Constructing a conformal map on the upper half plane.

Question

Let $S=\{z\in \mathbb{C}|\Re(z)>0, \Im(z)>0\}$. I want to construct a bijective conformal map $\phi:S\longrightarrow S$ that maps $1+i$ to $2+i$. I have tried all the conformal maps that I've known so far including Cayley's transform, the "swapping" transform $\Phi_{z_0}=\frac{z_0-z}{z-\overline{z_0}z}$, etc..., but none of them seems to be working. Can someone help me with this?

Answer 1

The quarter plane $S$ is mapped conformally on the half upper plane by the conformal mapping $z\mapsto z^2$. Now it is not difficult to find an homography $z \mapsto {az+b \over cz+d}$, $a,b,c,d \in {\bf R}$, $ad-bc=1$ of the half upper plane that maps the point $(1+i)^2 = 2i$ to the point $(2+i)^2 = 3+2i$.