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math185f09-hw4 - MATH 185 COMPLEX ANALYSIS FALL 2009/10...

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MATH 185: COMPLEX ANALYSIS FALL 2009/10 PROBLEM SET 4 1. Let Ω C be a region. Let f = u + iv be analytic on Ω. (a) If αu + βv is constant on Ω for some α, β C × , show that f is constant on Ω. (b) If u 2 + v 2 is constant on Ω, show that f is constant on Ω. (c) If u = v 2 , show that f is constant on Ω. (d) If u = ϕ v for some differentiable real function ϕ : R R , show that f is constant on Ω. (e) Determine all f for which g = u 2 + iv 2 (ie. g ( z ) := [ u ( x, y )] 2 + i [ v ( x, y )] 2 ) is also analytic on Ω. 2. Derive the following power series expansions and show that they must converge uniformly and absolutely in the respective given sets. (a) For all z C , e z = e + e X n =1 1 n ! ( z - 1) n . (b) For all z D (1 , 1), 1 z = X n =0 ( - 1) n ( z - 1) n . (c) For all z D ( - 1 , 1), 1 z 2 = 1 + X n =1 ( n + 1)( z + 1) n . 3. For each of the following functions, find a power series expansion about 0 and state its radius of convergence. e ( z ) = exp 1 1 - z , f ( z ) = sin 1 1 - z , g ( z ) = 1 1 - z - z 2 , h ( z ) = X n =0 z n 1 - z n .
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