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Hadamard regularization
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<h2 id="example">Example</h2> <p>Consider the divergent integral ∫ − 1 1 1 t 2 d t = ( lim a → 0 − ∫ − 1 a 1 t 2 d t ) + ( lim b → 0 + ∫ b 1 1 t 2 d t ) = lim a → 0 − ( − 1 a − 1 ) + lim b → 0 + ( − 1 + 1 b ) = + ∞ {\displaystyle \int _{-1}^{1}{\frac {1}{t^{2}}}\,dt=\left(\lim _{a\to 0^{-}}\int _{-1}^{a}{\frac {1}{t^{2}}}\,dt\right)+\left(\lim _{b\to 0^{+}}\int _{b}^{1}{\frac {1}{t^{2}}}\,dt\right)=\lim _{a\to 0^{-}}\left(-{\frac {1}{a}}-1\right)+\lim _{b\to 0^{+}}\left(-1+{\frac {1}{b}}\right)=+\infty } Its <a href="/facts/Cauchy_principal_value/Gm33Qevf">Cauchy principal value</a> also diverges since C ∫ − 1 1 1 t 2 d t = lim ε → 0 + ( ∫ − 1 − ε 1 t 2 d t + ∫ ε 1 1 t 2 d t ) = lim ε → 0 + ( 1 ε − 1 − 1 + 1 ε ) = + ∞ {\displaystyle {\mathcal {C}}\int _{-1}^{1}{\frac {1}{t^{2}}}\,dt=\lim _{\varepsilon \to 0^{+}}\left(\int _{-1}^{-\varepsilon }{\frac {1}{t^{2}}}\,dt+\int _{\varepsilon }^{1}{\frac {1}{t^{2}}}\,dt\right)=\lim _{\varepsilon \to 0^{+}}\left({\frac {1}{\varepsilon }}-1-1+{\frac {1}{\varepsilon }}\right)=+\infty } To assign a finite value to this divergent integral, we may consider H ∫ − 1 1 1 t 2 d t = H ∫ − 1 1 1 ( t − x ) 2 d t | x = 0 = d d x ( C ∫ − 1 1 1 t − x d t ) | x = 0 {\displaystyle {\mathcal {H}}\int _{-1}^{1}{\frac {1}{t^{2}}}\,dt={\mathcal {H}}\int _{-1}^{1}{\frac {1}{(t-x)^{2}}}\,dt{\Bigg |}_{x=0}={\frac {d}{dx}}\left({\mathcal {C}}\int _{-1}^{1}{\frac {1}{t-x}}\,dt\right){\Bigg |}_{x=0}} The inner Cauchy principal value is given by C ∫ − 1 1 1 t − x d t = lim ε → 0 + ( ∫ − 1 − ε 1 t − x d t + ∫ ε 1 1 t − x d t ) = lim ε → 0 + ( ln | ε + x 1 + x | + ln | 1 − x ε − x | ) = ln | 1 − x 1 + x | {\displaystyle {\mathcal {C}}\int _{-1}^{1}{\frac {1}{t-x}}\,dt=\lim _{\varepsilon \to 0^{+}}\left(\int _{-1}^{-\varepsilon }{\frac {1}{t-x}}\,dt+\int _{\varepsilon }^{1}{\frac {1}{t-x}}\,dt\right)=\lim _{\varepsilon \to 0^{+}}\left(\ln \left|{\frac {\varepsilon +x}{1+x}}\right|+\ln \left|{\frac {1-x}{\varepsilon -x}}\right|\right)=\ln \left|{\frac {1-x}{1+x}}\right|} Therefore, H ∫ − 1 1 1 t 2 d t = d d x ( ln | 1 − x 1 + x | ) | x = 0 = 2 x 2 − 1 | x = 0 = − 2 {\displaystyle {\mathcal {H}}\int _{-1}^{1}{\frac {1}{t^{2}}}\,dt={\frac {d}{dx}}\left(\ln \left|{\frac {1-x}{1+x}}\right|\right){\Bigg |}_{x=0}={\frac {2}{x^{2}-1}}{\Bigg |}_{x=0}=-2} Note that this value does not represent the area under the curve <i>y</i>(<i>t</i>) = 1/<i>t</i>2, which is clearly always positive. However, it can be seen where this comes from. Recall the Cauchy principal value of this integral, when evaluated at the endpoints, took the form lim ε → 0 + ( 1 ε − 1 − 1 + 1 ε ) = + ∞ {\displaystyle \lim _{\varepsilon \to 0^{+}}\left({\frac {1}{\varepsilon }}-1-1+{\frac {1}{\varepsilon }}\right)=+\infty } </p><p>If one removes the infinite components, the pair of 1 ε {\displaystyle {\frac {1}{\varepsilon }}} terms, that which remains is lim ε → 0 + ( − 1 − 1 ) = − 2 {\displaystyle \lim _{\varepsilon \to 0^{+}}\left(-1-1\right)=-2} </p><p>which equals the value derived above. </p> <ul><li>Ang, Whye-Teong (2013), <a href="https://books.google.com/books?id=rDGTlAEACAAJ"><i>Hypersingular Integral Equations in Fracture Analysis</i></a>, Oxford: <a href="/facts/Woodhead_Publishing/rWgKfdx9">Woodhead Publishing</a>, pp. 19–24, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-85709-479-7.</li> <li>Ang, Whye-Teong, <a href="https://personal.ntu.edu.sg/mwtang/anghypercorri.pdf"><i>Errata Sheet for Hypersingular Integral Equations in Fracture Analysis</i></a> (PDF).