1. References

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 {{harvs|txt|authorlink=Jacques Hadamard|last=Hadamard|year1=1923|loc1=book III, chapter I|year2=1932}}. {{harvs|txt|last=Riesz|year1=1938|year2=1949}} 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 {{mvar|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 {{math|a < x < b}}) may also be given by the following equivalent definitions:

The definitions above may be derived by assuming that the function {{math|f (t)}} is differentiable infinitely many times at {{math|t {{=}} x for a < x < b}}, that is, by assuming that {{math|f (t)}} can be represented by its Taylor series about {{math|t {{=}} x}}. For details, see {{harvs|txt|last=Ang|year1=2013}}. (Note that the term {{math|− {{sfrac|f (x)|2}}({{sfrac|1|b − x}} − {{sfrac|1|a − x}})}} in the second equivalent definition above is missing in {{harvs|txt|last=Ang|year1=2013}} but this is corrected in the errata sheet of the book.)

Integral equations containing Hadamard finite part integrals (with {{math|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.

References

• {{Citation

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• {{Citation | last1=Blanchet | first1=Luc | last2=Faye | first2=Guillaume | title=Hadamard regularization | doi=10.1063/1.1308506 | year=2000 | journal=Journal of Mathematical Physics | issn=0022-2488 | volume=41 | issue=11 | pages=7675–7714 | mr=1788597 | zbl = 0986.46024 | arxiv=gr-qc/0004008 | bibcode=2000JMP....41.7675B }}.
• {{Citation | last1=Hadamard | first1=Jacques | title=Lectures on Cauchy's problem in linear partial differential equations | url=https://books.google.com/books?id=B25O-x21uqkC | publisher=Dover Publications, New York | series=Dover Phoenix editions | pages = 316 | year=1923 | isbn=978-0-486-49549-1 | jfm=49.0725.04 | mr=0051411 | zbl = 0049.34805}}.
• {{Citation | last1=Hadamard | first1=J. | title=Le problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques | publisher= Hermann & Cie.|place=Paris | language=French | zbl=0006.20501 | year=1932 | pages=542}}.
• {{Citation

• {{Citation

• {{Citation | last1=Riesz | first1=Marcel | author1-link=Marcel Riesz | title=L'intégrale de Riemann-Liouville et le problème de Cauchy | doi=10.1007/BF02395016 | year=1949 | journal=Acta Mathematica | issn=0001-5962 | volume=81 | pages=1–223 | mr=0030102 | zbl = 0033.27601}}

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