Expressing the solutions of the equation $ \tan(x) = x $ in closed form.

I know that the equation $ \tan(x) = x $ can be solved using numerical methods, but I’m looking for a closed form of the solutions. In my opinion, having only numerical solutions means that we don’t know the problem, and sooner or later, we’ll be able to find a closed-form solution or at least a power-series solution.

I’m looking for an explicit form of a sequence $ (x_{n})_{n \in \mathbb{Z}} $ in $ \mathbb{R} $ such that

  • $ 0 < x_{0} < \dfrac{\pi}{2} $,
  • $ \tan(x_{n}) = x_{n} $ for all $ n \in \mathbb{Z} $, and
  • $ (2 n – 1) \dfrac{\pi}{2} < x_{n} < (2 n + 1) \dfrac{\pi}{2} $ for all $ n \in \mathbb{Z} \setminus \{ 0 \} $.

The existence of such a sequence follows from the continuity of $ x \mapsto \tan(x) – x $ over its domain. Its uniqueness follows from the monotonicity of $ \tan(x) – x $ over the intervals
$$
\left( (2 n – 1) \frac{\pi}{2},(2 n + 1) \frac{\pi}{2} \right), \quad n \in \mathbb{Z}.
$$

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As Hagen has succinctly mentioned in his comment above, whether an equation has a closed-form solution or not depends on the functions that you wish to admit as ‘elementary’. Questions about the existence of closed-form solutions are typically answered using differential Galois theory.

I thus cannot answer your question, but I can at least quote closed-form formulas for some infinite sums whose terms are fixed negative-integer powers of the positive real solutions of $ \tan(x) = x $.

Theorem: If $ (\lambda_{n})_{n \in \mathbb{N}} $ denotes the sequence of positive real solutions of $ \tan(x) = x $ in increasing order, then
\begin{align}
\sum_{n=1}^{\infty} \frac{1}{\lambda_{n}} &= \infty, \\
\sum_{n=1}^{\infty} \frac{1}{\lambda_{n}^{2}} &= \frac{1}{10}, \\
\sum_{n=1}^{\infty} \frac{1}{\lambda_{n}^{4}} &= \frac{1}{350}.
\end{align}

Reference

L. Hermia & N. Saito. On Rayleigh-Type Formulas for a Non-local Boundary Value Problem Associated with an Integral Operator Commuting with the Laplacian, preprint submitted to Journal of Mathematical Analysis and Applications (2010).