So I have a map from the unit disc to the upper half plane:
$$f(z) = \frac{z+i}{iz+1}$$
My logic is to now rotate this map by multiplying by $-i$ to arrive at the right half plane – though this does not seem to be a correct answer – what is the mistake I am making here?
Thanks
Best Answer
I find it simpler to remember what the conformal mappings from a half plane to a disk are. As an example, the right half plane is the locus of points which are closer to $+1$ than to $-1$, i.e. the points $w$ for which $|w-1| < |w+1|$. It follows that $$ T(w) = \frac{w-1}{w+1} $$ is the Möbius transformation which maps the right half plane onto the unit disk. Now you can solve the equation $z = \frac{w-1}{w+1}$ for $w$ to get the inverse mapping $$ S(z) = T^{-1}(z) = \frac{1+z}{1-z} $$ which maps the unit disk onto the right half plane.
This is related to your result via $$ -i f(z) = -i \frac{z+i}{iz+1} = \frac{-iz+1}{iz+1} = S(-iz) $$ which is therefore correct as well: $-if$ is a rotation by $-i$ which maps the unit disk onto itself, followed by $S$ which maps the unit disk onto the right half plane.