Dirac delta function

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class="vector-toc-text"> 3 Definitions </div> </a> <button aria-controls="toc-Definitions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Definitions subsection </button> <ul id="toc-Definitions-sublist" class="vector-toc-list"> <li id="toc-As_a_measure" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#As_a_measure"> <div class="vector-toc-text"> 3.1 As a measure </div> </a> <ul id="toc-As_a_measure-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-As_a_distribution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#As_a_distribution"> <div class="vector-toc-text"> 3.2 As a distribution </div> </a> <ul id="toc-As_a_distribution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> 3.3 Generalizations </div> </a> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> 4 Properties </div> </a> <button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Properties subsection </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Scaling_and_symmetry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Scaling_and_symmetry"> <div class="vector-toc-text"> 4.1 Scaling and symmetry </div> </a> <ul id="toc-Scaling_and_symmetry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebraic_properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebraic_properties"> <div class="vector-toc-text"> 4.2 Algebraic properties </div> </a> <ul id="toc-Algebraic_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Translation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Translation"> <div class="vector-toc-text"> 4.3 Translation </div> </a> <ul id="toc-Translation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Composition_with_a_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Composition_with_a_function"> <div class="vector-toc-text"> 4.4 Composition with a function </div> </a> <ul id="toc-Composition_with_a_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Indefinite_integral" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Indefinite_integral"> <div class="vector-toc-text"> 4.5 Indefinite integral </div> </a> <ul id="toc-Indefinite_integral-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties_in_n_dimensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Properties_in_n_dimensions"> <div class="vector-toc-text"> 4.6 Properties in n dimensions </div> </a> <ul id="toc-Properties_in_n_dimensions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Fourier_transform" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Fourier_transform"> <div class="vector-toc-text"> 5 Fourier transform </div> </a> <ul id="toc-Fourier_transform-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivatives" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Derivatives"> <div class="vector-toc-text"> 6 Derivatives </div> </a> <button aria-controls="toc-Derivatives-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Derivatives subsection </button> <ul id="toc-Derivatives-sublist" class="vector-toc-list"> <li id="toc-Higher_dimensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Higher_dimensions"> <div class="vector-toc-text"> 6.1 Higher dimensions </div> </a> <ul id="toc-Higher_dimensions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Representations_of_the_delta_function" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Representations_of_the_delta_function"> <div class="vector-toc-text"> 7 Representations of the delta function </div> </a> <button aria-controls="toc-Representations_of_the_delta_function-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Representations of the delta function subsection </button> <ul id="toc-Representations_of_the_delta_function-sublist" class="vector-toc-list"> <li id="toc-Approximations_to_the_identity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Approximations_to_the_identity"> <div class="vector-toc-text"> 7.1 Approximations to the identity </div> </a> <ul id="toc-Approximations_to_the_identity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Probabilistic_considerations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Probabilistic_considerations"> <div class="vector-toc-text"> 7.2 Probabilistic considerations </div> </a> <ul id="toc-Probabilistic_considerations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Semigroups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Semigroups"> <div class="vector-toc-text"> 7.3 Semigroups </div> </a> <ul id="toc-Semigroups-sublist" class="vector-toc-list"> <li id="toc-The_heat_kernel" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#The_heat_kernel"> <div class="vector-toc-text"> 7.3.1 The heat kernel </div> </a> <ul id="toc-The_heat_kernel-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_Poisson_kernel" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#The_Poisson_kernel"> <div class="vector-toc-text"> 7.3.2 The Poisson kernel </div> </a> <ul id="toc-The_Poisson_kernel-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Oscillatory_integrals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Oscillatory_integrals"> <div class="vector-toc-text"> 7.4 Oscillatory integrals </div> </a> <ul id="toc-Oscillatory_integrals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Plane_wave_decomposition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Plane_wave_decomposition"> <div class="vector-toc-text"> 7.5 Plane wave decomposition </div> </a> <ul id="toc-Plane_wave_decomposition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fourier_kernels" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fourier_kernels"> <div class="vector-toc-text"> 7.6 Fourier kernels </div> </a> <ul id="toc-Fourier_kernels-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hilbert_space_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hilbert_space_theory"> <div class="vector-toc-text"> 7.7 Hilbert space theory </div> </a> <ul id="toc-Hilbert_space_theory-sublist" class="vector-toc-list"> <li id="toc-Sobolev_spaces" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Sobolev_spaces"> <div class="vector-toc-text"> 7.7.1 Sobolev spaces </div> </a> <ul id="toc-Sobolev_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spaces_of_holomorphic_functions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Spaces_of_holomorphic_functions"> <div class="vector-toc-text"> 7.7.2 Spaces of holomorphic functions </div> </a> <ul id="toc-Spaces_of_holomorphic_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Resolutions_of_the_identity" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Resolutions_of_the_identity"> <div class="vector-toc-text"> 7.7.3 Resolutions of the identity </div> </a> <ul id="toc-Resolutions_of_the_identity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Infinitesimal_delta_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Infinitesimal_delta_functions"> <div class="vector-toc-text"> 7.8 Infinitesimal delta functions </div> </a> <ul id="toc-Infinitesimal_delta_functions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Dirac_comb" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Dirac_comb"> <div class="vector-toc-text"> 8 Dirac comb </div> </a> <ul id="toc-Dirac_comb-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sokhotski–Plemelj_theorem" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Sokhotski–Plemelj_theorem"> <div class="vector-toc-text"> 9 Sokhotski–Plemelj theorem </div> </a> <ul id="toc-Sokhotski–Plemelj_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relationship_to_the_Kronecker_delta" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Relationship_to_the_Kronecker_delta"> <div class="vector-toc-text"> 10 Relationship to the Kronecker delta </div> </a> <ul id="toc-Relationship_to_the_Kronecker_delta-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> 11 Applications </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Applications subsection </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Probability_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Probability_theory"> <div class="vector-toc-text"> 11.1 Probability theory </div> </a> <ul id="toc-Probability_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_mechanics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_mechanics"> <div class="vector-toc-text"> 11.2 Quantum mechanics </div> </a> <ul id="toc-Quantum_mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Structural_mechanics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Structural_mechanics"> <div class="vector-toc-text"> 11.3 Structural mechanics </div> </a> <ul id="toc-Structural_mechanics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 12 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 13 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 14 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 15 External links </div> </a> <ul id="toc-External_links-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="Contents" 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="Toggle the table of contents" > <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" > Toggle the table of contents </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">Dirac delta function</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="Go to an article in another language. Available in 45 languages" > <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" > 45 languages </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 – Arabic" lang="ar" hreflang="ar" data-title="دالة ديراك" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</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 – Bangla" lang="bn" hreflang="bn" data-title="ডিরাক ডেল্টা অপেক্ষক" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</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 – Belarusian" lang="be" hreflang="be" data-title="Дэльта-функцыя" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</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 – Bulgarian" lang="bg" hreflang="bg" data-title="Делта-функция" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</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 – Catalan" lang="ca" hreflang="ca" data-title="Delta de Dirac" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Diracovo_delta" title="Diracovo delta – Czech" lang="cs" hreflang="cs" data-title="Diracovo delta" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Diracs_deltafunktion" title="Diracs deltafunktion – Danish" lang="da" hreflang="da" data-title="Diracs deltafunktion" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Delta-Distribution" title="Delta-Distribution – German" lang="de" hreflang="de" data-title="Delta-Distribution" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Diraci_deltafunktsioon" title="Diraci deltafunktsioon – Estonian" lang="et" hreflang="et" data-title="Diraci deltafunktsioon" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</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 – Greek" lang="el" hreflang="el" data-title="Κρουστική συνάρτηση" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</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 – Spanish" lang="es" hreflang="es" data-title="Delta de Dirac" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</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">Esperanto</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 – Persian" lang="fa" hreflang="fa" data-title="تابع دلتای دیراک" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</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 – French" lang="fr" hreflang="fr" data-title="Distribution de Dirac" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</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 – Korean" lang="ko" hreflang="ko" data-title="디랙 델타 함수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</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">हिन्दी</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 – Indonesian" lang="id" hreflang="id" data-title="Fungsi delta Dirac" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Deltufalli%C3%B0" title="Deltufallið – Icelandic" lang="is" hreflang="is" data-title="Deltufallið" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Delta_di_Dirac" title="Delta di Dirac – Italian" lang="it" hreflang="it" data-title="Delta di Dirac" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</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 – Hebrew" lang="he" hreflang="he" data-title="פונקציית דלתא של דיראק" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</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 – Georgian" lang="ka" hreflang="ka" data-title="დირაკის დელტა ფუნქცია" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Delta_funkcija" title="Delta funkcija – Latvian" lang="lv" hreflang="lv" data-title="Delta funkcija" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Dirac-delta" title="Dirac-delta – Hungarian" lang="hu" hreflang="hu" data-title="Dirac-delta" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Diracdelta" title="Diracdelta – Dutch" lang="nl" hreflang="nl" data-title="Diracdelta" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</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 – Japanese" 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="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Diracs_deltafunksjon" title="Diracs deltafunksjon – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Diracs deltafunksjon" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Delta-funksiya" title="Delta-funksiya – Uzbek" lang="uz" hreflang="uz" data-title="Delta-funksiya" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Delta_Diraca" title="Delta Diraca – Polish" lang="pl" hreflang="pl" data-title="Delta Diraca" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</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 – Portuguese" lang="pt" hreflang="pt" data-title="Delta de Dirac" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</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 – Romanian" lang="ro" hreflang="ro" data-title="Funcția lui Dirac" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</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 – Russian" lang="ru" hreflang="ru" data-title="Дельта-функция" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</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 – Albanian" lang="sq" hreflang="sq" data-title="Funksioni i deltës së Dirakut" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</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 – Sinhala" lang="si" hreflang="si" data-title="ඩිරැක් ඩෙල්ටා ශ්‍රිතය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target">සිංහල</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">Simple English</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Porazdelitev_delta" title="Porazdelitev delta – Slovenian" lang="sl" hreflang="sl" data-title="Porazdelitev delta" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</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 – Serbian" lang="sr" hreflang="sr" data-title="Dirakova delta funkcija" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Diracin_deltafunktio" title="Diracin deltafunktio – Finnish" lang="fi" hreflang="fi" data-title="Diracin deltafunktio" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</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 – Swedish" lang="sv" hreflang="sv" data-title="Diracs delta-funktion" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</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 – Tatar" lang="tt" hreflang="tt" data-title="Дельта-функция" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target">Татарча / tatarça</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 – Thai" lang="th" hreflang="th" data-title="ดิแรกเดลตาฟังก์ชัน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</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 – Turkish" lang="tr" hreflang="tr" data-title="Dirac delta fonksiyonu" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</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 – Ukrainian" lang="uk" hreflang="uk" data-title="Дельта-функція Дірака" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</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 – Vietnamese" lang="vi" hreflang="vi" data-title="Hàm delta Dirac" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%8B%84%E6%8B%89%E5%85%8B%CE%B4%E5%87%BD%E6%95%B0" title="狄拉克δ函数 – Wu" lang="wuu" hreflang="wuu" data-title="狄拉克δ函数" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-zh badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a 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Click here for more information."><img alt="This is a good article. Click here for more information." src="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/19px-Symbol_support_vote.svg.png" decoding="async" width="19" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/29px-Symbol_support_vote.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/39px-Symbol_support_vote.svg.png 2x" data-file-width="180" data-file-height="185" /></a></div></div> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Generalized function whose value is zero everywhere except at zero</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Delta function" redirects here. For other uses, see <a href="/wiki/Delta_function_(disambiguation)" class="mw-disambig" title="Delta function (disambiguation)">Delta function (disambiguation)</a>.</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Dirac_distribution_PDF.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dirac_distribution_PDF.svg/325px-Dirac_distribution_PDF.svg.png" decoding="async" width="325" height="244" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dirac_distribution_PDF.svg/488px-Dirac_distribution_PDF.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dirac_distribution_PDF.svg/650px-Dirac_distribution_PDF.svg.png 2x" data-file-width="1300" data-file-height="975" /></a><figcaption>Schematic representation of the Dirac delta function by a line surmounted by an arrow. The height of the arrow is usually meant to specify the value of any multiplicative constant, which will give the area under the function. The other convention is to write the area next to the arrowhead.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:Dirac_function_approximation.