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 . Prove that f is constant. 5. Suppose that f is a non-constant analytic function on an open set Ω containing
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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 ) | < 1, show that f (Ω) 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 Ω minus two points a 1 and a 2 in Ω. 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 Ω-{ a 1 ,a 2 } such that F ( z ) = f ( z )-R 1 z-a 1-R 2 z-a 2 ....
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