Show that every meromorphic function \( f \) on the Riemann sphere \( \hat{\mathbb{C}} \) is rational, i.e. of the form \( p \over q \), where \( p \) and \( q \) are coprime polynomials.
Solution
The rational functions are certainly meromorphic. Let $f$ be a meromorphic function in $\hat{\mathbb{C}}$. The set of poles of a meromorphic function is discrete , hence finite since $\hat{\mathbb{C}}$ is compact.
Let $z_1, \dots, z_n \in \mathbb{C}$ be the poles and $d_1, \dots, d_n$ their degrees. Then $p=f\cdot \prod (z-z_i)^{d_i}$ does not have any roots in $\mathbb{C}$ and has at most one pole at $\infty$. Hence $p$ is a polynomial and thus $f$ is rational.
The exercise can also be found in mathimatikoi.org
Solution
The rational functions are certainly meromorphic. Let $f$ be a meromorphic function in $\hat{\mathbb{C}}$. The set of poles of a meromorphic function is discrete , hence finite since $\hat{\mathbb{C}}$ is compact.
Let $z_1, \dots, z_n \in \mathbb{C}$ be the poles and $d_1, \dots, d_n$ their degrees. Then $p=f\cdot \prod (z-z_i)^{d_i}$ does not have any roots in $\mathbb{C}$ and has at most one pole at $\infty$. Hence $p$ is a polynomial and thus $f$ is rational.
The exercise can also be found in mathimatikoi.org
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