On the Majorana solution to the Thomas-Fermi equation

TL;DR

This paper examines Majorana's solution to the Thomas-Fermi equation, demonstrating that a series expansion for the slope at the origin yields highly accurate results and analyzing the series' convergence properties.

Contribution

The paper introduces a series expansion method for solving the Thomas-Fermi equation that surpasses existing techniques in accuracy and provides insights into the series' convergence and singularity structure.

Findings

Series for the slope at origin achieves high accuracy

Estimated the radius of convergence of the series

Conjectured the nature of the nearest singularity as a square-root branch point

Abstract

We analyse the solution to the Thomas-Fermi equation discovered by Majorana. We show that the series for the slope at origin enables one to obtain results of accuracy far beyond those provided by available methods. We also estimate the radius of convergence of this series and conjecture that the singularity closest to origin is a square-root branch point.