= Solution
Let the finite <poles> be $a_1,\ldots,a_N$. 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: $h=f-\sum_jP_j$ 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. Consequently
$$
f(z)=P(z)+c+\sum_{j=1}^N\sum_{k=1}^{m_j}\frac{c_{jk}}{(z-a_j)^k}.
$$
Combining denominators gives \b[a <rational function> $p/q$]. This proves that <meromorphic functions on the sphere are rational> without assuming rationality at the start.
Back to article page
