Urgent! Please answer all parts with justification, thank you.

Question 3 Part (a). Let u(r,y) = my+ 3.7: and day) = 9:- 2y. (i) Determine all z E C such that f(z) = f(a: + g) = 1.:(3, y) + iv(a:, y) is differentiable in the complex sense, with z = a: + iy E C , a: E R , y E R . (ii) Find a real valued function w(z,y) such that g(z) = 9(a: + iy) = u(a:, y) + 1'74\"\": y) is differentiable in the complex sense for all z E C . (iii) Express g(z) in terms of z = a: -l- iy . Part (b). Let 50(2) = (z 1)(z+ 32' 2) and C1 , Cg , 03 , 04 be the contours given as follows; - C1 is the line segment with the initial point 21' and the end point 721'; . C2 is the line segment with the initial point 721' and the end point 72; - C3 is the line segment with the initial point 2 and the end point 21'; - C4 is the circle with centre at the origin and radius 3, oriented positively, as shown in the gure. iR xv Calculate the following integrals: (i) f0: go(z)dz ; 1 #2) dz ' (in fat (iii) fc'lLJCgUCg $0\