Map the region inside the circle $|z| = 1$ and outside the circle $|z-1/2| = 1/2$ conformally onto the unit disk.
I was thinking of using some scaling and shifting to get from the unit circle to the circle $|z-1/2| = 1/2$. but I'm not sure how that might be useful since it doesn't help with getting the region we want. The other idea I had was to work backwards starting with the unit circle to get to the region we want.
Source: Spring 1992