In my textbook, it is said that $z+1\over z-1$ maps the left half plane to the unit disk. So since it is its self-inverse, (right?) the unit disc should be mapped to the left half plane. But on another page it says that this maps the upper semidisk to the first quadrant which is in the right half plane... what is happening? please help, thanks.
So I have tried applying this map to the first quadrant and found that it is mapped to the lower half plane with the unit circle cut out of it, m i right? If not, please explain, there might be some fundamental concepts I have not realized.