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I would like to find a two-term or a three-term asymptotic expansion of $x_{n}$ the unique solution of $x_{n}=\frac{1}{\tan(x_{n})}$ on the interval $]n\pi,n\pi+\pi[ $

We have: $ x_{n}=n\pi+\arctan(\frac{1}{x_{n}})$

So $x_{n} \sim_{n\rightarrow \infty} n\pi$

What is the method to find the next terms of the asymptotic expansion?

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    Thanks! I should have thought more before asking...2012-05-28

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