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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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Related: math.stackexchange.com/q/806472/11127 –  Qmechanic May 26 '14 at 18:00

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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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