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.