Complex integral in finite contour at semiaxis

[Enrike]

Hi,

I have a difficult time trying to perform the following integral,

$$ j({T}, \Omega)=\int_0^{ T} d\tau \frac{\tau^2\exp(-i\Omega\tau)}{(\tau-i\epsilon)^2(\tau+i\epsilon)^2} $$

The problem is that the poles $\pm i\epsilon$ when taking the limit $\epsilon\rightarrow 0$ are located at the very origin of the contour of integrations $\tau\in(0,{T})$. I have to find the dependence on the "total time" ${T}$ so I have to maintain a finite interval in the integral and I can´t no make the limit ${ T}\rightarrow\infty$ from the beginning.

Please, any suggestion would be very appreciated.

Sincerely.

 

[jvicens]

Hi there,

Thank you for reaching out with your question. It seems like you are trying to evaluate a complex integral involving poles at the origin. This can be a challenging task, but there are some techniques that can help you out.

One approach you can take is to use the Residue Theorem. This theorem states that if you have a closed contour integral around a function with isolated singularities, the integral is equal to 2πi times the sum of the residues of the singularities inside the contour.

In your case, you can choose a contour that encloses the origin and evaluate the residues at the poles. This will give you a finite value for your integral, even when taking the limit ε→0.

Another approach you can try is to use the Cauchy Principal Value. This method involves splitting the integral into two parts, one around the origin and one away from the origin. Then, you can take the limit ε→0 for the integral around the origin and evaluate the other integral separately. This can help you avoid the issue of the poles being located at the origin.

I hope these suggestions are helpful in solving your integral. Good luck with your research!