math185f08-hw10 - MATH 185: COMPLEX ANALYSIS FALL 2008/09...

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MATH 185: COMPLEX ANALYSIS FALL 2008/09 PROBLEM SET 10 For a C , r > 0, we write D * ( a,r ) := { z C | 0 < | z - a | < r } . We write C × = C \{ 0 } . 1. Derive the following expansions. (a) For all z C , e z = e + e X n =1 1 n ! ( z - 1) n . (b) For all z D (1 , 1), 1 z = X n =0 ( - 1) n ( z - 1) n . (c) For all z D ( - 1 , 1), 1 z 2 = 1 + X n =1 ( n + 1)( z + 1) n . 2. Let n N be n 2. Determine all entire functions f that satisfy f ( z n ) = [ f ( z )] n for all z C . 3. Let z 0 C and r > 0. Let f : D ( z 0 ,r ) C be analytic and | f 0 ( z ) - f 0 ( z 0 ) | < | f 0 ( z 0 ) | for all z D ( z 0 ,r ). Prove that f is injective on D ( z 0 ,r ). 4. Let f : D (0 , 1) C be analytic and f ( z ) 6 = 0 for all z D (0 , 1). Show that there exist an analytic function g : D (0 , 1) C such that f ( z ) = e g ( z ) for all z D (0 , 1). 5. Let f and g be analytic functions on a region Ω. (a) Suppose
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math185f08-hw10 - MATH 185: COMPLEX ANALYSIS FALL 2008/09...

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