Lens Conformal Map Please help me find a conformal map of the set $ A = \left \{\; z: \; |z-1| < \sqrt{2} \; and \;  |z+1| < \sqrt{2} \;  \right \}$ one-to-one onto the open first quadrant.
First, I noticed that the circles meet at a right angle at $i$ and $-i$. So I know one of these need to be mapped to the origin. But I had know idea what to do next, so I tried shifting each circle so that they would be centered at the origin, then mapped one circle to the upper half plane and the other to the right half plane, each with an inverse Cayley Transform. 
For the left circle I got:
$$-i\frac{z+2}{z},$$ 
and for the right circle I got:
$$-\frac{z}{z-2}$$
Then I thought to myself, "Great. Now I have two functions...how am I going to combine them?" I peaked at the back of the book and they have:
$$f(z)=e^{-\frac{3xi}{4}}\left ( \frac{z-i}{z+i} \right )$$
I'm pretty sure the "x" is a typo. I think they meant  "$\pi$" instead. After seeing the answer, I lost hope in my strategy. There is a good chance there is something very important that I don't understand. Please, enlighten me. :(

As you say, the circles meet at right angles at $i$ and $-i$. You also know that one of them has to be mapped at $0$. The key is now to map the other to $\infty$. The circles will be mapped then to straight lines that meet at a right angle at $0$. The simplest function that doing this is
$$
\frac{z+i}{z-i},\quad -i\to0,\quad i\to\infty.
$$
Finally, do a rotation to get the first quadrant.
