The problem is simply to find the image of the upper half-plane under the inverse sine function.
The textbook defines the inverse sine in the following way. First, it defines arccos(w) = -i * log(w +/- sqrt(w^2 - 1)) and then it defines arcsin(w) = pi/2 - arccos(w).
The problem doesn't specify what branches of the logarithm and square root should be used. I'm going to assume the principal branches.
The Attempt at a Solution
I've attempted to just break the inverse sine into a sequence of compositions and to, so to speak, push the upper half-plane through this sequence, but I've not been able to identify a breakdown into suitably elementary functions allowing me to determine the images.
As for doing it directly, I'm just now seeing how to begin.