extra_7 - Extra Problems Sheet 7 Stephen Taylor 1 Prove...

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Extra Problems Sheet 7 Stephen Taylor May 15, 2005 1. Prove that if z 1 , z 2 , z 3 are distinct points and w 1 , w 2 , w 3 are distinct points, then the M¨ obius transformation T satisfying T ( z 1 ) = w 1 , T ( z 2 ) = w 2 , T ( z 3 ) = w 3 is unique. A M¨ obius transformation given is given by T ( z ) = az + b cz + d . Assuming a is not zero, we find the general form can be reduced to T ( z ) = a 1 z + b 1 c 1 z + d 1 = a - 1 1 a - 1 1 · a 1 z + b 1 c 1 z + d 1 = z + b 1 a - 1 1 c 1 a - 1 1 z + a 1 d 1 z + a bz + c Substituting the three transformed points into this equation, we obtain the system three equations w i = T ( z i ) = z i + a bz i + c where i = { 1 , 2 , 3 } So we have a system of three linear equations with three unknowns, all of which must be independent since all points are distinct. Therefore, provided a solution exists, it must be unique. 2. Find all M¨ obius transformations that map the unit disk onto the right half-plane. Let T ( z ) be a mapping that maps from the right half plane to the unit disk.
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