Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 !?

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.