What does holomorphic function on the Riemann sphere mean? Is that just $f\in H(\mathbb{C}\cup \{\infty\})$, so $f:\mathbb{C\cup \{\infty\}}\to \mathbb{C}$. And is $f$ entire? Since $f$ is holomorhpic over the whole complex plane.
I am attempting to prove the following:
All holomorphic functions on the Riemann shpere are constant
The way to do this is by Liouville's theorem, which said every bounded entire function is constant. So I just need to show that entire function on Riemann sphere is bounded.
Best Answer
$f$ is holomorphic at $\infty$. So $f$ if bounded as well in an open ball (say $B$) at $\infty$. Also $f$ is continuous imply $f$ is bounded in $\Bbb{C_\infty}\backslash\,B$ (as this set is compact).
So $f$ if bounded in ehole of $\Bbb{C_\infty}$.