TITLE: Nomenclature in complex analysis
QUESTION [7 upvotes]: I am having a little confusion on the naming of functions in complex analysis.
If $f$ is a holomorphic function on the complex plane and its domain is the complex plane then it's called an entire function.
If $g$ is holomorphic everywhere on the complex plane apart from its poles and its domain is the complex plane then it's a meromorphic function.
But...
If $g$'s domain is extended to the Riemann sphere (and it doesn't have a essential singularity at infinity) is it a "meromorphic function with domain of Riemann sphere" or "holomorphic function with domain of Riemann sphere" or a "rational function"? 
Thanks!

REPLY [12 votes]: Holomorphic functions are usually defined over an open set of the complex plane or Riemann sphere.
A holomorphic function defined over the whole complex plane is called entire. This is a definition.
A meromorphic function is one that is holomorphic except for isolated singularities, which must be poles. This is a definition.
A meromorphic function defined over the whole Riemann sphere must be a rational function. This is a theorem.
A holomorphic function defined over the whole Riemann sphere must be constant. This is a theorem.
The only entire functions that extend to meromorphic functions on the Riemann sphere are polynomials. (Thanks to Theo Buehler for pointing this out.)
See also http://en.wikipedia.org/wiki/Meromorphic#Meromorphic_functions_on_Riemann_surfaces.