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 Post subject: Meromorphic Functions On The Riemann Sphere Posted: Thu Feb 04, 2016 7:48 pm
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Show that every meromorphic function $f$ on the Riemann sphere $\tilde{\mathbb{C}}$ is rational, i.e. of the form $p \over q$, where $p$ and $q$ are coprime polynomials.

Top   Post subject: Re: Meromorphic Functions On The Riemann Sphere Posted: Fri Feb 05, 2016 10:19 pm
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Tsakanikas Nickos wrote:
Show that every meromorphic function $f$ on the Riemann sphere $\tilde{\mathbb{C}}$ is rational, i.e. of the form $p \over q$, where $p$ and $q$ are coprime polynomials.

Hello Nickos,

this is a standard result in complex analysis. Here is a 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.

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