# Abstract Nonsense

## Meromorphic Functions on the Riemann Sphere (Pt. II)

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

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Now, I want to use this nice characterization of $\mathcal{M}(\mathbb{C}_\infty)$ to make an interesting observation. In particular, take $r(z)\in\mathbb{C}(z)$, let’s say

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$\displaystyle r(z)=C\;\frac{\displaystyle \prod_{i=1}^{n}(z-\lambda_i)^{e_i}}{\displaystyle \prod_{j=1}^{m}(z-p_j)^{g_j}}$

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with $\lambda_i,p_j\in\mathbb{C}$ and $e_i,g_j\in\mathbb{N}$. Let us calculate $\text{ord}_p(r)$ for all $p\in\mathbb{C}_\infty$. Ok, now by definition if $p\in\mathbb{C}$ is not a zero or a pole of $r$ then $\text{ord}_p(r)=0$, and by almost definition we see that $\text{ord}_{\lambda_i}(r)=e_i$ and $\text{ord}_{p_j}(r)=g_j$. Thus, the only point left to check is $\infty$. Now, we merely check that with the standard chart $(\mathbb{C}_\infty-\{0\})$ at $\infty$ we get

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\begin{aligned}\displaystyle (r\circ\varphi^{-1})(z) &=C\; \frac{\displaystyle \prod_{i=1}^{n}\left(\frac{1}{z}-\lambda_i\right)^{e_i}}{\displaystyle \prod_{j=1}^{m}\left(\frac{1}{z}-p_j\right)^{g_j}}\\ &=C\;z^{k}\; \frac{\displaystyle \prod_{i=1}^{n}(1-z\lambda_i)^{e_i}}{\displaystyle \prod_{j=1}^{m}(1-zp_j)^{g_j}}\end{aligned}

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where $k$ is equal to

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$\displaystyle \sum_{j=1}^{m}g_j-\sum_{i=1}^{n}e_i$

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Now, it’s evident then that $r\circ\varphi^{-1}$ has a singuliarity of order

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$\displaystyle \sum_{j=1}^{m}g_j-\sum_{i=1}^{n}e_i$

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at $0$ and thus $r$ has a singularity of order

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$\displaystyle \sum_{j=1}^{m}g_j-\sum_{i=1}^{n}e_i$

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at $\infty$. Thus, we see that

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\displaystyle \begin{aligned}\sum_{p\in \mathbb{C}_\infty}\text{ord}_p(r) &= \sum_{i=1}^{n}\text{ord}_{\lambda_i}(r)+\sum_{j=1}^{m}\text{ord}_{p_j}(r)+\text{ord}_{\infty}(r)\\ &= \sum_{i=1}^{n}e_i+\sum_{j=1}^{m}(-g_j)+\left(\sum_{j=1}^{m}g_j-\sum_{i=1}^{n}e_i\right)\\ &=0\end{aligned}

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Thus, since every meromorphic function on $\mathbb{C}_\infty$ is rational the above calculation proves the following:

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Theorem: Let $f$ be a meromorphic function on $\mathbb{C}_\infty$. Then,

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$\displaystyle \sum_{p\in \mathbb{C}_\infty}\text{ord}_p(f)=0$

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This tells us that, if we count right (i.e. with multiplicity), that meromorphic functions on the Riemann sphere have the same number of zeros and poles. This gives us a definitive proof that Weierstrass’s theorem can’t hold for $\mathbb{C}_\infty$–we can’t pick meromorphic functions with arbitrary singularities, the number of poles and zeros must be equal. We shall see that this theorem will hold true more generally for any compact Riemann surface.

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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