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