Past exam of the mathematics course of the University of Cambridge / 2002 / ib / Paper 4 / 13G (Further Analysis)

Use the local coordinate $w=1/z$ around infinity. A pole at $w=0$ is isolated: for some $\epsilon >0$, $g(w)=f(1/w)$ is holomorphic at every point $0<|w|<\epsilon$. Taking $r=1/\epsilon$, the change of variable maps this punctured disk to $r<|z|<\infty$. Therefore there are no finite poles outside that disk. This is a property of the local meromorphic chart at infinity, not an assertion that infinity itself is regular.

At every regular point of the Riemann sphere, there is a neighborhood containing no poles; at each pole, there is a neighborhood containing no other poles. These neighborhoods cover the sphere. Its compactness gives a finite subcover, each member containing at most one pole. Hence the total number of poles, including a possible pole at infinity, is finite. Equivalently, an infinite pole set would have an accumulation point on the compact sphere, contrary to local regularity or isolation there.

Let the finite poles be $a_1,\dots ,a_N$. Their Laurent series have finite principal parts $P_j(z)={\sum }_{k=1}^{m_j}{c}_{jk}(z-a_j{)}^{-k}$. Subtracting them removes all finite singularities: $h=f-{\sum }_j{P}_j$ is entire. Each $P_j$ tends to zero at infinity, so $h$ still has at most a finite-order pole at infinity.

The Laurent expansion of $h(1/w)$ near $w=0$ has only finitely many negative powers. Thus $h(z)=P(z)+c+O(z^{-1})$ for large $|z|$, with $P$ a polynomial. The entire function $h-P$ is bounded outside a large disk and, by continuity, on that disk. Liouville's theorem makes it constant. ConsequentlyCombining denominators gives a rational function $p/q$. This proves that meromorphic functions on the sphere are rational without assuming rationality at the start.

Write the nonconstant bijective map as $p/q$ with coprime polynomials and set $d=\max (\mathrm{deg}p,\mathrm{deg}q)$. For a finite value $w$, its finite preimages solve $p(z)-wq(z)=0$. Except for a possible leading-coefficient cancellation value, this polynomial has degree $d$.

Multiple roots can occur only at zeros of $p^{\prime }q-p{q}^{\prime }$, since at a root $q\ne 0$ by coprimality. This derivative numerator is a nonzero polynomial for a nonconstant rational map. There are therefore only finitely many finite critical values to exclude. Choose $w$ outside them and outside the leading cancellation value. The fundamental theorem of algebra then gives exactly $d$ distinct finite preimages. Bijectivity forces $d=1$.

Hencewhich is a Möbius transformation. Conversely, the inverse is $(d_{0}w-b)/(a-cw)$, interpreted at its pole and at infinity, proving bijectivity on the Riemann sphere. The argument also establishes the generic-preimage interpretation of the degree of a rational map of the Riemann sphere.