Everyone: This is my first post. Sorry if I break some protocol.
I do know some complex analysis and how to tell when a function from $\mathbb C \rightarrow\mathbb C$ . But I am confused when I hear of analytic or meromorphic functions from (sorry, don't know the notation) Riemann Sphere to itself. I think this has something to see with Algebraic Geometry and Varieties, of which I know very little. Would someone please expand on how one determines if/when a function from the Riemann Sphere to itself is meromorphic or analytic? I have seen some Differential Geometry in which we compose functions with chart maps to determine if a function (from a real manifold to another real manifold) is differentiable, or $C^k$. Is that what we do for complex functions, and, if so, are there some theorems to avoid doing the chart composition? Thanks for any help.