Please answer to all questions with proper explanation step by step. Thankyou :)

3 _ 3:ry2 5. A function y) is defined by u, (a) [5 marks] Show that u is harmonic, i.e. it satisfies Laplace's equation 0211 o. Dx.2 (b) [7 marks] Determine the conjugate harmonic function of u and hence determine the function f (z) y) + y) as a function of z. 6. For the fimction f (z) (a) [7 marks] find the poles and determine the residues at each pole; (b) [5 marks] evaluate the integral f (z) (Iz, where C is the circle Izl 4. 7. Use contour integration to evalllate d.r (a) [8 marks] 27 (b) [8 marks]