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I was following the proof of the Bertrand theorem on Wikipedia, which is based on Goldstein "Classical mechanics" (2nd edition). The explanation was clear upto Eq (3). But then it assumes Fourier series expansion in the form $\eta(\theta)=\sum_{n=0}^\infty h_n\cos(n\beta\theta)$. My question is how to verify this form.

This frequency $\beta$ is derived from the analysis of an infinitesimal deviation from the perfect circular motion. What I do not understand is the assumption that more general deviations from the circle always have to come with the same period. I also wonder why $\sin(n\beta\theta)$ terms are absent.

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