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Complex Analysis Residues at Poles

  1. Dec 14, 2011 #1
    1. The problem statement, all variables and given/known data

    Find the residue at each pole of zsin(pi*z)/(4z^2 - 1)


    2. Relevant equations

    An isolated singular point z0 of f is a pole of order m if and only if f(z) can be written in the form:

    f(z) = phi(z)/(z-z0)^m

    where phi(z) is analytic and nonzero at z0. Moreover,

    Res(z=z0) f(z) = phi(z0) if m = 1

    and

    Res(z=z0) f(z) = phi^(m-1)(z0)/(m-1)! if m >= 2



    3. The attempt at a solution

    I don't know what I'm missing here, the problem seems really easy. I factored it to

    z*sin(pi*z)/[(2z+1)(2z-1)]

    so f(z) has simple poles at z = 1/2 and z = -1/2

    For z = 1/2, we have f(z) = phi(z)/(z-1/2) where phi(z) = z*sin(pi*z)/(z+1/2)

    Plugging in z = 1/2 in phi(z) I get a residue of 1/4.

    Similarly, I get a residue of -1/4 at the pole of z = -1/2.

    But the answer is -1/8 and 1/8 for the residues respectively, and I can't figure out what I'm doing wrong.
     
  2. jcsd
  3. Dec 14, 2011 #2

    micromass

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    I do not see where your phi comes from. You seem to have done

    [tex](2z+1)(2z-1)=(z+1/2)(z-1/2)[/tex]

    which is not true. You are missing a factor 4 there.

     
  4. Dec 14, 2011 #3
    Ohh haha that was stupid. Thanks!
     
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