= Power-law spherical projection kernel
{title2=$\rho(r)=\frac{A\Gamma(\alpha+1/2)}{\sqrt\pi\Gamma(\alpha)}(r^2+b^2)^{-\alpha-1/2}$}
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.
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