MA530_JAN07

MA530_JAN07 - ∑ ∞ n =0 a n z n has a radius of...

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MATH 530 Qualifying Exam January 2007, S. Bell, A. Weitsman Notation: D r ( a )= { z : | z - a | <r } . 1. (20 pts) i) Let t represent a non-zero real number. Find a linear fractional transformation T ( z ) such that T (0) = - i , T ( t ) = 1, and T ( )= i . ii) For which values of t does T map the upper half plane onto the unit disc D 1 (0)? Explain. 2. (15 pts) Suppose that f is analytic in a disc D r (0) = { z : | z | <r } , r> 0. Prove that f is an even function (i.e., f ( z )= f ( - z )) for z D r (0), if and only if the power series for f centered at the origin has only even powers. 3. (10 pts) Evaluate Z -∞ cos t 1+ t 4 dt. 4. (20 pts) Suppose that f is analytic in the disc D = D 2 (0) and continuous on its closure D . Prove that if | f ( z ) |≤| sin z | for all z in the boundary of D ,then | f ( π 2 ) |≤ 4 . 5. (20 pts) Suppose the power series
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Unformatted text preview: ∑ ∞ n =0 a n z n has a radius of convergence R 1 with < R 1 < ∞ , and the power series ∑ ∞ n =0 b n z n has a has a radius of convergence R 2 with 0 < R 2 < ∞ . Suppose further that b n 6 = 0 for all n . Prove that the power series ∞ X n =0 a n b n z n has a radius of convergence R 3 satisfying R 3 ≤ R 1 R 2 . 6. (15 pts) Suppose A is a finite set of points in the complex plane and let ? denote the union of the closed line segments joining each a ∈ A to the origin. If f is analytic on C-A and is such that 0 = X a ∈A Res a f, prove that f has an analytic anti-derivative on C-? ....
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This document was uploaded on 01/25/2012.

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