HW 8 solution - EGM 4313 S08 Homework Solution#8 EGM 4313...

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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 the period of 2 π . Sketch or graph the partial sums up to that including cos5x and sin5x. 2 ( ) , f x x x π π = < < Solution: From (6) on page 480, we calculate coefficients in Fourier series as 3 2 2 0 1 1 2 2 3 3 x a x dx π π π π π π π = = = ( ) 2 2 2 2 2 1 cos 1 sin 2 sin 1 sin 2 cos 2cos 4 1 n n a x nxdx x nx x nx dx n n x nx x nx nx dx n n n n π π π π π π π π π π π π π π π = = = + = 2 1 sin 0 n b x nxdx π π π = = (because x 2 is a even function, b n has to be zero) Therefore, ( ) 2 2 1 4 ( ) 1 cos 3 n n f x nx n π = = + 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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