Power-law spherical projection kernel ($\rho (r)=\frac{A\Gamma (\alpha +1/2)}{\sqrt{\pi }\Gamma (\alpha )}(r^2+b^2{)}^{-\alpha -1/2}$)

For $A,b>0$ and $\alpha >1/2$, the projected profile $F(R)=A(R^2+b^2{)}^{-\alpha }$ has the displayed spherical Abel transform inverse. Indeed its composed transform is $A(r^2+b^2{)}^{1/2-\alpha }{I}_{\alpha }$, with $I_{\alpha }={\int }_1^{\infty }{s}^{-\alpha }(s-1{)}^{-1/2}ds=B(1/2,\alpha -1/2)$. Differentiate in $r$, divide by $-2\pi r$, and use the Gamma function recurrence to obtain the coefficient. The same kernel reconstructs tracer number density or isotropic random pressure from their respective projected profiles.