Which mobius transformations map $|z-1|=1$ and $|z+1|=1$ onto the lines $Re(w)=1$ and $Re(w)=-1$, respectively, and the single point $z=2$ onto $w=1$?

Question:

Which mobius transformations map $|z-1|=1$ and $|z+1|=1$ onto the lines $Re(w)=1$ and $Re(w)=-1$, respectively, and the single point $z=2$ onto $w=1$?

Attempt:

I've been working on this for a while and I'm stuck. I understand that the solution will need some sort of inversion since we are mapping circles to lines. I know that a vertical line $z=c_1$ is mapped by $w=1/z$ to the circle $$-c_1 (u^2 + v^2) + u = 0$$ where $w=u + iv$. But I can't figure out how to use this to find the appropriate transformations for the question.

Any help understanding would be much appreciated...

The two circles share a point at 0. They are also perpendicular to the real axis. This suggests trying $z \to \frac{1}{z}$. The image of the circles under this must pass through the point $\infty$ and must be perpendicular to the image of the real axis. The real axis is preserved under inversion, so the images of the circles must be $Re(z) = \frac{1}{2}$ and $Re(z) = \frac{-1}{2}$. To put them in the right spots we need to dilate by a factor of 2. The desired map is therefore $$z \to \frac{2}{z}$$
• So this is equivalent to $w=2/z$. Is such a transformation unique? – mathjacks Nov 17 '14 at 3:14