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Thread: Radius of Convergence at a point?

  1. #1
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    Radius of Convergence at a point?

    Say I have a power series f(z) on the complex plane. Now I know what it means to find the radius of convergence, I know how to use the test, I know how to take the limit superior and use the formula. But what does it mean when one ask what is the radius of convergence of a power series at a particular point z, say z = 1+i?

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    MHF Contributor chisigma's Avatar
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    The taylor expansion of a complex $\displaystyle f(*)$ 'somewhere around' a point $\displaystyle z_{0}$ is...

    $\displaystyle f(z)= \sum_{n=0}^{\infty}\frac{f^{n}(z_{0})}{n!}\cdot (z-z_{0})^{n}$ (1)

    In general is $\displaystyle z_{0}= |z_{0}|\cdot e^{i\theta}$. If $\displaystyle z_{0}$ is not on real axis you can use the substitution $\displaystyle w=z\cdot e^{-i\theta}$ so that the series expansion (1) has only real terms. In particular the radious of cenvergence of the series for $\displaystyle f(z)$ and $\displaystyle f(w)$ is the same...

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    $\displaystyle \chi$ $\displaystyle \sigma$
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    So if a power series converges everywhere at a point say 0, then does that also means it converges everywhere at other points as well?
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    MHF Contributor chisigma's Avatar
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    If a power series converges 'everyhere around' a point $\displaystyle z_{0}$, that means by definition that it coverges 'everywhere around' any other point which is not $\displaystyle z_{0}$... obvious ...

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    $\displaystyle \chi$ $\displaystyle \sigma$
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