Accurate calculation of the solutions to the Thomas–Fermi equations

We obtain highly accurate solutions to the Thomas–Fermi equations for atoms and atoms in very strong magnetic fields. We apply the Padé–Hankel method, numerical integration, power series with Padé and Hermite–Padé approximants and Chebyshev polynomials. Both the slope at origin and the location of the right boundary in the magnetic-field case are given with unprecedented accuracy.