Delta di Dirac - Wikipedia

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href="#La_definizione_di_Dirac"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>La definizione di Dirac</span> </div> </a> <ul id="toc-La_definizione_di_Dirac-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-La_delta_come_distribuzione" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#La_delta_come_distribuzione"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>La delta come distribuzione</span> </div> </a> <ul id="toc-La_delta_come_distribuzione-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-La_delta_come_misura" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#La_delta_come_misura"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>La delta come misura</span> </div> </a> <ul id="toc-La_delta_come_misura-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalizzazioni" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generalizzazioni"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Generalizzazioni</span> </div> </a> <ul id="toc-Generalizzazioni-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Significato_fisico" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Significato_fisico"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Significato fisico</span> </div> </a> <ul id="toc-Significato_fisico-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applicazioni" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Applicazioni"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.6</span> <span>Applicazioni</span> </div> </a> <ul id="toc-Applicazioni-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Proprietà_e_operazioni_della_delta_di_Dirac" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Proprietà_e_operazioni_della_delta_di_Dirac"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Proprietà e operazioni della delta di Dirac</span> </div> </a> <button aria-controls="toc-Proprietà_e_operazioni_della_delta_di_Dirac-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Attiva/disattiva la sottosezione Proprietà e operazioni della delta di Dirac</span> </button> <ul id="toc-Proprietà_e_operazioni_della_delta_di_Dirac-sublist" class="vector-toc-list"> <li id="toc-Prodotto_per_uno_scalare" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Prodotto_per_uno_scalare"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Prodotto per uno scalare</span> </div> </a> <ul id="toc-Prodotto_per_uno_scalare-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Traslazione" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Traslazione"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Traslazione</span> </div> </a> <ul id="toc-Traslazione-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Riscalamento_(e_riflessione)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Riscalamento_(e_riflessione)"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Riscalamento (e riflessione)</span> </div> </a> <ul id="toc-Riscalamento_(e_riflessione)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Composizione_con_una_funzione" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Composizione_con_una_funzione"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Composizione con una funzione</span> </div> </a> <ul id="toc-Composizione_con_una_funzione-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Prodotto_per_una_funzione" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Prodotto_per_una_funzione"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Prodotto per una funzione</span> </div> </a> <ul id="toc-Prodotto_per_una_funzione-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivata_della_funzione_gradino" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivata_della_funzione_gradino"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Derivata della funzione gradino</span> </div> </a> <ul id="toc-Derivata_della_funzione_gradino-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivata_distribuzionale_della_delta" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivata_distribuzionale_della_delta"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7</span> <span>Derivata distribuzionale della delta</span> </div> </a> <ul id="toc-Derivata_distribuzionale_della_delta-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-La_delta_come_limite_di_una_successione" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#La_delta_come_limite_di_una_successione"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>La delta come limite di una successione</span> </div> </a> <button aria-controls="toc-La_delta_come_limite_di_una_successione-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Attiva/disattiva la sottosezione La delta come limite di una successione</span> </button> <ul id="toc-La_delta_come_limite_di_una_successione-sublist" class="vector-toc-list"> <li id="toc-Successioni_che_rappresentano_la_delta_di_Dirac" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Successioni_che_rappresentano_la_delta_di_Dirac"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Successioni che rappresentano la delta di Dirac</span> </div> </a> <ul id="toc-Successioni_che_rappresentano_la_delta_di_Dirac-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-La_delta_e_la_trasformata_di_Fourier" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#La_delta_e_la_trasformata_di_Fourier"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>La delta e la trasformata di Fourier</span> </div> </a> <button aria-controls="toc-La_delta_e_la_trasformata_di_Fourier-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Attiva/disattiva la sottosezione La delta e la trasformata di Fourier</span> </button> <ul id="toc-La_delta_e_la_trasformata_di_Fourier-sublist" class="vector-toc-list"> <li id="toc-Rappresentazione_di_Fourier_della_delta" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rappresentazione_di_Fourier_della_delta"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Rappresentazione di Fourier della delta</span> </div> </a> <ul id="toc-Rappresentazione_di_Fourier_della_delta-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-La_trasformata_della_delta" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#La_trasformata_della_delta"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>La trasformata della delta</span> </div> </a> <ul id="toc-La_trasformata_della_delta-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Note" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Note"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Note</span> </div> </a> <ul id="toc-Note-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voci_correlate" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voci_correlate"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Voci correlate</span> </div> </a> <ul id="toc-Voci_correlate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Altri_progetti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Altri_progetti"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Altri progetti</span> </div> </a> <ul id="toc-Altri_progetti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Collegamenti_esterni" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Collegamenti_esterni"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Collegamenti esterni</span> </div> </a> <ul id="toc-Collegamenti_esterni-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Indice" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Mostra/Nascondi l'indice" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Mostra/Nascondi l'indice</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Delta di Dirac</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Vai a una voce in un'altra lingua. Disponibile in 45 lingue" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-45" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">45 lingue</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AF%D8%A7%D9%84%D8%A9_%D8%AF%D9%8A%D8%B1%D8%A7%D9%83" title="دالة ديراك - arabo" lang="ar" hreflang="ar" data-title="دالة ديراك" data-language-autonym="العربية" data-language-local-name="arabo" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%94%D1%8D%D0%BB%D1%8C%D1%82%D0%B0-%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%8B%D1%8F" title="Дэльта-функцыя - bielorusso" lang="be" hreflang="be" data-title="Дэльта-функцыя" data-language-autonym="Беларуская" data-language-local-name="bielorusso" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%B5%D0%BB%D1%82%D0%B0-%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Делта-функция - bulgaro" lang="bg" hreflang="bg" data-title="Делта-функция" data-language-autonym="Български" data-language-local-name="bulgaro" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A1%E0%A6%BF%E0%A6%B0%E0%A6%BE%E0%A6%95_%E0%A6%A1%E0%A7%87%E0%A6%B2%E0%A7%8D%E0%A6%9F%E0%A6%BE_%E0%A6%85%E0%A6%AA%E0%A7%87%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A6%95" title="ডিরাক ডেল্টা অপেক্ষক - bengalese" lang="bn" hreflang="bn" data-title="ডিরাক ডেল্টা অপেক্ষক" data-language-autonym="বাংলা" data-language-local-name="bengalese" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Delta_de_Dirac" title="Delta de Dirac - catalano" lang="ca" hreflang="ca" data-title="Delta de Dirac" data-language-autonym="Català" data-language-local-name="catalano" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Diracovo_delta" title="Diracovo delta - ceco" lang="cs" hreflang="cs" data-title="Diracovo delta" data-language-autonym="Čeština" data-language-local-name="ceco" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Diracs_deltafunktion" title="Diracs deltafunktion - danese" lang="da" hreflang="da" data-title="Diracs deltafunktion" data-language-autonym="Dansk" data-language-local-name="danese" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Delta-Distribution" title="Delta-Distribution - tedesco" lang="de" hreflang="de" data-title="Delta-Distribution" data-language-autonym="Deutsch" data-language-local-name="tedesco" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CF%81%CE%BF%CF%85%CF%83%CF%84%CE%B9%CE%BA%CE%AE_%CF%83%CF%85%CE%BD%CE%AC%CF%81%CF%84%CE%B7%CF%83%CE%B7" title="Κρουστική συνάρτηση - greco" lang="el" hreflang="el" data-title="Κρουστική συνάρτηση" data-language-autonym="Ελληνικά" data-language-local-name="greco" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en badge-Q17437798 badge-goodarticle mw-list-item" title="voce di qualità"><a href="https://en.wikipedia.org/wiki/Dirac_delta_function" title="Dirac delta function - inglese" lang="en" hreflang="en" data-title="Dirac delta function" data-language-autonym="English" data-language-local-name="inglese" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Diraka_delta_funkcio" title="Diraka delta funkcio - esperanto" lang="eo" hreflang="eo" data-title="Diraka delta funkcio" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Delta_de_Dirac" title="Delta de Dirac - spagnolo" lang="es" hreflang="es" data-title="Delta de Dirac" data-language-autonym="Español" data-language-local-name="spagnolo" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Diraci_deltafunktsioon" title="Diraci deltafunktsioon - estone" lang="et" hreflang="et" data-title="Diraci deltafunktsioon" data-language-autonym="Eesti" data-language-local-name="estone" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%A7%D8%A8%D8%B9_%D8%AF%D9%84%D8%AA%D8%A7%DB%8C_%D8%AF%DB%8C%D8%B1%D8%A7%DA%A9" title="تابع دلتای دیراک - persiano" lang="fa" hreflang="fa" data-title="تابع دلتای دیراک" data-language-autonym="فارسی" data-language-local-name="persiano" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Diracin_deltafunktio" title="Diracin deltafunktio - finlandese" lang="fi" hreflang="fi" data-title="Diracin deltafunktio" data-language-autonym="Suomi" data-language-local-name="finlandese" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Distribution_de_Dirac" title="Distribution de Dirac - francese" lang="fr" hreflang="fr" data-title="Distribution de Dirac" data-language-autonym="Français" data-language-local-name="francese" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%99%D7%AA_%D7%93%D7%9C%D7%AA%D7%90_%D7%A9%D7%9C_%D7%93%D7%99%D7%A8%D7%90%D7%A7" title="פונקציית דלתא של דיראק - ebraico" lang="he" hreflang="he" data-title="פונקציית דלתא של דיראק" data-language-autonym="עברית" data-language-local-name="ebraico" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A1%E0%A4%BF%E0%A4%B0%E0%A5%88%E0%A4%95_%E0%A4%A1%E0%A5%87%E0%A4%B2%E0%A5%8D%E0%A4%9F%E0%A4%BE_%E0%A4%AB%E0%A4%B2%E0%A4%A8" title="डिरैक डेल्टा फलन - hindi" lang="hi" hreflang="hi" data-title="डिरैक डेल्टा फलन" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Dirac-delta" title="Dirac-delta - ungherese" lang="hu" hreflang="hu" data-title="Dirac-delta" data-language-autonym="Magyar" data-language-local-name="ungherese" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Fungsi_delta_Dirac" title="Fungsi delta Dirac - indonesiano" lang="id" hreflang="id" data-title="Fungsi delta Dirac" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiano" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Deltufalli%C3%B0" title="Deltufallið - islandese" lang="is" hreflang="is" data-title="Deltufallið" data-language-autonym="Íslenska" data-language-local-name="islandese" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%87%E3%82%A3%E3%83%A9%E3%83%83%E3%82%AF%E3%81%AE%E3%83%87%E3%83%AB%E3%82%BF%E9%96%A2%E6%95%B0" title="ディラックのデルタ関数 - giapponese" lang="ja" hreflang="ja" data-title="ディラックのデルタ関数" data-language-autonym="日本語" data-language-local-name="giapponese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%93%E1%83%98%E1%83%A0%E1%83%90%E1%83%99%E1%83%98%E1%83%A1_%E1%83%93%E1%83%94%E1%83%9A%E1%83%A2%E1%83%90_%E1%83%A4%E1%83%A3%E1%83%9C%E1%83%A5%E1%83%AA%E1%83%98%E1%83%90" title="დირაკის დელტა ფუნქცია - georgiano" lang="ka" hreflang="ka" data-title="დირაკის დელტა ფუნქცია" data-language-autonym="ქართული" data-language-local-name="georgiano" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%94%94%EB%9E%99_%EB%8D%B8%ED%83%80_%ED%95%A8%EC%88%98" title="디랙 델타 함수 - coreano" lang="ko" hreflang="ko" data-title="디랙 델타 함수" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Delta_funkcija" title="Delta funkcija - lettone" lang="lv" hreflang="lv" data-title="Delta funkcija" data-language-autonym="Latviešu" data-language-local-name="lettone" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Diracdelta" title="Diracdelta - olandese" lang="nl" hreflang="nl" data-title="Diracdelta" data-language-autonym="Nederlands" data-language-local-name="olandese" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Diracs_deltafunksjon" title="Diracs deltafunksjon - norvegese bokmål" lang="nb" hreflang="nb" data-title="Diracs deltafunksjon" data-language-autonym="Norsk bokmål" data-language-local-name="norvegese bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Delta_Diraca" title="Delta Diraca - polacco" lang="pl" hreflang="pl" data-title="Delta Diraca" data-language-autonym="Polski" data-language-local-name="polacco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Delta_de_Dirac" title="Delta de Dirac - portoghese" lang="pt" hreflang="pt" data-title="Delta de Dirac" data-language-autonym="Português" data-language-local-name="portoghese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Func%C8%9Bia_lui_Dirac" title="Funcția lui Dirac - rumeno" lang="ro" hreflang="ro" data-title="Funcția lui Dirac" data-language-autonym="Română" data-language-local-name="rumeno" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B5%D0%BB%D1%8C%D1%82%D0%B0-%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Дельта-функция - russo" lang="ru" hreflang="ru" data-title="Дельта-функция" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%A9%E0%B7%92%E0%B6%BB%E0%B7%90%E0%B6%9A%E0%B7%8A_%E0%B6%A9%E0%B7%99%E0%B6%BD%E0%B7%8A%E0%B6%A7%E0%B7%8F_%E0%B7%81%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B7%92%E0%B6%AD%E0%B6%BA" title="ඩිරැක් ඩෙල්ටා ශ්‍රිතය - singalese" lang="si" hreflang="si" data-title="ඩිරැක් ඩෙල්ටා ශ්‍රිතය" data-language-autonym="සිංහල" data-language-local-name="singalese" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Dirac_delta_function" title="Dirac delta function - Simple English" lang="en-simple" hreflang="en-simple" data-title="Dirac delta function" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Porazdelitev_delta" title="Porazdelitev delta - sloveno" lang="sl" hreflang="sl" data-title="Porazdelitev