Question

(20 points) Consider the function f(z) z in the interval [0, 2π). (a) Derive the Fourier coefficients ck fork = 0, 1,土2, (b) Derive the Fourier coefficients ao, ak, bk for k 1,2,... .. (c) Plot the partial Fourier series, along with the function f, by retaining 1, 10, 50, 100 terms in the summation (use the second form involving cosines and sines) d) Comment on the convergence of the partial Fourier series. Note: you should submit only 1 plot for this problem.

Answer 1

Given f(x) = x in[0, 2 pi) C_k = 1/2 pi integral^2 pi_0 (f(x)e^- I k t)dx C_k = 1/2 pi integral^2 pi_0 x e^- I k x dx C_k = 1/2 pi [x e^- i k x/ - i k]^2 pi_0 - integral^2 pi_0 e^-I k x/ - I k dx] C_k = 1/2 pi {2 pi k e^-i2 pi k/ - ik + [e^- I k x/k^2]^2 pi_0 C_k = 1/2 pi {-2 pi e^- I 2 pi k/ik + e^-I 2 pi k/k^2 - 1/k^2} C_k = 1/2 pi {e^-2 pi I k/k(1/k - 2 pi/i) - 1/k^2} b a_0 = 1/2 pi integral^2 pi_0 f(x) dx = 1/2 pi integral^2 pi_0 x dx = 1/2 pi [x^2/2]^2 pi_0 = pi a_k = 2/2 pi integral^2 pi_0 x cos (2 pi k x/2 pi)dx = 1/ pi integral^2 pi_0 x cos(k x)dx = 1/ pi {integral^2 pi_0 x sin k x/k]