This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: ASSIGNMENT 4 Â· SOLUTIONS MAT 572 A Â· FALL 2007 Problem 1 ( cf. [1, Exercise III.3.10]) . Find all MÂ¨ obius transformations which map the disc D = { z   z  < 1 } onto itself. Solution. First, any such MÂ¨ obius transformation T must also take D c onto D c , since MÂ¨ obius transformations are invertible, and it follows that T takes the unit circle Î“ to itself. To see this, fix any z âˆˆ Î“, and choose sequences ( z n ) âŠ† D with z n â†’ z , and ( w n ) âŠ† D c with w n â†’ z . Then ( T ( z n )) âŠ† D and ( T ( w n )) âŠ† D c , so by continuity of T (and  Â·  ), we then have  T ( z )  =  lim n T ( z n )  = lim n  T ( z n )  â‰¤ 1 â‰¤ lim n  T ( w n )  =  lim n T ( w n )  =  T ( z )  , so  T ( z )  = 1. So suppose T is a MÂ¨ obius transformation which takes Î“ to Î“, 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 âˆž to a point not in the disc. Thus, if also c 6 = 0, we must have (1.1)  T (0)  = b d < 1 <  T ( âˆž )  = a c , and therefore  ad  >  bc  . If c = 0, then neither d nor a may be zero (otherwise...
View
Full Document
 Spring '09
 Transformations, Konrad Zuse, Conformal map, Unit disk, MÂ¨bius transformation

Click to edit the document details