# Integral representation of Thomas-Fermi Equation

The Thomas-Fermi equation with dimensionless variables is identified as; $$\frac{d^2\phi}{dx^2} = \frac{\phi^{3/2}}{x^{1/2}}$$ with the boundary conditions as $$\phi(0) = 1 \\ \phi(\infty) = 0.$$ There are many series approximation solutions available to this equation. I am interested in finding the integral representation for the solution of this differential equation. Are there any literature sources available in this direction?

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## 1 Answer

You have lectures notes on Density Functional Theory by Andrei Postnikov that may answer your question, in particular chapter 1 eq (1.16) http://www.home.uni-osnabrueck.de/apostnik/lectures.html it is

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