In Stein's Complex Analysis book, he wrote 'If $f$ is holomorphic for all large values of $z$, we consider $F(z)=f(1/z)$, which is now holomorphic in a deleted neighborhood of the origin. Similarly, we can speak of $f$ having an essential singularity at infinity, or a removable singularity (hence holomorphic) at infinity in terms of of the corresponding behavior of $F$ at $0$. A meromorphic function in the complex plane that is either holomorphic at infinity or has a pole at infinity is said to be meromorphic in the extended complex plane.
What does the author mean by saying 'hence holomorphic'?
Best Answer
Let $U$ be a neighbourhood of $0$ and $F$ be holomorphic in $U \setminus \{ 0 \}$. $F$ has a removable singularity at $z=0$ if it can be extended over $0$, i.e. if there is holomorphic function $\tilde F$ on $U$ with $\left. \tilde F \right|_{U \setminus \{ 0 \}} = F$.
In that case one sometimes says (in a slight abuse of language) that “$F$ is holomorphic at $0$” or “$F$ is holomorphic in $U$”, even though $F$ is not defined at $z=0$. In other words, the extension $\tilde F$ is again called $F$.
In the same sense, $f$ is said to be “holomorphic at infinity” if $F(z) = f(1/z)$ is holomorphic at $0$, i.e. if $F$ has a removable singularity at $z=0$.