= Meromorphic functions on the sphere are rational
<Poles> are isolated, including in the coordinate $w=1/z$ around infinity. <Compactness> of the <Riemann sphere> therefore leaves only finitely many <poles>. Subtract the finite principal parts at finite <poles> and the <polynomial> principal part at infinity. The remainder is entire with a finite limit at infinity, hence bounded and constant by <Liouville's theorem>. The original function is a sum of a <polynomial> and finitely many partial fractions, so is rational.
Back to article page
