= Rotating Mestel disk distribution function
{title2=$f=A L_z^q e^{-E/s},\quad q=v_c^2/s-1$}
In the midplane <Mestel disk> potential $\Phi=v_c^2\log(R/R_0)$, take specific <energy> $E=(v_R^2+v_\phi^2)/2+\Phi$ and $L_z=Rv_\phi$. A prograde <galactic distribution function> $f=A L_z^qe^{-E/s}$ for $L_z>0$, with $s>0$, has <surface density> proportional to $R^{q-v_c^2/s}$. Matching $\Sigma=\Sigma_0R_0/R$ gives $q=v_c^2/s-1>-1$ and
$$
A=\frac{\Sigma_0R_0^{-q}}{\sqrt\pi\,2^{q/2}s^{(q+2)/2}\Gamma((q+1)/2)}.
$$
The denominator is the product of the full radial Gaussian integral and the positive-azimuthal half-line moment. Here $s$ is a squared-velocity scale, not a <velocity dispersion> itself. Changing the potential's additive constant changes $A$. With positive exponent $e^{\varepsilon/s}$, $\varepsilon$ must instead be the negative of the specific total energy.
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