Conformal map

1. Nov 29, 2011

murmillo

1. The problem statement, all variables and given/known data
Show that every conformal self-map of the complex plane C has the form f(z) = az + b, where a ≠ 0. (Hint: The isolated singularity of f(z) at ∞ must be a simple pole.)

2. Relevant equations

3. The attempt at a solution
I know about conformal self-maps of the open unit disk, but this is about the complex plane. I don't know why the isolated singularity of f at ∞ must be a simple pole. I really don't know how I should start.

2. Nov 29, 2011

micromass

Staff Emeritus
All we really need to prove is that $\infty$ is a pole. To prove this, we must show that $\infty$ is neither a removable and neither an essential singularity. Do you know equivalent characterizations of being a removable and an essential singularity to make it easier to see if $\infty$ is one??