I am looking for references that discuss solutions of the equation
$\tan x=-a x$
(for $x,a\in \mathbb{R}$). I know about the graphical approaches, and any number of numerical solution approaches, but I was wondering if there's a semi-closed form solution, which gives something like
$x = f(a,c)$
where $c$ is a (set of) numerically solved constants which are independent of $a$ (i.e. a set of constants corresponding to the set of roots).
I've come up with an approach (i.e. a function $f$), but I figure this has been solved already, so I'd prefer to reference that ``known'' solution.