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Unformatted text preview: Key to Complex Analysis Homework 2 September 27, 2011 Chapter 3 (in Greene & Krantz’s book) 4. Use Morera’s theorem to give another proof of Theorem 3.5.1: If { f j } is a sequence of holomorphic functions on a domain U and if the sequence converges unifromly on compact subsets of U to a limit function f , then f is holomorphic on U . Proof. Let γ be a closed piecewise C 1 curve in U , then by the generalized Cauchy’s theorem, we have that Z γ f n dz = 0 , for each n ∈ N . Then, we have that Z γ fdz = Z γ f n fdz ≤ Z γ  f n f  dz  By the given condition that f n converges uniformly to f on a compact subset of U which contains the curve γ and its interior , for each > 0, we can choose an N ∈ N , such that  f n f  < , n > N. Hence, it follows that Z γ fdz ≤ Z γ  f n f  dz  < · length ( γ ) , n > N. Letting → 0, we conclude Z γ fdz = 0 . Since our γ is arbitratily chosen, by the Morerea’s theorem, we know that f is holomorphic on U . 1 2 10. Find the complex power series expansion for z 2 / (1 z 2 ) 3 about 0 and determine the radius of convergence(do not use Taylor’s formula). Proof. We use the formula, for  z  < 1, 1 1 z = ∞ X n =0 z n . By differentiating above, we get 1 (1 z ) 2 = ∞ X n =1 nz n 1 ....
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This note was uploaded on 02/21/2012 for the course MATH 4377 taught by Professor Staff during the Summer '08 term at University of Houston.
 Summer '08
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