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 $w=1$?
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...