delta" data-language-autonym="Slovenščina" data-language-local-name="sloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Funksioni_i_delt%C3%ABs_s%C3%AB_Dirakut" title="Funksioni i deltës së Dirakut - albanese" lang="sq" hreflang="sq" data-title="Funksioni i deltës së Dirakut" data-language-autonym="Shqip" data-language-local-name="albanese" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Dirakova_delta_funkcija" title="Dirakova delta funkcija - serbo" lang="sr" hreflang="sr" data-title="Dirakova delta funkcija" data-language-autonym="Српски / srpski" data-language-local-name="serbo" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Diracs_delta-funktion" title="Diracs delta-funktion - svedese" lang="sv" hreflang="sv" data-title="Diracs delta-funktion" data-language-autonym="Svenska" data-language-local-name="svedese" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%94%E0%B8%B4%E0%B9%81%E0%B8%A3%E0%B8%81%E0%B9%80%E0%B8%94%E0%B8%A5%E0%B8%95%E0%B8%B2%E0%B8%9F%E0%B8%B1%E0%B8%87%E0%B8%81%E0%B9%8C%E0%B8%8A%E0%B8%B1%E0%B8%99" title="ดิแรกเดลตาฟังก์ชัน - thailandese" lang="th" hreflang="th" data-title="ดิแรกเดลตาฟังก์ชัน" data-language-autonym="ไทย" data-language-local-name="thailandese" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Dirac_delta_fonksiyonu" title="Dirac delta fonksiyonu - turco" lang="tr" hreflang="tr" data-title="Dirac delta fonksiyonu" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%94%D0%B5%D0%BB%D1%8C%D1%82%D0%B0-%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Дельта-функция - tataro" lang="tt" hreflang="tt" data-title="Дельта-функция" data-language-autonym="Татарча / tatarça" data-language-local-name="tataro" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%B5%D0%BB%D1%8C%D1%82%D0%B0-%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%94%D1%96%D1%80%D0%B0%D0%BA%D0%B0" title="Дельта-функція Дірака - ucraino" lang="uk" hreflang="uk" data-title="Дельта-функція Дірака" data-language-autonym="Українська" data-language-local-name="ucraino" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Delta-funksiya" title="Delta-funksiya - uzbeco" lang="uz" hreflang="uz" data-title="Delta-funksiya" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbeco" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%C3%A0m_delta_Dirac" title="Hàm delta Dirac - vietnamita" lang="vi" hreflang="vi" data-title="Hàm delta Dirac" data-language-autonym="Tiếng Việt" 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decoding="async" width="310" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Dirac_distribution_PDF.png/465px-Dirac_distribution_PDF.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/78/Dirac_distribution_PDF.png/620px-Dirac_distribution_PDF.png 2x" data-file-width="1300" data-file-height="975" /></a><figcaption>Grafico della delta di Dirac</figcaption></figure> <p>In <a href="/wiki/Matematica" title="Matematica">matematica</a>, la funzione <b>delta di Dirac</b>, anche detta <b>impulso di Dirac</b>, <b>distribuzione di Dirac</b> o <b>funzione <i>δ</i></b>, è una <a href="/wiki/Distribuzione_(matematica)" title="Distribuzione (matematica)">distribuzione</a> la cui introduzione formale ha spianato la strada per lo studio della <a href="/wiki/Distribuzione_(matematica)" title="Distribuzione (matematica)">teoria delle distribuzioni</a>. </p><p>Introdotta da <a href="/wiki/Paul_Dirac" title="Paul Dirac">Paul Dirac</a>, anche se già presente nei lavori di <a href="/wiki/Oliver_Heaviside" title="Oliver Heaviside">Oliver Heaviside</a>, è una funzione generalizzata che dipende da un parametro reale in modo tale che sia nulla per tutti i valori del parametro ad eccezione dello zero, ed il suo <a href="/wiki/Integrale" title="Integrale">integrale</a> sul parametro tra <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bddbb0e4420a7e744cf71bd71216e11b0bf88831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }"></span> sia uguale a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span>. </p><p>Viene utilizzata per rappresentare approssimativamente fenomeni come i picchi alti e stretti di alcune funzioni o le loro discontinuità: è lo stesso tipo di astrazione che si fa per la <a href="/wiki/Carica_elettrica" title="Carica elettrica">carica</a> puntiforme, la <a href="/wiki/Massa_(fisica)" title="Massa (fisica)">massa</a> puntiforme, l'<a href="/wiki/Elettrone" title="Elettrone">elettrone</a> puntiforme. L'analogo discreto è il <a href="/wiki/Delta_di_Kronecker" title="Delta di Kronecker">delta di Kronecker</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Descrizione">Descrizione</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delta_di_Dirac&veaction=edit&section=1" title="Modifica la sezione Descrizione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Delta_di_Dirac&action=edit&section=1" title="Edit section's source code: Descrizione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="La_definizione_di_Dirac">La definizione di Dirac</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delta_di_Dirac&veaction=edit&section=2" title="Modifica la sezione La definizione di Dirac" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Delta_di_Dirac&action=edit&section=2" title="Edit section's source code: La definizione di Dirac"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Prima ancora della definizione formale di Dirac, i matematici del passato avevano la necessità di definire una funzione di tipo <i>impulsivo</i>, che rappresentasse cioè un fenomeno fisico di durata infinitesima. Inizialmente la delta fu definita come una <a href="/wiki/Funzione_(matematica)" title="Funzione (matematica)">funzione</a> nulla per <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e528af863fd84d0edc808c3d9f53be7687a83d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.101ex; height:2.676ex;" alt="{\displaystyle t\neq 0}"></span>, con integrale pari a 1 integrando sull'intero <a href="/wiki/Asse_delle_ascisse" class="mw-redirect" title="Asse delle ascisse">asse delle ascisse</a>, e anche come il limite di opportune <a href="/wiki/Successione_(matematica)" title="Successione (matematica)">successioni</a>. </p><p>Formalmente la delta di Dirac viene definita dalla seguente notazione: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{+\infty }\delta (x-x_{0})\phi (x)\,\mathop {} \!\mathrm {d} x=\phi (x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{+\infty }\delta (x-x_{0})\phi (x)\,\mathop {} \!\mathrm {d} x=\phi (x_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81ef0b52f66154b1bed3a1f6edf1df8c3fafbda7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.015ex; height:6.176ex;" alt="{\displaystyle \int _{-\infty }^{+\infty }\delta (x-x_{0})\phi (x)\,\mathop {} \!\mathrm {d} x=\phi (x_{0})}"></span></dd></dl> <p>valida per ogni <a href="/wiki/Funzione_continua" title="Funzione continua">funzione continua</a> in un intorno di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span>. Questa definizione fu introdotta per la prima volta da Dirac alla fine degli <a href="/wiki/Anni_1920" title="Anni 1920">anni venti</a> nelle sue ricerche sulla <a href="/wiki/Meccanica_quantistica" title="Meccanica quantistica">meccanica quantistica</a>. Si noti che, pur utilizzando il simbolo dell'<a href="/wiki/Integrale" title="Integrale">integrale</a>, l'operazione non è di integrazione, ma di applicazione di un <a href="/wiki/Funzionale" title="Funzionale">funzionale</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }"></span> appunto) ad una <a href="/wiki/Funzione_test" class="mw-redirect" title="Funzione test">funzione test</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span>. La delta di <a href="/wiki/Paul_Adrien_Maurice_Dirac" class="mw-redirect" title="Paul Adrien Maurice Dirac">Dirac</a> è dunque la funzione generalizzata (definita con la simbologia di cui sopra) che trasforma la funzione test <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23781b983d21d78467b65e7e32b9e7bc05d625f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.034ex; height:2.843ex;" alt="{\displaystyle \phi (t)}"></span> nel numero <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (x_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aea05216ac448823d1929e457d262c4356725de4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.579ex; height:2.843ex;" alt="{\displaystyle \phi (x_{0})}"></span>. </p><p>Nonostante sia facilmente dimostrabile che non può esistere alcuna funzione con le proprietà della delta di Dirac, questa definizione si rivelò operativamente molto utile e fu presto adottata in molti ambiti della <a href="/wiki/Fisica" title="Fisica">fisica</a> e delle scienze applicate. Anche per Dirac era chiaro che la delta non era una funzione nel senso usuale; la sua idea era che il valore della delta nel punto 0 fosse un <a href="/wiki/Stima_asintotica" title="Stima asintotica">infinito</a> di grado "abbastanza elevato" da permettere la proprietà definitoria. Una formalizzazione matematicamente corretta della delta fu possibile solo molti anni dopo nell'ambito della <a href="/wiki/Teoria_delle_distribuzioni" class="mw-redirect" title="Teoria delle distribuzioni">teoria delle distribuzioni</a>. </p><p>In generale la delta di Dirac può essere definita sia come <a href="/wiki/Distribuzione_(matematica)" title="Distribuzione (matematica)">distribuzione</a>, sia come <a href="/wiki/Misura_(matematica)" title="Misura (matematica)">misura</a>. </p> <div class="mw-heading mw-heading3"><h3 id="La_delta_come_distribuzione">La delta come distribuzione</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delta_di_Dirac&veaction=edit&section=3" title="Modifica la sezione La delta come distribuzione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Delta_di_Dirac&action=edit&section=3" title="Edit section's source code: La delta come distribuzione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La delta di Dirac può essere definita come una <a href="/wiki/Distribuzione_(matematica)" title="Distribuzione (matematica)">distribuzione</a>, vale a dire un <a href="/wiki/Funzionale_lineare" title="Funzionale lineare">funzionale lineare</a> <a href="/wiki/Funzione_continua" title="Funzione continua">continuo</a> su un opportuno spazio di funzioni dette <a href="/wiki/Funzione_di_test" title="Funzione di test">funzioni di test</a> o "di prova". Si consideri come spazio delle funzioni di prova lo <a href="/wiki/Spazio_di_Schwartz" title="Spazio di Schwartz">spazio di Schwartz</a>, ovvero lo spazio delle funzioni a decrescenza rapida <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\mathbb {R} ^{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(\mathbb {R} ^{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b7c34a68f4cf179202afbf9a038399a95fd9198" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.205ex; height:2.843ex;" alt="{\displaystyle S(\mathbb {R} ^{n})}"></span> all'infinito e infinitamente derivabili, le cui derivate parziali sono ancora a decrescenza rapida. </p><p>Lo spazio delle distribuzioni temperate è definito come lo <a href="/wiki/Spazio_duale" title="Spazio duale">spazio duale</a> dello spazio di Schwartz. La distribuzione delta di Dirac associata alla funzione di prova <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {\phi } \in S(\mathbb {R} ^{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi>ϕ</mi> </mrow> <mo>∈</mo> <mi>S</mi> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {\phi } \in S(\mathbb {R} ^{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14ece70ddf5251cc010a24083cea58f71f72f2dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.431ex; height:2.843ex;" alt="{\displaystyle \operatorname {\phi } \in S(\mathbb {R} ^{n})}"></span> è definita come:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{a}[\phi ]=\phi (a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>ϕ</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{a}[\phi ]=\phi (a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0a150aacd2c18ba59f109d1d7ad1b1b585aea74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.336ex; height:2.843ex;" alt="{\displaystyle \delta _{a}[\phi ]=\phi (a)}"></span></dd></dl> <p>ossia la delta di una funzione in un punto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> è un funzionale che associa alla funzione il suo valore nel punto. </p> <div class="mw-heading mw-heading3"><h3 id="La_delta_come_misura">La delta come misura</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delta_di_Dirac&veaction=edit&section=4" title="Modifica la sezione La delta come misura" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Delta_di_Dirac&action=edit&section=4" title="Edit section's source code: La delta come misura"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Uno dei modi per definire la delta di Dirac è quello di considerarla una <a href="/wiki/Misura_(matematica)" title="Misura (matematica)">misura</a> che, per ogni sottinsieme <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> dei <a href="/wiki/Numero_reale" title="Numero reale">numeri reali</a>, restituisce <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (A)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (A)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55126e6ab679dd25c12b32bfd898b923dfb9223a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.862ex; height:2.843ex;" alt="{\displaystyle \delta (A)=1}"></span> se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f0a44a271c9cd47d6bc48ccf20be46f9fabb480" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.746ex; height:2.176ex;" alt="{\displaystyle 0\in A}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (A)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (A)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce36b96e803b5896596be7341f7c6a2f1b511113" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.862ex; height:2.843ex;" alt="{\displaystyle \delta (A)=0}"></span> altrimenti. L'<a href="/wiki/Integrale_di_Lebesgue" title="Integrale di Lebesgue">integrale di Lebesgue</a> permette di definire l'integrazione rispetto alla misura <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{+\infty }f(x)\,\delta \{\mathop {} \!\mathrm {d} x\}=f(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>δ</mi> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{+\infty }f(x)\,\delta \{\mathop {} \!\mathrm {d} x\}=f(0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b30570651eb7e7b375f4dca1250bca7b2daa115" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.154ex; height:6.176ex;" alt="{\displaystyle \int _{-\infty }^{+\infty }f(x)\,\delta \{\mathop {} \!\mathrm {d} x\}=f(0)}"></span></dd></dl> <p>per ogni funzione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> continua a supporto compatto. Questa misura è singolare, e non è quindi <a href="/wiki/Continuit%C3%A0_assoluta" title="Continuità assoluta">assolutamente continua</a> rispetto alla <a href="/wiki/Misura_di_Lebesgue" title="Misura di Lebesgue">misura di Lebesgue</a>. Di conseguenza, la delta di Dirac non ha <a href="/wiki/Teorema_di_Radon-Nikodym" title="Teorema di Radon-Nikodym">derivata di Radon-Nikodym</a>, ovvero non esiste nessuna funzione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }"></span> tale che: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{+\infty }f(x)\delta (x)\,\mathop {} \!\mathrm {d} x=f(0).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{+\infty }f(x)\delta (x)\,\mathop {} \!\mathrm {d} x=f(0).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ec8f7df00b216829f4715b295bf11d197ebbcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.002ex; height:6.176ex;" alt="{\displaystyle \int _{-\infty }^{+\infty }f(x)\delta (x)\,\mathop {} \!\mathrm {d} x=f(0).}"></span></dd></dl> <p>L'uso di quest'ultima notazione per la delta è un <a href="/wiki/Abuso_di_notazione" title="Abuso di notazione">abuso di notazione</a>, poiché la delta non è una distribuzione regolare (non esiste cioè una funzione che risulti la sua rappresentazione tramite l'integrale di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>). </p><p>Tuttavia la notazione integrale è largamente utilizzata, e nonostante <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (x-x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (x-x_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1e19714ba7092deaab9ad37aea46eb5057d6a4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.412ex; height:2.843ex;" alt="{\displaystyle \delta (x-x_{0})}"></span> non sia una funzione si usa scrivere:<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \delta _{x_{0}}|f\rangle =\int _{-\infty }^{+\infty }\delta (x-x_{0})f(x)\,\mathop {} \!