Vertical-energy Abel inversion of a stellar distribution ($F(E)=-\frac{1}{\pi \sqrt{2}}{\int }_E^{\infty }\frac{d\nu /d\psi }{\sqrt{\psi -E}}d\psi$)

For a phase-mixed one-dimensional collisionless stellar system, $E=v_z^2/2+\psi (z)$ is an integral of motion and $f_z=F(E)$. Integrating both signs of velocity gives $\nu (\psi )=\sqrt{2}{\int }_{\psi }^{\infty }F(E)(E-\psi {)}^{-1/2}dE$. Applying the Abel transform a second time gives ${\int }_E^{\infty }\nu (\psi )(\psi -E{)}^{-1/2}d\psi =\pi \sqrt{2}{\int }_E^{\infty }F(t)dt$; differentiating proves the inverse. Sufficient convergence, regularity, and correct escape-energy boundary conditions are essential. An arbitrary tracer-density/potential pair is physical only if the reconstructed phase-space distribution function is nonnegative. Applying this inversion to a galactic disk requires a locally separable vertical dynamics or a justified approximation to it, and control of radial mixing and the tilted velocity ellipsoid.