HW5s - MATH 6322, Complex Analysis Fall 2011, Key to HW#5...

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Due date: Wed., Oct 5, 2011 5. Let f j : D (0 , 1) C be holomorphic and suppose that each f j has at least k roots in D (0 , 1), counting multiplicities. Suppose that f j f uniformly on compact subsets. Show by example that it does not follow that f has at least k roots in D (0 , 1), counting multiplicities. In particular, construct examples, for each fixed k and each l , 0 l k , where f has exactly l roots. What simple hypothesis can you add that will guarantee that f does have at least k roots in D (0 , 1), counting multiplicities? Solution . Let f j = z j + k - 1 2 and f = 1 / 2. Then f j has at least k roots in D (0 , 1). But f = 1 2 has no zeros on D (0 , 1). Similarly, let f j ( z ) = z j + k + l - 1 2 z l . Then f j has at least k roots in D (0 , 1). But f = 1 2 z l has l zeros, counting multiplicities on D (0 , 1). The simple hypothesis we can add is as follows: we assume that f j be e holomorphic on D (0 , 1) U and suppose that each f j has at least k roots in D (0 , 1), counting multiplicities. Suppose that f j f uniformly on compact subsets of U . Assume that f has no zeros on ∂D (0 , 1). Then f has at least k roots in D (0 , 1), counting multiplicities. This can be proved by using Rouche’s theorem (similar to the proof of Problem 8 and extra problem 1 below): Since f has no no zeros on ∂D (0 , 1), by the compactness of ∂D (0 , 1), δ := min z ∂D (0 , 1) | f ( z ) | > 0 . Since f j f uniformly on ∂D (0 , 1), there exists N > 0 such that for j > N , for all z ∂D (0 , 1), | f j ( z ) - f ( z ) | < δ ≤ | f ( z ) | . Rouch’s theorem implies that f and f n have same number of zeros, counting multiplicities on D (0 , 1). Remark : In above, we compare the zeros of f and f n = f - f n , so in order to apply Rouche’s theorem, we need a lower bound of f and an upper bound fo f - f 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.

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HW5s - MATH 6322, Complex Analysis Fall 2011, Key to HW#5...

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