I am trying to find the image of the interior of the half disk {|z|<1, Im z>0} under the mapping
$$z + \frac{1}{z}$$
and the problem statement also asks to find the images of specific points A, B, C,D,E along the half disk (in a clockwise direction).
How could I start on this problem? Is it best to first figure out where the points A, B, C, D,E map to, perhaps notice a closed curve image in the w-plane, and then from there conclude where the interior of the half-disk must go to, based on the fact that conformal mappings will preserve the orientation of curves. So, in this case, since the half circle is chosen to be oriented clockwise, this means that the interior is to the right of the curve. Then in the w-plane, if I find some close curve formed by the images of A,B, C, D, and E, then the interior of the half disk we know must map again to the right of the image curve and we can conclude from here.
But the concern with proceeding in this way is that ... I am finding am image curve, where there is a blow-up point in the preimage half-circle, namely at z=0.
So, would this be a valid strategy? Or should I head in another direction and perhaps study some specific properties of the mapping $z + \frac{1}{z}$?
I would prefer the first, brute-force approach, since this would be good practice to handle other types of conformal mappings which aren't linear fractional transformations. But I'm afraid it may not work, because of the singularity at 0.
Thanks,