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June 24th, 2011, 12:46 AM  #1 
Senior Member Joined: Jan 2009 From: Russia Posts: 113 Thanks: 0  Upper halfplane to lower halfplane
Is there an easy way to observe that the image of the upper halfplane under the birational map is the lower half plane? 
June 29th, 2011, 02:05 AM  #2 
Member Joined: Nov 2009 Posts: 72 Thanks: 0  Re: Upper halfplane to lower halfplane
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 halfplane into the lower halfplane. That it maps the upper halfplane onto the lower halfplane would be shown by writing the inverse of f, and showing as above that if z=x+iy with y<0, then 
July 2nd, 2011, 09:51 PM  #3 
Senior Member Joined: Jan 2009 From: Russia Posts: 113 Thanks: 0  Re: Upper halfplane to lower halfplane
Indeed, just looking at the expression for imaginary part makes the job. Thank you martexel!


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