$$g(w) = (4+2/w) \tan \left( \frac{\pi}{4+2/w} \right)$$
which has a series at $w=0$ of
$$g(w) = \pi +\frac{\pi ^3 w^2}{12}-\frac{\pi ^3 w^3}{3}+\left(\pi ^3+\frac{\pi ^5}{120}\right) w^4+ \cdots$$
But I don't know how to derive the series form around w=0 for g(w) as shown above. The result is from wolfram alpha and from this Laurent Series for Trig functions previous post I made. A detailed derivation is greatly appreciated.