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MA530_JAN04 - z and ends at z a Prove that C-A is connected...

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QUALIFYING EXAMINATION JANUARY 2004 MATH 530 - Prof. Bell 1. (10 pts) Suppose f and g are analytic functions on a domain Ω and that f and g satisfy the identity f 0 ( z ) = g 0 ( z ) f ( z ) for all z on a closed line segment contained in Ω. Prove that f ( z ) = ce g ( z ) on Ω for some constant c . You must explain your steps carefully. 2. (30 pts) Suppose that w 1 , . . ., w N are points on the unit circle. Prove that there is a point z on the unit circle such that the product of the distances from z to the points w j , j = 1 , . . ., N is exactly equal to one. Hint: Use an analytic function, not analytic geometry. 3. (30 pts) Suppose that f is an analytic function on the complex plane minus a discrete set of points A . Suppose further that for each pont a in A , f has a simple pole with residue equal to a positive integer n ( a ). Let z 0 denote a point in C -A . For a point z in C - A , let γ z denote a contour in
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Unformatted text preview: z and ends at z . a) Prove that C-A is connected. b) Prove that the formula F ( z ) = exp ±Z γ z f ( w ) dw ² yields a well defined analytic function on C-A . c) Prove that F has a removable singularity at each point in A . 4. (30 pts) Suppose f and g are analytic in a disc D R (0) with R > 1 and suppose that f has a simple zero at z = 0 and has no other zeroes in the set { z : | z | ≤ 1 } . Let H ± ( z ) = f ( z ) + ±g ( z ) . Prove that there is a radius r with 0 < r < 1 such that H ± ( z ) has a unique zero z ± in the unit disc if 0 < ± < r . Finally, prove that the mapping ± 7→ z ± is a continuous map from (0 ,r ) into the unit disc. Hint: What is the residue of z H ± ( z ) H ± ( z ) at z ± ?...
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