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en>Aldnonymous: /* External links */
2014-01-31T20:40:31Z
<p><span class="autocomment">External links</span></p>
<p><b>New page</b></p><div>{{Renormalization and regularization}}<br />
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.<br />
<br />
If the [[Cauchy principal value]] integral<br />
<br />
: <math>\int_a^b \frac{f(t)}{t-x} \, dt</math><br />
<br />
exists, then the Hadamard finite part integral can be defined as<br />
<br />
: <math>\int_a^b \frac{f(t)}{(t-x)^2}\, dt = \frac{d}{dx} \int_{a}^{b} \frac{f(t)}{t-x} \,dt.</math><br />
<br />
Also it can be calculated from the definition<br />
<br />
: <math>\int_a^b \frac{f(t)}{(t-x)^2}\, dt = \lim_{\varepsilon \to 0} \left\{ \int_a^{x-\varepsilon}\frac{f(t)}{(t-x)^2}\,dt + \int_{x+\varepsilon}^b\frac{f(t)}{(t-x)^2}\,dt -\frac{2f(x)}{\varepsilon}\right\}.</math><br />
<br />
==References==<br />
<br />
*{{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 }}.<br />
*{{Citation | last1=Hadamard | first1=Jacques | title=Lectures on Cauchy's problem in linear partial differential equations | url=http://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}}.<br />
*{{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}}.<br />
*{{Citation <br />
| last1=Riesz <br />
| first1=Marcel <br />
| author1-link=Marcel Riesz <br />
| title=Intégrales de Riemann-Liouville et potentiels. <br />
| language=French <br />
| year=1938 <br />
| url = http://acta.fyx.hu/acta/showCustomerArticle.action?id=5634&dataObjectType=article<br />
| journal = [[Acta Scientiarum Mathematicarum|Acta Litt. Ac Sient. Univ. Hung. Francisco-Josephinae, Sec. Sci. Math.]] ([[Szeged]])<br />
| issue = 1–1<br />
| volume=9 <br />
| pages=1–42<br />
| jfm = 64.0476.03<br />
| zbl=0018.40704}}.<br />
*{{Citation <br />
| last1=Riesz <br />
| first1=Marcel <br />
| author1-link=Marcel Riesz <br />
| title=Rectification au travail "Intégrales de Riemann-Liouville et potentiels" <br />
| language=French <br />
| year=1938 <br />
| url = http://acta.fyx.hu/acta/showCustomerArticle.action?id=5667&dataObjectType=article<br />
| journal = [[Acta Scientiarum Mathematicarum|Acta Litt. Ac Sient. Univ. Hung. Francisco-Josephinae, Sec. Sci. Math.]] ([[Szeged]])<br />
| issue = 2–2<br />
| volume= 9 <br />
| pages=116–118<br />
| jfm = 65.1272.03<br />
| zbl=0020.36402}}.<br />
*{{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}}<br />
<br />
[[Category:Integrals]]<br />
[[Category:Summability methods]]</div>
en>Aldnonymous