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/b4/Dirac_function_approximation.gif" decoding="async" width="200" height="335" class="mw-file-element" data-file-width="200" data-file-height="335" /></a><figcaption>The Dirac delta as the limit as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\to 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad1af22c1cd2d9f30958c076d8e63a44fbb80cd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.006ex; height:2.176ex;" alt="{\displaystyle a\to 0}"> (in the sense of <a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">distributions</a>) of the sequence of zero-centered <a href="/wiki/Normal_distribution" title="Normal distribution">normal distributions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{a}(x)={\frac {1}{\left|a\right|{\sqrt {\pi }}}}e^{-(x/a)^{2}}}"> <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>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow> <mo>|</mo> <mi>a</mi> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{a}(x)={\frac {1}{\left|a\right|{\sqrt {\pi }}}}e^{-(x/a)^{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fdc14b952358515e6ee3244cae716023efdea67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:22.723ex; height:6.176ex;" alt="{\displaystyle \delta _{a}(x)={\frac {1}{\left|a\right|{\sqrt {\pi }}}}e^{-(x/a)^{2}}}"> </figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title" style="background:#ccccff;display:block;margin-bottom:0.2em;"><a href="/wiki/Differential_equation" title="Differential equation">Differential equations</a></th></tr><tr><th class="sidebar-heading" style="background:#ddddff;font-size:105%;display:block;margin-bottom:0.4em;"> Scope</th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Fields</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="padding-bottom:0;"> <div class="hlist"><ul><li><a href="/wiki/Natural_science" title="Natural science">Natural sciences</a></li><li><a href="/wiki/Engineering" title="Engineering">Engineering</a></li></ul></div></th></tr><tr><td class="sidebar-content hlist" style="padding-bottom:0.6em;"> <ul><li><a href="/wiki/Astronomy" title="Astronomy">Astronomy</a></li> <li><a href="/wiki/Physics" title="Physics">Physics</a></li> <li><a href="/wiki/Chemistry" title="Chemistry">Chemistry</a></li> <li> <a href="/wiki/Biology" title="Biology">Biology</a></li> <li><a href="/wiki/Geology" title="Geology">Geology</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="padding-bottom:0;"> <a href="/wiki/Applied_mathematics" title="Applied mathematics">Applied mathematics</a></th></tr><tr><td class="sidebar-content hlist" style="padding-bottom:0.6em;"> <ul><li><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></li> <li><a href="/wiki/Chaos_theory" title="Chaos theory">Chaos theory</a></li> <li><a href="/wiki/Dynamical_systems" class="mw-redirect" title="Dynamical systems">Dynamical systems</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="padding-bottom:0;"> <a href="/wiki/Social_science" title="Social science">Social sciences</a></th></tr><tr><td class="sidebar-content hlist" style="padding-bottom:0.6em;;padding-bottom:0;"> <ul><li><a href="/wiki/Economics" title="Economics">Economics</a></li> <li><a href="/wiki/Population_dynamics" title="Population dynamics">Population dynamics</a></li></ul></td> </tr></tbody></table> <hr /> <a href="/wiki/List_of_named_differential_equations" title="List of named differential equations">List of named differential equations</a></div></div></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;font-size:105%;display:block;margin-bottom:0.4em;;display:block;margin-top:0.1em;"> Classification</th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Types</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary</a></li> <li><a href="/wiki/Partial_differential_equation" title="Partial differential equation">Partial</a></li> <li><a href="/wiki/Differential-algebraic_system_of_equations" title="Differential-algebraic system of equations">Differential-algebraic</a></li> <li><a href="/wiki/Integro-differential_equation" title="Integro-differential equation">Integro-differential</a></li> <li><a href="/wiki/Fractional_differential_equations" class="mw-redirect" title="Fractional differential equations">Fractional</a></li> <li><a href="/wiki/Linear_differential_equation" title="Linear differential equation">Linear</a></li> <li><a href="/wiki/Non-linear_differential_equation" class="mw-redirect" title="Non-linear differential equation">Non-linear</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading"> By variable type</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Dependent_and_independent_variables" title="Dependent and independent variables">Dependent and independent variables</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Autonomous_differential_equation" class="mw-redirect" title="Autonomous differential equation">Autonomous</a></li> <li>Coupled / Decoupled</li> <li><a href="/wiki/Exact_differential_equation" title="Exact differential equation">Exact</a></li> <li><a href="/wiki/Homogeneous_differential_equation" title="Homogeneous differential equation">Homogeneous</a> / <a href="/wiki/Non-homogeneous_differential_equation" class="mw-redirect" title="Non-homogeneous differential equation">Nonhomogeneous</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading"> Features</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Ordinary_differential_equation#Definitions" title="Ordinary differential equation">Order</a></li> <li><a href="/wiki/Differential_operator" title="Differential operator">Operator</a></li></ul> </div> <ul><li><a href="/wiki/Notation_for_differentiation" title="Notation for differentiation">Notation</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Relation to processes</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"> <ul><li><a href="/wiki/Difference_equation" class="mw-redirect" title="Difference equation">Difference (discrete analogue)</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic</a> <ul><li><a href="/wiki/Stochastic_partial_differential_equation" title="Stochastic partial differential equation">Stochastic partial</a></li></ul></li> <li><a href="/wiki/Delay_differential_equation" title="Delay differential equation">Delay</a></li></ul> </div></div></div></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;font-size:105%;display:block;margin-bottom:0.4em;"> Solution</th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Existence and uniqueness</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"> <ul><li><a href="/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem" title="Picard–Lindelöf theorem">Picard–Lindelöf theorem </a></li> <li><a href="/wiki/Peano_existence_theorem" title="Peano existence theorem">Peano existence theorem</a></li> <li><a href="/wiki/Carath%C3%A9odory%27s_existence_theorem" title="Carathéodory's existence theorem">Carathéodory's existence theorem</a></li> <li><a href="/wiki/Cauchy%E2%80%93Kowalevski_theorem" class="mw-redirect" title="Cauchy–Kowalevski theorem">Cauchy–Kowalevski theorem</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">General topics</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><div class="hlist"> <ul><li><a href="/wiki/Initial_condition" title="Initial condition">Initial conditions</a></li> <li><a href="/wiki/Boundary_value_problem" title="Boundary value problem">Boundary values</a> <ul><li><a href="/wiki/Dirichlet_boundary_condition" title="Dirichlet boundary condition">Dirichlet</a></li> <li><a href="/wiki/Neumann_boundary_condition" title="Neumann boundary condition">Neumann</a></li> <li><a href="/wiki/Robin_boundary_condition" title="Robin boundary condition">Robin</a></li> <li><a href="/wiki/Cauchy_problem" title="Cauchy problem">Cauchy problem</a></li></ul></li> <li><a href="/wiki/Wronskian" title="Wronskian">Wronskian</a></li> <li><a href="/wiki/Phase_portrait" title="Phase portrait">Phase portrait</a></li> <li><a href="/wiki/Lyapunov_stability" title="Lyapunov stability">Lyapunov</a> / <a href="/wiki/Asymptotic_stability" class="mw-redirect" title="Asymptotic stability">Asymptotic</a> / <a href="/wiki/Exponential_stability" title="Exponential stability">Exponential stability</a></li> <li><a href="/wiki/Rate_of_convergence" title="Rate of convergence">Rate of convergence</a></li> <li><a href="/wiki/Power_series_solution_of_differential_equations" title="Power series solution of differential equations">Series</a> / Integral solutions</li> <li><a href="/wiki/Numerical_integration" title="Numerical integration">Numerical integration</a></li> <li><a class="mw-selflink selflink">Dirac delta function</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Solution methods</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><div class="hlist"> <ul><li>Inspection</li> <li><a href="/wiki/Method_of_characteristics" title="Method of characteristics">Method of characteristics</a></li> <li> <a href="/wiki/Euler_method" title="Euler method">Euler</a></li> <li><a href="/wiki/Exponential_response_formula" title="Exponential response formula">Exponential response formula</a></li> <li><a href="/wiki/Finite_difference_method" title="Finite difference method">Finite difference</a> (<a href="/wiki/Crank%E2%80%93Nicolson_method" title="Crank–Nicolson method">Crank–Nicolson</a>)</li> <li><a href="/wiki/Finite_element_method" title="Finite element method">Finite element</a> <ul><li><a href="/wiki/Infinite_element_method" title="Infinite element method">Infinite element</a></li></ul></li> <li><a href="/wiki/Finite_volume_method" title="Finite volume method">Finite volume</a></li> <li><a href="/wiki/Galerkin_method" title="Galerkin method">Galerkin</a> <ul><li><a href="/wiki/Petrov%E2%80%93Galerkin_method" title="Petrov–Galerkin method">Petrov–Galerkin</a></li></ul></li> <li><a href="/wiki/Green%27s_function" title="Green's function">Green's function</a></li> <li><a href="/wiki/Integrating_factor" title="Integrating factor">Integrating factor</a></li> <li><a href="/wiki/Integral_transform" title="Integral transform">Integral transforms</a></li> <li><a href="/wiki/Perturbation_theory" title="Perturbation theory">Perturbation theory</a></li> <li><a href="/wiki/Runge%E2%80%93Kutta_methods" title="Runge–Kutta methods">Runge–Kutta</a></li></ul> </div> <ul><li><a href="/wiki/Separation_of_variables" title="Separation of variables">Separation of variables</a></li> <li><a href="/wiki/Method_of_undetermined_coefficients" title="Method of undetermined coefficients">Undetermined coefficients</a></li> <li><a href="/wiki/Variation_of_parameters" title="Variation of parameters">Variation of parameters</a></li></ul></div></div></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;font-size:105%;display:block;margin-bottom:0.4em;"> People</th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">List</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><div class="hlist" style="padding-top:0.5em"> <ul><li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li> <li><a href="/wiki/Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Gottfried Leibniz</a></li> <li><a href="/wiki/Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a></li> <li><a href="/wiki/J%C3%B3zef_Maria_Hoene-Wro%C5%84ski" title="Józef Maria Hoene-Wroński">Józef Maria Hoene-Wroński</a></li> <li><a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Joseph