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

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Unformatted text preview: MATH 185: COMPLEX ANALYSIS FALL 2009/10 PROBLEM SET 4 1. Let C be a region. Let f = u + iv be analytic on . (a) If u + v is constant on for some , C , show that f is constant on . (b) If u 2 + v 2 is constant on , show that f is constant on . (c) If u = v 2 , show that f is constant on . (d) If u = v for some differentiable real function : R R , show that f is constant on . (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 . 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.of convergence....
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This note was uploaded on 02/17/2010 for the course MATH 185 taught by Professor Lim during the Fall '07 term at University of California, Berkeley.

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

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