\mathrm {d} x=f(x_{0}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \delta _{x_{0}}|f\rangle =\int _{-\infty }^{+\infty }\delta (x-x_{0})f(x)\,\mathop {} \!\mathrm {d} x=f(x_{0}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce2e0d7ee0add69ff7eb0c7b31429dbeb1d437e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.318ex; height:6.176ex;" alt="{\displaystyle \langle \delta _{x_{0}}|f\rangle =\int _{-\infty }^{+\infty }\delta (x-x_{0})f(x)\,\mathop {} \!\mathrm {d} x=f(x_{0}).}"></span></dd></dl> <p>Come <a href="/wiki/Misura_di_probabilit%C3%A0" title="Misura di probabilità">misura di probabilità</a> sui reali, la delta di Dirac è caratterizzata dalla sua <a href="/wiki/Funzione_di_ripartizione" title="Funzione di ripartizione">funzione di ripartizione</a> che non è altro che la <a href="/wiki/Funzione_gradino_di_Heaviside" title="Funzione gradino di Heaviside">funzione di Heaviside</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x)={\begin{cases}1&{\text{se }}x\geq 0\\0&{\text{se }}x<0.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>se </mtext> </mrow> <mi>x</mi> <mo>≥</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>se </mtext> </mrow> <mi>x</mi> <mo><</mo> <mn>0.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(x)={\begin{cases}1&{\text{se }}x\geq 0\\0&{\text{se }}x<0.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ca82a34bbfa4474c7a9356abb2ab9825c0111e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.047ex; height:6.176ex;" alt="{\displaystyle H(x)={\begin{cases}1&{\text{se }}x\geq 0\\0&{\text{se }}x<0.\end{cases}}}"></span></dd></dl> <p>Ciò significa che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11f241aa7195bebab9d0a3c248ea97ef0c78b1ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.203ex; height:2.843ex;" alt="{\displaystyle H(x)}"></span> è l'integrale della funzione indicatrice di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1} _{(-\infty ,x]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} _{(-\infty ,x]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dafa48be55018296d800398517050739e51f31de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.985ex; height:3.009ex;" alt="{\displaystyle \mathbf {1} _{(-\infty ,x]}}"></span> rispetto alla misura <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }"></span>. Ovvero: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x)=\int _{\mathbb {R} }\mathbf {1} _{(-\infty ,x]}(t)\,\delta \{\mathop {} \!\mathrm {d} t\}=\delta (-\infty ,x].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>δ</mi> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(x)=\int _{\mathbb {R} }\mathbf {1} _{(-\infty ,x]}(t)\,\delta \{\mathop {} \!\mathrm {d} t\}=\delta (-\infty ,x].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/091d0d781810e306eda3a38ff4d8b6e0a5ad76ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:39.767ex; height:5.676ex;" alt="{\displaystyle H(x)=\int _{\mathbb {R} }\mathbf {1} _{(-\infty ,x]}(t)\,\delta \{\mathop {} \!\mathrm {d} t\}=\delta (-\infty ,x].}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Generalizzazioni">Generalizzazioni</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delta_di_Dirac&veaction=edit&section=5" title="Modifica la sezione Generalizzazioni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Delta_di_Dirac&action=edit&section=5" title="Edit section's source code: Generalizzazioni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La funzione delta può essere definita in uno <a href="/wiki/Spazio_euclideo" title="Spazio euclideo">spazio euclideo</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> di dimensione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> come una misura tale che: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\mathbb {R} ^{n}}f(\mathbf {x} )\,\delta \{\mathop {} \!\mathrm {d} \mathbf {x} \}=f(\mathbf {0} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>δ</mi> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\mathbb {R} ^{n}}f(\mathbf {x} )\,\delta \{\mathop {} \!\mathrm {d} \mathbf {x} \}=f(\mathbf {0} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ba5dddbe7af047c9f5c8dd1db81a81aaa68dd37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.549ex; height:5.676ex;" alt="{\displaystyle \int _{\mathbb {R} ^{n}}f(\mathbf {x} )\,\delta \{\mathop {} \!\mathrm {d} \mathbf {x} \}=f(\mathbf {0} )}"></span></dd></dl> <p>per ogni funzione continua <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> a supporto compatto. Nel caso <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-dimensionale la delta è il prodotto delle singole delta in una dimensione, ossia se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} =(x_{1},x_{2},\dots ,x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} =(x_{1},x_{2},\dots ,x_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b371e78d6dd619594418b49d28696eaef5f06531" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.847ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} =(x_{1},x_{2},\dots ,x_{n})}"></span>, si ha: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (\mathbf {x} )=\delta (x_{1})\delta (x_{2})\dots \delta (x_{n}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>…</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (\mathbf {x} )=\delta (x_{1})\delta (x_{2})\dots \delta (x_{n}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a41c3db96374122d58bf0b849ddd273d9454551a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.401ex; height:2.843ex;" alt="{\displaystyle \delta (\mathbf {x} )=\delta (x_{1})\delta (x_{2})\dots \delta (x_{n}).}"></span></dd></dl> <p>Tale scrittura vale anche nella definizione della delta come distribuzione, ma tale prodotto può essere definito solamente sotto determinate e restrittive ipotesi. </p><p>Il concetto di <a href="/wiki/Misura_deltiforme" title="Misura deltiforme">misura deltiforme</a> ha invece senso su ogni insieme. Sia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> un insieme, sia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b79e955b57dd7aada93b8afd459996ae941d480" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.205ex; height:2.509ex;" alt="{\displaystyle x_{0}\in X}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1f558f53cda207614abdf90162266c70bc5c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }"></span> una <a href="/wiki/Sigma_algebra" class="mw-redirect" title="Sigma algebra">sigma algebra</a> dei sottoinsiemi di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>, allora la misura definita sugli insiemi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∈</mo> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\in \Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18496f0ae7c4589763102f30f1dd3bb02c0008c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.262ex; height:2.176ex;" alt="{\displaystyle A\in \Sigma }"></span> dalla relazione: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{x_{0}}(A)={\begin{cases}1&{\text{se }}x_{0}\in A\\0&{\text{se }}x_{0}\notin A\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>se </mtext> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <mi>A</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>se </mtext> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∉</mo> <mi>A</mi> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{x_{0}}(A)={\begin{cases}1&{\text{se }}x_{0}\in A\\0&{\text{se }}x_{0}\notin A\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97c2f7fb44e96f3cc8486db09195162292f1fede" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.164ex; height:6.176ex;" alt="{\displaystyle \delta _{x_{0}}(A)={\begin{cases}1&{\text{se }}x_{0}\in A\\0&{\text{se }}x_{0}\notin A\end{cases}}}"></span></dd></dl> <p>è la misura di Dirac in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span>. </p><p>Un'altra generalizzazione molto diffusa riguarda infine le <a href="/wiki/Variet%C3%A0_differenziabile" title="Varietà differenziabile">varietà differenziabili</a>, in cui molte delle proprietà della delta come distribuzione possono essere sfruttate grazie alla struttura differenziabile. La funzione delta su una varietà <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> nel punto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\in M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}\in M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ffbdb59406dc64aa6769cecf0e9ee109d181119" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.667ex; height:2.509ex;" alt="{\displaystyle x_{0}\in M}"></span> è definita come la distribuzione: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{x_{0}}[\phi ]=\phi (x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">[</mo> <mi>ϕ</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{x_{0}}[\phi ]=\phi (x_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f5dad5e0dc93400e2bade5902814177784264c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.393ex; height:3.009ex;" alt="{\displaystyle \delta _{x_{0}}[\phi ]=\phi (x_{0})}"></span></dd></dl> <p>per ogni funzione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {\phi } }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi>ϕ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {\phi } }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa6ae843bc55d1864431ed3b2c0c8807433723f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \operatorname {\phi } }"></span> reale, liscia e a supporto compatto su <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span>. Un caso particolare molto utilizzato è il caso in cui <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> sia un <a href="/wiki/Insieme_aperto" title="Insieme aperto">insieme aperto</a> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Significato_fisico">Significato fisico</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delta_di_Dirac&veaction=edit&section=6" title="Modifica la sezione Significato fisico" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Delta_di_Dirac&action=edit&section=6" title="Edit section's source code: Significato fisico"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La funzione delta può essere pensata come la <a href="/wiki/Densit%C3%A0" title="Densità">densità</a> di un punto. Consideriamo, ad esempio, un <a href="/wiki/Corpo_(fisica)" title="Corpo (fisica)">corpo</a> con <a href="/wiki/Massa_(fisica)" title="Massa (fisica)">massa</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> finita, esteso in una certa regione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> dello <a href="/wiki/Spazio_(fisica)" title="Spazio (fisica)">spazio</a> tridimensionale. Possiamo associare ad ogni punto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> dello spazio una quantità <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f32a26d8795457b2f5c2bdc078758dcbbc71b30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.341ex; height:2.843ex;" alt="{\displaystyle \rho (x)}"></span> che rappresenti la densità del corpo. La funzione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> sarà nulla al di fuori della regione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> e, all'interno, assumerà valori tali che l'integrale: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{V}\rho (x)\,\operatorname {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{V}\rho (x)\,\operatorname {d} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/431b8a8a06ba07b08b9386c67be1ab2d7842dd32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.3ex; height:5.676ex;" alt="{\displaystyle \int _{V}\rho (x)\,\operatorname {d} x}"></span></dd></dl> <p>converga a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span>. Essendo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f901253e394be3099a73020fb947cebf27c38de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.602ex; height:2.843ex;" alt="{\displaystyle \rho (x)=0}"></span> al di fuori di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> l'integrale può essere esteso a tutto lo spazio e si può quindi scrivere: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \rho (x)\,\operatorname {d} x=M.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>M</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \rho (x)\,\operatorname {d} x=M.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11573500f8122f78658324c8c560da3a471d553" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.893ex; height:5.676ex;" alt="{\displaystyle \int \rho (x)\,\operatorname {d} x=M.}"></span></dd></dl> <p>Ora, se immaginiamo di restringere la regione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> senza variare la massa del corpo, la densità di questo dovrà conseguentemente aumentare e tenderà all'<a href="/wiki/Infinito_(matematica)" title="Infinito (matematica)">infinito</a> al tendere di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> al singolo punto: vogliamo, quindi, trovare un'espressione come <i>densità limite</i> per la densità del corpo puntiforme. </p><p>Per semplicità consideriamo un corpo con massa costante e una regione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> sferica con raggio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>; il <a href="/wiki/Volume" title="Volume">volume</a> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> sarà: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {4}{3}}\pi R^{3},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mi>π</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {4}{3}}\pi R^{3},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/086041b534f6de09bd4994fd3288e34e9950bc99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.796ex; height:5.176ex;" alt="{\displaystyle {\frac {4}{3}}\pi R^{3},}"></span></dd></dl> <p>e la corrispondente densità: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{R}(x)={\frac {M}{V}}={\frac {3M}{4\pi R^{3}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>M</mi> <mi>V</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>M</mi> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{R}(x)={\frac {M}{V}}={\frac {3M}{4\pi R^{3}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65acbd519772ded18b5ad33ba9949800f7af4960" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:22.091ex; height:5.509ex;" alt="{\displaystyle \rho _{R}(x)={\frac {M}{V}}={\frac {3M}{4\pi R^{3}}},}"></span></dd></dl> <p>e in questo modo: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \rho _{R}(x)\,\mathop {} \!\mathrm {d} x=M,\quad \forall R.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>M</mi> <mo>,</mo> <mspace width="1em" /> <mi mathvariant="normal">∀</mi> <mi>R</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \rho _{R}(x)\,\mathop {} \!\mathrm {d} x=M,\quad \forall R.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1df3920ca7ed1a9aac86081e63948b969fa4a0d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.398ex; height:5.676ex;" alt="{\displaystyle \int \rho _{R}(x)\,\mathop {} \!\mathrm {d} x=M,\quad \forall R.