= Radial-ratio independence of two half-normal variables
{title2=$U=X^2+Y^2,\quad V=Y/X$}
If $X,Y$ are independent with density $2e^{-x^2}/\sqrt\pi$ on the positive axis, the <Jacobian matrix> of the inverse transformation has determinant $1/[2(1+v^2)]$. The joint density is $2e^{-u}/[\pi(1+v^2)]$ for $u,v>0$, factoring into an <exponential distribution> for $U$ and a unit-scale <half-Cauchy distribution> for $V$. The underlying radial symmetry makes radius and angle independent.
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