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Homework Help: Principle part of Laurent series

  1. Feb 12, 2009 #1
    1. The problem statement, all variables and given/known data

    Find the Principle part of the Laurent Expansion of f(z) about z=0 in the region
    0 < mod z < 1, where f(z) = exp(z) / [(z^2)*(z+1)]

    2. Relevant equations

    1/(1-z) = Summation (n = 0 to n = infinity) { z^n}


    3. The attempt at a solution

    First, by using partial fraction,
    I got f(z) = exp(z) {-1/z + 1/(z^2) + 1/(z+1)}

    Then f(z) = exp (z) {1/ (-1+1+z) + 1/ (-1+1+(z^2)) + (1/(z+1) }

    Since the question were only after the principle parts, so I ignore 1/(z+1) term

    Basically I need to evalute
    exp (z) { 1/ (-1+1+z) + 1/ (-1+1+(z^2)) }

    Is this step right?


    Then I tried to do the following,

    and I got something like

    exp (z) { - summation (1/(1+z))^(n+1) + summation (1/(1+z^2))^(n+1)}


    But is this right?

    Thanks a lot!
     
  2. jcsd
  3. Feb 12, 2009 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    You are making this harder than it is. Your function is 1/z^2 times e^z/(z+1). e^z/(z+1) is nonsingular at z=0. All you need is the first two terms in the series expansion of e^z/(z+1) around z=0. (Why only the first two?). Then multiply that series by 1/z^2.
     
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