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MA530_AUG07

# MA530_AUG07 - 5 Let f z = ∞ ∑ a n z n,g z = ∞ ∑ b n...

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Math 530 Qualifying Examination August 15, 2007 – Profs. Drasin and Weitsman (Please write on one side of the paper. Show work and details.) 1. (a) Find two conjugate functions to the harmonic function u ( z ) = log | z | with domain {| z - 1 | < 1 } . (b) Show there is no conjugate function to u ( z ) = log | z | in the punctured disk { 0 < | z | < 1 } . 2. Find all conformal self–maps of { 0 < | z | < 1 } . 3. Let f be analytic inside the square D having vertices at ( - 2 , ± 2) and (2 , ± 2), and suppose f is continuous on ∂D . Suppose < e ( f ( z )) = 0 at precisely four points of ∂D . Show that f has at most two zeros in D . (This is only an upper bound; use the argument principle.) 4. Compute Z 0 log x x 2 + 4 dx by residues.
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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 ( ζ-z ) 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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