Proposition

  1. Find a Mobius transformation that maps the unit circle to $\{ x + iy \in \mathbb{C} : x + y = 0 \}$.
  2. Find two Mobius transformations that map the unit disk $\{ z \in \mathbb{C} : \abs{z} < 1 \}$ to
    • $\{ x + iy \in \mathbb{C} : x + y > 0 \}$ and
    • $\{ x + iy \in \mathbb{C} : x + y < 0 \}$,

    respectively.

Solution

As shown previously,

\[\begin{align*} f(z) = \frac{(1 + i)z + (1 + i)}{(-2\sqrt{2})z + 2\sqrt{2}} \end{align*}\]

maps the unit circle to $\{ x + iy \in \mathbb{C} : x + y = 0 \}$, and the unit disk above the line $x + y = 0$.

Therefore, $f(1/z)$ maps the unit disk below the line $x + y = 0$.