04 - ASSIGNMENT 4 SOLUTIONS MAT 572 A FALL 2007 Problem 1 (...

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Unformatted text preview: ASSIGNMENT 4 SOLUTIONS MAT 572 A FALL 2007 Problem 1 ( cf. [1, Exercise III.3.10]) . Find all M obius transformations which map the disc D = { z | | z | < 1 } onto itself. Solution. First, any such M obius transformation T must also take D c onto D c , since M obius transformations are invertible, and it follows that T takes the unit circle to itself. To see this, fix any z , and choose sequences ( z n ) D with z n z , and ( w n ) D c with w n z . Then ( T ( z n )) D and ( T ( w n )) D c , so by continuity of T (and | | ), we then have | T ( z ) | = | lim n T ( z n ) | = lim n | T ( z n ) | 1 lim n | T ( w n ) | = | lim n T ( w n ) | = | T ( z ) | , so | T ( z ) | = 1. So suppose T is a M obius transformation which takes to , and write T ( z ) = az + b cz + d . If T maps D onto D , then it must take 0 to a point in the disc, so d 6 = 0; and T must also take to a point not in the disc. Thus, if also c 6 = 0, we must have (1.1) | T (0) | = b d < 1 < | T ( ) | = a c , and therefore | ad | > | bc | . If c = 0, then neither d nor a may be zero (otherwise...
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04 - ASSIGNMENT 4 SOLUTIONS MAT 572 A FALL 2007 Problem 1 (...

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