Conformal Mapping of Upper Half Plane from Unit Half Disc | Homework Solution

[Spriteling]

Homework Statement

Find a conformal mapping f of the set V to the upper half plane H+ = {z | Im(z) > 0 where V = {z: |z| < 1 and Im(z) > 0}

Homework Equations

None, really. It's worth noting that V is the unit half disc.

The Attempt at a Solution

I have a Mobius transformation S that maps 1 to 0, -1 to infinity, and 0 to - 1. The question hints that I should consider the image of V under this mapping. I saw that S sends i to i. However, I don't really know where to proceed from there; it seems to me that S sends V to H+ straight away, so I'm rather confused.

 

[lippyka]

Is this the answer? Or is there something else I should consider? Any help would be greatly appreciated.