Studying from Conway's book for a qualifying exam, I found an exercise that ask you to prove that if $G$ is a region, the set of meromorphic functions in $G$ is a field.
When I tried to prove that $1/f$ is an inverse for $f$, I've stumbled on how to define multiplication of two meromorphic functions.
What happens if $z$ is a zero of $f$ and a pole of $g$?
Thanks in advance!
When at least one of the functions $f$ and $g$ is undefined at an isolated point $z_0\in G$, then we see whether or not the limit $\lim_{z\to z_0}f(z)g(z)$ exits. If it does, and if it is equal to $w_0$, then we extend the domain of $fg$ (which originally is not defined at $z_0$) in such a way that the domain contains $z_0$ (and we extend it in such a way that $(fg)(z_0)=w_0$, of course).
With this convenvention, if, say, $G=\mathbb C$, $f(z)=z$ and $g(z)=\frac1z$, then the domain of $fg$ is aain $\mathbb C$ and $fg$ is the constant function $1$.