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Residue of double pole

  1. Jun 13, 2016 #1
    1. The problem statement, all variables and given/known data
    How would I calculate the residue of the function

    ##\frac{1}{(x^4+1)^2}##

    2. Relevant equations


    3. The attempt at a solution
    So I have found that the poles are at

    ##z=e^{\frac{i \pi}{4}}##
    ##z=e^{\frac{3i \pi}{4}}##
    ##z=e^{\frac{5i \pi}{4}}##
    ##z=e^{\frac{7i \pi}{4}}##

    I tried calculating this by finding its laurent series around each of the poles, but it was very algebraically heavy and I could not get the correct answer of

    ## \frac{3}{16 \sqrt{2}}+\frac{3}{16 \sqrt{2}}i##

    I feel there must be an easier way. The standard residue formula also does not work here (i'm assuming because its a double pole)

    Any help would be extremely appreciated!!
     
  2. jcsd
  3. Jun 13, 2016 #2

    vela

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  4. Jun 13, 2016 #3
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