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Unformatted text preview: z ) g ( z ) is real for all z ∈ D . Prove that f = cg where c is a real constant, or g ≡ . 6. Let f be an analytic function in the region { z : 0 <  z  < 1 } , and suppose that it satisﬁes  f ( z )  ≤ A + B  z α for some positive numbers A,B and α . Prove that f is meromorphic in the unit disc (that is 0 is either a pole or a removable singularity of f ). 7. Let f = ∞ X n =1 a n z n be a conformal map from the unit disc onto the square { z : < z  < 1 , = z  < 1 } . Prove that a n = 0 for all n 6 = 4 k + 1 , k = 0 , 1 , 2 , 1 .... 1 8. Evaluate the integral Z ∞ log x 1 + x 2 . Hint: integrate log z/ (1 + z 2 ) over the boundary of the region { z : ± <  z  < R, = z > } . 2...
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 Spring '09
 Math, Holomorphic function, 4k, unit disc, A. Eremenko

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