Mapping unit disc onto upper half plane

How can I map the unit disc onto the upper half plane?

I tried mapping $(1,i,-1)\rightarrow(1,0,\infty)$ using cross-ratio: $z\rightarrow \frac{(z-z_3)(z_2-z_4)}{(z-z_4)(z_2-z_3)}$, but didn't give me the right answer...

$$z \to \frac{z-i}{z+i}$$
• Nope, I don't know how to ''select'' the right triples $(z_2,z_3,z_4)$. I think that's my problem Commented Oct 18, 2013 at 12:54
• I want to take the boundary of the upper half plane to the boundary of the disc, so I want everything on the vertical line to have norm $1$ in the image. I choose to send $i$ to $0$ as the new center of the circle, and since I want points on the horizontal line to go to norm $1$, this forces me to make the denominator $z+i$, since $-i$ is symmetric to $i$ across the line. Commented Oct 18, 2013 at 13:03
• I learned it like this: the cross ratio which maps $(z_2,z_3,z_4)\rightarrow (1,0,\infty)$ is your axiom and making use of inverse and composition, I can map anything to anything. I think your method is better, but I don't ''see'' it happening. E.g. wat do you define as the boundary of the upper half plane? And what do you mean with ''the vertical line'' Commented Oct 18, 2013 at 13:12