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"?