}"></span></dd></dl> <p>Se si considera il <a href="/wiki/Limite_(matematica)" title="Limite (matematica)">limite</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (x)=\lim _{R\to 0}\rho _{R}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (x)=\lim _{R\to 0}\rho _{R}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75d23e8bab3c90b9640a255fd95778c29e0584be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.359ex; height:4.009ex;" alt="{\displaystyle \rho (x)=\lim _{R\to 0}\rho _{R}(x)}"></span></dd></dl> <p>avverrà che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (x)=\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (x)=\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/301b2c833c551864d8abf5436659ddc109b4374b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.763ex; height:2.843ex;" alt="{\displaystyle \rho (x)=\infty }"></span> per <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f901253e394be3099a73020fb947cebf27c38de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.602ex; height:2.843ex;" alt="{\displaystyle \rho (x)=0}"></span> per <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\not =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\not =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24f21a554ec30ad79de0bdb34fcc519131ca84e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.591ex; height:2.676ex;" alt="{\displaystyle x\not =0}"></span>, da cui: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \rho (x)\,\operatorname {d} x=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \rho (x)\,\operatorname {d} x=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56590788db46ea0a9f2ecf4db38ddd0422840b61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.613ex; height:5.676ex;" alt="{\displaystyle \int \rho (x)\,\operatorname {d} x=0,}"></span></dd></dl> <p>e questo vuol dire che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f32a26d8795457b2f5c2bdc078758dcbbc71b30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.341ex; height:2.843ex;" alt="{\displaystyle \rho (x)}"></span> non è assimilabile alla densità di un punto di massa <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span>. </p><p>Consideriamo allora un diverso tipo di limite per le densità <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{R}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{R}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc632f89377634319b1eb466e1eb9bc83036941b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.682ex; height:2.176ex;" alt="{\displaystyle \rho _{R}}"></span>: il cosiddetto <a href="/w/index.php?title=Limite_debole&action=edit&redlink=1" class="new" title="Limite debole (la pagina non esiste)">limite debole</a>. Con pochi calcoli si nota che per ogni <a href="/wiki/Funzione_continua" title="Funzione continua">funzione continua</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{R\to 0}\int \rho _{R}(x)h(x)\,\operatorname {d} x=Mh(0).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mo>∫</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>M</mi> <mi>h</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{R\to 0}\int \rho _{R}(x)h(x)\,\operatorname {d} x=Mh(0).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55c999d96a2e0912f44c449e3830d1e0e643e8ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.261ex; height:5.676ex;" alt="{\displaystyle \lim _{R\to 0}\int \rho _{R}(x)h(x)\,\operatorname {d} x=Mh(0).}"></span></dd></dl> <p>Questa formula mostra che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> è il funzionale che associa alla funzione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{R}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{R}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc632f89377634319b1eb466e1eb9bc83036941b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.682ex; height:2.176ex;" alt="{\displaystyle \rho _{R}}"></span> il valore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Mh(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mi>h</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Mh(0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc4ff7c4a07041b3bc179434fc0b4532cc7ad470" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.753ex; height:2.843ex;" alt="{\displaystyle Mh(0)}"></span>. </p><p>Questo limite, che indichiamo simbolicamente <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\delta (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\delta (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df2591ae53beb4a399b7e22095e5fe77d5db1a55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.63ex; height:2.843ex;" alt="{\displaystyle M\delta (x)}"></span>, è la massa cercata; infatti, posto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(x)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(x)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6397dc3ddab50b85f5c07df2d057a05b65442572" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.739ex; height:2.843ex;" alt="{\displaystyle h(x)=1}"></span>, si ha: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int M\delta (x)\,\operatorname {d} x=\lim _{R\to 0}\int \rho _{R}(x)\,\operatorname {d} x=M,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>M</mi> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mo>∫</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>M</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int M\delta (x)\,\operatorname {d} x=\lim _{R\to 0}\int \rho _{R}(x)\,\operatorname {d} x=M,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46f3fc158de7005f508667b21e7a8d3fba457989" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.564ex; height:5.676ex;" alt="{\displaystyle \int M\delta (x)\,\operatorname {d} x=\lim _{R\to 0}\int \rho _{R}(x)\,\operatorname {d} x=M,}"></span></dd></dl> <p>dove il primo integrale è un'espressione simbolica con cui si sottintende il passaggio al limite. </p> <div class="mw-heading mw-heading3"><h3 id="Applicazioni">Applicazioni</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delta_di_Dirac&veaction=edit&section=7" title="Modifica la sezione Applicazioni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Delta_di_Dirac&action=edit&section=7" title="Edit section's source code: Applicazioni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La delta di Dirac può essere utilizzata per esprimere in maniera <i>impulsiva</i> una qualsiasi <a href="/wiki/Grandezza_fisica" title="Grandezza fisica">grandezza fisica</a> estensiva (ad es. tramite moltiplicazione della grandezza per tale funzione). In <a href="/wiki/Telecomunicazioni" class="mw-redirect" title="Telecomunicazioni">telecomunicazioni</a> ad esempio è utilizzata per esprimere un <a href="/wiki/Segnale_(fisica)" class="mw-redirect" title="Segnale (fisica)">segnale</a> di tipo impulsivo ossia della durata infinitesima di ampiezza <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> e per la formalizzazione del cosiddetto <a href="/wiki/Teorema_del_campionamento" class="mw-redirect" title="Teorema del campionamento">teorema del campionamento</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Proprietà_e_operazioni_della_delta_di_Dirac"><span id="Propriet.C3.A0_e_operazioni_della_delta_di_Dirac"></span>Proprietà e operazioni della delta di Dirac</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delta_di_Dirac&veaction=edit&section=8" title="Modifica la sezione Proprietà e operazioni della delta di Dirac" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Delta_di_Dirac&action=edit&section=8" title="Edit section's source code: Proprietà e operazioni della delta di Dirac"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Nel seguito si espongono le proprietà principali della delta. </p> <div class="mw-heading mw-heading3"><h3 id="Prodotto_per_uno_scalare">Prodotto per uno scalare</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delta_di_Dirac&veaction=edit&section=9" title="Modifica la sezione Prodotto per uno scalare" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Delta_di_Dirac&action=edit&section=9" title="Edit section's source code: Prodotto per uno scalare"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Per definizione di distribuzione si ha: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{+\infty }a\delta (t)\phi (t)\,\mathop {} \!\mathrm {d} t=a\int _{-\infty }^{+\infty }\delta (t)\phi (t)\,\mathop {} \!\mathrm {d} t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>a</mi> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> <mi>a</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{+\infty }a\delta (t)\phi (t)\,\mathop {} \!\mathrm {d} t=a\int _{-\infty }^{+\infty }\delta (t)\phi (t)\,\mathop {} \!\mathrm {d} t.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/243aefe080c669a8e9bae51a8e199c06103240a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.879ex; height:6.176ex;" alt="{\displaystyle \int _{-\infty }^{+\infty }a\delta (t)\phi (t)\,\mathop {} \!\mathrm {d} t=a\int _{-\infty }^{+\infty }\delta (t)\phi (t)\,\mathop {} \!\mathrm {d} t.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Traslazione">Traslazione</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delta_di_Dirac&veaction=edit&section=10" title="Modifica la sezione Traslazione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Delta_di_Dirac&action=edit&section=10" title="Edit section's source code: Traslazione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dalla definizione di distribuzione si ha che la delta di Dirac "tempo-ritardata" agisce come: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{+\infty }f(t)\delta (t-T)\,\mathop {} \!\mathrm {d} t=f(T).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{+\infty }f(t)\delta (t-T)\,\mathop {} \!\mathrm {d} t=f(T).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/223284193cdd0af0d03f5bc96f9bff463538a294" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.483ex; height:6.176ex;" alt="{\displaystyle \int _{-\infty }^{+\infty }f(t)\delta (t-T)\,\mathop {} \!\mathrm {d} t=f(T).}"></span></dd></dl> <p>Ossia la <a href="/wiki/Convoluzione" title="Convoluzione">convoluzione</a> di una funzione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}"></span> con la delta tempo-ritardata significa valutare la funzione al tempo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>, e da questo segue che: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f*\delta (t-T))=\int _{-\infty }^{+\infty }f(\tau )\cdot \delta (t-T-\tau )\,\mathop {} \!\mathrm {d} \tau =\int _{-\infty }^{+\infty }f(\tau )\cdot \delta (\tau -(t-T))\,\mathop {} \!\mathrm {d} \tau =f(t-T).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>T</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>τ</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>τ</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f*\delta (t-T))=\int _{-\infty }^{+\infty }f(\tau )\cdot \delta (t-T-\tau )\,\mathop {} \!\mathrm {d} \tau =\int _{-\infty }^{+\infty }f(\tau )\cdot \delta (\tau -(t-T))\,\mathop {} \!\mathrm {d} \tau =f(t-T).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d05ede7eb61dac5aa9bcf47c1e384b789b87908a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:88.531ex; height:6.176ex;" alt="{\displaystyle (f*\delta (t-T))=\int _{-\infty }^{+\infty }f(\tau )\cdot \delta (t-T-\tau )\,\mathop {} \!\mathrm {d} \tau =\int _{-\infty }^{+\infty }f(\tau )\cdot \delta (\tau -(t-T))\,\mathop {} \!\mathrm {d} \tau =f(t-T).}"></span></dd></dl> <p>Questo vale se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}"></span> è una distribuzione temperata, e come caso particolare si ha: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{+\infty }\delta (\xi -x)\delta (x-\eta )\,\mathop {} \!\mathrm {d} x=\delta (\xi -\eta ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>η</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo>−</mo> <mi>η</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{+\infty }\delta (\xi -x)\delta (x-\eta )\,\mathop {} \!\mathrm {d} x=\delta (\xi -\eta ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59e9e43df0a22c9a6b1ea13cc02cca1358d73524" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.3ex; height:6.176ex;" alt="{\displaystyle \int _{-\infty }^{+\infty }\delta (\xi -x)\delta (x-\eta )\,\mathop {} \!\mathrm {d} x=\delta (\xi -\eta ).}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Riscalamento_(e_riflessione)"><span id="Riscalamento_.28e_riflessione.29"></span>Riscalamento (e riflessione)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delta_di_Dirac&veaction=edit&section=11" title="Modifica la sezione Riscalamento (e riflessione)" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Delta_di_Dirac&action=edit&section=11" title="Edit section's source code: Riscalamento (e riflessione)"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dalla definizione di delta si ha: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (at)={1 \over |a|}\delta (t),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (at)={1 \over |a|}\delta (t),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44a680768fedf1eb0aadd6df1411eaf6dc6d79cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.73ex; height:6.009ex;" alt="{\displaystyle \delta (at)={1 \over |a|}\delta (t),}"></span></dd></dl> <p>infatti: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{+\infty }\delta (at)\operatorname {\phi } (t)\,\mathop {} \!\mathrm {d} t={1 \over |a|}\int _{-\infty }^{+\infty }\delta (t)\phi \left({t \over a}\right)\mathop {} \!\mathrm {d} t={1 \over |a|}\phi (0)=\int _{-\infty }^{+\infty }{1 \over |a|}\delta (t)\phi (t)\,\mathop {} \!\mathrm {d} t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi>ϕ</mi> </mrow> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{+\infty }\delta (at)\operatorname {\phi } (t)\,\mathop {} \!\mathrm {d} t={1 \over |a|}\int _{-\infty }^{+\infty }\delta (t)\phi \left({t \over a}\right)\mathop {} \!\mathrm {d} t={1 \over |a|}\phi (0)=\int _{-\infty }^{+\infty }{1 \over |a|}\delta (t)\phi (t)\,\mathop {} \!\mathrm {d} t.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e34dc202088213c0cf34e09be742a0bc1ae90e0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:79.151ex; height:6.343ex;" alt="{\displaystyle \int _{-\infty }^{+\infty }\delta (at)\operatorname {\phi } (t)\,\mathop {} \!\mathrm {d} t={1 \over |a|}\int _{-\infty }^{+\infty }\delta (t)\phi \left({t \over a}\right)\mathop {} \!\mathrm {d} t={1 \over |a|}\phi (0)=\int _{-\infty }^{+\infty }{1 \over |a|}\delta (t)\phi (t)\,\mathop {} \!\mathrm {d} t.}"></span></dd></dl> <p>Il primo passaggio è lecito se si considerano separatamente <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f34a80ea013edb56e340b19550430a8b6dfd7b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a>0}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5d7ca60f6ed64b99649dcee21847295fedf206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a<0}"></span>, e trovando che il risultato è definito a meno del segno <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bd52ce670743d3b61bec928a7ec9f47309eb36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle -}"></span>. </p><p>Segue come caso particolare che, vista come una funzione, la delta è <a href="/wiki/Funzioni_pari_e_dispari" title="Funzioni pari e dispari">pari</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (t)=\delta (-t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (t)=\delta (-t).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbfa8921c6b70523eafd5f49a35796afc20d3746" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.949ex; height:2.843ex;" alt="{\displaystyle \delta (t)=\delta (-t).}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Composizione_con_una_funzione">Composizione con una funzione</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delta_di_Dirac&veaction=edit&section=12" title="Modifica la sezione Composizione con una funzione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Delta_di_Dirac&action=edit&section=12" title="Edit section's source code: Composizione con una funzione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> è una funzione derivabile con derivata non nulla negli zeri <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}"></span> della funzione, allora: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (f(x))=\sum _{i}{\frac {\delta (x-x_{i})}{|f'(x_{i})|}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (f(x))=\sum _{i}{\frac {\delta (x-x_{i})}{|f'(x_{i})|}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d52b600144da69737c012755c13708c663a62fbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.756ex; height:6.843ex;" alt="{\displaystyle \delta (f(x))=\sum _{i}{\frac {\delta (x-x_{i})}{|f'(x_{i})|}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Prodotto_per_una_funzione">Prodotto per una funzione</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delta_di_Dirac&veaction=edit&section=13" title="Modifica la sezione Prodotto per una funzione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Delta_di_Dirac&action=edit&section=13" title="Edit section's source code: Prodotto per una funzione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Data una funzione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6cfa6e844c065b7c29c0a39cc4dcf40b237ea80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle \alpha (t)}"></span> di classe <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/971ed05871d69309df32efdfd2020128c9cf69d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.673ex; height:2.343ex;" alt="{\displaystyle C^{\infty }}"></span>, si ha: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha (t)\delta (t-t_{0})=\alpha (t_{0})\delta (t-t_{0}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>α</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha (t)\delta (t-t_{0})=\alpha (t_{0})\delta (t-t_{0}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90e5127cd8881472bfafa680b6d0c31a77e29834" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.936ex; height:2.843ex;" alt="{\displaystyle \alpha (t)\delta (t-t_{0})=\alpha (t_{0})\delta (t-t_{0}).