My Math Forum Upper half-plane to lower half-plane

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 June 24th, 2011, 12:46 AM #1 Senior Member   Joined: Jan 2009 From: Russia Posts: 113 Thanks: 0 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 $f(z)=\frac{az+b}{cz+d} \quad\quad ad-bc<0 \quad a,b,c,d \in \mathbb R=$ is the lower half plane?
 June 29th, 2011, 02:05 AM #2 Member   Joined: Nov 2009 Posts: 72 Thanks: 0 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 $\mathrm{Im}\,f(z)=\frac{(ad-bc)y}{(cx+d)^2+c^2y^2}<0=$ 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 $\mathrm{Im}\,f(z)>0.$
 July 2nd, 2011, 09:51 PM #3 Senior Member   Joined: Jan 2009 From: Russia Posts: 113 Thanks: 0 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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