Show that the only meromorphic functions on \(\CP^1\) are rational functions.
Solution
Compactness of \(\CP^1\) implies that \(f\) has only finitely many poles. Let \(a_1,\ldots,a_n\in\CC\) be the finite poles, and let \(P_j\) be the principal part of \(f\) at \(a_j\). Then \[ g(z)\da f(z)-\sum_{j=1}^nP_j(z) \] is entire on \(\CC\).
The function \(g\) is also meromorphic at \(\infty\), because both \(f\) and each rational principal part \(P_j\) are meromorphic on \(\CP^1\). Hence \(g(1/w)\) has at most a pole at \(w=0\), so its Laurent series has only finitely many negative powers: \[ g(1/w)=\sum_{k=-m}^{\infty}c_kw^k. \] Equivalently, \[ g(z)=\sum_{k=1}^m c_{-k}z^k+h(z), \] where \(h\) is entire and has a removable singularity at \(\infty\). Thus \(h\) extends holomorphically to the compact sphere, so it is constant. Therefore \(g\) is a polynomial, and \[ f(z)=g(z)+\sum_{j=1}^nP_j(z) \] is rational.