}"></span></dd></dl> <p>Infatti: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{+\infty }(\alpha (t)\delta (t-t_{0}))\phi (t)\,\mathop {} \!\mathrm {d} t=\int _{-\infty }^{+\infty }\delta (t-t_{0})(\alpha (t)\phi (t))\,\mathop {} \!\mathrm {d} t=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{+\infty }(\alpha (t)\delta (t-t_{0}))\phi (t)\,\mathop {} \!\mathrm {d} t=\int _{-\infty }^{+\infty }\delta (t-t_{0})(\alpha (t)\phi (t))\,\mathop {} \!\mathrm {d} t=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c130c1e45e3692fc7da9b705aec5bf8363c2e9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:59.812ex; height:6.176ex;" alt="{\displaystyle \int _{-\infty }^{+\infty }(\alpha (t)\delta (t-t_{0}))\phi (t)\,\mathop {} \!\mathrm {d} t=\int _{-\infty }^{+\infty }\delta (t-t_{0})(\alpha (t)\phi (t))\,\mathop {} \!\mathrm {d} t=}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha (t_{0})\phi (t_{0})=\int _{-\infty }^{+\infty }(\alpha (t_{0})\delta (t-t_{0}))\phi (t)\,\mathop {} \!\mathrm {d} t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha (t_{0})\phi (t_{0})=\int _{-\infty }^{+\infty }(\alpha (t_{0})\delta (t-t_{0}))\phi (t)\,\mathop {} \!\mathrm {d} t.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58a836a2d977e98a239eb2437da113009da3dc02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.015ex; height:6.176ex;" alt="{\displaystyle \alpha (t_{0})\phi (t_{0})=\int _{-\infty }^{+\infty }(\alpha (t_{0})\delta (t-t_{0}))\phi (t)\,\mathop {} \!\mathrm {d} t.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Derivata_della_funzione_gradino">Derivata della funzione gradino</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delta_di_Dirac&veaction=edit&section=14" title="Modifica la sezione Derivata della funzione gradino" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Delta_di_Dirac&action=edit&section=14" title="Edit section's source code: Derivata della funzione gradino"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La funzione delta è la derivata della <a href="/wiki/Funzione_gradino" title="Funzione gradino">funzione gradino</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {u} (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">u</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {u} (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/700093b797e7f58af74ddc48b87f2f64ce698ed9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.941ex; height:2.843ex;" alt="{\displaystyle \operatorname {u} (t)}"></span> (a volte indicata, con abuso di notazione, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {1} (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">1</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {1} (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7b1e0cecc1947ffed7ea2899e862c01be1e4a96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.811ex; height:2.843ex;" alt="{\displaystyle \operatorname {1} (t)}"></span>). Tale funzione viene anche chiamata <a href="/wiki/Funzione_gradino_di_Heaviside" title="Funzione gradino di Heaviside">funzione di Heaviside</a> e in questo caso viene indicata con il simbolo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {H} (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">H</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {H} (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaa7e95126a3fcfe6934b9d9eff19d233cfeb481" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.882ex; height:2.843ex;" alt="{\displaystyle \operatorname {H} (x)}"></span>. Il valore della funzione gradino è 0 per <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a4dbbf970b2d2863dcab589eafe006f08e727d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x<0}"></span> e 1 per <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80d24be5f0eb4a9173da6038badc8659546021d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x>0}"></span>. </p><p>La dimostrazione si ottiene eseguendo una integrazione per parti ed applicando le proprietà degli integrali e della funzione a gradino: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{+\infty }\operatorname {u} '(t)\phi (t)\,\mathop {} \!\mathrm {d} t=-\int _{-\infty }^{+\infty }\operatorname {u} (t)\phi '(t)\,\mathop {} \!\mathrm {d} t=-\int _{0}^{+\infty }\phi '(t)\,\mathop {} \!\mathrm {d} t=-[\phi (t)]_{0}^{+\infty }=\phi (0)=\int _{-\infty }^{+\infty }\delta (t)\phi (t)\,\mathop {} \!\mathrm {d} t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi mathvariant="normal">u</mi> <mo>′</mo> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi mathvariant="normal">u</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>ϕ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>ϕ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> <mo>−</mo> <mo stretchy="false">[</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msubsup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo>=</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{+\infty }\operatorname {u} '(t)\phi (t)\,\mathop {} \!\mathrm {d} t=-\int _{-\infty }^{+\infty }\operatorname {u} (t)\phi '(t)\,\mathop {} \!\mathrm {d} t=-\int _{0}^{+\infty }\phi '(t)\,\mathop {} \!\mathrm {d} t=-[\phi (t)]_{0}^{+\infty }=\phi (0)=\int _{-\infty }^{+\infty }\delta (t)\phi (t)\,\mathop {} \!\mathrm {d} t.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57dbcb37d380004dfaf0815bc077bf3a579d093b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:100.595ex; height:6.176ex;" alt="{\displaystyle \int _{-\infty }^{+\infty }\operatorname {u} '(t)\phi (t)\,\mathop {} \!\mathrm {d} t=-\int _{-\infty }^{+\infty }\operatorname {u} (t)\phi '(t)\,\mathop {} \!\mathrm {d} t=-\int _{0}^{+\infty }\phi '(t)\,\mathop {} \!\mathrm {d} t=-[\phi (t)]_{0}^{+\infty }=\phi (0)=\int _{-\infty }^{+\infty }\delta (t)\phi (t)\,\mathop {} \!\mathrm {d} t.}"></span></dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Dirac_distribution_CDF.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Dirac_distribution_CDF.svg/220px-Dirac_distribution_CDF.svg.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Dirac_distribution_CDF.svg/330px-Dirac_distribution_CDF.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Dirac_distribution_CDF.svg/440px-Dirac_distribution_CDF.svg.png 2x" data-file-width="512" data-file-height="384" /></a><figcaption>La funzione gradino di Heaviside, usando la convenzione della metà del massimo</figcaption></figure> <p>Tale definizione è il punto di partenza per calcolare la <a href="/w/index.php?title=Derivata_distribuzionale&action=edit&redlink=1" class="new" title="Derivata distribuzionale (la pagina non esiste)">derivata distribuzionale</a> di una funzione, ossia la sua derivata nel senso delle distribuzioni. Tale calcolo si effettua addizionando alla derivata ordinaria della funzione gli impulsi concentrati nei punti di discontinuità della funzione, con area pari al salto della funzione nei punti stessi. Tale approccio è fondamentale nello studio dei <a href="/wiki/Teoria_dei_segnali" title="Teoria dei segnali">segnali</a>. </p><p>Si può ottenere la dimostrazione inversa, ossia dimostrare che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {u} (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">u</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {u} (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/700093b797e7f58af74ddc48b87f2f64ce698ed9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.941ex; height:2.843ex;" alt="{\displaystyle \operatorname {u} (t)}"></span> è primitiva di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c050147a97868286252447ef73515c8108edd398" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.698ex; height:2.843ex;" alt="{\displaystyle \delta (t)}"></span>, osservando che: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}\delta (t)\,\mathop {} \!\mathrm {d} t=\left\{{\begin{matrix}1,\,{\text{se }}a<0<b\\0,\,{\text{se }}0\notin [a,b]\end{matrix}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>se </mtext> </mrow> <mi>a</mi> <mo><</mo> <mn>0</mn> <mo><</mo> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>se </mtext> </mrow> <mn>0</mn> <mo>∉</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}\delta (t)\,\mathop {} \!\mathrm {d} t=\left\{{\begin{matrix}1,\,{\text{se }}a<0<b\\0,\,{\text{se }}0\notin [a,b]\end{matrix}}\right.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dd1b6e21bc12faa7b8d3af18fe7218320242d65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.685ex; height:6.509ex;" alt="{\displaystyle \int _{a}^{b}\delta (t)\,\mathop {} \!\mathrm {d} t=\left\{{\begin{matrix}1,\,{\text{se }}a<0<b\\0,\,{\text{se }}0\notin [a,b]\end{matrix}}\right.}"></span></dd></dl> <p>Dalle proprietà dell'<a href="/wiki/Integrale_di_Riemann" title="Integrale di Riemann">integrale di Riemann</a> si ha che: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f'(t)\mathop {} \!\mathrm {d} t=[f(t)]_{a}^{b}=f(b)-f(a).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msubsup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f'(t)\mathop {} \!\mathrm {d} t=[f(t)]_{a}^{b}=f(b)-f(a).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a10fd19550ce1d1275f4e5846e2919c38f6feea9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.373ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f'(t)\mathop {} \!\mathrm {d} t=[f(t)]_{a}^{b}=f(b)-f(a).}"></span></dd></dl> <p>L'unica funzione che soddisfa tale vincolo è il gradino. </p> <div class="mw-heading mw-heading3"><h3 id="Derivata_distribuzionale_della_delta">Derivata distribuzionale della delta</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delta_di_Dirac&veaction=edit&section=15" title="Modifica la sezione Derivata distribuzionale della delta" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Delta_di_Dirac&action=edit&section=15" title="Edit section's source code: Derivata distribuzionale della delta"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La derivata distribuzionale della delta è la distribuzione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>δ</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta '}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aab2d3fbf71b804a6005c7546c058e15f4ea3685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.738ex; height:2.509ex;" alt="{\displaystyle \delta '}"></span> definita a partire da una funzione di test <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {\phi } }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi>ϕ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {\phi } }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa6ae843bc55d1864431ed3b2c0c8807433723f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \operatorname {\phi } }"></span> liscia e a supporto compatto: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta '[\phi ]=-\delta [\phi ']=-\phi '(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>δ</mi> <mo>′</mo> </msup> <mo stretchy="false">[</mo> <mi>ϕ</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo>−</mo> <mi>δ</mi> <mo stretchy="false">[</mo> <msup> <mi>ϕ</mi> <mo>′</mo> </msup> <mo stretchy="false">]</mo> <mo>=</mo> <mo>−</mo> <msup> <mi>ϕ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta '[\phi ]=-\delta [\phi ']=-\phi '(0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44139067721aaa8a9dac985ebd06be99ebf73fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.685ex; height:3.009ex;" alt="{\displaystyle \delta '[\phi ]=-\delta [\phi ']=-\phi '(0)}"></span></dd></dl> <p>In modo equivalente: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{+\infty }\delta '(x)\phi (x)\,\mathop {} \!\mathrm {d} x=-\int _{-\infty }^{+\infty }\delta (x)\phi '(x)\,\mathop {} \!\mathrm {d} x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>δ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>ϕ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{+\infty }\delta '(x)\phi (x)\,\mathop {} \!\mathrm {d} x=-\int _{-\infty }^{+\infty }\delta (x)\phi '(x)\,\mathop {} \!\mathrm {d} x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d49e1d7772dc150f5270460b72e844c0931e0ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.542ex; height:6.176ex;" alt="{\displaystyle \int _{-\infty }^{+\infty }\delta '(x)\phi (x)\,\mathop {} \!\mathrm {d} x=-\int _{-\infty }^{+\infty }\delta (x)\phi '(x)\,\mathop {} \!\mathrm {d} x.}"></span></dd></dl> <p>Infatti, integrando per parti: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{+\infty }{\frac {\mathop {} \!\mathrm {d} }{\operatorname {d} t}}\delta (t)\phi (t)\,\operatorname {d} t=\left[\delta (t)\phi (t)\right]_{-\infty }^{+\infty }-\int _{-\infty }^{+\infty }\delta (t)\;{\frac {\operatorname {d} }{\operatorname {d} t}}\phi (t)\,\operatorname {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi>t</mi> <mo>=</mo> <msubsup> <mrow> <mo>[</mo> <mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">d</mi> <mrow> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{+\infty }{\frac {\mathop {} \!\mathrm {d} }{\operatorname {d} t}}\delta (t)\phi (t)\,\operatorname {d} t=\left[\delta (t)\phi (t)\right]_{-\infty }^{+\infty }-\int _{-\infty }^{+\infty }\delta (t)\;{\frac {\operatorname {d} }{\operatorname {d} t}}\phi (t)\,\operatorname {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2cba8df2cabec1df4968de75fd1ca9c9e458658" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:59.535ex; height:6.176ex;" alt="{\displaystyle \int _{-\infty }^{+\infty }{\frac {\mathop {} \!\mathrm {d} }{\operatorname {d} t}}\delta (t)\phi (t)\,\operatorname {d} t=\left[\delta (t)\phi (t)\right]_{-\infty }^{+\infty }-\int _{-\infty }^{+\infty }\delta (t)\;{\frac {\operatorname {d} }{\operatorname {d} t}}\phi (t)\,\operatorname {d} t}"></span></dd></dl> <p>e il termine valutato si annulla grazie alla definizione della delta. </p><p>La derivata <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>-esima è la distribuzione definita in modo analogo: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta ^{(k)}[\phi ]=(-1)^{k}\phi ^{(k)}(0).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">[</mo> <mi>ϕ</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta ^{(k)}[\phi ]=(-1)^{k}\phi ^{(k)}(0).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/346600317f5ef8fe753b4e28c838147c7019e8ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.44ex; height:3.343ex;" alt="{\displaystyle \delta ^{(k)}[\phi ]=(-1)^{k}\phi ^{(k)}(0).}"></span></dd></dl> <p>La derivata prima della delta è il limite del <a href="/wiki/Rapporto_incrementale" title="Rapporto incrementale">rapporto incrementale</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta '(x)=\lim _{h\to 0}{\frac {\delta (x+h)-\delta (x)}{h}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>δ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta '(x)=\lim _{h\to 0}{\frac {\delta (x+h)-\delta (x)}{h}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e663507250524166118f700547c741e46d72dbd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.653ex; height:5.843ex;" alt="{\displaystyle \delta '(x)=\lim _{h\to 0}{\frac {\delta (x+h)-\delta (x)}{h}},}"></span></dd></dl> <p>e più precisamente si ha: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta '=\lim _{h\to 0}{\frac {1}{h}}(\tau _{h}\delta -\delta ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>δ</mi> <mo>′</mo> </msup> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>h</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mi>δ</mi> <mo>−</mo> <mi>δ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta '=\lim _{h\to 0}{\frac {1}{h}}(\tau _{h}\delta -\delta ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d69e97b47a309e8735752c60e1fcf527b2f6f6d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.4ex; height:5.343ex;" alt="{\displaystyle \delta '=\lim _{h\to 0}{\frac {1}{h}}(\tau _{h}\delta -\delta ),}"></span></dd></dl> <p>dove <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{h}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{h}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/867e45ea8ac80b41aff93fc173490481a2904c62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.195ex; height:2.009ex;" alt="{\displaystyle \tau _{h}}"></span> è l'operatore di traslazione, definito su una funzione da <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{h}\phi (x)=\phi (x+h)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{h}\phi (x)=\phi (x+h)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40fc0a5a292971c8788d8800acc924781e7be04a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.522ex; height:2.843ex;" alt="{\displaystyle \tau _{h}\phi (x)=\phi (x+h)}"></span> e su una distribuzione da: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\tau _{h}S)[\phi ]=S[\tau _{-h}\phi ].