Conformal map from two disks to the unit disc

Question

Find a conformal equivalence f from $U$ onto $ D = \{ z : |z| < 1 \} $, where $U=\{z: |z+1|<2\}\cup \{z: |z-1|<2\}.$

How to find such a map $f$? I think we need to map the two discs to two strips individually. I have no idea how to do this.

Answer 1

Let $P,Q$ be the two points of intersection of the two circles.

First apply a Möbius transformation that takes $P$ to $0$ and takes $Q$ to $\infty$. The image of your region is now an angle bounded by two rays.

Next, apply a power transformation $z \mapsto z^s$ where the exponent $s$ is chosen so that the union of the two rays is mapped to a straight line through $0$. The image of the region is now a half plane bounded by that line.

Now apply a Mobius transformation which takes the half plane to the disc $D$.