Planar isotropic distribution inversion
= Planar isotropic distribution inversion
{title2=$F(E)=-\frac1{2\pi}\frac{dS}{d\Phi}\big|_{\Phi=E}$}
The <surface density of a disk> associated with a <planar isotropic distribution> is $S(\Phi)=2\pi\int_\Phi^\infty F(E)dE$. Differentiation gives $F(E)=-(2\pi)^{-1}dS/d\Phi|_{\Phi=E}$. A vanishing upper-limit boundary value recovers the original integral; monotonic decrease of $S(\Phi)$ ensures a nonnegative distribution.