I am trying to find a conformal map from $D = \{|z| <1\} \cap \{|z-1/2|>1/2\}$ to the open unit disk. I know that $D$ is the intersection of two "disks" so I want to use some sort of LFT to map $D$ onto a sector (which I believe will actually turn out to be a vertical strip). The problem is that usually I would use the LFT $f(z) = \frac{z-a}{z-b}$ where $a$ and $b$ are the points of intersection of the boundaries of the two disks. However, the boundaries of the unit disk and $|z-1/2| >1/2$ only intersect at one point (z=1) so I am not sure what to do.
Find a conformal map onto the open disk
complex-analysis
Best Answer
Any Möbius transformation which maps the intersection point $z=1$ to $w=\infty$ and the real axis onto itself is a good start, because it maps the disc boundaries to vertical lines.
A possible choice is $T(z) = z/(z-1)$ which maps $1, 0, -1$ to $\infty, 0, 1/2$, respectively, and therefore the circle $\{ |z| = 1 \}$ to $\{ \operatorname{Re}(w) = 1/2 \}$ and the circle $\{ |z-1/2| = 1/2 \}$ to $\{ \operatorname{Re}(w) = 0 \}$, i.e. to the imaginary axis.
Now map one additional point to decide whether the image of your domain is the area between the vertical lines or the “outside.”