P: 99 consider the rational function : $$f(x,z)=\frac{z}{x^{z}-1}$$ $$x\in \mathbb{R}^{+}$$ $$z\in \mathbb{C}$$ We wish to find an expansion in z that is valid for all x and z. a Bernoulli-type expansion is only valid for : $$\left | z\ln x \right |<2\pi$$ Therefore, we consider an expansion around z=1 of the form : $$\frac{z}{x^{z}-1}=\sum_{n=0}^{\infty}f_{n}(x)(z-1)^{n}$$ Where $f_{n}(x)$ are suitable functions in x that make the expansion converge. the first two are given by : $$f_{0}(x)=\frac{1}{x-1}$$ $$f_{1}(x)=\frac{x-x\ln x -1}{(x-1)^{2}}$$ now i have two questions : 1-in the literature, is there a similar treatment to this specific problem !? and under what name !? 2- how can we find the radius of convergence for such an expansion !?
 P: 99 it's not so hard to prove that the functions $f_{n}(x)$ have the general form : $$f_{n}(x)=\frac{(-\ln x)^{n-1}}{n!}\left(n\text{Li}_{1-n}(x^{-1})-\ln x\;\text{Li}_{-n}(x^{-1})\right)$$