Unformatted text preview: 5. Let f ( z ) = ∞ ∑ a n z n ,g ( z ) = ∞ ∑ b n z n be analytic in U : { z  < 1 } and continuous on ∂U , oriented counterclockwise. Prove that 1 2 πi Z ∂U f ( ζ ) g ( z/ζ ) dζ ζ = ∞ X a n b n z n ( z ∈ U ) (You must justify all steps in your argument.) 6. Let R be the boundary of the rectangle with vertices (3 , ± 2) , (+3 , ± 2), and let F ( z ) = Z R ze 2  ζ  2 dζ ( ζz ) 2 . (a) Prove that F is analytic inside R . (b) Try to obtain as good an upper bound for  F 00 (0)  as possible....
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 Spring '09
 Math, Derivative, Holomorphic function, harmonic function, Harmonic conjugate, log z

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