Regularizing a divergent integral with a test function or convergence factor

In summary, the authors of the paper are using test functions, which are similar to convergence factors, to make integrals of previously unintegrable functions. This is considered acceptable depending on the context and there are theorems that determine its validity. The paper does not provide any references for these theorems. The use of test functions is often used in larger integrals, such as in quantum field theory, where they can be rearranged to make the integration process easier. However, these test functions are technically called "distributions" and do not converge on their own, but are valid in the larger integral due to the dampening effect of the rest of the integral.
  • #1
graphicsRat
3
0
I'm trying to understand a paper in which the authors use a number of test functions (are they the same as convergence factors) to make integrate unintegrable functions. Now here is my ignorant question: why is this acceptable? The product of the original function and the test function or convergence factor surely is an entirely new function. Why is the integral of this new function the same as the integral of the old function which I recall was unintegrable? In layperson terms, can we be sure that the integral of the new function is valid and or correct?

Thanks
 
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  • #2
graphicsRat said:
… why is this acceptable?

In layperson terms, can we be sure that the integral of the new function is valid and or correct?

Hi graphicsRat! :smile:

It depends on the context …

usually this is done to a subsidiary integral in the middle of a much larger integration …

there are theorems that tell you whether it's justified or not …

doesn't the paper give any references?
 
  • #3
tiny-tim said:
It depends on the context …

usually this is done to a subsidiary integral in the middle of a much larger integration …

there are theorems that tell you whether it's justified or not …

doesn't the paper give any references?

Thanks Tim.

No the paper didn't give any references. Which theorems are you referring to? I'd like to look them up. What do you mean by a "subsidiary integral in the middle of a much larger integration"?

I'm sorry to ask so many annoying questions.
 
  • #4
graphicsRat said:
Which theorems are you referring to?

dunno … i wasn't really concentrating when they went over that boring stuff :redface:
What do you mean by a "subsidiary integral in the middle of a much larger integration"?

in quantum field theory, there are really long integrals, and you can rearrange the order of the variables so as to get some easy integrals in the middle :smile:

(btw, this rearranging involves combinatorial additions of integrals, and Feynman diagrams are what keep track of the different combinations)

but those easy integrals are of functions (technically called "distributions", such as the Dirac delta "function" … see http://en.wikipedia.org/wiki/Distribution_(mathematics)) which don't actually converge on their own, because they oscillate, but are ok in the larger integral because the oscillations are "damped down" by the rest of the long integral
 

1. What is regularizing a divergent integral?

Regularizing a divergent integral refers to the process of modifying a divergent integral so that it becomes convergent. This is often done by multiplying the integrand with a test function or convergence factor.

2. Why do we need to regularize divergent integrals?

Divergent integrals arise when the integrand approaches infinity or becomes undefined at certain points. This makes it impossible to evaluate the integral using traditional methods. Regularizing the integral allows us to assign a finite value to it and make it convergent, thus allowing for a meaningful solution.

3. What is a test function in the context of regularizing a divergent integral?

A test function is a function that is used to modify the integrand in order to make a divergent integral convergent. It is usually chosen to have certain properties, such as being smooth and rapidly decreasing, to ensure that the modified integral still has a meaningful solution.

4. How does a convergence factor regularize a divergent integral?

A convergence factor is a function that is multiplied with the integrand to make it convergent. This factor is chosen in such a way that it cancels out the divergent behavior of the integrand, resulting in a finite value for the integral. It is often used when the integrand has a singularity at a certain point.

5. Can any divergent integral be regularized?

No, not all divergent integrals can be regularized. The success of regularizing a divergent integral depends on the type and behavior of the integrand. In some cases, it may not be possible to find a suitable test function or convergence factor to make the integral convergent. In these cases, other techniques such as analytic continuation may be used to evaluate the integral.

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