Numerical integration of the Thomas-Fermi equation from zero to infinity

Abstract

The Thomas-Fermi equation is expanded in a series at both x = 0 and x = ∞. Forward numerical integration from x = 0 determines an initial slope of −1.5880710226. Backward integration from x = ∞ depends on one parameter. By fitting the backward and forward integrations near x = 30, the parameter is determined and the numerical solution is obtained from zero to infinity with a high degree of accuracy.