Padé approximant approach to the Thomas-Fermi problem

Comment on: “Series solution to the Thomas–Fermi equation” [Phys. Lett. A 365 (2007) 111]

“…The fact that our original rational approximation (8) does not have the correct behaviour at infinity is not at all a problem because we may resort to a more convenient expansion [13] once we have an accurate value of the unknown slope at origin.…”

“…Some time ago Liao [16] proposed the application of a technique called homotopy analysis method (HAM) to the solution of the TF equation and stated that "it is the first time such an elegant and explicit analytic solution of the Thomas-Fermi equation is given". This claim is surprising because at first sight earlier analytical approaches are apparently simpler and seem to have produced much more accurate results [10][11][12][13]. Recently, Khan and Xu [17] improved Liao's HAM by the addition of adjustable parameters that improve the convergence of the perturbation series.…”

“…Present rational approximation to the TF function is completely different from previous application of the Padé approximants, where the slope at origin was determined by the asymptotic behaviour of u(x) at infinity [13]. Our approach applies to u(x) 1/2 and the slope at origin is determined by a local condition at that point (9) which results in the Hankel determinant (10).…”

“…In order to have the correct behaviour at infinity we choose N = M + 3 [13]. Table 1 shows values of u(x) and its first derivative for 1 < x < 1000 (the approximation is obviously much better for 0 < x < 1) given by the approximant [5/8].…”

“…The purpose of this Letter is to compare the improved HAM with a straightforward analytical procedure based on Padé approximants [13] supplemented with a method developed some time ago [18][19][20][21][22][23][24][25][26]. In Section 2 we outline the main ideas of the HAM, in Section 3 apply the Hankel-Padé method (HPM) to the TF equation, and in Section 4 we compare the HAM with the HPM and with other approaches.…”