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Math Help - Finding the residue of the following function

  1. #1
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    Finding the residue of the following function

    Hi all, I have the following function f(z) = \frac{1}{(z^2 + z + 1)^2} and I need to find the residue at the double pole z = (-1)^{\frac{2}{3}}. Any tips? I'm hoping to not have to use the Laurent expansion for this. I know that (z^2 + z + 1) = \frac{(1-z)}{(1-z^3)}. I'm aware of a limit formula for higher order poles but I don't know how to manipulate the function to get it in that form. Maybe that's not even needed. Thanks!
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  2. #2
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    Re: Finding the residue of the following function

    Quote Originally Posted by director View Post
    Hi all, I have the following function f(z) = \frac{1}{(z^2 + z + 1)^2} and I need to find the residue at the double pole z = (-1)^{\frac{2}{3}}. Any tips? I'm hoping to not have to use the Laurent expansion for this. I know that (z^2 + z + 1) = \frac{(1-z)}{(1-z^3)}. I'm aware of a limit formula for higher order poles but I don't know how to manipulate the function to get it in that form. Maybe that's not even needed. Thanks!
    Note that the (double) roots of f(z) are \frac{-1\pm i\sqrt{3}}{2} (which are equivalent to (-1)^{2/3}). Set z_1=\frac{-1+ i\sqrt{3}}{2} and z_2=\frac{-1- i\sqrt{3}}{2}, then Res(f,z_1) = \frac{1}{(n-1)!} \lim_{z->z_1}\frac{d^{n-1}}{dz^{n-1}}(z-z_1)^2 f(z) where n=2. Similarily for z_2.
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  3. #3
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    Re: Finding the residue of the following function

    I got the answer, thanks!
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