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Math Help - radius of convergence problem (Taylor expansion of a multivariable function)

  1. #1
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    radius of convergence problem (Taylor expansion of a multivariable function)

    consider the rational function :
    f(x,z)=\frac{z}{x^{z}-1}
    x\in \mathbb{R}^{+}/[0,1]
    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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  2. #2
    MHF Contributor
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    Re: radius of convergence problem (Taylor expansion of a multivariable function)

    Hey mmzaj.

    Have you looked at standard complex analytic techniques to get a taylor series expansion through differentiation?
    If the function is entire then all derivatives will exist and you can get a series expansion by using the standard definition of calculating a derivative and evaluate that at the right point.

    If there are only a fixed number of singularities for the mapping (in terms of the number of points that produce infinities or non-sensical values) then you can still obtain the series.
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  3. #3
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    Re: radius of convergence problem (Taylor expansion of a multivariable function)

    Hey chiro
    we can easily prove that the functions f_{n}(x) have the general form :
    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)
    and the question remains : what's the domain of convergence of such an expansion !?
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  4. #4
    MHF Contributor
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    Re: radius of convergence problem (Taylor expansion of a multivariable function)

    Have you tried using the supremum norm to find the radius and centre of convergence?

    Also as pointed out earlier, have you also tried to obtain an analytic power series for that function?
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