Find Möbius transformation for half-plane to unit disk $|w|<1$?

Question

Consider the half-plane depicted in the following figure

How can a Möbius transformation that takes that half-plane onto the unit disk $|w|<1$ be found?

What are the steps and things to think about?

Answer 1

Step 1: Map the half plane shown to the upper half plane $y > 0$ by an isometry of $\mathbb C$, written in the form $\frac{az+b}{0z+1}$.

Step 2: Map the upper half plane to the unit disc by a Möbius transformation, which I assume you know how to do, or you can look it up elsewhere such as here.

Step 3: Compose those two maps.

Answer 2

For an arbitrary (open) half-plane $H$ one can construct a Möbius transformation to the unit disk as follows: Find a pair of points $a \in H$ and $b \notin H$ which are symmetric with respect to the boundary of $H$. Then $$ T(z) = \frac{z-a}{z-b} $$ is such a Möbius transformation. The reason is that $H$ is the locus of all points which are closer to $a$ than to $b$, i.e. all points with $|z-a| < |z-b|$.

In your case, $a=0$ and $b=1-i$ is a possible choice, giving $$ T(z) = \frac{z}{z-1+i} $$