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

In summary, changing the limits of integration, using a change of variable, and adding or subtracting a constant from the integrand can sometimes make a non-convergent integral finite. There are also specific techniques and methods, such as integration by parts and partial fraction decomposition, that can be used to achieve this. However, a non-convergent integral cannot have a finite value without any modifications.
  • #1
Emmanuel Ortiz
2
0
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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  • #2
Something must be missing. Which of those F, k, k0 things are functions of x?
 
  • #3
Sorry is dk, I just corrected it
 

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

1. Can changing the limits of integration make a non-convergent integral finite?

Yes, changing the limits of integration can sometimes make a non-convergent integral finite. This is because the behavior of the function being integrated may differ at different limits, and choosing appropriate limits of integration can lead to a finite result.

2. Is it possible to use a change of variable to make a non-convergent integral finite?

Yes, using a change of variable can sometimes make a non-convergent integral finite. This is because a change of variable can change the behavior of the integrand and lead to a finite result.

3. Can adding or subtracting a constant from the integrand make a non-convergent integral finite?

Yes, adding or subtracting a constant from the integrand can sometimes make a non-convergent integral finite. This is because adding or subtracting a constant can change the behavior of the integrand and lead to a finite result.

4. Are there specific techniques or methods that can be used to make a non-convergent integral finite?

Yes, there are specific techniques and methods that can be used to make a non-convergent integral finite. These include using integration by parts, partial fraction decomposition, and other advanced integration techniques.

5. Is it possible for a non-convergent integral to have a finite value without any modifications?

No, a non-convergent integral cannot have a finite value without any modifications. The definition of a non-convergent integral is that it does not have a finite value, and some kind of modification or intervention is necessary to make it finite.

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