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**cribby** I want to take the upper half of the unit disc, $\displaystyle D^+$ and send it to the upper half of the plane$\displaystyle P^+$. 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 $\displaystyle D^+$, I was hoping that if I sent $\displaystyle i \mapsto \infty$ and left 1 and 0 fixed, that the boundary would "unfurl" to the real axis and the interior of $\displaystyle D^+$ would map to $\displaystyle P^+$ as a consequence. In terms of the cross-ratio, this translates to $\displaystyle z \mapsto \frac{\frac{w-0}{w-i}}{\frac{1-0}{1-i}} = \frac{-iw}{w-i}$.

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!!