The Thomas–Fermi problem and solutions of the Emden–Fowler equation

A two-point boundary value problem is considered for the Emden–Fowler equation, which is a singular nonlinear ordinary differential equation of the second order. Assuming that the exponent in the coefficient of the nonlinear term is rational, new parametric representations are obtained for the solution of the boundary value problem on the half-line and on the interval. For the problem on the half-line, a new efficient formula is given for the first term of the well-known Coulson–March expansion of the solution in a neighborhood of infinity, and generalizations of this representation and its analogues for the inverse of the solution are obtained. For the Thomas–Fermi model of a multielectron atom and a positively charged ion, highly efficient computational algorithms are constructed that solve the problem for an atom (that is, the boundary value problem on the half-line) and find the derivative of this solution with any prescribed accuracy at an arbitrary point of the half-line. The results are based on an analytic property of a special Abel equation of the second kind to which the original Emden–Fowler equation reduces, to be precise, the property of partially passing a modified Painlevé test at a nodal singular point.