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I have seen in wikipedia, that how the Cauchy's pricnipal value and Hadamard's finite part work in dimension one

My question is when can we (or if negative answer why can not ) generalize the Hadamard's finite part and Cauchy's principal value to include integration over several variables: surface integral volume integral and so on ?

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  • $\begingroup$ Move this question to math.stackexchange.com $\endgroup$
    – user21574
    Commented May 31, 2014 at 11:33
  • $\begingroup$ Both Cauchy's and Hadamard's prescriptions are ways of turning divergent integrals into well-defined distributions (aka generalized functions). Such regularizations are often treated in the mathematical literature under the name 'extension of distrubtions'. The solutions are highly non-unique, so the right one depends on the application in mind. A classic treatment can be found in Gel'fand & Shilov, Generalized Functions (vol.1). $\endgroup$
    – Igor Khavkine
    Commented May 31, 2014 at 15:19

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