Large roots near tangent poles ($x_n\sim (n+\frac{1}{2})\pi -[\mu (n+\frac{1}{2})\pi {]}^{-1}$)

For fixed real $\mu \ne 0$, roots of $\tan x=\mu x$ tending to positive infinity approach tangent poles $a_n=(n+1/2)\pi$. Expanding $-\cot (x-a_n)$ givesThe expansion works for either sign of $\mu$ when $|\mu |a_n\gg 1$, but is nonuniform at $\mu =0$, where the roots lie at $n\pi$ instead. The pole labeling need not equal the ordered root index at small $n$.