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mi>S</mi> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>ϕ</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>S</mi> <mo stretchy="false">[</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>h</mi> </mrow> </msub> <mi>ϕ</mi> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\tau _{h}S)[\phi ]=S[\tau _{-h}\phi ].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7a70024c4c1f98d1d38662e084192bd6f89a827" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.58ex; height:2.843ex;" alt="{\displaystyle (\tau _{h}S)[\phi ]=S[\tau _{-h}\phi ].}"></span></dd></dl> <p>Dalla derivata della delta si può recuperare la delta stessa tramite la formula: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\delta '(x)=-\delta (x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <msup> <mi>δ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\delta '(x)=-\delta (x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ef95344a964c3c185ef4b4252083a73f18a1f5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.948ex; height:3.009ex;" alt="{\displaystyle x\delta '(x)=-\delta (x).}"></span></dd></dl> <p>Inoltre, la convoluzione di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>δ</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta '}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aab2d3fbf71b804a6005c7546c058e15f4ea3685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.738ex; height:2.509ex;" alt="{\displaystyle \delta '}"></span> una funzione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> liscia e a supporto compatto è: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta '*f=\delta *f'=f',}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>δ</mi> <mo>′</mo> </msup> <mo>∗</mo> <mi>f</mi> <mo>=</mo> <mi>δ</mi> <mo>∗</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta '*f=\delta *f'=f',}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/144c0e65bbe33d5d01a3e16c247d03cd8522ada5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.309ex; height:2.843ex;" alt="{\displaystyle \delta '*f=\delta *f'=f',}"></span></dd></dl> <p>esplicitamente: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\delta '*f)(a)=\int _{-\infty }^{+\infty }\delta '(a-x)f(x)\,\mathop {} \!\mathrm {d} x=f'(a),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>δ</mi> <mo>′</mo> </msup> <mo>∗</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>δ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\delta '*f)(a)=\int _{-\infty }^{+\infty }\delta '(a-x)f(x)\,\mathop {} \!\mathrm {d} x=f'(a),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e83a00b89f8698dd16d074d3440eafcf22c3265" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.715ex; height:6.176ex;" alt="{\displaystyle (\delta '*f)(a)=\int _{-\infty }^{+\infty }\delta '(a-x)f(x)\,\mathop {} \!\mathrm {d} x=f'(a),}"></span></dd></dl> <p>che segue direttamente dalle proprietà della derivata di una convoluzione nel senso delle distribuzioni. </p> <div class="mw-heading mw-heading2"><h2 id="La_delta_come_limite_di_una_successione">La delta come limite di una successione</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delta_di_Dirac&veaction=edit&section=16" title="Modifica la sezione La delta come limite di una successione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Delta_di_Dirac&action=edit&section=16" title="Edit section's source code: La delta come limite di una successione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La funzione delta può essere considerata come il limite di alcune particolari successioni </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (x)=\lim _{\varepsilon \to 0^{+}}\eta _{\varepsilon }(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> <mo stretchy="false">→</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> </munder> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (x)=\lim _{\varepsilon \to 0^{+}}\eta _{\varepsilon }(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca3f21c2fc0d8e1124f8373521e0939c0713c0a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.046ex; height:4.343ex;" alt="{\displaystyle \delta (x)=\lim _{\varepsilon \to 0^{+}}\eta _{\varepsilon }(x).}"></span></dd></dl> <p>In modo equivalente è definita utilizzando la convergenza nel senso delle distribuzioni: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{\varepsilon \to 0^{+}}\int _{-\infty }^{+\infty }\eta _{\varepsilon }(x)f(x)\,\mathop {} \!\mathrm {d} x=f(0),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> <mo stretchy="false">→</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> </munder> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{\varepsilon \to 0^{+}}\int _{-\infty }^{+\infty }\eta _{\varepsilon }(x)f(x)\,\mathop {} \!\mathrm {d} x=f(0),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4bf87248c39e7ace9df028085c25da89413e56a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.928ex; height:6.176ex;" alt="{\displaystyle \lim _{\varepsilon \to 0^{+}}\int _{-\infty }^{+\infty }\eta _{\varepsilon }(x)f(x)\,\mathop {} \!\mathrm {d} x=f(0),}"></span></dd></dl> <p>per tutte le <a href="/wiki/Funzioni_continue" class="mw-redirect" title="Funzioni continue">funzioni continue</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> a supporto compatto. La successione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{\varepsilon }(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{\varepsilon }(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60ae9c18828a364c4e212b0a1ff5e1fa5160614a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.293ex; height:2.843ex;" alt="{\displaystyle \eta _{\varepsilon }(x)}"></span> si dice allora successione di <i>approssimanti</i> della delta. È da tener presente che si tratta di <a href="/wiki/Distribuzione_(matematica)#Convergenza_e_topologia_debole" title="Distribuzione (matematica)">convergenza debole</a> nel senso della teoria delle distribuzioni, cioè valida in senso ordinario solo per la successione degli integrali. Di fatto molte delle successioni di approssimanti non sono convergenti in senso ordinario. </p><p>È possibile dare un criterio generale per le approssimanti della delta. Una successione di funzioni <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\delta _{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\delta _{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/982477366dc9943ec4083ba13c4d15ec1185f2df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.251ex; height:2.676ex;" alt="{\displaystyle {\delta _{n}}}"></span> localmente integrabili reali converge debolmente alla delta, se: </p> <ul><li>per ogni <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568095ad3924314374a5ab68fae17343661f2a71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon >0}"></span>, le successioni:</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{-a}\delta _{n}(x)\,\mathop {} \!\mathrm {d} x,\qquad \int _{a}^{+\infty }\delta _{n}(x)\,\mathop {} \!\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>a</mi> </mrow> </msubsup> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>,</mo> <mspace width="2em" /> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{-a}\delta _{n}(x)\,\mathop {} \!\mathrm {d} x,\qquad \int _{a}^{+\infty }\delta _{n}(x)\,\mathop {} \!\mathrm {d} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efdc495f4e563a0793fa03895a734e8524fca182" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.488ex; height:6.176ex;" alt="{\displaystyle \int _{-\infty }^{-a}\delta _{n}(x)\,\mathop {} \!\mathrm {d} x,\qquad \int _{a}^{+\infty }\delta _{n}(x)\,\mathop {} \!\mathrm {d} x}"></span></dd></dl></dd></dl> <dl><dd><a href="/wiki/Convergenza_di_funzioni#Convergenza_uniforme" class="mw-redirect" title="Convergenza di funzioni">convergono uniformemente</a> a 0 per ogni <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in [\epsilon ,{+\infty }]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mi>ϵ</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in [\epsilon ,{+\infty }]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83a6406461e9647ea92b2d07ac3d6539b9ecefae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.474ex; height:2.843ex;" alt="{\displaystyle a\in [\epsilon ,{+\infty }]}"></span></dd></dl> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }\,\int _{-\infty }^{+\infty }\delta _{n}(x)\,\mathop {} \!\mathrm {d} x=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mspace width="thinmathspace" /> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }\,\int _{-\infty }^{+\infty }\delta _{n}(x)\,\mathop {} \!\mathrm {d} x=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/580b3c930378f59049ecf765444b13b3aed9f354" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.099ex; height:6.176ex;" alt="{\displaystyle \lim _{n\to \infty }\,\int _{-\infty }^{+\infty }\delta _{n}(x)\,\mathop {} \!\mathrm {d} x=1}"></span></li></ul> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\int _{-\infty }^{a}\delta _{n}(x)\,\mathop {} \!\mathrm {d} x\right|<K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> <mo>|</mo> </mrow> <mo><</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\int _{-\infty }^{a}\delta _{n}(x)\,\mathop {} \!\mathrm {d} x\right|<K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63440b2d9bee2df8e96d7b0278d86451c313fa2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.078ex; height:6.176ex;" alt="{\displaystyle \left|\int _{-\infty }^{a}\delta _{n}(x)\,\mathop {} \!\mathrm {d} x\right|<K}"></span></li></ul> <dl><dd>per ogni <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }"></span>, dove <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> è un numero reale positivo indipendente da <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Successioni_che_rappresentano_la_delta_di_Dirac">Successioni che rappresentano la delta di Dirac</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delta_di_Dirac&veaction=edit&section=17" title="Modifica la sezione Successioni che rappresentano la delta di Dirac" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Delta_di_Dirac&action=edit&section=17" title="Edit section's source code: Successioni che rappresentano la delta di Dirac"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Di seguito alcune tra le più note successioni che rappresentano la delta di Dirac: </p> <ul><li>Limite di una <a href="/wiki/Distribuzione_normale" title="Distribuzione normale">distribuzione normale</a> (per <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9702f04f2d0e5b887b99faeeffb0c4cfd8263eee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle n\rightarrow \infty }"></span>):</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{n}(x)={\sqrt {\frac {n}{\pi }}}e^{-nx^{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>n</mi> <mi>π</mi> </mfrac> </msqrt> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{n}(x)={\sqrt {\frac {n}{\pi }}}e^{-nx^{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a82aebaf3f3360a5615617ff74c22425571c898b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.042ex; height:6.343ex;" alt="{\displaystyle \delta _{n}(x)={\sqrt {\frac {n}{\pi }}}e^{-nx^{2}}.}"></span></dd></dl></dd></dl> <ul><li>Limite di una <a href="/wiki/Distribuzione_di_Cauchy" title="Distribuzione di Cauchy">distribuzione di Cauchy</a> (per <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\to 0^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\to 0^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d34f4546d35bfc39ca1143560025d2f14618e8f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.682ex; height:2.509ex;" alt="{\displaystyle n\to 0^{+}}"></span>):</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{n}(x)={\frac {1}{\pi }}{\frac {n}{n^{2}+x^{2}}}={\frac {1}{2\pi }}\int _{-\infty }^{+\infty }e^{ikx-|nk|}\,\mathop {} \!\mathrm {d} k.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>k</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{n}(x)={\frac {1}{\pi }}{\frac {n}{n^{2}+x^{2}}}={\frac {1}{2\pi }}\int _{-\infty }^{+\infty }e^{ikx-|nk|}\,\mathop {} \!\mathrm {d} k.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29a64d1529f2f94f049336c60dffc4dbbf662f6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.628ex; height:6.176ex;" alt="{\displaystyle \delta _{n}(x)={\frac {1}{\pi }}{\frac {n}{n^{2}+x^{2}}}={\frac {1}{2\pi }}\int _{-\infty }^{+\infty }e^{ikx-|nk|}\,\mathop {} \!\mathrm {d} k.}"></span></dd></dl></dd></dl> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> di Cauchy:</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{n}(x)={\frac {e^{-|x/n|}}{2n}}={\frac {1}{2\pi }}\int _{-\infty }^{+\infty }{\frac {e^{ikx}}{1+n^{2}k^{2}}}\,\mathop {} \!\mathrm {d} k.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>k</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{n}(x)={\frac {e^{-|x/n|}}{2n}}={\frac {1}{2\pi }}\int _{-\infty }^{+\infty }{\frac {e^{ikx}}{1+n^{2}k^{2}}}\,\mathop {} \!\mathrm {d} k.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc21fb3132c397e1982b74da9314cfed6d68021c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.881ex; height:6.343ex;" alt="{\displaystyle \delta _{n}(x)={\frac {e^{-|x/n|}}{2n}}={\frac {1}{2\pi }}\int _{-\infty }^{+\infty }{\frac {e^{ikx}}{1+n^{2}k^{2}}}\,\mathop {} \!\mathrm {d} k.}"></span></dd></dl></dd></dl> <ul><li>Limite di una <a href="/wiki/Funzione_rettangolo" title="Funzione rettangolo">funzione rettangolo</a>:</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{n}(x)={\frac {\operatorname {rect} (x/n)}{n}}={\frac {1}{2\pi }}\int _{-\infty }^{+\infty }\operatorname {sinc} \left({\frac {nk}{2\pi }}\right)e^{ikx}\,\mathop {} \!\mathrm {d} k.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>rect</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>sinc</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>k</mi> </mrow> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>k</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{n}(x)={\frac {\operatorname {rect} (x/n)}{n}}={\frac {1}{2\pi }}\int _{-\infty }^{+\infty }\operatorname {sinc} \left({\frac {nk}{2\pi }}\right)e^{ikx}\,\mathop {} \!\mathrm {d} k.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ccecdb407957b3823bf1a7f0e027c8e8439576e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:50.079ex; height:6.343ex;" alt="{\displaystyle \delta _{n}(x)={\frac {\operatorname {rect} (x/n)}{n}}={\frac {1}{2\pi }}\int _{-\infty }^{+\infty }\operatorname {sinc} \left({\frac {nk}{2\pi }}\right)e^{ikx}\,\mathop {} \!\mathrm {d} k.}"></span></dd></dl></dd></dl> <ul><li>Funzione rettangolare <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> (per <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\to 0^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\to 0^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d34f4546d35bfc39ca1143560025d2f14618e8f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.682ex; height:2.509ex;" alt="{\displaystyle n\to 0^{+}}"></span>):<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{n}(x)={\frac {1}{\pi x}}\sin \left({\frac {x}{n}}\right)={\frac {1}{2\pi }}\int _{-1/n}^{1/n}\cos(kx)\,\mathop {} \!\mathrm {d} k.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>π</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msubsup> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>k</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{n}(x)={\frac {1}{\pi x}}\sin \left({\frac {x}{n}}\right)={\frac {1}{2\pi }}\int _{-1/n}^{1/n}\cos(kx)\,\mathop {} \!\mathrm {d} k.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ab7340e94f268329b842e54da9ef70a5add4b46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:44.258ex; height:6.676ex;" alt="{\displaystyle \delta _{n}(x)={\frac {1}{\pi x}}\sin \left({\frac {x}{n}}\right)={\frac {1}{2\pi }}\int _{-1/n}^{1/n}\cos(kx)\,\mathop {} \!\mathrm {d} k.