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Radius of convergence problem 
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#1
Nov612, 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 Bernoullitype 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)(z1)^{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}{x1}[/tex] [tex] f_{1}(x)=\frac{xx\ln x 1}{(x1)^{2}}[/tex] now i have two questions : 1in 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 !? 


#2
Nov912, 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)^{n1}}{n!}\left(n\text{Li}_{1n}(x^{1})\ln x\;\text{Li}_{n}(x^{1})\right) [/tex] 


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