Fourier</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li> <li><a href="/wiki/George_Green_(mathematician)" title="George Green (mathematician)">George Green</a></li> <li><a href="/wiki/Carl_David_Tolm%C3%A9_Runge" class="mw-redirect" title="Carl David Tolmé Runge">Carl David Tolmé Runge</a></li> <li><a href="/wiki/Martin_Kutta" title="Martin Kutta">Martin Kutta</a></li> <li><a href="/wiki/Rudolf_Lipschitz" title="Rudolf Lipschitz">Rudolf Lipschitz</a></li> <li><a href="/wiki/Ernst_Lindel%C3%B6f" class="mw-redirect" title="Ernst Lindelöf">Ernst Lindelöf</a></li> <li><a href="/wiki/%C3%89mile_Picard" title="Émile Picard">Émile Picard</a></li> <li><a href="/wiki/Phyllis_Nicolson" title="Phyllis Nicolson">Phyllis Nicolson</a></li> <li><a href="/wiki/John_Crank" title="John Crank">John Crank</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Differential_equations" title="Template:Differential equations"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Differential_equations" title="Template talk:Differential equations"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Differential_equations" title="Special:EditPage/Template:Differential equations"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> In <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, the Dirac delta function (or δ distribution), also known as the unit impulse,<a href="#cite_note-FOOTNOTEatis2013unit_impulse-1">[1]</a> is a <a href="/wiki/Generalized_function" title="Generalized function">generalized function</a> on the <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a>, whose value is zero everywhere except at zero, and whose <a href="/wiki/Integral" title="Integral">integral</a> over the entire real line is equal to one.<a href="#cite_note-FOOTNOTEArfkenWeber200084-2">[2]</a><a href="#cite_note-FOOTNOTEDirac1930§22_The_''δ''_function-3">[3]</a><a href="#cite_note-FOOTNOTEGelfandShilov1966–1968Volume_I,_§1.1-4">[4]</a> Thus it can be represented heuristically as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (x)={\begin{cases}0,&x\neq 0\\{\infty },&x=0\end{cases}}}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mi>x</mi> <mo>≠</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> <mo>,</mo> </mtd> <mtd> <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 \delta (x)={\begin{cases}0,&x\neq 0\\{\infty },&x=0\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91e3bc73bcd8fd2d467cf2af7aef0e2e014ae6e2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.664ex; height:6.176ex;" alt="{\displaystyle \delta (x)={\begin{cases}0,&x\neq 0\\{\infty },&x=0\end{cases}}}"> such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }\delta (x)dx=1.}"> <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"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }\delta (x)dx=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/501e41bca83b0b6f5d744ae653c282b2d290bc5a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.475ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }\delta (x)dx=1.}"> Since there is no function having this property, modelling the delta "function" rigorously involves the use of <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limits</a> or, as is common in mathematics, <a href="/wiki/Measure_theory" class="mw-redirect" title="Measure theory">measure theory</a> and the theory of <a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">distributions</a>. The delta function was introduced by physicist <a href="/wiki/Paul_Dirac" title="Paul Dirac">Paul Dirac</a>, and has since been applied routinely in physics and engineering to model point masses and instantaneous impulses. It is called the delta function because it is a continuous analogue of the <a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a> function, which is usually defined on a discrete domain and takes values 0 and 1. The mathematical rigor of the delta function was disputed until <a href="/wiki/Laurent_Schwartz" title="Laurent Schwartz">Laurent Schwartz</a> developed the theory of distributions, where it is defined as a linear form acting on functions. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Motivation_and_overview">Motivation and overview</h2>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=1" title="Edit section: Motivation and overview">edit</a>]</div> The <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph</a> of the Dirac delta is usually thought of as following the whole x-axis and the positive y-axis.<a href="#cite_note-5">[5]</a>: 174  The Dirac delta is used to model a tall narrow spike function (an impulse), and other similar <a href="/wiki/Abstraction" title="Abstraction">abstractions</a> such as a <a href="/wiki/Point_charge" class="mw-redirect" title="Point charge">point charge</a>, <a href="/wiki/Point_mass" class="mw-redirect" title="Point mass">point mass</a> or <a href="/wiki/Electron" title="Electron">electron</a> point. For example, to calculate the <a href="/wiki/Dynamics_(mechanics)" class="mw-redirect" title="Dynamics (mechanics)">dynamics</a> of a <a href="/wiki/Billiard_ball" title="Billiard ball">billiard ball</a> being struck, one can approximate the <a href="/wiki/Force" title="Force">force</a> of the impact by a Dirac delta. In doing so, one not only simplifies the equations, but one also is able to calculate the <a href="/wiki/Motion_(physics)" class="mw-redirect" title="Motion (physics)">motion</a> of the ball, by only considering the total impulse of the collision, without a detailed model of all of the elastic energy transfer at subatomic levels (for instance). To be specific, suppose that a billiard ball is at rest. At time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=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=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43469ec032d858feae5aa87029e22eaaf0109e9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}"> it is struck by another ball, imparting it with a <a href="/wiki/Momentum" title="Momentum">momentum</a> P, with units kg⋅m⋅s−1. The exchange of momentum is not actually instantaneous, being mediated by elastic processes at the molecular and subatomic level, but for practical purposes it is convenient to consider that energy transfer as effectively instantaneous. The <a href="/wiki/Force" title="Force">force</a> therefore is P δ(t); the units of δ(t) are s−1. To model this situation more rigorously, suppose that the force instead is uniformly distributed over a small time interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t=[0,T]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t=[0,T]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9e2d96fc8758cc490227383eb4eadb234578795" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11ex; height:2.843ex;" alt="{\displaystyle \Delta t=[0,T]}">. That is, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\Delta t}(t)={\begin{cases}P/\Delta t&0<t\leq T,\\0&{\text{otherwise}}.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</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> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mtd> <mtd> <mn>0</mn> <mo><</mo> <mi>t</mi> <mo>≤</mo> <mi>T</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise</mtext> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\Delta t}(t)={\begin{cases}P/\Delta t&0<t\leq T,\\0&{\text{otherwise}}.\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9659e9a928b5f961fcae78af091c75e42055fa1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.419ex; height:6.176ex;" alt="{\displaystyle F_{\Delta t}(t)={\begin{cases}P/\Delta t&0<t\leq T,\\0&{\text{otherwise}}.\end{cases}}}"> Then the momentum at any time t is found by integration: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(t)=\int _{0}^{t}F_{\Delta t}(\tau )\,d\tau ={\begin{cases}P&t\geq T\\P\,t/\Delta t&0\leq t\leq T\\0&{\text{otherwise.}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>P</mi> </mtd> <mtd> <mi>t</mi> <mo>≥</mo> <mi>T</mi> </mtd> </mtr> <mtr> <mtd> <mi>P</mi> <mspace width="thinmathspace" /> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mtd> <mtd> <mn>0</mn> <mo>≤</mo> <mi>t</mi> <mo>≤</mo> <mi>T</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise.</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(t)=\int _{0}^{t}F_{\Delta t}(\tau )\,d\tau ={\begin{cases}P&t\geq T\\P\,t/\Delta t&0\leq t\leq T\\0&{\text{otherwise.}}\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4773a28889268dcfbd5fbc2476031db8bb527696" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; margin-left: -0.089ex; width:45.562ex; height:8.509ex;" alt="{\displaystyle p(t)=\int _{0}^{t}F_{\Delta t}(\tau )\,d\tau ={\begin{cases}P&t\geq T\\P\,t/\Delta t&0\leq t\leq T\\0&{\text{otherwise.}}\end{cases}}}"> Now, the model situation of an instantaneous transfer of momentum requires taking the limit as Δt → 0, giving a result everywhere except at 0: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(t)={\begin{cases}P&t>0\\0&t<0.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>t</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> <mi>P</mi> </mtd> <mtd> <mi>t</mi> <mo>></mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>t</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 p(t)={\begin{cases}P&t>0\\0&t<0.\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e797a75e81faa6b8e49ec30d1a3ba73a428175b2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.089ex; width:19.316ex; height:6.176ex;" alt="{\displaystyle p(t)={\begin{cases}P&t>0\\0&t<0.