HW3s - Key to Complex Analysis Homework 3 September 27,...

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Unformatted text preview: Key to Complex Analysis Homework 3 September 27, 2011 Chapter 1 (in Greene & Krantzs book) 29. Suppose that f : D (0 , 1) C is holomorphic and that | f | 2. Derive an estimate for 3 z 3 f i 3 . Proof. Choose 0 < r < 2 / 3 so that D ( i 3 ,r ) D (0 , 1). By Cauchys esti- mate, 3 z 3 f i 3 2 * 3! r 3 . Letting r 2 / 3, we get 3 z 3 f i 3 2 * 3! ( 2 3 ) 3 = 81 2 . 2 32. Suppose that f is bounded and holomorphic on C \ { } . Prove that f is constant. [ Hint: Consider the function g ( z ) = z 2 f ( z ) and endeavor to apply Theorem 3.4.4] Proof. We consider the following auxiliary function, g ( z ) = z 2 f ( z ) , if z 6 = 0; , if z = 0 . Since f is holomorphic on C \{ } , so is g on C \{ } . However, at z = 0, we need to check whether g (0) exists. By the condition that f is bounded on C \ { } , we obtain g (0) = lim z g ( z )- g (0) z- = lim z z 2 f ( z ) z = lim z zf ( z ) = 0 . 1 Hence, we know that g is holomorphic on entire C . Choose M such that | f | M , on C \ { } , we have | g ( z ) | M | z | 2 , for all z C this implies, by Theorem 3.4.4, g is a complex polynomial of at most 2 degree....
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This note was uploaded on 02/21/2012 for the course MATH 4377 taught by Professor Staff during the Summer '08 term at University of Houston.

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HW3s - Key to Complex Analysis Homework 3 September 27,...

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