</li> <li>Blanchet, Luc; Faye, Guillaume (2000), "Hadamard regularization", <i><a href="/facts/Journal_of_Mathematical_Physics/hfGFBt5M">Journal of Mathematical Physics</a></i>, 41 (11): 7675–7714, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/gr-qc/0004008">gr-qc/0004008</a>, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2000JMP....41.7675B">2000JMP....41.7675B</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1063%2F1.1308506">10.1063/1.1308506</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0022-2488">0022-2488</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1788597">1788597</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0986.46024">0986.46024</a>.</li> <li>Hadamard, Jacques (1923), <a href="https://books.google.com/books?id=B25O-x21uqkC"><i>Lectures on Cauchy's problem in linear partial differential equations</i></a>, Dover Phoenix editions, Dover Publications, New York, p. 316, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-49549-1, <a href="/facts/JFM_(identifier)/P6rFxKKx">JFM</a> <a href="https://zbmath.org/?format=complete&q=an:49.0725.04">49.0725.04</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0051411">0051411</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0049.34805">0049.34805</a>.</li> <li>Hadamard, J. (1932), <i>Le problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques</i> (in French), Paris: Hermann & Cie., p. 542, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0006.20501">0006.20501</a>.</li> <li><a href="/facts/Marcel_Riesz/9jWL9fQz">Riesz, Marcel</a> (1938), <a href="https://web.archive.org/web/20160305004701/http://acta.fyx.hu/acta/showcustomerarticle.action?dataobjecttype=article&id=5634">"Intégrales de Riemann-Liouville et potentiels."</a>, <i>Acta Litt. AC Sient. Univ. Hung. Francisco-Josephinae, Sec. Sci. Math. (Szeged)</i> (in French), 9 (1): 1–42, <a href="/facts/JFM_(identifier)/P6rFxKKx">JFM</a> <a href="https://zbmath.org/?format=complete&q=an:64.0476.03">64.0476.03</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0018.40704">0018.40704</a>, archived from <a href="http://acta.fyx.hu/acta/showCustomerArticle.action?id=5634&dataObjectType=article">the original</a> on 2016-03-05, retrieved 2012-06-22.</li> <li><a href="/facts/Marcel_Riesz/9jWL9fQz">Riesz, Marcel</a> (1938), <a href="https://web.archive.org/web/20160304052904/http://acta.fyx.hu/acta/showCustomerArticle.action?id=5667&dataObjectType=article">"Rectification au travail "Intégrales de Riemann-Liouville et potentiels""</a>, <i>Acta Litt. AC Sient. Univ. Hung. Francisco-Josephinae, Sec. Sci. Math. (Szeged)</i> (in French), 9 (2): 116–118, <a href="/facts/JFM_(identifier)/P6rFxKKx">JFM</a> <a href="https://zbmath.org/?format=complete&q=an:65.1272.03">65.1272.03</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0020.36402">0020.36402</a>, archived from <a href="http://acta.fyx.hu/acta/showCustomerArticle.action?id=5667&dataObjectType=article">the original</a> on 2016-03-04, retrieved 2012-06-22.</li> <li><a href="/facts/Marcel_Riesz/9jWL9fQz">Riesz, Marcel</a> (1949), "L'intégrale de Riemann-Liouville et le problème de Cauchy", <i><a href="/facts/Acta_Mathematica/11kbuhDT">Acta Mathematica</a></i>, 81: 1–223, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF02395016">10.1007/BF02395016</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0001-5962">0001-5962</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0030102">0030102</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0033.27601">0033.27601</a></li></ul>