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I am trying to come up with a diffeomorphism of the upper half plane $y> 0$ onto the first quadrant $x> 0 , y> 0$

Can anyone come up with such a diffeomorphism?

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    Try the squareroot function in $\mathbb{C}$, with branch cut on the $x$-axis.2012-03-02
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    @Brett Frankel: I didn't quite understand how your function look like. Can you write it explicitly?2012-03-02
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    $re^{i\theta}$ maps to $\sqrt{r}e^{i\theta/2}$ where $\theta \in (0, \pi)$2012-03-02
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    If you prefer polar coordinates to complex analysis, how about the map sending $(r,\theta)$ to $(r,\theta/2)$?2012-03-02
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    @Brett Frankel: I think your map definitely maps the upper half plane to the first quadrant, but I can't figure out how you can prove that it is a diffeomorphism. Can you elaborate the diffeomorphism part?2012-03-02
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    I suggest you write it down, show it's differentiable. Then, write down its inverse and show it's differentiable. This is one of those cases where you should just sort of do it. If you encounter trouble, edit your question with your attempt and we can go from there perhaps.2012-03-02

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Why not $(x,y)\mapsto (e^x, y)$?