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Math Help - Laurent expansion of principal root

  1. #1
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    Laurent expansion of principal root

    How do I find the Laurent expansion of a function containing the principal branch cut of the nth root?

    Example:
    f(z)=-iz\cdot\mathrm{pv}\sqrt[4]{1-\frac{1}{z^{4}}}
    Last edited by bernardbb; May 2nd 2009 at 07:52 AM.
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  2. #2
    MHF Contributor chisigma's Avatar
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    Remembering the binomial series expansion...

    (1+x)^{\alpha}= \sum_{k=0}^{\infty} \frac{\alpha\cdot (\alpha-1)\dots (\alpha-k+1)}{k!}\cdot x^{k}

    ... for x=-\frac{1}{z^{4}} and \alpha=\frac{1}{4} we have...

    (1-\frac{1}{z^{4}})^{\frac{1}{4}}= 1 - \frac{1}{4}\cdot z^{-4} - \frac{3}{4\cdot 4\cdot 2!}\cdot z^{-8} - \frac{3\cdot 7}{4\cdot 4\cdot 4\cdot 3!}\cdot z^{-12} + \dots

    ... and then...

    f(z)= -i\cdot z \cdot (1-\frac{1}{z^{4}})^{\frac{1}{4}}= -i\cdot (z - \frac{1}{4}\cdot z^{-3} - \frac{3}{4\cdot 4\cdot 2!}\cdot z^{-7} - \frac{3\cdot 7}{4\cdot 4\cdot 4\cdot 3!}\cdot z^{-11} + \dots)

    Kind regards

    \chi \sigma
    Last edited by chisigma; May 2nd 2009 at 10:00 AM. Reason: added factorials
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  3. #3
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    Thanks, that seems helpful.
    I think you forgot the (k!)?
    Last edited by bernardbb; May 3rd 2009 at 10:11 AM.
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  4. #4
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    Edit: Nevermind, figured it all out.
    Last edited by bernardbb; May 4th 2009 at 03:27 PM.
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