4.II.13G

A function $f$ is said to be analytic at $\infty$ if there exists a real number $r>0$ such that $f$ is analytic for $|z|>r$ and $\lim _{z \rightarrow 0} f(1 / z)$ is finite (i.e. $f(1 / z)$ has a removable singularity at $z=0)$. $f$ is said to have a pole at $\infty$ if $f(1 / z)$ has a pole at $z=0$. Suppose that $f$ is a meromorphic function on the extended plane $\mathbb{C}_{\infty}$, that is, $f$ is analytic at each point of $\mathbb{C}_{\infty}$ except for poles.

(a) Show that if $f$ has a pole at $z=\infty$, then there exists $r>0$ such that $f(z)$ has no poles for $r<|z|<\infty$.

(b) Show that the number of poles of $f$ is finite.

(c) By considering the Laurent expansions around the poles show that $f$ is in fact a rational function, i.e. of the form $p / q$, where $p$ and $q$ are polynomials.

(d) Deduce that the only bijective meromorphic maps of $\mathbb{C}_{\infty}$ onto itself are the Möbius maps.

*Typos? Please submit corrections to this page on GitHub.*