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Unformatted text preview: to prove that such map does not exist. In your proof, you can use any fact, proved in Ahlfors’ book, but you have to state this fact precisely. 5. a) What is the radius of convergence of the series in the right hand side of this equation: Log z z1 = ∞ X n =0 a n ( z2) n ? 1 Here Log is the principal branch of the logarithm, Log( z ) = Log  z  + i Arg z, where  Arg z  < π. b) Will your answer change, if we consider some other branch of the logarithm in the same region  Arg z  < π ? 6. Does there exist an analytic function f in the unit disc, with the following property f (1 /n ) = n 2 + n for n = 1 , 2 , 3 ,.... Give an example or prove that it does not exist. 2...
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 Spring '09
 Complex Numbers, Complex number, surjective holomorphic map, cj bn

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