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Finding residue of a function(complex analysis)

  1. May 12, 2012 #1
    1. The problem statement, all variables and given/known data
    I need to find the residue of [itex]\frac{e^{iz}}{(1+9z^{2})^{2}}[/itex] so I can use it as part of the residue theorem for a problem.


    2. Relevant equations
    Laurent Series
    R(z[itex]_{0}[/itex]) = [itex]\frac{g(z_{0})}{h^{'}(z_{0})}[/itex]


    3. The attempt at a solution
    I tried using the laurent series but after expanding I got a 0 in the denominator.
    For the second equation I used I also got a 0 in the denominator and I don't believe it converges. Any help would be appreciated. Thanks!
     
  2. jcsd
  3. May 12, 2012 #2
    Look at the inverse of the function (call it f(z)), you have a zero of order 2. So you have a pole of order 2 for your function. The formula for this goes like


    ## \text{Res}(f,i/3) = \lim_{z \to a}\frac{d}{dz}\left( (z-i/3)^2 \frac{\exp (i z)}{(1+9z^2)^2} \right) ##

    you can do the others

    ## \text{Res}(f,a) = \lim_{z \to a} \frac{1}{(m-1)!}\frac{d^{m-1}}{dz^{m-1}} (z-a)^m f(z) ##

    If the pole is of order m. I think. Maybe look that up.
     
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