Calculating the Residue of ##\frac{1}{(x^4+1)^2}## at Double Poles

[Physgeek64]

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!

 

[vela]

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)

See https://en.wikipedia.org/wiki/Residue_(complex_analysis)#Limit_formula_for_higher_order_poles

 

[Physgeek64]

See https://en.wikipedia.org/wiki/Residue_(complex_analysis)#Limit_formula_for_higher_order_poles

I've attempted to use this, but it doesn't make the algebra any easier. Its okay- I've figured it out now anyway :) Thank you though :)