Can you provide me by means examples, one or two examples, or an example and hints to get a different example/application, an explanation of the mathematical meaning of the Hadamard regularization?

Question. For example, can you provide me an explicit example of the second paragraph in Wikipedia, I say the differentiation of a Cauchy Principal value? Many thanks.

I am asking it, because I believe that there are few references in this site, and I want to learn about this tool.

If you know what are the main applications in analytic number theory, I say in few words a view from the top of feasible applications of this regularization in analytic number theory, please tell me as an addemdum of your answer. I am asking this because perhaps it is obvious if one read previous article, but I don't know how interpret such information. That I am saying is that if you know how express it in easy words as: 'well, the regularization is useful to do this... in analytic number theory'. Of course is welcome an explanation more detailed in few paragraphs. I want to encourage this addemdum, as companion of previous example(s). If it ins't interesting in analytic number theory tell us.


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