#4 - Solution of Homework 1 Problem (6.6): Solution: The...

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Solution of Homework 1 Problem (6.6): Solution: The domain Ω is conformally equivalent to D (0 , 1) \{ 0 } . f ( z ) = 1 /z is a conformal map from D (0 , 1) \{ 0 } to Ω . Suppose g is a conformal mapping from Ω to Ω . Then f - 1 g f is a biholomorphc-self map of D (0 , 1) \{ 0 } . And Aut ( D (0 , 1) \{ 0 } ) = { e z | ψ R } So we know that g = e - z . Above all: Aut (Ω) = { e z | ϕ R } ¥ Problem (6.11): Solution: Observe that φ 1 φ - 1 2 : D D is a conformal map. According to Aut ( D (0 , 1)) = { e a - z 1 - ¯ az | a D (0 , 1) [0 , 2 π ) } So there exist a θ [0 , 2 π ) , ϕ Aut ( D (0 , 1)) such that φ 1 = ϕ φ 2 . Then we know φ 1 ( z ) = e a - φ 2 ( z ) 1 - ¯ 2 ( z ) ¥ Problem (6.17) Solution: Since φ ( P ) = P , φ ( P ) = 1 , from the hint in the book, we know that φ ( z ) = P + ( z - P ) + h ( z ) , where h ( z ) contains the higher order terms.

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This document was uploaded on 12/08/2010.

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#4 - Solution of Homework 1 Problem (6.6): Solution: The...

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