Meromorphic Functions on Riemann Surfaces (Pt. III)
Point of Post: This is a continuation of this post.
Meromorphic Functions as Functions to the Riemann Sphere
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.
Let’s start by assuming that we have a meromorphic function , and suppose that are the poles of . Define then the function by
The rigorous formulation of “meromorphic functions are just holomorphic functions to the Riemann sphere” is given by the following:
Theorem: The assignment defines a bijection , where is the constant function .
Proof: The first thing to verify is that 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 is not identically infinity then we know that is discrete and closed. Thus, we see that is holomorphic on , an open set, is discrete, and evidently for any . Thus, we see that the assignment with is a mapping , which is easily seen to be the inverse of .
Because of this correspondence we shall not really differentiate between elements of and .
 Varolin, Dror. Riemann Surfaces by Way of Complex Analytic Geometry. Providence, RI: American Mathematical Society, 2011. Print.
 Miranda, Rick. Algebraic Curves and Riemann Surfaces. Providence, RI: American Mathematical Society, 1995. Print.
 Forster, Otto. Lectures on Riemann Surfaces. New York: Springer-Verlag, 1981. Print.
 Conway, John B. Functions of One Complex Variable. New York: Springer-Verlag, 1978. Print.
 Gong, Sheng. Concise Complex Analysis. Singapore: World Scientific, 2001. Print
 Rudin, Walter. Real and Complex Analysis. New York: McGraw-Hill, 1966. Print
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