= Spherical Abel deprojection
{title2=$\rho(r)=-\pi^{-1}\int_r^\infty\Sigma'(R)(R^2-r^2)^{-1/2}\,dR$}
Put $F(u)=\Sigma(\sqrt u)$ and $g(v)=\rho(\sqrt v)$. Then $F(u)=\int_u^\infty g(v)/\sqrt{v-u}\,dv$. Composing this <Abel transform> with itself and exchanging convergent <integrals> gives $\int_t^\infty F(u)/\sqrt{u-t}\,du=\pi\int_t^\infty g(v)\,dv$, because the inner integral from $t$ to $v$ is $\pi$. Differentiate after writing $u=t+s$ to obtain $g(t)=-\pi^{-1}\int_t^\infty F'(u)/\sqrt{u-t}\,du$. Substituting $t=r^2,u=R^2$ yields the displayed reconstruction. Decay and regularity sufficient for the exchanges and derivatives are required.
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