consider the rational function :(adsbygoogle = window.adsbygoogle || []).push({});

[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 !?

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Radius of convergence problem

Loading...

Similar Threads for Radius convergence problem | Date |
---|---|

Taylor series radius converge | Jul 15, 2012 |

Convergence radius of a perturbation series | Mar 29, 2012 |

Extending radius of convergence by analytic continuation | Mar 1, 2012 |

Complex Radius of Convergence (without limsup formula) | Feb 16, 2012 |

Question about radius of convergence of fractional power series | Dec 4, 2011 |

**Physics Forums - The Fusion of Science and Community**