Integrals | Summability methods

Hadamard regularization

In mathematics, Hadamard regularization (also called Hadamard finite part or Hadamard's partie finie) is a method of regularizing divergent integrals by dropping some divergent terms and keeping the finite part, introduced by Hadamard . Riesz showed that this can be interpreted as taking the meromorphic continuation of a convergent integral. If the Cauchy principal value integral exists, then it may be differentiated with respect to x to obtain the Hadamard finite part integral as follows: Note that the symbols and are used here to denote Cauchy principal value and Hadamard finite-part integrals respectively. The Hadamard finite part integral above (for a < x < b) may also be given by the following equivalent definitions: The definitions above may be derived by assuming that the function f (t) is differentiable infinitely many times at t = x for a < x < b, that is, by assuming that f (t) can be represented by its Taylor series about t = x. For details, see Ang. (Note that the term − f (x)/2(1/b − x − 1/a − x) in the second equivalent definition above is missing in Ang but this is corrected in the errata sheet of the book.) Integral equations containing Hadamard finite part integrals (with f (t) unknown) are termed hypersingular integral equations. Hypersingular integral equations arise in the formulation of many problems in mechanics, such as in fracture analysis. (Wikipedia).

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Related pages

Cauchy principal value