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?
[Math] Find Möbius transformation for half-plane to unit disk $|w|<1$
complex-analysiscomplex-geometryconformal-geometrymobius-transformation
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Best Answer
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 $$ T(z) = \frac{z}{z-1+i} $$