Let $f(z) = z^2 + 1$ be defined on $\mathbb{C}$. How does one extend $f$ to the Riemann Sphere? Also I would like to find any singularities of the extension if any.
Is there a simple method to find such extension for any function $f$? I am completely lost how to extend it to the Riemann sphere to begin with.
On
The Riemann sphere adds a point called $\infty$ to $\mathbb{C}$, along with the topology such that the formal limit $z\to\infty$ is the same as the real limit $|z|\to+\infty$.
If we want the extension of $f$ to be continuous we should have $$f(\infty)= \lim_{z\to\infty} f(z) = \lim_{|z|\to+\infty} f(z).$$
I think this is the same process as the projective space answer.
First the Riemann sphere is $\mathbb{C}\mathbb{P}^1=(\mathbb{C}^2\setminus \{(0,0)\})/\sim$
where $\sim$ is defined as $z\sim z' \iff $ there exist $\lambda\neq 0 \in \mathbb{C}$ such that $z=\lambda z'.$
Now the extension of $f$ is
$F:\mathbb{C}\mathbb{P}^1\rightarrow \mathbb{C}$ such that $F(z,z')={(\frac{z}{z'})}^2+1$ on $U=\mathbb{C}\mathbb{P}^1\setminus \{ \infty\}=\{(z,1):z\in \mathbb{C} \}$.
We have $z'=1$, thus it is clear that
$F|_U(z,1)=z^2+1=f(z).$ Now note that identifying $(1,0)=+\infty$, we get $F(1,0)=+\infty$.
Hence the sigularity is at $(1,0).$