How to find a möbius transformation mapping $B(p, r)$ to $B(0, 1)$?

Question

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

Answer 1

You can do that even with a linear transformation: We have $$ z \in B(p, r) \iff |z-p| < r \iff | \frac {z-p}r - 0 | < 1 \iff \frac {z-p}r \in B(0, 1) $$ so that $f(z) = \frac {z-p}r$ maps $B(p, r)$ conformally onto $B(0, 1)$.

All Möbius transformations mapping $B(p, r)$ onto $B(0, 1)$ are given by $T \circ f$, where $T$ is an automorphism of the unit disk.