{ a 2). (a) (i) Calculate the Fourier transform F{g(x)} = ģ(k) of 1 if -1<<1 g(x) = 0 if |_| > 1. (ii) Show that the inv

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A 2 A I Calculate The Fourier Transform F G X G K Of 1 If 1 1 G X 0 If 1 Ii Show That The Inv 1 (117.47 KiB) Viewed 22 times

{ a 2). (a) (i) Calculate the Fourier transform F{g(x)} = ģ(k) of 1 if -1<<1 g(x) = 0 if |_| > 1. (ii) Show that the inverse Fourier Transform F-1{ħ(k)} = h(x) of ħ(k) = e-alkl, for a > 0, is 2 h(x) = h a2 + x2 b) The function u(x, y) satisfies Laplace's equation vều = 0 on the semi- 0 infinite plane y < 1, subject to u + 0 as y → – and u(x,1) = g(x), where g(x) was defined in part (a). (i) Determine an expression for the Fourier Transform ū(k, y) of u in terms of ģ(k). (ii) Use the convolution theorem to show that u can be written in the form pax,y) dp u(x, y) B(x,y) 1 + p2 where the functions a(x, y) and B(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~a/(1 – y) as y →-oo, for some a that you should determine. = al 7T