math185-hw10

math185-hw10 - C be an open set. Let f be continuous on ....

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MATH 185: COMPLEX ANALYSIS FALL 2007/08 PROBLEM SET 10 For a C , 0 < r < R , we write A ( a ; r,R ) := { z C | r < | z - a | < R } . We also write C × = A (0; r, ) and D * ( a,r ) = A ( a ;0 ,r ). The number of zeroes, counted with multiplicity, of f in Ω is denoted Z f (Ω). You may use without proof any results that had been proved in the lectures. 1. Let the Laurent expansion of cot( πz ) on A (0;1 , 2) be cot( πz ) = X n = -∞ a n z n . Compute a n for n < 0. 2. (a) For n = 0 , 1 , 2 ,..., compute 1 2 πi Z Γ n dz z 3 sin z where Γ n is the circle ∂D (0 ,r n ) traversed once counter-clockwise and r n = ( n + 1 2 ) π . (b) Evaluate 1 2 πi Z Γ z 11 12 z 12 - 4 z 9 + 2 z 6 - 4 z 3 + 1 dz where Γ = ∂D (0 , 1) traversed once counter-clockwise. 3. Determine Z f i i ) for i = a,b,c . (a) f a ( z ) = z 87 + 36 z 57 + 71 z 4 + z 3 - z + 1 and Ω a = D (0 , 1). (b) f b ( z ) = z 87 + 36 z 57 + 71 z 4 + z 3 - z + 1 and Ω b = D (0 , 2). (c) f c ( z ) = 2 z 5 - 6 z 2 + z + 1 and Ω c = A (0;1 , 2). 4. (a) Prove the following form of Morera’s Theorem: Let Ω
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Unformatted text preview: C be an open set. Let f be continuous on . Show that if Z R f = 0 for every R , boundary of a rectangle R C (note that may not be simply connected and R may not be contained in ), then f has an antiderivative on , ie. there exists an analytic F on such that F = f . (b) Does the following function have an antiderivative on A (0;4 , )? z ( z-1)( z-2)( z-3) (c) Does the following function have an antiderivative on A (0;4 , )? z 2 ( z-1)( z-2)( z-3) Date : December 6, 2007 (Version 1.0); posted: December 6, 2007; due: December 10, 2007. 1...
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