# domain of convergence of a multivariable taylor series

consider the rational function :

$$f(x,z)=\frac{z}{x^{z}-1}$$ $x\in \mathbb{R}^{+}/[0,1]\;\;$, $z\in \mathbb{C}\;\;$ .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 $$f_{0}(x)=\frac{1}{x-1}$$

$$f_{1}(x)=\frac{x-x\ln x -1}{(x-1)^{2}}$$ in general : $$f_{n}(x)=\frac{(-\ln x)^{n-1}}{n!}\left(n\text{Li}_{1-n}(x^{-1})-\ln x\;\text{Li}_{-n}(x^{-1})\right)$$ Where $\text{Li}_{m}(x)$ is the Polylogarithm function of the mth order. 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 domain of convergence for such an expansion !?

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