(b) The function u(x, y) satisfies Laplace's equation vều = 0 on the semi- infinite plane y < 1, subject to u → 0 as y →

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(b) The function u(x, y) satisfies Laplace's equation vều = 0 on the semi- 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 ğ(k). (ii) Use the convolution theorem to show that u can be written in the form pa(x,y) dp (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 una/(1 – y) as y →-00, for some a that you should determine. uls, y) = à Salons in 1 = TT