Unformatted text preview: f is an entire function such that for every compact set K ⊂ C , the inverse image f1 ( K ) is also compact. Prove that f ( C ) = C . 6. Suppose that f is a nonvanishing analytic function on the complex plane with the two points ± 1 deleted. Let γ 1 denote the curve given by z 1 ( t ) = 1+ e it where ≤ t ≤ 2 π and let γ 2 denote the curve given by z 2 ( t ) =1+ e it where 0 ≤ t ≤ 2 π . Suppose that 1 2 πi Z γ j f ( z ) f ( z ) dz is divisible by 2 for j = 1 , 2. First, explain why 1 2 πi Z γ f ( z ) f ( z ) dz must be divisible by 2 for any closed curve in C{± 1 } . Next, prove that f has an analytic square root on C{± 1 } , i.e., show that there is an analytic function g on C{± 1 } such that g 2 = f ....
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This document was uploaded on 01/25/2012.
 Spring '09
 Math

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