MA530_AUG04

MA530_AUG04 - the closed unit disc. If | f ( z ) | 1...

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QUALIFYING EXAMINATION AUGUST 2004 MATH 530 - Prof. Bell Each problem is worth 20 points Notation: D r ( a )= { z : | z - a | <r } 1. How many zeros does the function f ( z )= e z + z 11 + 2004 have in the annulus { z C :1 < | z | < 2 } ? Explain. 2. State the Schwarz Lemma. Use it to prove that if f is a one-to-one analytic map of the unit disc onto itself such that f (0) = 0, then f ( z )= λz for some constant λ with | λ | =1. 3. Suppose that f ( z ) is analytic in a neighborhood of the origin and X n =1 f ( n ) (0) converges. Prove that f ( z ) extends to be an entire function. 4. Let f be an entire function such that | f ( z ) |≤ 2004 + p | z | for all z C .P r o v e that f is constant.
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Unformatted text preview: the closed unit disc. If | f ( z ) | 1 whenever | z | = 1 and there exists a point z D 1 (0) such that | f ( z ) | &lt; 1, show that f () contains D 1 (0). 6. Evaluate Z sin x x dx by complex variable methods. 7. Suppose that f ( z ) is analytic on a simply connected domain minus two points a 1 and a 2 in . If the residue of f at a 1 is R 1 and the residue of f at a 2 is R 2 , prove that there is an analytic function F ( z ) on -{ a 1 ,a 2 } such that F ( z ) = f ( z )-R 1 z-a 1-R 2 z-a 2 ....
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