Self-mappings of the sphere IC.
(a) Show that the composition of two linear fractional transformations (22.50) is a linear fractional transformation.
(b) A fixed point of a transformation w = f ( z ) is a point zo such that f ( zo ) = z 0 .Prove that each linear fractional transformation, with the exception of the identity transformation w = z , has at most two fixed points.
(c) Show that (22.59) is the unique linear fractional transformation that maps the three different points z 1 , z2, and z 3 into the three different points w 1 , w2, and w3 , respectively. [Hint: Let S and T be two distinct linear fractional transformations which satisfy this property, and consider the transformation obtained by composing S with the inverse of T.]