1. [4 marks] Verify that the real-valued function Mm, y) = log(:z:2 + y2) is harmonic for all (m, y) E R2, except at (0,0). 2. [6 marks] Is there an analytic function f = u+iv with Mm, y) 2 2y3 6m2y+ 5y 4 and f(1 i) 2 (5+ 122')? If so, nd it and explain why it is analytic; if not, explain why not. Locate the critical points of both u(a:, y) and your Mm, y), remarking on what you nd. 3. [4 marks] Find all values of (1 i)\". 4. [6 marks] Evaluate the following contour/ path integrals, justifying your answers. l (a) f dz where the closed contour C is the circle {2 : |z + 27] = 1} C 2 + 27, traveling in the anti-clockwise direction. (b) / (62 51') eggLW+1 dz where C' is any closed contour in the complex plane (C. O