= Scale-free Kuzmin superposition
{title2=$\mu(b)=Bb^\beta,\quad0<\beta<1$}
For $B>0$ and $0<\beta<1$, the unreferenced <relative potential> of the <Kuzmin reflection method> diverges at large $b$. Referencing each component to the origin before integrating gives the finite potential difference
$$
\psi(R,z)=GB\int_0^\infty b^\beta\left[\frac1{\sqrt{R^2+(|z|+b)^2}}-\frac1b\right]db.
$$
At small $b$ the subtraction is integrable because $\beta>0$; at large $b$ the difference is $O(b^{-2})$, integrable because $\beta<1$. Its plane force is $\partial_R\psi=-CR^{\beta-1}$, where $C=(GB/2)B((1+\beta)/2,1-\beta/2)$. A <circular speed> $V=KR^{\beta/2}$ fixes $C=K^2$, giving
$$
\Sigma(R)=\frac{K^2R^{\beta-1}}{2\pi G}\frac{\Gamma(1+\beta/2)\Gamma((1-\beta)/2)}{\Gamma((1+\beta)/2)\Gamma(1-\beta/2)}.
$$
The <beta function> substitutions for the force and <surface density> have different exponents. Its central mass is locally finite, but its total mass diverges at large radius; a potential zero at infinity cannot be imposed. The force and density approach those of the <Mestel disk> as $\beta\downarrow0$.
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