How to find a conformal mapping that transfers the first quadrant
$D_0 = \{(r, \theta) \mid 0 \leq r < \infty\}$ onto the upper half plane
$D_1 = \{w \mid \operatorname{Im}(w) \geq 0\}$ such that $z_0 = \sqrt{2}i$, $z_1 = 0$, $z_2 = 1$ are mapped to $w_0 = 0$, $w_1 = \infty$, $w_2 = -1$ respectively?
This is what I got after using the generic form for conformal mapping. I am not sure where should I plug the numbers z0, z1, z2, w0, w1, w2.
Best Answer
Take a conformal making of the first quadrant onto the upper half plane, and then use the fact that Möbius transformations are 3-transitive on the upper half plane.