Can a meromorphic function be written as ratio of holomorphic function?

Well, I want to know whether a meromorphic function can be written as ratio of two holomorphic function on $\mathbb{C}$ or on a Riemann surface. Thank you for help.

Solution 1:

a) On a compact Riemann surface $X$ holomorphic functions are constant so that the the quotients of holomorphic functions are just the constants too. In formulas: $$\mathcal O(X)=\mathbb C \; ,\quad \text{Frac} (\mathcal O(X))=\mathbb C$$ However a deep theorem (Riemann's Existence Theorem) assures us that there exists a non-constant meromorphic function on $X$ and that these meromorphic functions form a finitely generated field of transcendence degree one over $\mathbb C$ ($\;trdeg_ \mathbb C\mathcal M(X)=1$), so that the answer to your question is negative for compact Riemann surfaces: $$\text{Frac} (\mathcal O(X))=\mathbb C \subsetneq \mathcal M(X)$$

b) On a non-compact Riemann surface $Y$ however another difficult theorem, first proved only in 1948 by Behnke and Stein, says that indeed every meromorphic function is the quotient of two holomorphic functions . In formula: $$ \text{Frac} (\mathcal O(Y))= \mathcal M(Y) $$
The modern point of view is that this is an easy consequence of the difficult result that $Y$ is a Stein manifold, the analogue in complex-analytic geometry of an affine algebraic variety.

Bibliography As usual, Forster's Lectures on Riemann Surfaces is the best reference for these questions.