math185-hw9

math185-hw9 - Let be a region containing D (0 , 1). Let be...

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MATH 185: COMPLEX ANALYSIS FALL 2007/08 PROBLEM SET 9 For a C , r > 0, we write D * ( a,r ) := { z C | 0 < | z - a | < r } . We write C × = C \{ 0 } . You may use without proof any results that had been proved in the lectures. 1. Evaluate the integral Z Γ i f i for i = a,b . (a) f a : C × C is given by f a ( z ) = e e 1 z and Γ a is the boundary ∂D (0 , 2) traversed once counter-clockwise. (b) f b : D * (0 ) C is given by f b ( z ) = 1 (sin z ) 3 and Γ b is the boundary ∂D (0 , 1) traversed once counter-clockwise. 2. Let f : D (0 , 1) C be an analytic function satisfying | f ( z ) | < 1 whenever | z | ≤ 1. Show that f has a fixed point in D (0 , 1), ie. there exists a point a D (0 , 1) such that f ( a ) = a . 3.
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Unformatted text preview: Let be a region containing D (0 , 1). Let be the boundary D (0 , 1) traversed once counter-clockwise. Suppose f : C is analytic, f ( z ) 6 = 0 for all z D (0 , 1), and 1 2 i Z f ( z ) f ( z ) dz = 2 , (3.1) 1 2 i Z z f ( z ) f ( z ) dz = 0 , (3.2) 1 2 i Z z 2 f ( z ) f ( z ) dz = 1 2 . (3.3) Find all zeros of f in D (0 , 1). Date : November 28, 2007 (Version 1.0); posted: November 29, 2007; due: December 5, 2007. 1...
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