I have an open connected $\Omega$ of the complex plane and for a point $p \in \Omega$ and $r > 0$ we have a ball $B(p, r) \subset \Omega$. Is there a general möbius transformation to map this to B(0, 1)?
I know I can find a möbius transformation between two sets circles if I have three distinct points from each, but in this case I don't have any points to work with from $B(p, r)$. Or could I go with something like $p_1 = (a+r) + p$, $p_2 = (a-r) + p$, and $p_3 = (a+ir) + p$? If yes, how would I proceed?
I am stuck.
Thank you for your time, Isak