Q:

Biholomorphic function between given set and open unit disk

Let $A=\{z\in\mathbb{C}:|z|<1, |z-1|<1\}$.
From Riemann mapping theorem we know that there exist biholomorphic function between $A$ and open unit disk.
How to construct these function ?
I tried to apply Möbius transformation and classical mappings ( exponent, $z^2$, etc.) but I doesn't work. I need only hints for this problem.

A:

The circles $\{|z|=1\}$ and $\{|z-1|=1\}$ intersect at two points $a,\bar a\in\mathbb{C}$, $\operatorname{Im}a>0$. The Möbius mapping
$$
\phi(z)=\frac{z-a}{z-\bar a}
$$
transforms $A$ into an angle with vertex at $0$. A power function transforms this into a half plane, and another Möbius mapping transforms it in the open unit disk.

