## 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 .

**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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