}"></span></dd></dl></dd></dl> <ul><li>Derivata della <a href="/wiki/Sigmoide" class="mw-redirect" title="Sigmoide">sigmoide</a> (o <a href="/wiki/Statistica_di_Fermi-Dirac" title="Statistica di Fermi-Dirac">Statistica di Fermi-Dirac</a>):</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{n}(x)=\partial _{x}{\frac {1}{1+e^{-x/n}}}=-\partial _{x}{\frac {1}{1+e^{x/n}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{n}(x)=\partial _{x}{\frac {1}{1+e^{-x/n}}}=-\partial _{x}{\frac {1}{1+e^{x/n}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13f37a85e2478796ef5fb100dde65718afc62ca5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:37.94ex; height:5.676ex;" alt="{\displaystyle \delta _{n}(x)=\partial _{x}{\frac {1}{1+e^{-x/n}}}=-\partial _{x}{\frac {1}{1+e^{x/n}}}.}"></span></dd></dl></dd></dl> <ul><li>Limite della <a href="/wiki/Funzioni_di_Airy" title="Funzioni di Airy">funzione di Airy</a>:</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{n}(x)={\frac {1}{n}}A_{i}\left({\frac {x}{n}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{n}(x)={\frac {1}{n}}A_{i}\left({\frac {x}{n}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b0534117eec3f343d902277beb41cabf8a6c1d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.689ex; height:5.176ex;" alt="{\displaystyle \delta _{n}(x)={\frac {1}{n}}A_{i}\left({\frac {x}{n}}\right).}"></span></dd></dl></dd></dl> <ul><li>Limite della <a href="/wiki/Funzione_di_Bessel" class="mw-redirect" title="Funzione di Bessel">funzione di Bessel</a>:</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{n}(x)={\frac {1}{n}}J_{1/n}\left({\frac {x+1}{n}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{n}(x)={\frac {1}{n}}J_{1/n}\left({\frac {x+1}{n}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27b1bed04fe6638b00bc0ad9e881f13ddc0b8042" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.883ex; height:6.176ex;" alt="{\displaystyle \delta _{n}(x)={\frac {1}{n}}J_{1/n}\left({\frac {x+1}{n}}\right).}"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading2"><h2 id="La_delta_e_la_trasformata_di_Fourier">La delta e la trasformata di Fourier</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delta_di_Dirac&veaction=edit&section=18" title="Modifica la sezione La delta e la trasformata di Fourier" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Delta_di_Dirac&action=edit&section=18" title="Edit section's source code: La delta e la trasformata di Fourier"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r130657691">body:not(.skin-minerva) .mw-parser-output .vedi-anche{font-size:95%}</style><style data-mw-deduplicate="TemplateStyles:r139142988">.mw-parser-output .hatnote-content{align-items:center;display:flex}.mw-parser-output .hatnote-icon{flex-shrink:0}.mw-parser-output .hatnote-icon img{display:flex}.mw-parser-output .hatnote-text{font-style:italic}body:not(.skin-minerva) .mw-parser-output .hatnote{border:1px solid #CCC;display:flex;margin:.5em 0;padding:.2em .5em}body:not(.skin-minerva) .mw-parser-output .hatnote-text{padding-left:.5em}body.skin-minerva .mw-parser-output .hatnote-icon{padding-right:8px}body.skin-minerva .mw-parser-output .hatnote-icon img{height:auto;width:16px}body.skin--responsive .mw-parser-output .hatnote a.new{color:#d73333}body.skin--responsive .mw-parser-output .hatnote a.new:visited{color:#a55858}</style> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Trasformata_di_Fourier" title="Trasformata di Fourier">Trasformata di Fourier</a></b>.</span></div> </div> <div class="mw-heading mw-heading3"><h3 id="Rappresentazione_di_Fourier_della_delta">Rappresentazione di Fourier della delta</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delta_di_Dirac&veaction=edit&section=19" title="Modifica la sezione Rappresentazione di Fourier della delta" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Delta_di_Dirac&action=edit&section=19" title="Edit section's source code: Rappresentazione di Fourier della delta"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ogni funzione appartenente ad <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{1}(\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9a86223fba658a5daf612b39d3e25495d548bdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.124ex; height:3.176ex;" alt="{\displaystyle L^{1}(\mathbb {R} )}"></span> può essere scritta come: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {1}{2\pi }}\int _{-\infty }^{+\infty }e^{ikx}\left(\int _{-\infty }^{+\infty }e^{-iky}f(y)\,\mathop {} \!\mathrm {d} y\right)\mathop {} \!\mathrm {d} k.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>k</mi> <mi>y</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>k</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {1}{2\pi }}\int _{-\infty }^{+\infty }e^{ikx}\left(\int _{-\infty }^{+\infty }e^{-iky}f(y)\,\mathop {} \!\mathrm {d} y\right)\mathop {} \!\mathrm {d} k.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1096fd2885e2b383cdf88989f9f1ee95f5e7cefc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:46.571ex; height:6.176ex;" alt="{\displaystyle f(x)={\frac {1}{2\pi }}\int _{-\infty }^{+\infty }e^{ikx}\left(\int _{-\infty }^{+\infty }e^{-iky}f(y)\,\mathop {} \!\mathrm {d} y\right)\mathop {} \!\mathrm {d} k.}"></span></dd></dl> <p>Non è possibile scambiare l'ordine di integrazione, tuttavia è possibile scrivere: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\lim _{N\to \infty }{\frac {1}{2\pi }}\int _{-N}^{+N}e^{ikx}\ \left(\int _{-\infty }^{+\infty }e^{-iky}f(y)\,\mathop {} \!\mathrm {d} y\right)\mathop {} \!\mathrm {d} k.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi>N</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> <mtext> </mtext> <mrow> <mo>(</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>k</mi> <mi>y</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>k</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\lim _{N\to \infty }{\frac {1}{2\pi }}\int _{-N}^{+N}e^{ikx}\ \left(\int _{-\infty }^{+\infty }e^{-iky}f(y)\,\mathop {} \!\mathrm {d} y\right)\mathop {} \!\mathrm {d} k.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50e260115978d30f3aca25689579a7bf209f1a65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:52.1ex; height:6.343ex;" alt="{\displaystyle f(x)=\lim _{N\to \infty }{\frac {1}{2\pi }}\int _{-N}^{+N}e^{ikx}\ \left(\int _{-\infty }^{+\infty }e^{-iky}f(y)\,\mathop {} \!\mathrm {d} y\right)\mathop {} \!\mathrm {d} k.}"></span></dd></dl> <p>Il primo termine dell'integrale equivale alla successione: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{N}(t)={\frac {1}{\pi }}{\frac {\sin Nt}{t}}={\frac {1}{2\pi }}\int _{-N}^{+N}e^{ikt}\,\mathop {} \!\mathrm {d} k.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>N</mi> <mi>t</mi> </mrow> <mi>t</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi>N</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>k</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{N}(t)={\frac {1}{\pi }}{\frac {\sin Nt}{t}}={\frac {1}{2\pi }}\int _{-N}^{+N}e^{ikt}\,\mathop {} \!\mathrm {d} k.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b17240e1c2a6535f5c257bdaaf20735ee71fefc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.517ex; height:6.343ex;" alt="{\displaystyle \delta _{N}(t)={\frac {1}{\pi }}{\frac {\sin Nt}{t}}={\frac {1}{2\pi }}\int _{-N}^{+N}e^{ikt}\,\mathop {} \!\mathrm {d} k.}"></span></dd></dl> <p>Si nota che tale successione gode delle proprietà: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{t\to {\pm \infty }}\delta _{N}(t)=0,\qquad \int _{-\infty }^{+\infty }\delta _{N}(t)\,\mathop {} \!\mathrm {d} t=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> <mi mathvariant="normal">∞</mi> </mrow> </mrow> </munder> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="2em" /> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{t\to {\pm \infty }}\delta _{N}(t)=0,\qquad \int _{-\infty }^{+\infty }\delta _{N}(t)\,\mathop {} \!\mathrm {d} t=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7273e3634c17b907a1fc930af5f1e461e4f03d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.404ex; height:6.176ex;" alt="{\displaystyle \lim _{t\to {\pm \infty }}\delta _{N}(t)=0,\qquad \int _{-\infty }^{+\infty }\delta _{N}(t)\,\mathop {} \!\mathrm {d} t=1}"></span></dd></dl> <p>che sono le proprietà richieste alla delta di Dirac. </p><p>Inserendo tale rappresentazione nella precedente scrittura, e sapendo che il <a href="/wiki/Teorema_di_Fubini" title="Teorema di Fubini">teorema di Fubini</a> Tonelli permette di scambiare l'ordine di integrazione, si ottiene infatti: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\lim _{N\to \infty }\int _{-\infty }^{+\infty }\delta _{N}(x-y)f(y)\,\mathop {} \!\mathrm {d} y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\lim _{N\to \infty }\int _{-\infty }^{+\infty }\delta _{N}(x-y)f(y)\,\mathop {} \!\mathrm {d} y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f0d654a8d2f024d9376227288bf4dc8dd590383" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.625ex; height:6.176ex;" alt="{\displaystyle f(x)=\lim _{N\to \infty }\int _{-\infty }^{+\infty }\delta _{N}(x-y)f(y)\,\mathop {} \!\mathrm {d} y.}"></span></dd></dl> <p>Ossia la delta di Dirac è definita come il limite della successione: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (t)=\lim _{N\to \infty }{\frac {1}{\pi }}{\frac {\sin Nt}{t}}=\lim _{N\to \infty }{\frac {1}{2\pi }}\int _{-N}^{+N}e^{ikt}\,\mathop {} \!\mathrm {d} k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>N</mi> <mi>t</mi> </mrow> <mi>t</mi> </mfrac> </mrow> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi>N</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (t)=\lim _{N\to \infty }{\frac {1}{\pi }}{\frac {\sin Nt}{t}}=\lim _{N\to \infty }{\frac {1}{2\pi }}\int _{-N}^{+N}e^{ikt}\,\mathop {} \!\mathrm {d} k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eddfb835b44fb9c32b8507bdfa25f639a5001ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.46ex; height:6.343ex;" alt="{\displaystyle \delta (t)=\lim _{N\to \infty }{\frac {1}{\pi }}{\frac {\sin Nt}{t}}=\lim _{N\to \infty }{\frac {1}{2\pi }}\int _{-N}^{+N}e^{ikt}\,\mathop {} \!\mathrm {d} k}"></span></dd></dl> <p>e dunque la rappresentazione di Fourier della delta è: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (t)={\frac {1}{2\pi }}\int _{-\infty }^{+\infty }e^{ikt}\,\mathop {} \!\mathrm {d} k.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>k</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (t)={\frac {1}{2\pi }}\int _{-\infty }^{+\infty }e^{ikt}\,\mathop {} \!\mathrm {d} k.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/465e7bdfd4c4811e14bac48ecfbf142dd3fd911d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.777ex; height:6.176ex;" alt="{\displaystyle \delta (t)={\frac {1}{2\pi }}\int _{-\infty }^{+\infty }e^{ikt}\,\mathop {} \!\mathrm {d} k.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="La_trasformata_della_delta">La trasformata della delta</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delta_di_Dirac&veaction=edit&section=20" title="Modifica la sezione La trasformata della delta" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Delta_di_Dirac&action=edit&section=20" title="Edit section's source code: La trasformata della delta"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La rappresentazione di Fourier rende evidente che la delta è l'antitrasformata della funzione costante <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea78f54e69b72f398cf6077e61c50a05b532d4c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.678ex; height:2.843ex;" alt="{\displaystyle f(x)=1}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{+\infty }1\cdot e^{i2\pi xk}\,\mathop {} \!\mathrm {d} k=\delta (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mn>1</mn> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>x</mi> <mi>k</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>k</mi> <mo>=</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{+\infty }1\cdot e^{i2\pi xk}\,\mathop {} \!\mathrm {d} k=\delta (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80794b748b7792d0b30f4f31e13a4578b6a346c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.855ex; height:6.176ex;" alt="{\displaystyle \int _{-\infty }^{+\infty }1\cdot e^{i2\pi xk}\,\mathop {} \!\mathrm {d} k=\delta (x)}"></span></dd></dl> <p>e dunque: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\delta }}(k)=\int _{-\infty }^{+\infty }e^{-i2\pi xk}\delta (x)\,\mathop {} \!\mathrm {d} x=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>δ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>x</mi> <mi>k</mi> </mrow> </msup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\delta }}(k)=\int _{-\infty }^{+\infty }e^{-i2\pi xk}\delta (x)\,\mathop {} \!\mathrm {d} x=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eac53fef21183b95ffefaa2818332e821bf46d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.64ex; height:6.176ex;" alt="{\displaystyle {\hat {\delta }}(k)=\int _{-\infty }^{+\infty }e^{-i2\pi xk}\delta (x)\,\mathop {} \!\mathrm {d} x=1.}"></span></dd></dl> <p>La dimostrazione si può ottenere anche a partire dalla definizione di trasformata di Fourier delle distribuzioni: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathcal {F}}[\delta ],\phi )=\int _{-\infty }^{+\infty }{\mathcal {F}}[\delta ](\omega )\phi (\omega )\,\mathop {} \!\mathrm {d} \omega =\int _{-\infty }^{+\infty }\delta (\omega ){\mathcal {F}}[\phi ](\omega )\,\mathop {} \!\mathrm {d} \omega ={\mathcal {F}}[\phi ](0)=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>δ</mi> <mo stretchy="false">]</mo> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>δ</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ω</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>ϕ</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>ϕ</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathcal {F}}[\delta ],\phi )=\int _{-\infty }^{+\infty }{\mathcal {F}}[\delta ](\omega )\phi (\omega )\,\mathop {} \!\mathrm {d} \omega =\int _{-\infty }^{+\infty }\delta (\omega ){\mathcal {F}}[\phi ](\omega )\,\mathop {} \!\mathrm {d} \omega ={\mathcal {F}}[\phi ](0)=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79586c4a7f0c89a742e8fd2564ab239f3a1de64e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:71.188ex; height:6.176ex;" alt="{\displaystyle ({\mathcal {F}}[\delta ],\phi )=\int _{-\infty }^{+\infty }{\mathcal {F}}[\delta ](\omega )\phi (\omega )\,\mathop {} \!\mathrm {d} \omega =\int _{-\infty }^{+\infty }\delta (\omega ){\mathcal {F}}[\phi ](\omega )\,\mathop {} \!\mathrm {d} \omega ={\mathcal {F}}[\phi ](0)=}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\left[\int _{-\infty }^{+\infty }\phi (t)e^{-i\omega t}\,\mathop {} \!\mathrm {d} t\right]_{\omega =0}=\int _{-\infty }^{+\infty }\phi (t)\,\mathop {} \!\mathrm {d} t=(1,\phi ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <msub> <mrow> <mo>[</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\left[\int _{-\infty }^{+\infty }\phi (t)e^{-i\omega t}\,\mathop {} \!\mathrm {d} t\right]_{\omega =0}=\int _{-\infty }^{+\infty }\phi (t)\,\mathop {} \!\mathrm {d} t=(1,\phi ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e4bfac72068f2306167bff1cb57561413038fdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:51.168ex; height:6.343ex;" alt="{\displaystyle =\left[\int _{-\infty }^{+\infty }\phi (t)e^{-i\omega t}\,\mathop {} \!\mathrm {d} t\right]_{\omega =0}=\int _{-\infty }^{+\infty }\phi (t)\,\mathop {} \!\mathrm {d} t=(1,\phi ).}"></span></dd></dl> <p>La trasformata <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\delta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>δ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\delta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2658f21ffb6333f21dc9c0685cd3c2eebe320008" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.302ex; height:2.843ex;" alt="{\displaystyle {\hat {\delta }}}"></span> della delta è definita come l'unica distribuzione temperata tale che: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle {\hat {\delta }},\phi \rangle =\langle \delta ,{\hat {\phi }}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>δ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>δ</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle {\hat {\delta }},\phi \rangle =\langle \delta ,{\hat {\phi }}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/078199534ba88bea9d93a343888ef5a330b9f510" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.989ex; height:3.343ex;" alt="{\displaystyle \langle {\hat {\delta }},\phi \rangle =\langle \delta ,{\hat {\phi }}\rangle }"></span></dd></dl> <p>per ogni funzione di Schwartz <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span>. </p><p>Segue inoltre che la delta fornisce la condizione di ortogonalizzazione delle autofunzioni degli operatori di derivazione e integrazione, che costituiscono il nucleo della <a href="/wiki/Trasformata_integrale" title="Trasformata integrale">trasformata integrale</a> di Fourier su <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{+\infty }e^{i2\pi \xi _{1}t}\left[e^{i2\pi \xi _{2}t}\right]^{*}\,\mathop {} \!