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homework108c6

# homework108c6 - | f j z | 2 on a disc centered around z(and...

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HOMEWORK 6 FOR MA108C: COMPLEX ANALYSIS DUE DATE: 4PM, MAY 19, 2009 (1) Determine all conformal maps of D (0 , 1) \{ 0 } → D (0 , 1) \{ 0 } . (2) Define the automorphism group Aut ( H ) = { f : H H | f holomorphic and bijective } . Here H = { z | = ( z ) > 0 } denotes the upper half plane. Now show that Aut ( H ) is isomorphic to PSL 2 ( R ) = { M = a b c d | a, b, c, d R , ad - bc = 1 } / where M N if and only if N = M or N = - M . Finally show that Aut ( D (0 , 1)) is also isomorpic to PSL 2 ( R ). (3) Let U C be a bounded domain. Let { f j } be a sequence of holomorphic functions U C such that there exists a uniform bound C as in ZZ | f j ( z ) | 2 dxdy < C. Show that the f j form a normal family. Hint: Use the Cauchy inequalities to show | f j ( z ) | 2 is less than the mean of | f
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Unformatted text preview: | f j ( z ) | 2 on a disc centered around z (and included in U ). Then show the f j are uniformly bounded on compact sets. (4) Consider f ( z ) = 1 2 ( z + 1 /z ). Show that f gives a conformal equivalence between D (0 , 1) and C ∪{∞}\ [-1 , 1]. 1 What happens with circles ∂D (0 ,r ) under this map? And what with the linesections ω [0 , 1) originating from the origin (for ω ∈ ∂D (0 , 1))? Hint: conic sections. 1 In a coincidence, this conformal map was used in a talk I went to today (after I made the problem set). 1...
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