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

In summary, the conversation discusses how to calculate the residue of the function ##\frac{1}{(x^4+1)^2}## and the difficulty in using the standard residue formula due to the double pole. The conversation suggests using the limit formula for higher order poles, but notes that it does not make the algebra any easier. Ultimately, the person has figured out the solution on their own.
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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!
 
Physics news on Phys.org

1. What is the residue of a double pole?

The residue of a double pole is the value that is left over after a function has been integrated over a closed path and any singularities within that path have been accounted for.

2. How is the residue of a double pole calculated?

The residue of a double pole can be calculated by taking the limit of the function as it approaches the pole, multiplied by the difference between the function and the pole.

3. What is the significance of the residue of a double pole in complex analysis?

The residue of a double pole is important in complex analysis because it helps us to determine the behavior of a function near a pole, and can be used to calculate integrals over closed paths.

4. Can the residue of a double pole be negative?

Yes, the residue of a double pole can be negative. This occurs when the pole is located in the lower half plane of the complex plane.

5. How is the residue of a double pole used in practical applications?

The residue of a double pole is used in a variety of practical applications, such as calculating the residues of poles in circuit analysis, evaluating complex integrals, and solving differential equations in physics and engineering.

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