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Course: MAT 328, Fall 2009
School: Paul Quinn
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328 Math Homework 2 due on Thursday 2/17/05 Problem 1. Find the Fourier series of the following functions: (i) f (x) = cos x, < x < +. (ii) f (x) = cos x, < x < , extended as a -periodic function on the real line. 2 2 (iii) f (x) = 0 1 < x < 0 , extended as a 2-periodic function on the real line. x 0<x<1 Problem 2. Without computing the coecients,...

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328 Math Homework 2 due on Thursday 2/17/05 Problem 1. Find the Fourier series of the following functions: (i) f (x) = cos x, < x < +. (ii) f (x) = cos x, < x < , extended as a -periodic function on the real line. 2 2 (iii) f (x) = 0 1 < x < 0 , extended as a 2-periodic function on the real line. x 0<x<1 Problem 2. Without computing the coecients, explain why the Fourier series of the function 0 L < x < 0 f (x) = 1 0<x<L has no cosine term in it. Problem 3. (i) Show that if p(x) is a polynomial of degree and n g(x) is a continuous function, then p(x)g(x) dx = p(x)G1 (x) p (x)G2 (x) + p (x)G3 (x) + (1)n p(n) (x)Gn+1 (x). Here, G1 is an antiderivative of g, G2 is an antiderivative of G1 , etc. (Hint: Dierentiate both sides with respect to x). (ii) Use (i) to show that for any constant = 0, sin(x) cos(x) sin(x) cos(x) x3...

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