Final Exam Solutions - Complex Analysis I | MATH 535, Exams of Mathematics

Download Final Exam Solutions - Complex Analysis I | MATH 535 and more Exams Mathematics in PDF only on Docsity! Math 535 - Complex Analysis - Spring 2010 Final Exam Advice: All of these problems have relatively “short” solutions. If you find yourself writing pages of complicated calculations, you might want to stop and reconsider your approach. (1) (a) What does it mean for a region Ω ⊂ C to be simply connected? (Give a definition.) (b) State Cauchy’s theorem for analytic functions on a simply connected region Ω. (2) Does there exist an analytic function f on the unit disk such that f(1/n) = f(−1/n) = sin(1/n) for all integers n ≥ 2? (Either construct such a function or prove that no such function exists.) (3) (a) What does it mean for two regions to be conformally equivalent? (Give a definition.) (b) Prove that the upper half-plane H = {z | Im(z) > 0} is not confor- mally equivalent to the complex plane C. (4) Suppose that Ω ⊂ C is a region and F ⊂ O(Ω) is a normal family of analytic functions on Ω. Define √ F = {f ∈ O(Ω) | f 2 ∈ F}. Show that √ F is a normal family. (5) Find a polynomial p(x, y) of degree 5 that is a harmonic function, and then find its harmonic conjugate. (6) Let A = {z | 1 < |z| < 535}. Prove that there does not exist an analytic function f ∈ O(A) such that exp(f(z)) = z for all z ∈ A. Math 535 - Complex Analysis - Spring 2010 Final Exam Solutions (1) (a) What does it mean for a region Ω ⊂ C to be simply connected? Ahlfors’ definition: A region Ω is simply connected if its complement in the Riemann sphere, Ĉ \ Ω, is connected. (Other accepted answers: • n(γ, a) = 0 for all closed curves γ ⊂ Ω and a /∈ Ω • Any correct statement of the usual topological definition of “simply connected”, e.g. all closed curves in Ω are null-homotopic.) (b) State Cauchy’s theorem for analytic functions on a simply connected region Ω. Theorem: If f is analytic on a simply connected region Ω and if γ ⊂ Ω is a cycle, then ∫ γ f(z) dz = 0. (Also acceptable if stated for a closed curve rather than a general cycle.) (2) Does there exist an analytic function f on the unit disk such that f(1/n) = f(−1/n) = sin(1/n) for all integers n ≥ 2? (Either construct such a function, or prove that no such function exists.) Suppose that f is such a function. Since {1/n | n ≥ 2} accumulates at 0, the condition f(1/n) = sin(1/n) for n ≥ 2 gives f(z) = sin(z) for all z ∈ ∆. But sin(−1/n) 6= sin(1/n), contradicting the condition f(−1/n) = sin(1/n). Therefore no such function exists. (3) (a) What does it mean for two regions to be conformally equivalent? The regions Ω and Ω′ are conformally equivalent if there is an analytic homeomorphism of Ω onto Ω′. (b) Prove that the upper half-plane H = {z | Im(z) > 0} is not conformally equivalent to the complex plane C. Suppose f : C→ H is a conformal equivalence. Then f is an entire func- tion with positive imaginary part, hence f is constant, a contradiction. (4) Suppose that Ω ⊂ C is a region and F ⊂ O(Ω) is a normal family of analytic functions on Ω. Define √ F = {f ∈ O(Ω) | f2 ∈ F}. Show that √ F is a normal family. Let {fn} ⊂ √ F be a sequence. Then {f2n} ⊂ F , so there is a subsequence {f2nk} that is either locally uniformly convergent or which tends to ∞ locally uniformly.