Unformatted text preview: z and ends at z . a) Prove that CA is connected. b) Prove that the formula F ( z ) = exp ±Z γ z f ( w ) dw ² yields a well deﬁned analytic function on CA . 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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 Spring '09
 Math, Analytic function, 10 pts, 30 pts, Prof. Bell, points A. Suppose

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