= Untruncated Eddington inversion
{title2=$f(E)=\frac1{\sqrt8\pi^2}\frac d{dE}\int_{-\infty}^E\frac{\rho'(\psi)}{\sqrt{E-\psi}}\,d\psi$}
If the <galactic distribution function> is not restricted to positive <relative energy>, isotropic <velocity> integration gives $\rho(\psi)=4\pi\sqrt2\int_{-\infty}^{\psi}f(E)\sqrt{\psi-E}\,dE$. The same <Abel transform> argument as <Eddington inversion> yields the displayed formula, with lower limit $-\infty$. For $\rho=\rho_0e^{\psi/\sigma_0^2}$, substitution $\psi=E-\sigma_0^2s^2$ gives $f(E)=\rho_0(2\pi\sigma_0^2)^{-3/2}e^{E/\sigma_0^2}$. An escape-energy truncation changes both the density law and the velocity tails.
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