hw9 - C \ L must have at least k bounded path connected...

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Mathematics 185 – Intro to Complex Analysis Fall 2005 – M. Christ Problem Set 9 Due Tuesday November 1: Solve these problems from Chapter 10: 4 (but don’t bother calculating c n explicitly for small n ), 16. Solve these problems from Chapter 11: 1 (parts 1,3,6), 2 (part 2; use partial fractions), 3 (parts 3,5), 5 (do only ( z 2 - 1) - 1 , and only in powers of z - i ), 6, 17. Hints and further instructions. Problem 4 of Chapter 10: Think about multiplying together the Taylor series for cos and sec; Theorem 3.9 can be used to justify the manipulation of this product of infinite series. Problem 16 of Chapter 10: It is a true fact that if the level set L = { z : | p ( z ) | = R } has k connected components, then
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Unformatted text preview: C \ L must have at least k bounded path connected components. Assume this fact without proof. (It is also true that each connected component of L is a closed path, that is, equals the range of some closed path. This furnishes some intuition for the other fact.) Now show rigorously, using something we have learned, that each bounded com-ponent of C \ S must contain at least one zero of p . Deduce the stated conclusion from this. Problem 5 of Chapter 11: Even though we’re only discussing one function and ex-pansions in terms of z-i , your answer should consist of two different expansions, valid in two disjoint regions....
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This note was uploaded on 07/31/2009 for the course MATH 185 taught by Professor Lim during the Spring '07 term at Berkeley.

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