Regularizing a divergent integral with a test function or convergence factor

AI Thread Summary
The discussion centers on the use of test functions or convergence factors to regularize divergent integrals in mathematical contexts, particularly in quantum field theory. Participants question the validity of integrating a product of an unintegrable function and a test function, wondering if the resulting integral is correct. It is noted that this practice often occurs within larger integrals, where certain theorems justify the approach. The conversation highlights that these integrals can involve distributions, such as the Dirac delta function, which may not converge independently but are acceptable within the broader context. Overall, the validity of the integral depends on the specific mathematical framework and theorems applied.
graphicsRat
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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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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?
 
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.
 
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
 
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