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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 !?

[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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