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Unformatted text preview: 4. ind a fractional linear transformation that maps the two circles { z :  z  = 1 } and { z :  z1  = 3 } onto some concentric circles. 1 5. a) Prove that every function of the form az + bm s n =1 c k zt k , (1) where b and t k are real, a 0 and c k > 0, maps the upper halfplane into itself and the lower halfplane into itself. b) Prove that every rational function that maps the upper halfplane into itself and the lower halfplane into itself has the form (1). 6. Let f be an analytic function in a neighborhood of 0, that satisFes the functional equation f ( z ) = qf ( q 2 z ) , f (0) = 1 , with some q,  q  1. ind explicitly the Taylor coecients of f at 0 and then determine the radius of convergence of its Taylor series. 2...
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This document was uploaded on 01/25/2012.
 Spring '09
 Math

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