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PHZ 3113 Fall 2010 Homework #9, Due Monday, December 6 1. Let the differential operator d d (1 − x2 ) dx dx act on functions defined on the interval (−1, 1). D= (a) Suppose DPm (x) = −m(m + 1) Pm (x) and DPn (x) = −n(n + 1) Pn (x) Show that for n 6= m Pm (x) and Pn (x) are orthogonal on (−1, 1). (b) Verify that P0 = 1 and P1 = x are solutions to DPm = −m(m + 1) Pm . What are the corresponding values of m? (c) Let P2 = c0 + c1 x + c2 x2 be a polynomial of degree 2. Choose c0 and c1 so that P2 is orthogonal to P0 and P1 . Choose c2 so that P2 (1) = 1. Write your resulting P2 (x). Is your P2 a solution of DP2 = −m(m + 1) P2 for some m? If so, what is the your value for m? 2. Let f (x) = sin x 0 < x < π , 0 π < x < 2π . (a) Compute the Fourier series coefficients an , bn . (b) Compute the value of the sum (c) Compute the value of the sum ∞ X (−1)k+1 . (2k)2 − 1 k=1 ∞ X k=1 3. Let f (x) = 1 [(2k)2 − 1]2 [Hint: what is f ( π2 )?] . cos(k0 x) −a < x < a, 0 otherwise. with k0 a = π2 . (a) Compute the Fourier transform f̃ (k). What happens at k → k0 ? (b) Compute the integral (there is an easy way and an easier way to do this) Z ∞ cos y dy . π 2 2 −∞ [( 2 ) − y ] (c) Compute the integral Z ∞ cos2 y dy . π 2 22 −∞ [( 2 ) − y ]