MA530_JAN02

MA530_JAN02 - QUALIFYING EXAMINATION JANUARY 2002 MATH 530...

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QUALIFYING EXAMINATION JANUARY 2002 MATH 530 - Prof. Catlin (15 pts) 1. Let Ω = { z C ; | z | > 1 , Re z> 0 , Im z> 0 } . Find an explicit conformal map f of Ω onto the unit disk. You may represent f as a finite composition of maps. (15 pts) 2. Prove that the radius of convergence of the power series X n =0 a n z n is R = ± lim n →∞ sup | a n | 1 n ² - 1 . (20 pts) 3. Evaluate Z 0 log x ( x 2 +1) 2 dx . (15 pts) 4. Find the number of zeros of f ( z )=2
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