Deriving the Pollaczek - Khintchine formula

Recall that $g^*(s)=E[e^{-sx}]$ and that the Taylor series for $g^*(s)$ around $s=0$ is given by $g^*(s)=g^*(0)+\frac{\mathrm{d}g^*}{\mathrm{d}s}|_{s=0}+\frac{1}{2}\frac{\mathrm{d}^2g^*}{\mathrm{d}s^2}|_{s=0}+...$

It is a straightforward exercise to calculate the first few terms of the Taylor series of $g^*(s)$; $g^*(s)=1-\mu(s)+\frac{\mu^2(1+c^2)}{2}s^2 + O(s^3)$.

From this we get $h(s)=1-\frac{\mu s}{2}(1+c^2) + O(s^2)$ and $\frac{\mathrm{d}h}{\mathrm{d}s}=\frac{-\mu}2(1+c^2)+O(s)$.

We know the Laplace-Stieltjes transform (or "moment generating function") $M(s)=\frac{1-\rho}{1-\rho h^*(s)}$ and, substituting in these series for $h^*$ and $\frac{\mathrm{d}h^*}{\mathrm{d}s}$, we have that $\frac{\mathrm{d}M}{\mathrm{d}s}=\frac{\rho(\rho-1)\frac{\mathrm{d}h^*}{\mathrm{d}s}}{(1-\rho h)^2}$ can be approximated as $\frac{\mathrm{d}M}{\mathrm{d}s}=\frac{-\rho(1-\rho)\frac{\mu}{2}(1+c^2)+O(s)}{(1-\rho)^2+O(s)}$. When $s=0$ we can conclude that $\frac{\mathrm{d}M}{\mathrm{d}s}=\frac{-\rho\mu(1+c^2))}{2(1-\rho)}$.