609S09Ex1s - Signature Math 609 March 5 2009 Printed Name...

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Signature Printed Name Math 609 Jerry L. Kazdan March 5, 2009 10:30- 11:50 Complex Analysis Exam I Directions This exam has two parts, Part A has 7 short answer problems (35 points) while Part B has 5 traditional problems (65 points). Closed book but you may use one 3 × 5 card with notes (on both sides). All contour integrals are assumed to be in the positive sense (counterclockwise). Short Answer Problems 7 problems [5 points each] (35 points total) For A1–A5 let f ( z ) be holomorphic for 0 < | z | < . What can you say about f ( z ) if you are told the following? Briefly justify your assertions. A1. | z 2 f ( z ) | < 5. A2. | f ( z ) | → ∞ as | z | → 0. A3. f ′′ ( z ) + f ( z ) = 0 for all real rational z negationslash = 0. A4. | f ( z ) | ≤ | z | + 1 and f ( 1 n ) = 0, n = 1 , 2 , . . . . A5. | f ( z ) | ≤ | f (3) | for | z 3 | < 2. A6. If n =0 a n z n represents the function sin z z 2 + 2 , whch of the following are true? (Why?) (A). Converges for z = 1. (B). Converges absolutely for z = 1. (C). Converges absolutely for z = 2. A7. Describe the singularities of ϕ ( z ) := 1 cos( z 5 ) sin 3 z at z = 0, z = π , and z = .
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Traditional Problems [13 points each] (65 points total) B1. Prove the Fundamental Theorem of Algebra: that any polynomial p ( z ) = z n + a n 1
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