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June 24th, 2011, 12:46 AM   #1
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Upper half-plane to lower half-plane

Is there an easy way to observe that the image of the upper half-plane under the birational map

is the lower half plane?
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June 29th, 2011, 02:05 AM   #2
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Re: Upper half-plane to lower half-plane

This isn't very elegant but it works: take z=x+iy with x,y real. Then it's not hard to check that

as long as y>0, so that f maps the upper half-plane into the lower half-plane. That it maps the upper half-plane onto the lower half-plane would be shown by writing the inverse of f, and showing as above that if z=x+iy with y<0, then
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July 2nd, 2011, 09:51 PM   #3
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Re: Upper half-plane to lower half-plane

Indeed, just looking at the expression for imaginary part makes the job. Thank you martexel!
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