consider the rational function :

$\displaystyle f(x,z)=\frac{z}{x^{z}-1}$

$\displaystyle x\in \mathbb{R}^{+}/[0,1]$

$\displaystyle z\in \mathbb{C}$

We wish to find an expansion in z that is valid for all x and z. a Bernoulli-type expansion is only valid for :

$\displaystyle \left | z\ln x \right |<2\pi$

Therefore, we consider an expansion around z=1 of the form :

$\displaystyle \frac{z}{x^{z}-1}=\sum_{n=0}^{\infty}f_{n}(x)(z-1)^{n}$

Where $\displaystyle f_{n}(x)$ are suitable functions in x that make the expansion converge. the first two are given by :

$\displaystyle f_{0}(x)=\frac{1}{x-1}$

$\displaystyle f_{1}(x)=\frac{x-x\ln x -1}{(x-1)^{2}}$

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 !?