Question

Problem 2. (18 points) (a) Find a fractional linear transformation that maps the right half-plane to the unit disk such that the origin is mapped to -1. (b) A fixed point of a transformation T is one where T(2) = 2. Let T be a fractional linear transformation. Assume T is not the identity map. Show T has a most two fixed points. (c) Let S be a circle and 21 a point not on the circle. Show that there is exactly one point Z2 such that and 22 are symmetric with respect to S. (Hint: start by proving this for S a line.)

Answer 1

C710 Scht (LLLLLLLLLLLLLLL Problema Let we azob be the fonctional 0 liner lang hos version that maps the right half plass Re(2) 70 to the comit dost 12/21. Then we can write El 77 dy! s which c70. Otherwise priats at infinity would Correspond in the two planes. Since transforms lives inte Ciodes. therefore points symmetrical with respect lines in 2-plane are toons hoored into the inverse points of Irel=1. Here the prints 2 and 3 are Sysmitirere about Rel2)=0 will Correspond to w=0 and wr=0 which are the inverse points of /w/=/ But w=0 x = -2 and w=00 2= - d. This We have a bad and de C

Usich in O, wo set W= (2-0) -- 772 The Point 2-0 on the boundary of Re12) 20 losses poonds to w=-) given in the question) -- ) COC Honce , we get usich in But weo gives Agais 7-8 = 0. W=0. But w=0 is interior point of /w/= 1. Hence 2=4 must be interior point of the right hatt plane e Re (670. d e Thus W = F (2-4), Reed) 70 is the required fraction lines transformation.. catd 6 Let T(z) = azeb be the fractional linear loans formation such that T (2) #2 il TC2) is not identically equal to 2. then for T(2) = 2 f 9776 = 7 card