MATH 113 Final: MATH 113 Harvard 113 Fall 01113FinalSolution
Document Summary
Final examination solutions: (15 points, state the cauchy-riemann equations for the real and imaginary parts of an analytic function. Suppose f = u + iv is analytic. Then it satis es the cauchy-riemann equations ux = vy, uy = vx. Equivalently, fy = ifx: de ne the winding number of a closed curve around a point a 6 . 2 i z dz z a: state the schwarz lemma (including the condition for equality). Let be the open unit disk |z| < 1. Suppose f : is analytic, and f (0) = 0. Then |f (z)| |z| for all z and |f (0)| 1. We have equality in either case i f is a rotation: (10 points, determine and classify all singularities of f (z) = z3e1/z + and calculate the residues at each. Solution: z2 (z 1)3 z2 f (z) has an essential singularity at z = 0 and has a triple pole at z = 1.
