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Homework Help: Residue of e^(az)/(1+e^z)^2 at I Pi

  1. Apr 1, 2010 #1
    1. The problem statement, all variables and given/known data

    I need to find the residue of e^(az)/(1+e^z)^2 at I Pi. For some reason this is such much harder than I thought it was going to be. Mathematica is not even helping :(.

    2. Relevant equations

    Cauchy's kth Integral formula.

    3. The attempt at a solution

    I made an attempt at doing a u substitution of u=e^z, but I ended up with a residue of 0, which was not what I was expecting. All this lead me to believe that you probably can't use u-substitution strategies with line integrals and expect them to work.
     
  2. jcsd
  3. Apr 1, 2010 #2

    vela

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    Try expanding the function as a Laurent series. Use the fact that

    [tex]e^{az} = e^{a(z-i\pi+i\pi)} = e^{ia\pi}e^{a(z-i\pi)}[/tex]

    Don't forget to expand both exponentials (in the original function) as series. Remember that all you're interested in is the coefficient of 1/(z-iπ), so just concentrate on the terms that will contribute to that.
     
    Last edited: Apr 1, 2010
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