I want to take the upper half of the unit disc, and send it to the upper half of the plane . My instructor hinted that if I can specify three distinct points that go to 1, 0, and the point at infinity under the transformation that I can use the cross-ratio to determine a bilinear transformation that will "almost work".
I'm totally lost in this course right now, so I wasn't even sure what the hint meant, but I tried *something* (even if ignorantly).
Looking at the , I was hoping that if I sent and left 1 and 0 fixed, that the boundary would "unfurl" to the real axis and the interior of would map to as a consequence. In terms of the cross-ratio, this translates to .
But I don't even know how to tell if this works. And what might my instructor mean by "almost work"? Thanks ahead of time for any advice!!