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Fourier-Taylor series for the figure eight solution of the three body problem

We provide an analytical approximation of a periodic solution of the three body problem in celestial mechanics, the so-called figure eight solution, discovered 1993 by C. Moore. This approximation has the form of a Fourier series whose components are in turn Taylor series w. r. t. some parameter. The method is first illustrated by application to two other problems, (1) the problem of oscillations of a particle in a cubic potential that has a well-known analytic solution in terms of elliptic functions and (2) periodic solutions and corresponding eigenvalues of a generalized Mathieu equation that cannot be solved analytically. When applied to the three body problem it turns out that the Fourier-Taylor series, evaluated up to 30th order, represents un-physical solutions except for a particular value of the series parameter. For this value the series approximates the numerical solution known from the literature up to a relative error of $1.6\times 10^{-3}$.