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Unformatted text preview: n1   a n  z  ≤ a R n + a R n1 + · · · + a R ≤ na R < 1 for all  z  ≥ R . Thus, we have  P ( z )  < 2  a n  z n  whenever  z  ≥ R . (3) (50. 7)(40pts) Let g ( z ) = e f ( z ) , then g ( z ) is analytic on a closed bounded region R and not a constant on R . Hence the maximum and minimum modulus of g ( z ) must be taken on the boundary of R , so  g ( z )  = e u ( x,y ) has its minimum value on the boundary of R . Since z = e x is a monotonic increasing function of x ∈ R , u ( x,y ) has its minimum value on the boundary of R . 1...
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This note was uploaded on 09/01/2010 for the course MATH 185 taught by Professor Lim during the Fall '07 term at Berkeley.
 Fall '07
 Lim
 Math

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