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what is the image of Q by the mapping [itex]f(z)=z^2[/itex]

by trial and error with various points, my answer is that it takes Q to the semicircle [itex]\{ z: Re(z)>0, |z|<1 \}[/itex]

but can't how this explicitly as it's not a mobius transformation with which I am used to dealing with.