Home_6 - Homework 6 Stephen Taylor June 4, 2005 Page 55 :...

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Unformatted text preview: Homework 6 Stephen Taylor June 4, 2005 Page 55 : 9. If T ( z ) = az + b cz + d , find necessary and sufficient conditions that T () = where is the unit circle { z : | z | = 1 } . We note the general form of a function that maps the unit circle to the unit circle is f ( z ) = a- z 1- az where z and | a | < 1. It is obvious that any map that is a composition of rotation and inversions maps map the unit circle to itself. However, the above map also has a translation associated with it. We do not yet have the sufficient results to show that any function taking the above form is both necessary and sufficient to map the unit circle to itself. 12. Prove the following theorem: Theorem 1. If f : G C is analytic then f preserves angles at each point z of G where f ( z ) 6 = 0 . Define the angle between two curves at a point a to be the angle between the tangent vectors at a . Let ( t ) and ( t ) be two parametrically defined curves in G that intersect at z . After applying....
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Home_6 - Homework 6 Stephen Taylor June 4, 2005 Page 55 :...

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