Given is the following equation:
$\cot(w \tau) = w$
Using the taylor series of cotangens, one can approximate
$\cot(w \tau) \approx \frac{1}{w \tau} - w \tau$
under the assumption $w \tau \ll 1$.
Thus, we get
$w \approx \frac{1}{w \tau} - w \tau$,
which can be solved to
$w \approx \frac{1}{\sqrt{\tau (\frac{1}{3} \tau + 1)}}$,
considering only the positive solution.
My question:
The equation $\cot(w \tau) = w$ has infinitely many solutions $w(\tau)$. With the approximation above, we get only one relation. Is there any possibility to gain Information on the other solutions? Numerically, I can find that their algebraic structure is similar to $w \approx \frac{1}{\sqrt{\tau (\frac{1}{3} \tau + 1)}}$.