\end{cases}}}"> Here the functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\Delta t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\Delta t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/066097534d064bf09e1ada80287198283cb082c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.689ex; height:2.509ex;" alt="{\displaystyle F_{\Delta t}}"> are thought of as useful approximations to the idea of instantaneous transfer of momentum. The delta function allows us to construct an idealized limit of these approximations. Unfortunately, the actual limit of the functions (in the sense of <a href="/wiki/Pointwise_convergence" title="Pointwise convergence">pointwise convergence</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lim _{\Delta t\to 0^{+}}F_{\Delta t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">→</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> </munder> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \lim _{\Delta t\to 0^{+}}F_{\Delta t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88c4469f9c83b22d4b7d17bb2286baa97a110d9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.168ex; height:2.676ex;" alt="{\textstyle \lim _{\Delta t\to 0^{+}}F_{\Delta t}}"> is zero everywhere but a single point, where it is infinite. To make proper sense of the Dirac delta, we should instead insist that the property <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }F_{\Delta t}(t)\,dt=P,}"> <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"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mi>P</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }F_{\Delta t}(t)\,dt=P,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d27685cedc70dbae6103b1c8af44d728e782d8d8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.105ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }F_{\Delta t}(t)\,dt=P,}"> which holds for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce341dd9e80a38fa8665b0be0ae5e1b42f16e70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.036ex; height:2.176ex;" alt="{\displaystyle \Delta t>0}">, should continue to hold in the limit. So, in the equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle F(t)=P\,\delta (t)=\lim _{\Delta t\to 0}F_{\Delta t}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mspace width="thinmathspace" /> <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 mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle F(t)=P\,\delta (t)=\lim _{\Delta t\to 0}F_{\Delta t}(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1130a882e8233e8f15ab035dbf67ccde2968bcdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.032ex; height:2.843ex;" alt="{\textstyle F(t)=P\,\delta (t)=\lim _{\Delta t\to 0}F_{\Delta t}(t)}">, it is understood that the limit is always taken outside the integral. In applied mathematics, as we have done here, the delta function is often manipulated as a kind of limit (a <a href="/wiki/Weak_limit" class="mw-redirect" title="Weak limit">weak limit</a>) of a <a href="/wiki/Sequence" title="Sequence">sequence</a> of functions, each member of which has a tall spike at the origin: for example, a sequence of <a href="/wiki/Gaussian_distribution" class="mw-redirect" title="Gaussian distribution">Gaussian distributions</a> centered at the origin with <a href="/wiki/Variance" title="Variance">variance</a> tending to zero. The Dirac delta is not truly a function, at least not a usual one with domain and range in <a href="/wiki/Real_number" title="Real number">real numbers</a>. For example, the objects f(x) = δ(x) and g(x) = 0 are equal everywhere except at x = 0 yet have integrals that are different. According to <a href="/wiki/Lebesgue_integral#Basic_theorems_of_the_Lebesgue_integral" title="Lebesgue integral">Lebesgue integration theory</a>, if f and g are functions such that f = g <a href="/wiki/Almost_everywhere" title="Almost everywhere">almost everywhere</a>, then f is integrable <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> g is integrable and the integrals of f and g are identical. A rigorous approach to regarding the Dirac delta function as a <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical object</a> in its own right requires <a href="/wiki/Measure_theory" class="mw-redirect" title="Measure theory">measure theory</a> or the theory of <a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">distributions</a>. <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=2" title="Edit section: History">edit</a>]</div> <a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Joseph Fourier</a> presented what is now called the <a href="/wiki/Fourier_integral_theorem" class="mw-redirect" title="Fourier integral theorem">Fourier integral theorem</a> in his treatise Théorie analytique de la chaleur in the form:<a href="#cite_note-Fourier-6">[6]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\ \ d\alpha \,f(\alpha )\ \int _{-\infty }^{\infty }dp\ \cos(px-p\alpha )\ ,}"> <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"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mtext> </mtext> <mtext> </mtext> <mi>d</mi> <mi>α</mi> <mspace width="thinmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>d</mi> <mi>p</mi> <mtext> </mtext> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mi>x</mi> <mo>−</mo> <mi>p</mi> <mi>α</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\ \ d\alpha \,f(\alpha )\ \int _{-\infty }^{\infty }dp\ \cos(px-p\alpha )\ ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/becd8ed273bf37371fddee5f994a03acfbbd797f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:48.193ex; height:6.009ex;" alt="{\displaystyle f(x)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\ \ d\alpha \,f(\alpha )\ \int _{-\infty }^{\infty }dp\ \cos(px-p\alpha )\ ,}"> which is tantamount to the introduction of the δ-function in the form:<a href="#cite_note-Kawai-7">[7]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (x-\alpha )={\frac {1}{2\pi }}\int _{-\infty }^{\infty }dp\ \cos(px-p\alpha )\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>α</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"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>d</mi> <mi>p</mi> <mtext> </mtext> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mi>x</mi> <mo>−</mo> <mi>p</mi> <mi>α</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (x-\alpha )={\frac {1}{2\pi }}\int _{-\infty }^{\infty }dp\ \cos(px-p\alpha )\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/855cfbd62c5dd8b13927d93d9cfafc3a52600288" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.663ex; height:6.009ex;" alt="{\displaystyle \delta (x-\alpha )={\frac {1}{2\pi }}\int _{-\infty }^{\infty }dp\ \cos(px-p\alpha )\ .}"> Later, <a href="/wiki/Augustin_Cauchy" class="mw-redirect" title="Augustin Cauchy">Augustin Cauchy</a> expressed the theorem using exponentials:<a href="#cite_note-Myint-U-8">[8]</a><a href="#cite_note-Debnath-9">[9]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\ e^{ipx}\left(\int _{-\infty }^{\infty }e^{-ip\alpha }f(\alpha )\,d\alpha \right)\,dp.}"> <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"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>p</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"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>p</mi> <mi>α</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>α</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>p</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\ e^{ipx}\left(\int _{-\infty }^{\infty }e^{-ip\alpha }f(\alpha )\,d\alpha \right)\,dp.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74e95846a406cc87f80d3576531863dd2ba6d2b6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.453ex; height:6.176ex;" alt="{\displaystyle f(x)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\ e^{ipx}\left(\int _{-\infty }^{\infty }e^{-ip\alpha }f(\alpha )\,d\alpha \right)\,dp.}"> Cauchy pointed out that in some circumstances the order of integration is significant in this result (contrast <a href="/wiki/Fubini%27s_theorem" title="Fubini's theorem">Fubini's theorem</a>).<a href="#cite_note-Grattan-Guinness-10">[10]</a><a href="#cite_note-Cauchy-11">[11]</a> As justified using the <a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">theory of distributions</a>, the Cauchy equation can be rearranged to resemble Fourier's original formulation and expose the δ-function as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}f(x)&={\frac {1}{2\pi }}\int _{-\infty }^{\infty }e^{ipx}\left(\int _{-\infty }^{\infty }e^{-ip\alpha }f(\alpha )\,d\alpha \right)\,dp\\[4pt]&={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\left(\int _{-\infty }^{\infty }e^{ipx}e^{-ip\alpha }\,dp\right)f(\alpha )\,d\alpha =\int _{-\infty }^{\infty }\delta (x-\alpha )f(\alpha )\,d\alpha ,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <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"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>p</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"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>p</mi> <mi>α</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>α</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>p</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <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"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <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 mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>p</mi> <mi>x</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>p</mi> <mi>α</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>p</mi> </mrow> <mo>)</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>α</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>α</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f(x)&={\frac {1}{2\pi }}\int _{-\infty }^{\infty }e^{ipx}\left(\int _{-\infty }^{\infty }e^{-ip\alpha }f(\alpha )\,d\alpha \right)\,dp\\[4pt]&={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\left(\int _{-\infty }^{\infty }e^{ipx}e^{-ip\alpha }\,dp\right)f(\alpha )\,d\alpha =\int _{-\infty }^{\infty }\delta (x-\alpha )f(\alpha )\,d\alpha ,\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/656c821d528199232d832d301bbec290a815fe63" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:69.351ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}f(x)&={\frac {1}{2\pi }}\int _{-\infty }^{\infty }e^{ipx}\left(\int _{-\infty }^{\infty }e^{-ip\alpha }f(\alpha )\,d\alpha \right)\,dp\\[4pt]&={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\left(\int _{-\infty }^{\infty }e^{ipx}e^{-ip\alpha }\,dp\right)f(\alpha )\,d\alpha =\int _{-\infty }^{\infty }\delta (x-\alpha )f(\alpha )\,d\alpha ,\end{aligned}}}"> where the δ-function is expressed as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (x-\alpha )={\frac {1}{2\pi }}\int _{-\infty }^{\infty }e^{ip(x-\alpha )}\,dp\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>α</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"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>p</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (x-\alpha )={\frac {1}{2\pi }}\int _{-\infty }^{\infty }e^{ip(x-\alpha )}\,dp\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a17bef7344e179c88573e40d7dc2f10b17b881f4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.425ex; height:6.009ex;" alt="{\displaystyle \delta (x-\alpha )={\frac {1}{2\pi }}\int _{-\infty }^{\infty }e^{ip(x-\alpha )}\,dp\ .