How to make this integral (which does not converge) be finite?

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    Finite Integral
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SUMMARY

The integral $$\int_{0}^{\infty} \frac{ k^4 e^{-2F^2k^2} }{ (k-k_0)^2 }dk$$ does not converge due to a double pole at ##k_0##. To address this issue, methods from Quantum Field Theory (QFT) can be employed to regularize divergent integrals. The discussion emphasizes the need for physical arguments, restrictions, approximations, or assumptions to render the integral finite. Participants suggest exploring various regularization techniques to achieve convergence.

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Emmanuel Ortiz
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I have to deal with this integral in my work, $$\int_{0}^{\infty} \frac{ k^4 e^{-2F^2k^2} }{ (k-k_0)^2 }dk$$ where ##F^2>0 , k_0>0## Is important to mention that it has a double pole in ##k_0## and as a consequence mathematically doesn’t converge. However I have seen before some methods in Quantum Field Theory to regularise divergent integrals with poles, unfortunally I have not had success in solving it. I’like to solve this integral in some reasonable way, perhaps with a physical argument, restrictions, approximation or under some assumptions. There exist a way to solve this Integral? Please help me.
 
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Something must be missing. Which of those F, k, k0 things are functions of x?
 
Sorry is dk, I just corrected it
 

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