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Residue of double pole

  • Thread starter Physgeek64
  • Start date
  • #1
247
11

Homework Statement


How would I calculate the residue of the function

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

Homework Equations




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!!
 

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