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Math · Advanced Math
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a) (10 %) Say the periodic function f(t) has the Fourier cosine series n 2t 9-1 where q is some constant real number. The Fourier series converges to f(t). From the ex pression for the Fourier series, determine the period of the function f(t). b) (10 %) Say the function h(z) is an odd function. That is, h(- ) -h(z). Without knowing anything more about the function, what can you infer about its Fourier coefficients an and bn if the fumction has the following Fourier series? COS RT m-1 Briefly explain why. Also, sketch and/or explain in words what an odd function looks like (a sentence or two would be enough) c) (60 %) Determine the Fourier series for the periodic function g(t) shown below, defined as follows by its basic period: 0 for-1t<0 t for 0<t<1 g(t) = If you had enough time and a calculator, how would you check your answer for this question? Periodic function gft) 1 0 4 Figure 1: A periodic function.d) (10 %) Say you are given a well-behaved function p(1) in the interval 0 < 1-2x. It is not known what the value of the function is outside of this interval. In particular, you do not know if it is periodic. In a couple of sentences, outline how you would find a Fourier sine series2 that converges to p(1) in the interval (0, 2π). A Fourier cosine series has only cosine terms, no sine terms. A Fourier sine series has only the sine terms, no cosine terms. ystem Dynamics & Vibration Ph.D. Qualifying Examination Student ID # roblem # M11 (contd) Is this Fourier sine series unique? (That is, is there only one way to get a Fourier sine series and only one Fourier sine series that converges to the function in the given interval?) e) (5 %) For the function p(1) fron part-d, is it also possible to find a Fourier cosine series that converges to p(t) in the interval (0, 2T)? If no, why? If yes, how? f) (5 %) what is the period of the periodic function/2(1) given by: A(1) = 4 cos 35mt + 3 sin 21.

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