Abstract Nonsense

Crushing one theorem at a time

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

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October 7, 2012 - Posted by | Complex Analysis, Riemann Surfaces | , , , , , , ,

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