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

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?

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