}"> A rigorous interpretation of the exponential form and the various limitations upon the function f necessary for its application extended over several centuries. The problems with a classical interpretation are explained as follows:<a href="#cite_note-Mitrović-12">[12]</a> <dl><dd>The greatest drawback of the classical Fourier transformation is a rather narrow class of functions (originals) for which it can be effectively computed. Namely, it is necessary that these functions <a href="/wiki/Rapidly_decreasing" class="mw-redirect" title="Rapidly decreasing">decrease sufficiently rapidly</a> to zero (in the neighborhood of infinity) to ensure the existence of the Fourier integral. For example, the Fourier transform of such simple functions as polynomials does not exist in the classical sense. The extension of the classical Fourier transformation to distributions considerably enlarged the class of functions that could be transformed and this removed many obstacles.</dd></dl> Further developments included generalization of the Fourier integral, "beginning with <a href="/wiki/Michel_Plancherel" title="Michel Plancherel">Plancherel's</a> pathbreaking L2-theory (1910), continuing with <a href="/wiki/Norbert_Wiener" title="Norbert Wiener">Wiener's</a> and <a href="/wiki/Salomon_Bochner" title="Salomon Bochner">Bochner's</a> works (around 1930) and culminating with the amalgamation into <a href="/wiki/Laurent_Schwartz" title="Laurent Schwartz">L. Schwartz's</a> theory of <a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">distributions</a> (1945) ...",<a href="#cite_note-Kracht-13">[13]</a> and leading to the formal development of the Dirac delta function. An <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a> formula for an infinitely tall, unit impulse delta function (infinitesimal version of <a href="/wiki/Cauchy_distribution" title="Cauchy distribution">Cauchy distribution</a>) explicitly appears in an 1827 text of <a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a>. <a href="#cite_note-FOOTNOTELaugwitz1989230-14">[14]</a> <a href="/wiki/Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Siméon Denis Poisson</a> considered the issue in connection with the study of wave propagation as did <a href="/wiki/Gustav_Kirchhoff" title="Gustav Kirchhoff">Gustav Kirchhoff</a> somewhat later. Kirchhoff and <a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">Hermann von Helmholtz</a> also introduced the unit impulse as a limit of <a href="/wiki/Gaussian_distribution" class="mw-redirect" title="Gaussian distribution">Gaussians</a>, which also corresponded to <a href="/wiki/Lord_Kelvin" title="Lord Kelvin">Lord Kelvin</a>'s notion of a point heat source. At the end of the 19th century, <a href="/wiki/Oliver_Heaviside" title="Oliver Heaviside">Oliver Heaviside</a> used formal <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a> to manipulate the unit impulse.<a href="#cite_note-15">[15]</a> The Dirac delta function as such was introduced by <a href="/wiki/Paul_Dirac" title="Paul Dirac">Paul Dirac</a> in his 1927 paper The Physical Interpretation of the Quantum Dynamics<a href="#cite_note-16">[16]</a> and used in his textbook <a href="/wiki/The_Principles_of_Quantum_Mechanics" title="The Principles of Quantum Mechanics">The Principles of Quantum Mechanics</a>.<a href="#cite_note-FOOTNOTEDirac1930§22_The_''δ''_function-3">[3]</a> He called it the "delta function" since he used it as a continuous analogue of the discrete <a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a>. <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=3" title="Edit section: Definitions">edit</a>]</div> The Dirac delta function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4457507451c205a7e6adda92d919ee4c4a369cea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.188ex; height:2.843ex;" alt="{\displaystyle \delta (x)}"> can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (x)\simeq {\begin{cases}+\infty ,&x=0\\0,&x\neq 0\end{cases}}}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> </mtd> <mtd> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <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 \delta (x)\simeq {\begin{cases}+\infty ,&x=0\\0,&x\neq 0\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a839e28bf4397d834d571fbe69960e6d7b8d37" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.473ex; height:6.176ex;" alt="{\displaystyle \delta (x)\simeq {\begin{cases}+\infty ,&x=0\\0,&x\neq 0\end{cases}}}"> and which is also constrained to satisfy the identity<a href="#cite_note-FOOTNOTEGelfandShilov1966–1968Volume_I,_§1.1,_p._1-17">[17]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }\delta (x)\,dx=1.}"> <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"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }\delta (x)\,dx=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/007285dfa02f992dc4148ef7f74c8d461c2ca0b8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.862ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }\delta (x)\,dx=1.}"> This is merely a <a href="/wiki/Heuristic" title="Heuristic">heuristic</a> characterization. The Dirac delta is not a function in the traditional sense as no <a href="/wiki/Extended_real_number" class="mw-redirect" title="Extended real number">extended real number</a> valued function defined on the real numbers has these properties.<a href="#cite_note-FOOTNOTEDirac193063-18">[18]</a> <div class="mw-heading mw-heading3"><h3 id="As_a_measure">As a measure</h3>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=4" title="Edit section: As a measure">edit</a>]</div> One way to rigorously capture the notion of the Dirac delta function is to define a <a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">measure</a>, called <a href="/wiki/Dirac_measure" title="Dirac measure">Dirac measure</a>, which accepts a subset A of the real line R as an argument, and returns δ(A) = 1 if 0 ∈ A, and δ(A) = 0 otherwise.<a href="#cite_note-Rudin_1966_loc=§1.20-19">[19]</a> If the delta function is conceptualized as modeling an idealized point mass at 0, then δ(A) represents the mass contained in the set A. One may then define the integral against δ as the integral of a function against this mass distribution. Formally, the <a href="/wiki/Lebesgue_integral" title="Lebesgue integral">Lebesgue integral</a> provides the necessary analytic device. The Lebesgue integral with respect to the measure δ satisfies <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }f(x)\,\delta (dx)=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"> <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 stretchy="false">(</mo> <mi>d</mi> <mi>x</mi> <mo 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 (dx)=f(0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a9b6ffbd48d5e84f6bbba55c16c6384b8b9adc6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.391ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }f(x)\,\delta (dx)=f(0)}"> for all continuous compactly supported functions f. The measure δ is not <a href="/wiki/Absolutely_continuous" class="mw-redirect" title="Absolutely continuous">absolutely continuous</a> with respect to the <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a>—in fact, it is a <a href="/wiki/Singular_measure" title="Singular measure">singular measure</a>. Consequently, the delta measure has no <a href="/wiki/Radon%E2%80%93Nikodym_derivative" class="mw-redirect" title="Radon–Nikodym derivative">Radon–Nikodym derivative</a> (with respect to Lebesgue measure)—no true function for which the property <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }f(x)\,\delta (x)\,dx=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"> <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 stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <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 (x)\,dx=f(0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76c4fd51ea21ec903e1ca7ea11d48f2d8592ea7d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.107ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }f(x)\,\delta (x)\,dx=f(0)}"> holds.<a href="#cite_note-FOOTNOTEHewittStromberg1963§19.61-20">[20]</a> As a result, the latter notation is a convenient <a href="/wiki/Abuse_of_notation" title="Abuse of notation">abuse of notation</a>, and not a standard (<a href="/wiki/Riemann_integral" title="Riemann integral">Riemann</a> or <a href="/wiki/Lebesgue_integral" title="Lebesgue integral">Lebesgue</a>) integral. As a <a href="/wiki/Probability_measure" title="Probability measure">probability measure</a> on R, the delta measure is characterized by its <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a>, which is the <a href="/wiki/Unit_step_function" class="mw-redirect" title="Unit step function">unit step function</a>.