I am trying to map the region G, {|z|<1, |z+i|> (2)^0.5} to the infinite vertical strip at x = +/- pi.
I have started by using a Mobius Map which sends the two common points of the circles to 0 and infinity (z=+/- 1), namely, using the map f(z)=(z-1)/(z+1). This maps the unit circle to the positive imaginary axis and should map the other circle to a circle. However, when I calculate the image of the other circle, I don't get a region that is one of the axes as one would expect...
If Mobius maps take circlines to circlines, why is this the case?