- #1
mmzaj
- 107
- 0
consider the rational function :
f(x,z)=\frac{z}{x^{z}-1}
x\in \mathbb{R}^{+}
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}}
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 !?
f(x,z)=\frac{z}{x^{z}-1}
x\in \mathbb{R}^{+}
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}}
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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