<a href="#cite_note-21">[21]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x)={\begin{cases}1&{\text{if }}x\geq 0\\0&{\text{if }}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>if </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>if </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{if }}x\geq 0\\0&{\text{if }}x<0.\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0018ddb859cf54620a6169d5e79c18a45915951e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.457ex; height:6.176ex;" alt="{\displaystyle H(x)={\begin{cases}1&{\text{if }}x\geq 0\\0&{\text{if }}x<0.\end{cases}}}"> This means that H(x) is the integral of the cumulative <a href="/wiki/Indicator_function" title="Indicator function">indicator function</a> 1(−∞, x] with respect to the measure δ; to wit, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x)=\int _{\mathbf {R} }\mathbf {1} _{(-\infty ,x]}(t)\,\delta (dt)=\delta \!\left((-\infty ,x]\right),}"> <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="bold">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 stretchy="false">(</mo> <mi>d</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>δ</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(x)=\int _{\mathbf {R} }\mathbf {1} _{(-\infty ,x]}(t)\,\delta (dt)=\delta \!\left((-\infty ,x]\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a67b29a323434685200c36791fa963803be0c31" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:41.602ex; height:5.676ex;" alt="{\displaystyle H(x)=\int _{\mathbf {R} }\mathbf {1} _{(-\infty ,x]}(t)\,\delta (dt)=\delta \!\left((-\infty ,x]\right),}"> the latter being the measure of this interval. Thus in particular the integration of the delta function against a continuous function can be properly understood as a <a href="/wiki/Riemann%E2%80%93Stieltjes_integral" title="Riemann–Stieltjes integral">Riemann–Stieltjes integral</a>:<a href="#cite_note-FOOTNOTEHewittStromberg1963§9.19-22">[22]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }f(x)\,\delta (dx)=\int _{-\infty }^{\infty }f(x)\,dH(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"> <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 stretchy="false">(</mo> <mi>d</mi> <mi>x</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"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }f(x)\,\delta (dx)=\int _{-\infty }^{\infty }f(x)\,dH(x).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61af3e728d5a694eaae1b010f1c718bf1b49f012" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.844ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }f(x)\,\delta (dx)=\int _{-\infty }^{\infty }f(x)\,dH(x).}"> All higher <a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">moments</a> of δ are zero. In particular, <a href="/wiki/Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">characteristic function</a> and <a href="/wiki/Moment_generating_function" class="mw-redirect" title="Moment generating function">moment generating function</a> are both equal to one. <div class="mw-heading mw-heading3"><h3 id="As_a_distribution">As a distribution</h3>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=5" title="Edit section: As a distribution">edit</a>]</div> In the theory of <a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">distributions</a>, a generalized function is considered not a function in itself but only through how it affects other functions when "integrated" against them.<a href="#cite_note-FOOTNOTEHazewinkel2011[httpsbooksgooglecombooksid_YPtCAAAQBAJpgPA41_41]-23">[23]</a> In keeping with this philosophy, to define the delta function properly, it is enough to say what the "integral" of the delta function is against a sufficiently "good" test function φ. Test functions are also known as <a href="/wiki/Bump_function" title="Bump function">bump functions</a>. If the delta function is already understood as a measure, then the Lebesgue integral of a test function against that measure supplies the necessary integral. A typical space of test functions consists of all <a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">smooth functions</a> on R with <a href="/wiki/Compact_support" class="mw-redirect" title="Compact support">compact support</a> that have as many derivatives as required. As a distribution, the Dirac delta is a <a href="/wiki/Linear_functional" class="mw-redirect" title="Linear functional">linear functional</a> on the space of test functions and is defined by<a href="#cite_note-FOOTNOTEStrichartz1994§2.2-24">[24]</a> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta [\varphi ]=\varphi (0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta [\varphi ]=\varphi (0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df9006764a21dbc1593891d76b94e4a6c13cdd06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.453ex; height:2.843ex;" alt="{\displaystyle \delta [\varphi ]=\varphi (0)}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(1)</td></tr></tbody></table> for every test function φ. For δ to be properly a distribution, it must be continuous in a suitable topology on the space of test functions. In general, for a linear functional S on the space of test functions to define a distribution, it is necessary and sufficient that, for every positive integer N there is an integer MN and a constant CN such that for every test function φ, one has the inequality<a href="#cite_note-FOOTNOTEHörmander1983Theorem_2.1.5-25">[25]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|S[\varphi ]\right|\leq C_{N}\sum _{k=0}^{M_{N}}\sup _{x\in [-N,N]}\left|\varphi ^{(k)}(x)\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mi>S</mi> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">]</mo> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </munderover> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mi>N</mi> <mo>,</mo> <mi>N</mi> <mo stretchy="false">]</mo> </mrow> </munder> <mrow> <mo>|</mo> <mrow> <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> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|S[\varphi ]\right|\leq C_{N}\sum _{k=0}^{M_{N}}\sup _{x\in [-N,N]}\left|\varphi ^{(k)}(x)\right|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8921d57621545fc5640742ebe3f0a611bfd5998" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.501ex; height:7.676ex;" alt="{\displaystyle \left|S[\varphi ]\right|\leq C_{N}\sum _{k=0}^{M_{N}}\sup _{x\in [-N,N]}\left|\varphi ^{(k)}(x)\right|}"> where sup represents the <a href="/wiki/Infimum_and_supremum" title="Infimum and supremum">supremum</a>. With the δ distribution, one has such an inequality (with CN = 1) with MN = 0 for all N. Thus δ is a distribution of order zero. It is, furthermore, a distribution with compact support (the <a href="/wiki/Support_(mathematics)" title="Support (mathematics)">support</a> being {0}). The delta distribution can also be defined in several equivalent ways. For instance, it is the <a href="/wiki/Distributional_derivative" class="mw-redirect" title="Distributional derivative">distributional derivative</a> of the <a href="/wiki/Heaviside_step_function" title="Heaviside step function">Heaviside step function</a>. This means that for every test function φ, one has <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta [\varphi ]=-\int _{-\infty }^{\infty }\varphi '(x)\,H(x)\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">[</mo> <mi>φ</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>φ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta [\varphi ]=-\int _{-\infty }^{\infty }\varphi '(x)\,H(x)\,dx.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d867f4262298fa1f65e8b5ddc95df0f4d509f4a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.503ex; height:6.009ex;" alt="{\displaystyle \delta [\varphi ]=-\int _{-\infty }^{\infty }\varphi '(x)\,H(x)\,dx.}"> Intuitively, if <a href="/wiki/Integration_by_parts" title="Integration by parts">integration by parts</a> were permitted, then the latter integral should simplify to <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }\varphi (x)\,H'(x)\,dx=\int _{-\infty }^{\infty }\varphi (x)\,\delta (x)\,dx,}"> <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"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>H</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }\varphi (x)\,H'(x)\,dx=\int _{-\infty }^{\infty }\varphi (x)\,\delta (x)\,dx,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/632d4f66470a99e5399f193ab495214fe43c7dc6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.485ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }\varphi (x)\,H'(x)\,dx=\int _{-\infty }^{\infty }\varphi (x)\,\delta (x)\,dx,}"> and indeed, a form of integration by parts is permitted for the Stieltjes integral, and in that case, one does have <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\int _{-\infty }^{\infty }\varphi '(x)\,H(x)\,dx=\int _{-\infty }^{\infty }\varphi (x)\,dH(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>φ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\int _{-\infty }^{\infty }\varphi '(x)\,H(x)\,dx=\int _{-\infty }^{\infty }\varphi (x)\,dH(x).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4819960dd4de8df289c0599b08ce3aef662343dc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.939ex; height:6.009ex;" alt="{\displaystyle -\int _{-\infty }^{\infty }\varphi '(x)\,H(x)\,dx=\int _{-\infty }^{\infty }\varphi (x)\,dH(x).