This preview shows page 1. Sign up to view the full content.
Unformatted text preview: n-1 | | a n | z | ≤ a R n + a R n-1 + · · · + 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...
View Full Document
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