Question

Question:

Required formulae from Question 4:

Other formulae:

8. (a) If f(t) /2 show thattf. Use formulae from Question 4 to show thatpwF (the same equation in the transformed variables). It follows that F(w) - Ae-/2; evaluate the arbitrary constant A by putting w 0. Deduce that F(w) f(w) (i.e., this function is equal to its Fourier transform) (b)" Using Question 4(i), show that Fe-t2/202)-ơe_ơ2w2/2. There is a general theorem that the more widely spread out a function is, the more concentrated will its Fourier transform be, and vice versa. Question 6(ii) illustrates one extreme (the function is infinitely concentrated and its transform is infinitely spread out). The functions here (multiples of Gaussian normal distributions) illustrate the general situation as σ ranges rove the following results for Fourier transforms (where F(f(t)) F(w)) by modifying the corresponding prools for Laplace transforms. (i) F(f(at)) F() for a0 (ii) F(f'(t)) -iwF(w) a ta As I → 00approxi becomes valid for all t, and we obtain oo, the approximation to the integral becomes exact, the expression for f(t)edt F(w) f(t) =-/= 2n J-00 where F(u)dM da, The function () is called the Fourier transform of f(t), and is denoted F(f(t)). We have therefore proved Fourier Inversion Theorem. At points where f(t) is continuous f(t) =- where

Answer 1

dlu Cond dF ヲFD-A , e

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