UMD Analysis Qualifying Exam/Aug08 Complex

Problem 2 Compute Solution 2 We will compute the general case: Find Poles of f(z) The poles of are just the zeros of , so we can compute them in the following manner: If is a solution of , then and , k=0,1,2,...,n-1. Thus, the poles of are of the form with Choose Path of Contour Integral In order to get obtain the integral of from 0 to , let us consider the path consisting in a line going from 0 to , then the arc of radius from the angle 0 to and then the line joining the end point of and the initial point of , File:pi_n_contour.jpg where is a fixed positive number such that the pole is inside the curve . Then , we need to estimate the integral Compute Residues of f at z0= exp{i\pi /n} Bound Arc Portion (B) of Integral Hence as , Parametrize (C) in terms of (A) Let where is real number. Then Apply Cauchy Integral Formula From Cauchy Integral Formula, we have, As , . Also can be written in terms of . Hence We then have, Problem 4 Suppose and there is an entire function with . If and , prove that Solution 4 Lemma: Two fixed points imply identity Lemma. Let be analytic on the unit , and assume that on the disc. Prove that if there exist two distinct points and in the disc which are fixed points, that is, and , then . Proof Let be the automorphism defined as Consider now . Then, F has two fixed points, namely . Since , (since is different to ), and , by Schwarz Lemma, . But, replacing into the last formula, we get . Therefore, , which implies Shift Points to Create Fixed Points Let . Then and . Notice that is an infinite horizontal strip centered around the real axis with height . Since is a unit horizontal shift left, . Use Riemann Mapping Theorem From the Riemann mapping theorem, there exists a biholomorphic (bijective and holomorphic) mapping , from the open unit disk to . Define Composition Function Let . Then maps to . From the lemma, since has two fixed points, which implies which implies . Problem 6 Let be the family of functions analytic on so that Prove that is a normal family on Solution 6 Choose any compact set K in D Choose any compact set in the open unit disk . Since is compact, it is also closed and bounded. We want to show that for all and all , is bounded i.e. where is some constant dependent on the choice of . Apply Maximum Modulus Principle to find |f(z0)| Choose that is the shortest distance from the boundary of the unit disk . From the maximum modulus principle, . Note that is independent of the choice of . Apply Cauchy's Integral Formula to f^2(z0) We will apply Cauchy's Integral formula to (instead of ) to take advantage of the hypothesis. Choose sufficiently small so that Integrate with respect to r Integrating the left hand side, we have Hence, Bound |f(z0)| by using hypothesis Apply Montel's Theorem Then, since any is uniformly bounded in every compact set, by Montel's Theorem, it follows that is normal