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 za 1R 2 za 2 ....
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 Spring '09
 Math, unit disc, onetoone analytic map, nonconstant analytic function

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