Planar isotropic distribution inversion ($F(E)=-\frac{1}{2\pi }\frac{dS}{d\Phi }{|}_{\Phi =E}$)

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.