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Is this the answer for what is the residue of f(z)=e^z/[(z-1)^2(z+2)]

  1. Jun 15, 2007 #1
    1. The problem statement, all variables and given/known data
    This was an exam question I had today:

    Give the location and order of each pole of
    f(z)=e^z/[(z-1)^2(z+2)]
    and evaluate the reside at these points


    2. Relevant equations

    Res(f(z)) = g(p-1)(a) / (p-1)!



    3. The attempt at a solution

    g(z) = e^z/(z+2)
    g'(z) = -e^z/(z+2)^2
    let a=1
    g'(1) = -e^1/9

    So res(f,1) = -e/9

    Then I did
    g(z) = e^z/(z-1)^2
    (this is the part I think I might have stuffed up, because I did use the formula above but I had m=1 for f(z) which was 1/(z+2) so I thought it was right??? anyways I got

    g(-2) = e^(-2) / (-3)^2 = res(f,-2)=1/(9e^2)

    It would mean a lot to me if someone could let me know because I am trying to work out if I should be depressed because I failed or if I can be happy because I passed :)
     
  2. jcsd
  3. Jun 15, 2007 #2
    At first glance I agree with your first answer for res(f,-2) but not for res(f,1).

    [tex]
    g\left( z \right) = \frac{{e^z }}{{\left( {z + 2} \right)}} \Rightarrow g'\left( z \right) = \frac{{e^z }}{{\left( {z + 2} \right)}} - \frac{{e^z }}{{\left( {z + 2} \right)^2 }}
    [/tex]
     
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