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ComplexAnalysis2009jan

# ComplexAnalysis2009jan - f extends holomorphically to the...

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QUALIFYING EXAM COMPLEX ANALYSIS Thursday, January 8, 2009 Show ALL your work. Write all your solutions in clear, logical steps. Good luck! Your Name: Problem Score Max 1 20 2 20 3 30 4 30 Total 100

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Problem 1 . Let f = f ( z ) be analytic in the unit disk, f (0) = 0. Show that the inﬁnite series n =1 f ( z n ) is converging and represents an analytic function in the unit disk.
Problem 2 . Consider an analytic function deﬁned in the unit disk by the following power series f ( z ) = n =1 a n z n , where the coeﬃcients are real numbers such that n - 2009 6 a n 6 n Show that f does not extend analytically near the point z = 1.

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Problem 3 . (Cauchy Formula) Let F be a countable compact subset of a domain Ω C . Suppose we are given a bounded holomorphic function f : Ω \ F C Show that
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Unformatted text preview: f extends holomorphically to the entire domain Ω. a) First try a simple case when F is ﬁnite b) Try the case when F has ﬁnite number of accumulation points c) Try the general case. d) The problem still remains valid if F is a compact set of zero length (1-dimensional Hausdorﬀ measure), try to extend your proof to this general case. Recall that F has zero length if it can be covered by a ﬁnite number of disks whose diameters sum up to a number as small as we wish. Problem 4 . Compute the following integral ∫ ∞ cos x (1 + x 2 ) 2 dx Hint. Consider the following complex function in the upper half plane f ( z ) = e iz (1 + z 2 ) 2...
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ComplexAnalysis2009jan - f extends holomorphically to the...

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