Extending holomorphic function on punctured plane to Riemann sphere

Question

I have a holomorphic function $f: \mathbb{C} \setminus \{0\} \rightarrow \mathbb{C}$, and I wish to extend $f$ to a holomorphic function $\tilde{f}: \widehat{\mathbb{C} \setminus \{0\} } \rightarrow \hat{\mathbb{C}}$ where $\widehat{\mathbb{C} \setminus \{0\} }$ is homeomorphic to the Riemann sphere by adding two points at infinity.

I believe that it suffices to determine where these two points at infinity go, so it seems that I should map both of them to $\infty$ in $\hat{\mathbb{C}}$. (In the more specific context of the problem I'm working on, I also have to show that $\tilde{f}$ is a double branched cover.) However, I'm struggling to show that $\tilde{f}$ is holomorphic that at these two points at infinity. I've tried computing the derivative directly by definition, but I'm running to issues involving arithmetic with $\infty$.

Please let me know in the comments if you need the complete context of the problem in case this is not enough information.

Answer 1

If either of those two isolated singularities is an essential singularity, then there is no extension possible. If they are both (proper) poles, then you do have an extension, and the values are $\infty$. If either is a removable singularity, then you can determine the (uniquely-determined) value there.

For the missing point at $\infty$, switch to coordinate $1/z$, which converts the question of the nature of the singularity to the question near $z=0$.

You do not need to attempt to do arithmetic with $\infty$, fortunately. :)