Weird conformal map problem Construct a conformal map from the region $\omega$ = open disk of radius 1 centered at 0 minus the closed disk of radius 0.5 centered at 0.5 to $\mathbb{D}$ = disk radius 1 centered at 0.
I really have no clue where to begin.  I am alright at standard conformal map examples, but this one is hard.  Some help would be awesome.

(The deleted answer seemed basically fine, but, now that it's deleted, maybe there is reason to add something...) As in the comments, it is reasonable to think of lines-and-circles together, so the crescent-shaped region is a "generalized polygon" with just two (curvy) sides. It is not a generic bi-gon, because the two vertices are the same point. If the two vertices had been distinct, mapping one to $0$ and the other to $\infty$ by a linear fractional transformation, and then applying a suitable power map $z\to z^r$, converts any such to a half-plane, and then a disk by a Cayley transform. In contrast, in your special case, mapping the single vertex to $\infty$ by a linear fractional transformation gives you a strip (bounded by two parallel lines, thus intersecting just at $\infty$). 
(As in the deleted answer), a suitable exponentiation maps a (bounded, open) strip to a half-plane, and then by Cayley to an open disk.
