A function is said to be analytic at if there exists a real number such that is analytic for and is finite (i.e. has a removable singularity at . is said to have a pole at if has a pole at . Suppose that is a meromorphic function on the extended plane , that is, is analytic at each point of except for poles.
(a) Show that if has a pole at , then there exists such that has no poles for .
(b) Show that the number of poles of is finite.
(c) By considering the Laurent expansions around the poles show that is in fact a rational function, i.e. of the form , where and are polynomials.
(d) Deduce that the only bijective meromorphic maps of onto itself are the Möbius maps.