Math 5283, HW Set 3, Fall 2006
Due: Wednesday, September 27, 2006
1. Page 72, Problem 3 in Ahlfors.
2. Page 83, Problem 4 in Ahlfors.
3. Page 96, Problem 1 in Ahlfors.
4. Page 97, Problem 6 in Ahlfors.
5. Show that under the stereographic projection the angle between any two
curves on the Riemann sphere (with their intersection not at the north pole)
equals the angle between the images of the curves in the complex plane.
6. (a) Assume that w=T z is a linear fractional transformation (also called a
M¨obius transformation) with a single finite fixed point z0. Prove that
1
w−z0
=1
z−z0
+h
for some h6= 0.
(b) Next assume that w=T z is a M¨obius transformation with two distinct
finite fixed points z1and z2. Prove that
w−z1
w−z2
=kz−z1
z−z2
for some k.
7. Show that z7→ 1/z and z7→ 1−zgenerate a finite subgroup in the group
of M¨obius transformations.
1