[Math] Mapping unit disc onto upper half plane

Question

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…

Answer 1

Do you know how to conformally map the upper half plane to the unit disc? If so, you can take the inverse of this using matrix algebra.

An example of conformal map from the upper half plane to the unit disc is:

$$z \to \frac{z-i}{z+i}$$