Find a Möbius transformation which maps the region $|z| > 1$, $\Im z > 0$ onto the first quadrant of the complex plane. a) Find a Möbius transformation which maps the region $|z| > 1$, $\Im z > 0$ onto the first quadrant of the complex plane.
b) Find all Möbius transformations satisfying the above requirement.
I know how to find Möbius transformations in general, using the cross-ratio, but in this case I don't even know how to start. Any input is appreciated! I am studying for an exam in complex analysis.
 A: Since each generalized circle (circle or line) is mapped to another generalized circle, the image of $|z|=1$ is $\Re z = 0, \Im z \ge 0$ and the image of $\Im z = 0$ is $\Im z = 0, \Re z \ge 0$, or vice versa. In both cases either $-1 \mapsto 0$, $1 \mapsto \infty$ or $-1 \mapsto \infty$, $1 \mapsto 0$. So our mapping must be of the form
$$z \mapsto a\frac{z+1}{z-1}$$
or
$$z \mapsto a\frac{z-1}{z+1}.$$
The point $0$ is on the real line, but outside our region. So it is mapped either to negative real halfline or negative imaginary halfline. Therefore $a=\alpha$ or $a=i\alpha$, where $\alpha > 0$. We also know that $i$ is mapped to the boundary of the first quadrant. This means that we have an imaginary $a$ for the mapping of the first form and a real $a$ for the mapping of the second form. So the general form of Möbius transformations satisfying the requirement is

$$z \mapsto i\alpha\frac{z+1}{z-1}, \quad \alpha >0, \qquad \text{ or } \qquad z \mapsto \alpha\frac{z-1}{z+1}, \quad \alpha >0.$$

