Extensions and applications of the Bogema–Davidson inversion theorem

For a single particle species confined by a uniform magnetic field B, the Bogema–Davidson inversion theorem determines a self-consistent rigid-rotor equilibrium distribution function F(H⊥−ωrPθ) as the derivative of the given radial density profile n(r) with respect to the corresponding generalized potential ψ(r). Aside from the quadratic density profile, most forms one can choose for n(r), even if integrable, yield potentials ψ(r) that cannot be inverted. The function F can be found parametrically, however, by writing the derivative of n with respect to ψ as the ratio of the radial derivatives of n and ψ. Several nontrivial examples of physical interest are studied. Numerical methods for calculating F are presented, its asymptotic form and integration are investigated, and applications to experiment are discussed.