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MA530_JAN06 - ∑ ∞ n =0 a n z n be analytic in the unit...

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QUALIFYING EXAMINATION JANUARY 2006 MATH 530 - Prof. Weitsman 1. (20) Find the radii of convergence of the following. i ) X n =0 1 (1 + 2 i ) n z n ii ) X n =0 z n 2 n ! 2. (20) Show that Z -∞ x sin(2 x ) x 4 + 16 dx = πe - 2 2 sin(2 2) 4 . Show all work. 3. (20) Expand f ( z ) = 1 z ( z + 2) as a Laurent series i) for 0 < | z | < 2, ii) for | z - 3 | < 3. 4. (20) Prove that all roots of the equation z 6 - 5 z 2 +10 = 0 lie in the annulus { z : 1 < | z | < 2 } . 5. (20) Find all linear fractional transformations T ( z ) which map the upper half plane H + = { z : = mz > 0 } onto the unit disk U = { z : | z | < 1 } such that T ( i ) = 0. 6. (20) Let f ( z ) =
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Unformatted text preview: ∑ ∞ n =0 a n z n be analytic in the unit disk U = { z : | z | < 1 } with f (0) = 0 and f (0) = 1. Prove that if ∑ ∞ n =2 n | a n | ≤ 1, then f is one-to-one in U . 7. (20) Suppose that f ( z ) is analytic in the disk { z : | z | < R } ( R > 1) except for a simple pole at z = 1 with residue -1. If its Taylor expansion in the unit disk is f ( z ) = ∞ X n =0 a n z n , prove that a n → 1 as n → ∞ . 1...
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