-1
$\begingroup$

Consider the half-plane depicted in the following figure

Half-plane

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?

$\endgroup$
1
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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 $$ T(z) = \frac{z}{z-1+i} $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.