Unformatted text preview: ∑ ∞ n =0 a n z n be analytic in the unit disk U = { z :  z  < 1 } with f (0) = 0 and f (0) = 1. Prove that if ∑ ∞ n =2 n  a n  ≤ 1, then f is onetoone in U . 7. (20) Suppose that f ( z ) is analytic in the disk { z :  z  < R } ( R > 1) except for a simple pole at z = 1 with residue 1. If its Taylor expansion in the unit disk is f ( z ) = ∞ X n =0 a n z n , prove that a n → 1 as n → ∞ . 1...
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 Spring '09
 Math, Taylor Series, Laurent, Unit disk, linear fractional transformations, Prof. Weitsman

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