Give an explicit description of all the biholomorphic self maps of $\Omega$, for:
- $\Omega = \mathbb{C} \setminus \{0 \}$.
- $\Omega = \mathbb{C} \setminus \{P_1,\ldots,P_k \}$.
(Greene and Krantz, Function Theory of One Complex Variable (3rd ed.), Ch 6, Problem 5, rephrased)
My thoughts: For part 1, one may reason as follows. We know that the biholomorphic self maps of $\mathbb{C}$ are the linear transformations $ f(z) = az+b, \; (a\ne 0).$ Of these, we only want those that preserve $0$, which means functions of the form $z\mapsto az$.
Edit: Having seen Arthur's remark, there are additional biholomorphic self map in the form $ z \mapsto \frac{a}{z}, \;\; (a\ne 0).$
In part 2, trying to generalize to $k$ points, it's clear that the only linear function that preserves even two points is just the identity. But we may find linear functions that permute the set $\{P_1,\ldots,P_k\}$. Any ideas on how to continue ?