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Math Help - laurent serie

  1. #1
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    laurent serie

    hi, i have to find out what is the principal part of the laurent serie of f(z)=z^4/(z^2-1) about infinity.
    is z^2 true?
    regards
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  2. #2
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    Re: laurent serie

    Yes.

    Let f(z) = \frac{z^4}{z^2 - 1}, let w = 1/z, z = 1/w, and get

    Let g(w) = f(1/w) = \frac{(1/w)^4}{(1/w)^2 - 1} = \frac{1}{w^2 - w^4} = \frac{1}{w^2}\frac{1}{1 - w^2}.

    = \frac{1}{w^2}\left(1 + (-w^2) + (-w^2)^2 + (-w^2)^3 + ... \right) = \frac{1}{w^2}\left(1 - w^2 + w^4 - w^6 +- ... \right)

    = \frac{1}{w^2} - 1 + w^2 - w^4 +- ... \right) which is holomorphic on 0<\lVert w \rVert < 1, hence after w \leftrightarrow 1/z, for 1 < \lVert z \rVert < \infty.

    That \frac{1}{w^2} is the principlal part of g(w) = f(1/w) at w = 0, and so is the principle part, after w \leftrightarrow 1/z, of f at infinity.

    Thus \frac{1}{(1/z)^2} = z^2 is the principle part of f at infinity.
    Last edited by johnsomeone; September 24th 2012 at 02:34 PM.
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