# Abstract Nonsense

## Meromorphic Functions on Riemann Surfaces (Pt. III)

Point of Post: This is a continuation of this post.

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Meromorphic Functions as Functions to the Riemann Sphere

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As promised, we now discuss how meromorphic functions on Riemann surfaces are the same thing as holomorphic functions from that surface to the Riemann sphere.

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Let’s start by assuming that we have a meromorphic function $f:X\to\mathbb{C}$, and suppose that $P=\{x\in X:\text{ord}_p(f)<0\}$ are the poles of $f$. Define then the function $F_f:X\to\mathbb{C}_\infty$ by

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$\displaystyle F_f(x)=\begin{cases}f(x) & \mbox{if}\quad x\notin P\\ \infty &\mbox{if}\quad x\in P\end{cases}$

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The rigorous formulation of “meromorphic functions are just holomorphic functions to the Riemann sphere” is given by the following:

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Theorem: The assignment $f\mapsto F_f$ defines a bijection $\mathcal{M}(X)\to\text{Hol}(X,\mathbb{C}_\infty)-\{\infty\}$, where $\infty$ is the constant function $x\mapsto\infty$.

Proof: The first thing to verify is that $F_f$ really is holomorphic, but this is done precisely as the case for rational functions on the Riemann sphere. It’s obvious that this is a bijection, because if $F\in\text{Hol}(X,\mathbb{C}_\infty)$ is not identically infinity then we know that $P:= F^{-1}(\infty)$ is discrete and closed. Thus, we see that $F$ is holomorphic on $X-P$, an open set, $P$ is discrete, and evidently $\displaystyle \lim_{z\to p}|f(z)|=\infty$ for any $p\in P$. Thus, we see that the assignment $F\mapsto F\mid_{X-P}$ with $P=F^{-1}(\infty)$ is a mapping $\text{Hol}(X,\mathbb{C}_\infty)-\{\infty\}\to\mathcal{M}(X)$, which is easily seen to be the inverse of $f\mapsto F_f$. $\blacksquare$

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Because of this correspondence we shall not really differentiate between elements of $\mathcal{M}(X)$ and $\text{Hol}(X,\mathbb{C}_\infty)-\{\infty\}$.

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References:

[1] Varolin, Dror. Riemann Surfaces by Way of Complex Analytic Geometry. Providence, RI: American Mathematical Society, 2011. Print.

[2] Miranda, Rick. Algebraic Curves and Riemann Surfaces. Providence, RI: American Mathematical Society, 1995. Print.

[3] Forster, Otto. Lectures on Riemann Surfaces. New York: Springer-Verlag, 1981. Print.

[4] Conway, John B. Functions of One Complex Variable. New York: Springer-Verlag, 1978. Print.

[5] Gong, Sheng. Concise Complex Analysis. Singapore: World Scientific, 2001. Print

[6] Rudin, Walter. Real and Complex Analysis. New York: McGraw-Hill, 1966. Print