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Radius of convergence problem

by mmzaj
Tags: convergence, radius
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mmzaj
#1
Nov6-12, 06:19 PM
P: 99
consider the rational function :

[tex]f(x,z)=\frac{z}{x^{z}-1}[/tex]
[tex]x\in \mathbb{R}^{+}[/tex]
[tex]z\in \mathbb{C}[/tex]

We wish to find an expansion in z that is valid for all x and z. a Bernoulli-type expansion is only valid for :
[tex] \left | z\ln x \right |<2\pi[/tex]
Therefore, we consider an expansion around z=1 of the form :
[tex] \frac{z}{x^{z}-1}=\sum_{n=0}^{\infty}f_{n}(x)(z-1)^{n}[/tex]
Where [itex] f_{n}(x)[/itex] are suitable functions in x that make the expansion converge. the first two are given by :
[tex] f_{0}(x)=\frac{1}{x-1}[/tex]

[tex] f_{1}(x)=\frac{x-x\ln x -1}{(x-1)^{2}}[/tex]
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 !?
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mmzaj
#2
Nov9-12, 03:37 AM
P: 99
it's not so hard to prove that the functions [itex]f_{n}(x) [/itex] have the general form :
[tex]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) [/tex]


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