I am trying to find a conformal map that maps a circle in the $\zeta$ plane to a circle in the $z$ plane. As far as I know, a Mobius transformation is appropriate for this. These are the conditions that I am trying to satisfy: 1) The circles should remain as unit disks around the origin 2) There should be a point, m, inside the circle in the $\zeta$ plane that maps to $\infty$ in the $z$ plane.
So far, I can only get this to work for $m = 0$ but not for some arbitrary $m < 1$.
So for example I start with the Mobius transformation \begin{equation} z(\zeta) = \frac{a\zeta + b}{c \zeta + d} \end{equation}
And then if I apply the conditions: $z(\infty) = 0$, $z(1) = 1$ and $z(m) = \infty$ to solve for $a, b, c$ and $d$, I get the map
\begin{equation} z(\zeta) = \frac{1 - m}{\zeta - m} \end{equation}
This map seems to satisfy both 1) and 2) only when $m = 0$. Otherwise, 1) is not satisfied (for example $z(-1) \neq -1$ so it isn't a unit disk around the origin in the $z$ plane). I realize that there are holes to this approach since there's nothing that I've done in my solving for $a,b,c,d$ that would ensure this condition but I am not really sure how else to go about doing this.