03 - ASSIGNMENT 3 SOLUTIONS MAT 572 A FALL 2007 Problem 1 (...

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Unformatted text preview: ASSIGNMENT 3 SOLUTIONS MAT 572 A FALL 2007 Problem 1 ( cf. [1, Exercise III.2.13]) . Let G = C \ { z | z = Re z } , and let n be a positive integer. Find all analytic functions f : G C such that z = ( f ( z )) n for all z G . Solution. For each positive integer n , let f n be the principal branch of the n th root function: f n ( z ) = exp ( 1 n log z ) , where log is the principal branch of the logarithm. Then f is analytic, and for any z G , ( f n ( z )) n = (exp( 1 n log z )) n = exp( n 1 n log z ) = exp(log z ) = z. (The familiar identity (exp w ) n = exp( nw ) is established using (exp a )(exp b ) = exp( a + b ) and induction.) I claim that the functions in question are precisely those of the form f ( z ) = n f n ( z ) , where n is an n th root of unity. Proof. Any such function is clearly analytic, and for any z G we have ( f ( z )) n = n n ( f n ( z )) n = z....
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03 - ASSIGNMENT 3 SOLUTIONS MAT 572 A FALL 2007 Problem 1 (...

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