Abstract Nonsense

Crushing one theorem at a time

Meromorphic Functions on Riemann Surfaces (Pt. II)


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

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Let X be a Riemann surface and suppose that f:X\to\mathbb{C} is meromorphic at p. We know then that for each chart (U,\varphi) at p one has that f\circ\varphi^{-1}:\varphi(U)\to\mathbb{C} is meromorphic at p. Thus, from basic complex analysis we can form the Laurent series at p relative to \varphi \displaystyle \sum_{n\in\mathbb{Z}}a_n(z-\varphi(p))^n, which is defined on a punctured disc D-\{p\}\subseteq\varphi(U). We then define the order of f relative \varphi to be \inf\{n:a_n\ne 0\}. As one would hope, despite the fact that the Laurent series of f varies greatly depending on which chart one takes it relative to, the order of f does not!

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Theorem: Let X be a Riemann surface and f:X\to\mathbb{C} be meromorphic at p. Then, the order of f is independent of chart.

Proof: Let (U,\varphi) and (V,\psi) be any two charts at p. We first consider the Laurent series of f relative \varphi, so that on some disc D_{r_1}(\varphi(p))-\{\varphi(p)\}\subseteq U we have that

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\displaystyle (f\circ\varphi^{-1})(z)=\sum_{n=m}^{\infty}a_n (z-\varphi(p))^n

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where m\in\mathbb{Z} is the order of f relative \varphi (note that this order is finite by the assumption that f\circ\varphi^{-1} is meromorphic at \varphi(p)). Now, we consider a disc D_r(\varphi(p))-\{\varphi(p)\} contained in \varphi(U)\cap\psi(U). We see then that on this disc

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f\circ\psi^{-1}=(f\circ\varphi^{-1})\circ\underbrace{(\varphi\circ\psi^{-1})}_{T}

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Now, since T is biholomorphic on a neighborhood of \psi(p) we have that we can extend T into it’s Taylor series

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\displaystyle T(z)=\sum_{k=0}^{\infty}b_n(z-\psi(p))^k

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Now, note that b_0=T(\psi(p))=\varphi(p), and b_1=T'(\psi(p))\ne 0. This second fact is because if S=T^{-1} [remember that T is biholomorphic!) then S(T(w))=w for all w in some neighborhood of \psi(p), and so S'(T(w))T'(w)=1, so T'(w)\ne 0. Thus, we see that

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\displaystyle \begin{aligned}(f\circ\psi^{-1})(z) &= (f\circ \varphi^{-1})\circ T\\ &= \sum_{n=m}^{\infty}a_n(T(z)-\varphi(p))^n\\ &= \sum_{n=m}^{\infty}\left(\sum_{k=0}^{\infty}(z-\psi(p))^k-\varphi(p)\right)\\ &= \sum_{n=m}^{\infty}\left(\sum_{k=1}^{\infty}(z-\psi(p))^k\right)^n\\ &= a_mb_1(z-\psi(p))^m+\sum_{n=m+1}^{\infty}c_n(z-\psi(p))^n\end{aligned}

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where c_n are some coefficients in \mathbb{C}. Now, by the uniqueness of the Laurent series on a punctured disc, this must be the Laurent series of f at p relative \psi. But, note that a_m and b_1 are both nonzero so that a_mb_1 is non-zero. It follows easily that the order of f at p relative \psi is also m. Since (U,\varphi) and (V,\psi) were arbitrary the conclusion follows. \blacksquare

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Thus, if f:X\to\mathbb{C} is a meromorphic function at p, we may unabashedly define the order of f at p to be its order at p relatively any chart at p. We denote the order of f at p by \text{ord}_p(f) or sometimes (although rarely, if at all) by v_p(f) (this is supposed to be suggestive to anyone familiar with some parts of algebra!).

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Let’s give an example of a meromorphic function (finally, right?). Consider the extended complex plane \mathbb{C}_\infty and a rational function \displaystyle h(z)=\frac{f(z)}{g(z)}\in\mathbb{C}(z). It’s trivial that h is meromorphic at each point of \mathbb{C}, and checking the h is meromorphic at \infty is equivalent to checking that \displaystyle h\left(\frac{1}{z}\right) is meromorphic at 0–a classical exercise in basic analysis. This should be expected considering our intuition that meromorphic functions are just holomorphic functions to the Riemann sphere, since we have already shown that h is a holomorphic function \mathbb{C}_\infty\to\mathbb{C}_\infty.

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Now, having the notion of order allows us to classify singularities. Namely, let f:X\to\mathbb{C} be meromorphic at p, then we say that p is a removable singularity if \text{ord}_p(f)\geqslant 0 and that p is a pole if \text{ord}_p(f)<0. Note that the name ‘removable singularity’ makes sense since evidently if \text{ord}_p(f)\geqslant 0 then |f| is bounded in a neighborhood of p and so Riemann’s Removable Singularity Theorem allows us to holomorphically extend f to p. In particular, we see that f is holomorphic at p if and only if \text{ord}_p(f)\geqslant 0. Along with this, it’s easy to see that f has a pole at p if and only if \displaystyle \lim_{z\to p}|f(z)|=\infty.

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The other thing to notice about this definition is that the poles of f are necessarily discrete in X. Indeed, if f is meromorphic at p then we know that f\circ\varphi^{-1} is meromorphic at \varphi(p) for any chart (U,\varphi) at p. Thus, we know there exists a neighborhood p\in O\subseteq U such that f\circ\varphi^{-1} is holomorphic on O-\{p\} and thus we see that \varphi^{-1}(O)-\{p\} is a neighborhood of p which intersects the poles of f only at p–and thus discreteness follows. This is comforting since we always had the discreteness of poles for meromorphic functions on subsets of \mathbb{C}. In fact, not only is this comforting, but it should be expected. For, if meromorphic f:X\to\mathbb{C} are really just holomorphic F:X\to\mathbb{C}_\infty then the poles of f should be F^{-1}(\infty) and fibers are discrete!

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This above analysis gives us an equivalent way of defining holomorphic functions on a Riemann surface X–by specifying poles. Namely, suppose that X is a Riemann surface and W\subseteq X is open. Then, we say that a function f:W\to\mathbb{C} is a meromorphic function on X if it is a holomorphic function on W, if X-W is discrete, and if \displaystyle \lim_{z\to p}|f(z)|=\infty for each p\in X-W. In essence, this says that a meromorphic function on X is a holomorphic function with a discrete set of poles.

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Now, we leave it to the reader to verify the following simple properties of the order:

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Theorem: Let f,g:X\to\mathbb{C} be meromorphic at p, then:

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\begin{aligned}&\mathbf{(1)}\quad \text{ord}_p(fg)=\text{ord}_p(f)+\text{ord}_p(g)\\ &\mathbf{(2)}\quad \text{ord}_p(f\pm g)\geqslant \inf\{\text{ord}_p(f),\text{ord}_p(g)\}\end{aligned}

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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, Uncategorized | , , , , , , ,

2 Comments »

  1. […] Meromorphic Functions on Riemann Surfaces (Pt. III) Point of Post: This is a continuation of this post. […]

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  2. […] Meromorphic Functions on Riemann Surfaces (Pt. III) Point of Post: This is a continuation of this post. […]

    Pingback by Meromorphic Functions on Riemann Surfaces (Pt. III) « Abstract Nonsense | October 7, 2012 | Reply


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