In this question f(t) is a real valued odd square integrable function. a) What does it mean for f(t) to be odd. b) Prove that the Fourier transformation of f(t) is a purely imaginary function and may be obtained by either of the following: Full Soln F(w) 2j Im -jwt F(w) -2j f(t) sin(wt)dt. 8. c) Prove that F(w) is odd, and that f(t) F(w) sin(wt)duw. Full Soln. d) Suppose that (Full Soln) |t| 1 / f(t) (I) Prove that the Fourier transformation of f(t) is F(w) : 2j (w cos(w)- sin(w)). The following may help: / x sin(ax)dx = sin(ax) - cos(a.x)+ K. (II) Prove that w cos(w) sin(w) 2 sin du = w? 3 6. w cos(w) sin(w) w2 sin (3w) dw 0.