Is there an explicit set of solutions for the equation:
$$ \cot(x) = \frac 1 2 \left ( x - \frac 1 x \right ) $$ ?
I am reading a paper in which they state that this is standard without giving solutions neither references.
Thank you all very much in advance,
Looking for the zeros of function $$f(x)=\cot(x) - \frac 1 2 \left ( x - \frac 1 x \right )$$ is the same as looking for the zeros of function $$g(x)=\left(x^2-1\right) \sin (x)-2 x \cos (x)$$ which is much better conditioned.
Beside the first root, the zeros of $g(x)$ are closer and closer to $n\pi$ because of the $x^2$ term. Using series expansion and power series reversion, $$x_ {(n)}=n\pi+\sum_{k=1}^\infty (-1)^{k+1} a_k\, t^k\qquad \text{where}\qquad t=\frac{2 \pi n}{\pi ^2 n^2-3}$$
The first coefficients are $$\left\{1,\frac{3 \pi n}{\pi ^2 n^2-3},\frac{\pi ^4 n^4+92 \pi ^2 n^2+39}{6 \left(\pi ^2 n^2-3\right)^2},\frac{5 \pi n \left(5 \pi ^4 n^4+234 \pi ^2 n^2+225\right)}{12 \left(\pi ^2 n^2-3\right)^3}\right\} $$
Using only the terms given in the table, some results
$$\left( \begin{array}{ccc} n & \text{estimate} & \text{solution} \\ 2 & 6.5800222 & 6.5846200 \\ 3 & 9.6314973 & 9.6316846 \\ 4 & 12.723216 & 12.723241 \\ 5 & 15.834100 & 15.834105 \\ 6 & 18.954970 & 18.954971 \\ 7 & 22.081659 & 22.081660 \\ 8 & 25.212027 & 25.212027 \\ \end{array} \right)$$