Conformal mapping of region between two circles

Question

Suppose that one circle is contained inside another and that they are tangent at the point $a$. Let $G$ be the region between the two circles and map $G$ conformally onto the open unit disk. Hint: first try $(z-a)^{-1}$.

My confusion on this question comes from the fact that the only example of conformal mapping shown in class was from the left side of a line to the outside of the unit circle, which seems much easier and more concrete. I don't understand how to map something that's not entirely to one side of a line or circle. I'm also not sure what the hint means, is that supposed to represent a Mobius transformation? Please explain as much as possible.

Answer 1

A Moebius transformation maps circles $\cup$ lines to circles $\cup$ lines. If you map $a$ using $z\mapsto{1\over z-a}$ to $\infty$ the two touching circles will be mapped onto two parallel lines, and the shape between them will be mapped onto the infinite strip between these lines.

Mapping such a strip to $D$ is another matter, and requires more expertise than is available to you at the moment. Maybe you find such a map on the net.