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ComplexAnalysis2006Jan - Qualifying Exam Complex Analysis 1...

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Unformatted text preview: / Qualifying Exam, Complex Analysis, January 28, 2006 1. Find a conformal map from the half—disk {z 2 [z — 1] < 1, Imz > 0} onto the upper half-plane {Imw > O}. 2. Find znel/‘dz, where n is an integer. [z[=1 , 3." Let f be a holomorphic function on U \ {0}, where U is the open unit disk, such that f(1/2) = 2 and the function 9(2) = E |f(Z)|2 is holomorphic on U \ {0}. Find f. 4. Let f be a holomorphic function in U \ {0}, Where U is the open unit disk, which satisfies |f(z)l S ~10glzl, Vz E U\{0}~ Prove that f 2 0. ...
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