\mathrm {d} t=\int _{-\infty }^{+\infty }e^{-i2\pi (\xi _{2}-\xi _{1})t}\,\mathop {} \!\mathrm {d} t=\delta (\xi _{1}-\xi _{2}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>t</mi> </mrow> </msup> <msup> <mrow> <mo>[</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>t</mi> </mrow> </msup> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mo stretchy="false">(</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{+\infty }e^{i2\pi \xi _{1}t}\left[e^{i2\pi \xi _{2}t}\right]^{*}\,\mathop {} \!\mathrm {d} t=\int _{-\infty }^{+\infty }e^{-i2\pi (\xi _{2}-\xi _{1})t}\,\mathop {} \!\mathrm {d} t=\delta (\xi _{1}-\xi _{2}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c09f7e1b0300dd15c693358351e38673cc34c4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:60.27ex; height:6.176ex;" alt="{\displaystyle \int _{-\infty }^{+\infty }e^{i2\pi \xi _{1}t}\left[e^{i2\pi \xi _{2}t}\right]^{*}\,\mathop {} \!\mathrm {d} t=\int _{-\infty }^{+\infty }e^{-i2\pi (\xi _{2}-\xi _{1})t}\,\mathop {} \!\mathrm {d} t=\delta (\xi _{1}-\xi _{2}).}"></span></dd></dl> <p>Tramite <a href="/wiki/Prolungamento_analitico" title="Prolungamento analitico">prolungamento analitico</a> è anche possibile definire la <a href="/wiki/Trasformata_di_Laplace" title="Trasformata di Laplace">trasformata di Laplace</a> della delta nel seguente modo: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{+\infty }\delta (t-a)e^{-st}\,\mathop {} \!\mathrm {d} t=e^{-sa}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>a</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{+\infty }\delta (t-a)e^{-st}\,\mathop {} \!\mathrm {d} t=e^{-sa}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de0327bb57906bb5c126f7c531441cef88b63c5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.619ex; height:6.009ex;" alt="{\displaystyle \int _{0}^{+\infty }\delta (t-a)e^{-st}\,\mathop {} \!\mathrm {d} t=e^{-sa}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delta_di_Dirac&veaction=edit&section=21" title="Modifica la sezione Note" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Delta_di_Dirac&action=edit&section=21" title="Edit section's source code: Note"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFreed">Reed, Simon</a>, pag. 135</cite>.</span> </li> <li id="cite_note-2"><a href="#cite_ref-2"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFnasa">F. Farassat</a>, pag. 4</cite>.</span> </li> <li id="cite_note-3"><a href="#cite_ref-3"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFreed">Reed, Simon</a>, pag. 136</cite>.</span> </li> <li id="cite_note-4"><a href="#cite_ref-4"><b>^</b></a> <span class="reference-text">Se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (n,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (n,x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c1f9db0493c9a0d22ebed838de7b34e3d1d4519" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.616ex; height:2.843ex;" alt="{\displaystyle \delta (n,x)}"></span> è una <a href="/wiki/Distribuzione_di_probabilit%C3%A0" class="mw-redirect" title="Distribuzione di probabilità">distribuzione di probabilità</a> su tutto l'asse reale (per esempio non è negativa tra <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bddbb0e4420a7e744cf71bd71216e11b0bf88831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }"></span>), allora un'altra <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\phi }(n,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\phi }(n,x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eaf1165f7ed9cf74ed9685f2c64dcc30a526232" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.812ex; height:3.009ex;" alt="{\displaystyle \delta _{\phi }(n,x)}"></span> può essere costruita sulla sua <a href="/wiki/Funzione_caratteristica_(teoria_della_probabilit%C3%A0)" title="Funzione caratteristica (teoria della probabilità)">funzione caratteristica</a> come segue: <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\varphi }(a,x)={\frac {1}{2\pi }}~{\frac {\varphi (1/a,x)}{\delta (1/a,0)}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\varphi }(a,x)={\frac {1}{2\pi }}~{\frac {\varphi (1/a,x)}{\delta (1/a,0)}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9df1a165881fbf9deae9e4a3e1d13458a696b53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.483ex; height:6.509ex;" alt="{\displaystyle \delta _{\varphi }(a,x)={\frac {1}{2\pi }}~{\frac {\varphi (1/a,x)}{\delta (1/a,0)}},}"></span></dd></dl></dd></dl> <dl><dd>dove:</dd></dl> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (a,k)=\int _{-\infty }^{+\infty }\delta (a,x)e^{-ikx}\,\mathop {} \!\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP"> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (a,k)=\int _{-\infty }^{+\infty }\delta (a,x)e^{-ikx}\,\mathop {} \!\mathrm {d} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6ba85799d863e54a25d4c149406786f3696341a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.714ex; height:6.176ex;" alt="{\displaystyle \varphi (a,k)=\int _{-\infty }^{+\infty }\delta (a,x)e^{-ikx}\,\mathop {} \!\mathrm {d} x}"></span></dd></dl></dd></dl> <dl><dd>è la funzione caratteristica di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (n,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (n,x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c1f9db0493c9a0d22ebed838de7b34e3d1d4519" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.616ex; height:2.843ex;" alt="{\displaystyle \delta (n,x)}"></span>. Questo risultato è collegato alla proprietà di località della <a href="/wiki/Trasformata_di_Fourier" title="Trasformata di Fourier">trasformata di Fourier</a>.</dd></dl> </span></li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delta_di_Dirac&veaction=edit&section=22" title="Modifica la sezione Bibliografia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Delta_di_Dirac&action=edit&section=22" title="Edit section's source code: Bibliografia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite id="CITEREFreed" class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Michael Reed, Barry Simon, <span style="font-style:italic;">Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis</span>, 2ª ed., San Diego, California, Academic press inc., 1980, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/0-12-585050-6" title="Speciale:RicercaISBN/0-12-585050-6">0-12-585050-6</a>.</cite></li> <li><cite id="CITEREFnasa" class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) F. Farassat, <span style="font-style:italic;"><a rel="nofollow" class="external text" href="https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19940029887_1994029887.pdf">Introduction to Generalized Functions With Applications in Aerodynamics and Aeroacoustics</a></span>, Langley Research Center, Hampton, Virginia, NASA Technical Paper 3428, 1994.</cite></li> <li><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) JB Fourier, <a rel="nofollow" class="external text" href="http://books.google.com/books?id=-N8EAAAAYAAJ&pg=PA408&dq=%22when+the+integrals+are+taken+between+infinite+limits%22+%22that+is+to+say,+that+we+have+the+equation%22&hl=en&sa=X&ei=rFe-T96cEIzKiQKcyfDtDQ&ved=0CD4Q6AEwAA#v=onepage&q=%22when%20the%20integrals%20are%20taken%20between%20infinite%20limits%22%20%22that%20is%20to%20say%2C%20that%20we%20have%20the%20equation%22&f=false"><span style="font-style:italic;">The Analytical Theory of Heat</span></a>, English translation by Alexander Freeman, 1878, The University Press, 1822.</cite></li> <li><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Hikosaburo Komatsu, <a rel="nofollow" class="external text" href="http://books.google.com/books?id=8GwKzEemrIcC&pg=PA200&dq=%22Fourier+introduced+the%22+%22+-function+much+earlier%22&hl=en&sa=X&ei=oJa6T5L2O6SriQKGloCUBw&ved=0CDQQ6AEwAA#v=onepage&q=%22Fourier%20introduced%20the%22%20%22%20-function%20much%20earlier%22&f=false"><span style="font-style:italic;">Fourier's hyperfunctions and Heaviside's pseudodifferential operators</span></a>, in Takahiro Kawai, Keiko Fujita, eds (a cura di), <span style="font-style:italic;">Microlocal Analysis and Complex Fourier Analysis</span>, World Scientific, 2002, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/981-238-161-9" title="Speciale:RicercaISBN/981-238-161-9">981-238-161-9</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Tyn Myint-U., Lokenath Debnath, <a rel="nofollow" class="external text" href="http://books.google.com/books?id=Zbz5_UvERIIC&pg=PA4&dq=%22It+was+the+work+of+Augustin+Cauchy%22&hl=en&sa=X&ei=RnW6T52LNovYiQLa9-mABw&ved=0CDgQ6AEwAA#v=onepage&q=%22It%20was%20the%20work%20of%20Augustin%20Cauchy%22&f=false"><span style="font-style:italic;">Linear Partial Differential Equations for Scientists And Engineers</span></a>, 4th, Springer, 2007, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/0-8176-4393-1" title="Speciale:RicercaISBN/0-8176-4393-1">0-8176-4393-1</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Lokenath Debnath, Dambaru Bhatta, <a rel="nofollow" class="external text" href="http://books.google.com/books?id=WbZcqdvCEfwC&pg=PA2&dq=%22It+was+the+work+of+Cauchy+that+contained%22&hl=en&sa=X&ei=Jym9T8L-NK6OigK-m_GYDg&ved=0CDQQ6AEwAA#v=onepage&q=%22It%20was%20the%20work%20of%20Cauchy%20that%20contained%22&f=false"><span style="font-style:italic;">Integral Transforms And Their Applications</span></a>, 2nd, CRC Press, 2007, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/1-58488-575-0" title="Speciale:RicercaISBN/1-58488-575-0">1-58488-575-0</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Ivor Grattan-Guinness, <a rel="nofollow" class="external text" href="http://books.google.com/books?id=_GgioErrbW8C&pg=PA653&dq=%22Further,+in+a+double+integral%22&hl=en&sa=X&ei=4gC9T7KVDvDRiALq-dTLDQ&ved=0CDgQ6AEwAA#v=onepage&q=%22Further%2C%20in%20a%20double%20integral%22&f=false"><span style="font-style:italic;">Convolutions in French Mathematics, 1800-1840: From the Calculus and Mechanics to Mathematical Analysis and Mathematical Physics, Volume 2</span></a>, Birkhäuser, 2009, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/3-7643-2238-1" title="Speciale:RicercaISBN/3-7643-2238-1">3-7643-2238-1</a>.</cite></li></ul> <div class="mw-heading mw-heading2"><h2 id="Voci_correlate">Voci correlate</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delta_di_Dirac&veaction=edit&section=23" title="Modifica la sezione Voci correlate" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Delta_di_Dirac&action=edit&section=23" title="Edit section's source code: Voci correlate"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Delta_di_Kronecker" title="Delta di Kronecker">Delta di Kronecker</a></li> <li><a href="/wiki/Distribuzione_(matematica)" title="Distribuzione (matematica)">Distribuzione (matematica)</a></li> <li><a href="/wiki/Funzione_gradino" title="Funzione gradino">Funzione gradino</a></li> <li><a href="/wiki/Funzione_indicatrice" title="Funzione indicatrice">Funzione indicatrice</a></li> <li><a href="/wiki/Misura_deltiforme" title="Misura deltiforme">Misura deltiforme</a></li> <li><a href="/wiki/Soluzione_fondamentale" title="Soluzione fondamentale">Soluzione fondamentale</a></li> <li><a href="/wiki/Trasformata_di_Fourier" title="Trasformata di Fourier">Trasformata di Fourier</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Altri_progetti">Altri progetti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delta_di_Dirac&veaction=edit&section=24" title="Modifica la sezione Altri progetti" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Delta_di_Dirac&action=edit&section=24" title="Edit section's source code: Altri progetti"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div id="interProject" class="toccolours" style="display: none; clear: both; margin-top: 2em"><p id="sisterProjects" style="background-color: #efefef; color: black; font-weight: bold; margin: 0"><span>Altri progetti</span></p><ul title="Collegamenti verso gli altri progetti Wikimedia"> <li class="" title=""><span class="plainlinks" title="commons:Category:Dirac distribution"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Dirac_distribution?uselang=it">Wikimedia Commons</a></span></li></ul></div> <ul><li><span typeof="mw:File"><a href="https://commons.wikimedia.org/wiki/?uselang=it" title="Collabora a Wikimedia Commons"><img alt="Collabora a Wikimedia Commons" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png" decoding="async" width="18" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/27px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/36px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> <span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/?uselang=it">Wikimedia Commons</a></span> contiene immagini o altri file su <b><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Dirac_distribution?uselang=it">Delta di Dirac</a></span></b></li></ul> <div class="mw-heading mw-heading2"><h2 id="Collegamenti_esterni">Collegamenti esterni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delta_di_Dirac&veaction=edit&section=25" title="Modifica la sezione Collegamenti esterni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Delta_di_Dirac&action=edit&section=25" title="Edit section's source code: Collegamenti esterni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li class="mw-empty-elt"></li> <li><cite id="CITEREFMathWorld" class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Eric W. Weisstein, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/DeltaFunction.html"><span style="font-style:italic;">Delta di Dirac</span></a>, su <span style="font-style:italic;"><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></span>, Wolfram Research.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q209675#P2812" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="https://www.physicsforums.com/showthread.php?t=73447">The Dirac Delta function</a>, a tutorial on the Dirac delta function.</li> <li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/lecture-23-use-with-impulse-inputs">Video Lectures - Lecture 23</a>, a lecture by <a href="/w/index.php?title=Arthur_Mattuck&action=edit&redlink=1" class="new" title="Arthur Mattuck (la pagina non esiste)">Arthur Mattuck</a>.</li> <li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="https://planetmath.org/encyclopedia/DiracDeltaFunction.html">Dirac Delta Function</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20040813080641/http://planetmath.org/encyclopedia/DiracDeltaFunction.html">Archiviato</a> il 13 agosto 2004 in <a href="/wiki/Internet_Archive" title="Internet Archive">Internet Archive</a>. on <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>.</li></ul> <div class="noprint" style="width:100%; padding: 3px 0; display: flex; flex-wrap: wrap; row-gap: 4px; column-gap: 8px; box-sizing: border-box;"><div style="flex-grow: 1"><style data-mw-deduplicate="TemplateStyles:r140555418">.mw-parser-output .itwiki-template-occhiello{width:100%;line-height:25px;border:1px solid #CCF;background-color:#F0EEFF;box-sizing:border-box}.mw-parser-output .itwiki-template-occhiello-progetto{background-color:#FAFAFA}@media screen{html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}</style><div class="itwiki-template-occhiello"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Crystal128-kmplot.svg" class="mw-file-description" title="Matematica"><img alt=" " src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/25px-Crystal128-kmplot.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/38px-Crystal128-kmplot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/50px-Crystal128-kmplot.svg.png 2x" data-file-width="245" data-file-height="244" /></a></span> <b><a href="/wiki/Portale:Matematica" title="Portale:Matematica">Portale Matematica</a></b>: accedi alle voci di Wikipedia che trattano di matematica</div></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; 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Delta di Dirac - Wikipedia