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Unformatted text preview: Math 814 HW 3 October 16, 2007 p. 54: 9, 14, 18, 24, 25, 26 p.54, Exercise 9. If Tz = az + b cz + d , find necessary and sufficient conditions for T to preserve the unit circle. T preserves the unit circle iff  ae iθ + b  =  ce iθ + d  , for all θ ∈ [0 , 2 π ) . Squaring both sides and multiplying out, we get  a  2 +  b  2 + a ¯ be iθ + ¯ abe iθ =  c  2 +  d  2 + c ¯ de iθ + ¯ cde iθ . Comparing coefficients, we get two equations:  a  2 +  b  2 =  c  2 +  d  2 , a ¯ b = c ¯ d. Dividing the first by  d  2 and using the second, we get  c  2  b  2 +  b  2  d  2 =  c  2  d  2 + 1 , which can be written  c  2  b  2  b  2 =  c  2  b  2  d  2 . If the numerators are zero, this means  c  =  b  , hence  a  =  d  . If the numerators are nonzero, we have  b  =  d  , hence  a  =  c  . 1 In the first case, there are u,v with  u  =  v  = 1 such that c = ub, d = va. Then we have a ¯ b = c ¯ d = ub ¯ v ¯ a, so ub ¯ b = va ¯ a . This last number, call it λ , also has  λ  = 1 , and we have a b c d = 1 0 λ a b ¯ b ¯ a . (1) In the second case, there are u,v with  u  =  v  = 1 such that d = ub, c = va. This time, we have a ¯ b = c ¯ d = va ¯ u ¯ b, so v ¯ u = 1 . But ¯ u = u 1 , so v = u , and we have ad bc = aub bua = 0 , which is illegal for a M¨obius transformation. So the second case does not occur, and all such transformations T are of the form (1). p.54, Exercise 14. Let G be the region between two circles inside one another, tangent at the point a . Map G conformally to the open unit disk....
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This note was uploaded on 10/19/2009 for the course MATH 814 taught by Professor Cong during the Three '09 term at University of Adelaide.
 Three '09
 Cong
 Math, Unit Circle

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