I have two complex variables $u$ and $v$ with the constraints
$|u|^2+|v|^2=1\ \ \ \ \ \ \ $ and $\ \ \ \ \ \ \ \ uv=|uv|\ $.
With two constraints there are basically two degrees of freedom, and I should be able to express everything in terms of a single complex coordinate $z=u/v$. Then I'd like to write $u$ and $v$ in terms of $z$ but so far I am unable to.
I am able to get partly there using the equations above as follows:
$1+|z|^2=\frac{|v|^2+|u|^2}{|v|^2}=|v|^{-2}$
$uv=|uv|=\frac{|u/v|}{|v|^{-2}}=\frac{|z|}{1+|z|^2}$
$u^2=uv\frac{u}{v}=z\frac{|z|}{1+|z|^2}\ \ \ \Rightarrow\ \ \ u=\pm\ z^{1/2}\sqrt{\frac{|z|}{1+|z|^2}}$
$v^2=uv/\frac{u}{v}=z^{-1}\frac{|z|}{1+|z|^2}\ \ \ \Rightarrow\ \ \ v=\pm\ z^{-1/2}\sqrt{\frac{|z|}{1+|z|^2}}\ $.
But I need an expression that determines $u$ and $v$ completely and not just up to a sign. Can anyone see how to get there?
You can't. If $u$ and $v$ satsify your equations, then $-u$ and $-v$ do too (and give the same value of $z$).