Question

MECH2407: Multivariable Calculus and Partial Differential Equations 4. (a) Given two periodic functions as below in part (0) and part (iã) i) f2 1 -1st< (ii) f)-1/2 State the period of the two periodic functions respectively. Hence, sketch the two given perodic functions for 3 periods. Find the Fourier series for the two given perodic functions over the given interval and expand the series to give the partial sum up to the first three non-zero terms respectively. (16 marks) Use either the answer obtained in Q4(a)i) or Q4(a)(ii) to find the sum of the (b) es5 (2n-1) (4 marks)

Answer 1

Given that:

f(t) = {-4/pi t - pi/2 lessthanorequalto t lessthanorequalto 0 4/pi t 0 < t lessthanorequalto pi/2 f(t) = {1 -1 lessthanorequalto t < 0 1/2 t = 0 t 0 < t lessthanorequalto 1 a_K = 1/pi[integral^0_-pi/2 -4/pi te^-integral k 2t dt + integral^pi/2_0 4/pi t e^- integral k 2t dt] = 4/pi^2[integral^pi/2_0 te^-integral 2 k t dt - integral^0_-pi/2 te^-integral 2k dt] -4/pi^2 [integral t k pi e^- integral pi k - e^- integral pi k/4 j^2 k^2 + 1 - [-1 - pi e^pi j k jk/4j^2 k^2 + e^jk] = 4/pi^2 [2 - j k pi e^-j k pi - e^j pi k + pi e^pi j k jk - e^j k pi/4 j^2 k^2] = 4/pi^2 [2 + j k pi [e^pi jk - e^-j k pi/-4 k^2 + [e^j k pi + e^- j k pi]] a_0 = 1/2 f(t) = Sigma a_k e^- j k (2) t \r\n a_k = 1/2 [integral^0_1 (1)