Q4 (20 points) (a) (i) Calculate the Fourier transform {g(x)) = g(k) of ſi if -1

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Q4 20 Points A I Calculate The Fourier Transform G X G K Of Si If 1 X 1 G X 1x 21 Ii Show That The In 1 (37.62 KiB) Viewed 20 times

Q4 (20 points) (a) (i) Calculate the Fourier transform {g(x)) = g(k) of ſi if -1<x<1 g(x) = 1x 21. (ii) Show that the inverse Fourier Transform F-'{(k)) = h(x) of h(k) = euk To if for a > 0, is WVG) ( h(x) = (b) The function u(x, y) satisfies Laplace's equation Vļu = 0 on the semi-infinite plane y < 1. subject to u → as y → -0 and u(x, 1) = g(x), where g(x) was defined in part (a). (1) Determine an expression for the Fourier Transform ū(k, y) of u in terms of ğ(k). (ii) Use the convolution theorem to show that u can be written in the form {x.) dp # Santos 1 u(x, y) 1 + p2 where the functions a(x, y) and (x, y) are to be determined. (iii) Compute the integral and demonstrate explicitly how your answer recovers the boundary condition u(x, 1) = g(x). Show further that u ~ al(1 - y) as y → -00, for some a that you should determine.