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.
- Spring '09