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Math Help - laurent series and residues

  1. #1
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    laurent series and residues

    Hi
    Any help would be greatful!

    The question is: Find and classify all singularities of the following functions and where appropriate, find the residues?

    (c) exp(z)/(z-1)^2
    the singularity is at z=1 how do i calculate the residue via laurent series.

    (d) (z-i)^(-3/2)
    the singularity is at z=i how do i calculate the residue via laurent series.

    Thanks
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  2. #2
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    Quote Originally Posted by jules View Post
    (c) exp(z)/(z-1)^2
    the singularity is at z=1 how do i calculate the residue via laurent series.
    Why use Laurent here? Notice that z=1 is a double pole. To find the residue define f(z) = (z-1)^2 \cdot \frac{e^z}{(z-1)^2} = e^z and compute \frac{1}{(2-1)!}\cdot f'(1) = e.


    (d) (z-i)^(-3/2)
    the singularity is at z=i how do i calculate the residue via laurent series.
    Do it with derivatives like I did above. What what of pole is this? Meaning what is smallest n so that \lim_{z\to i} (z-i)^n (z-i)^{-3/2} exists?
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