}"> In the context of measure theory, the Dirac measure gives rise to distribution by integration. Conversely, equation (<a href="#math_1">1</a>) defines a <a href="/wiki/Daniell_integral" title="Daniell integral">Daniell integral</a> on the space of all compactly supported continuous functions φ which, by the <a href="/wiki/Riesz%E2%80%93Markov%E2%80%93Kakutani_representation_theorem" title="Riesz–Markov–Kakutani representation theorem">Riesz representation theorem</a>, can be represented as the Lebesgue integral of φ with respect to some <a href="/wiki/Radon_measure" title="Radon measure">Radon measure</a>. Generally, when the term Dirac delta function is used, it is in the sense of distributions rather than measures, the <a href="/wiki/Dirac_measure" title="Dirac measure">Dirac measure</a> being among several terms for the corresponding notion in measure theory. Some sources may also use the term Dirac delta distribution. <div class="mw-heading mw-heading3"><h3 id="Generalizations">Generalizations</h3>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=6" title="Edit section: Generalizations">edit</a>]</div> The delta function can be defined in n-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> Rn as the measure such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\mathbf {R} ^{n}}f(\mathbf {x} )\,\delta (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="bold">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 stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo 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 _{\mathbf {R} ^{n}}f(\mathbf {x} )\,\delta (d\mathbf {x} )=f(\mathbf {0} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55587e15eb61b3e98fcbf434ec818cdd0b1d3bc1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.187ex; height:5.676ex;" alt="{\displaystyle \int _{\mathbf {R} ^{n}}f(\mathbf {x} )\,\delta (d\mathbf {x} )=f(\mathbf {0} )}"> for every compactly supported continuous function f. As a measure, the n-dimensional delta function is the <a href="/wiki/Product_measure" title="Product measure">product measure</a> of the 1-dimensional delta functions in each variable separately. Thus, formally, with x = (x1, x2, ..., xn), one has<a href="#cite_note-FOOTNOTEBracewell1986Chapter_5-26">[26]</a> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (\mathbf {x} )=\delta (x_{1})\,\delta (x_{2})\cdots \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> <mspace width="thinmathspace" /> <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})\cdots \delta (x_{n}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4a7f32615b53f845fa7acf65d68b37bbdf5dc8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.789ex; height:2.843ex;" alt="{\displaystyle \delta (\mathbf {x} )=\delta (x_{1})\,\delta (x_{2})\cdots \delta (x_{n}).}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(2)</td></tr></tbody></table> The delta function can also be defined in the sense of distributions exactly as above in the one-dimensional case.<a href="#cite_note-FOOTNOTEHörmander1983§3.1-27">[27]</a> However, despite widespread use in engineering contexts, (<a href="#math_2">2</a>) should be manipulated with care, since the product of distributions can only be defined under quite narrow circumstances.<a href="#cite_note-FOOTNOTEStrichartz1994§2.3-28">[28]</a><a href="#cite_note-FOOTNOTEHörmander1983§8.2-29">[29]</a> The notion of a <a href="/wiki/Dirac_measure" title="Dirac measure">Dirac measure</a> makes sense on any set.<a href="#cite_note-FOOTNOTERudin1966§1.20-30">[30]</a> Thus if X is a set, x0 ∈ X is a marked point, and Σ is any <a href="/wiki/Sigma_algebra" class="mw-redirect" title="Sigma algebra">sigma algebra</a> of subsets of X, then the measure defined on sets A ∈ Σ by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{x_{0}}(A)={\begin{cases}1&{\text{if }}x_{0}\in A\\0&{\text{if }}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>if </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>if </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{if }}x_{0}\in A\\0&{\text{if }}x_{0}\notin A\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc893ea7c4c910b2794fce66bae49200ddca1047" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.574ex; height:6.176ex;" alt="{\displaystyle \delta _{x_{0}}(A)={\begin{cases}1&{\text{if }}x_{0}\in A\\0&{\text{if }}x_{0}\notin A\end{cases}}}"> is the delta measure or unit mass concentrated at x0. Another common generalization of the delta function is to a <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">differentiable manifold</a> where most of its properties as a distribution can also be exploited because of the <a href="/wiki/Differentiable_structure" class="mw-redirect" title="Differentiable structure">differentiable structure</a>. The delta function on a manifold M centered at the point x0 ∈ M is defined as the following distribution: <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{x_{0}}[\varphi ]=\varphi (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}}[\varphi ]=\varphi (x_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9219b75c2a59f85c4a495187a42b9f997c7547a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.662ex; height:3.009ex;" alt="{\displaystyle \delta _{x_{0}}[\varphi ]=\varphi (x_{0})}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(3)</td></tr></tbody></table> for all compactly supported smooth real-valued functions φ on M.<a href="#cite_note-FOOTNOTEDieudonné1972§17.3.3-31">[31]</a> A common special case of this construction is a case in which M is an <a href="/wiki/Open_set" title="Open set">open set</a> in the Euclidean space Rn. On a <a href="/wiki/Locally_compact_Hausdorff_space" class="mw-redirect" title="Locally compact Hausdorff space">locally compact Hausdorff space</a> X, the Dirac delta measure concentrated at a point x is the <a href="/wiki/Radon_measure" title="Radon measure">Radon measure</a> associated with the Daniell integral (<a href="#math_3">3</a>) on compactly supported continuous functions φ.<a href="#cite_note-32">[32]</a> At this level of generality, calculus as such is no longer possible, however a variety of techniques from abstract analysis are available. For instance, the mapping <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\mapsto \delta _{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> <mo 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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}\mapsto \delta _{x_{0}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/322a24e937ccf433716b71929b01812a46e26a87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.034ex; height:3.009ex;" alt="{\displaystyle x_{0}\mapsto \delta _{x_{0}}}"> is a continuous embedding of X into the space of finite Radon measures on X, equipped with its <a href="/wiki/Vague_topology" title="Vague topology">vague topology</a>. Moreover, the <a href="/wiki/Convex_hull" title="Convex hull">convex hull</a> of the image of X under this embedding is <a href="/wiki/Dense_set" title="Dense set">dense</a> in the space of probability measures on X.<a href="#cite_note-FOOTNOTEFederer1969§2.5.19-33">[33]</a> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=7" title="Edit section: Properties">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Scaling_and_symmetry">Scaling and symmetry</h3>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=8" title="Edit section: Scaling and symmetry">edit</a>]</div> The delta function satisfies the following scaling property for a non-zero scalar α:<a href="#cite_note-FOOTNOTEStrichartz1994Problem_2.6.2-34">[34]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }\delta (\alpha x)\,dx=\int _{-\infty }^{\infty }\delta (u)\,{\frac {du}{|\alpha |}}={\frac {1}{|\alpha |}}}"> <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"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }\delta (\alpha x)\,dx=\int _{-\infty }^{\infty }\delta (u)\,{\frac {du}{|\alpha |}}={\frac {1}{|\alpha |}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee8a19ea564a084622f5d05a399acd6904f70fd9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.282ex; height:6.176ex;" alt="{\displaystyle \int _{-\infty }^{\infty }\delta (\alpha x)\,dx=\int _{-\infty }^{\infty }\delta (u)\,{\frac {du}{|\alpha |}}={\frac {1}{|\alpha |}}}"> and so <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (\alpha x)={\frac {\delta (x)}{|\alpha |}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (\alpha x)={\frac {\delta (x)}{|\alpha |}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42c78ef0c2838ded4ffb5b247102b512fd7c8781" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.444ex; height:6.509ex;" alt="{\displaystyle \delta (\alpha x)={\frac {\delta (x)}{|\alpha |}}.}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(4)</td></tr></tbody></table> Scaling property proof: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \limits _{-\infty }^{\infty }dx\ g(x)\delta (ax)={\frac {1}{a}}\int \limits _{-\infty }^{\infty }dx'\ g\left({\frac {x'}{a}}\right)\delta (x')={\frac {1}{a}}g(0).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>d</mi> <mi>x</mi> <mtext> </mtext> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>a</mi> </mfrac> </mrow> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mtext> </mtext> <mi>g</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mo>′</mo> </msup> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>a</mi> </mfrac> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \limits _{-\infty }^{\infty }dx\ g(x)\delta (ax)={\frac {1}{a}}\int \limits _{-\infty }^{\infty }dx'\ g\left({\frac {x'}{a}}\right)\delta (x')={\frac {1}{a}}g(0).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9bf5cf309d766be8c809990237a1c206c230a3a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; margin-left: -0.447ex; width:52.052ex; height:8.843ex;" alt="{\displaystyle \int \limits _{-\infty }^{\infty }dx\ g(x)\delta (ax)={\frac {1}{a}}\int \limits _{-\infty }^{\infty }dx'\ g\left({\frac {x'}{a}}\right)\delta (x')={\frac {1}{a}}g(0).}"> where a change of variable x′ = ax is used. If a is negative, i.e., a = −|a|, then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \limits _{-\infty }^{\infty }dx\ g(x)\delta (ax)={\frac {1}{-\left\vert a\right\vert }}\int \limits _{\infty }^{-\infty }dx'\ g\left({\frac {x'}{a}}\right)\delta (x')={\frac {1}{\left\vert a\right\vert }}\int \limits _{-\infty }^{\infty }dx'\ g\left({\frac {x'}{a}}\right)\delta (x')={\frac {1}{\left\vert a\right\vert }}g(0).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>d</mi> <mi>x</mi> <mtext> </mtext> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo>−</mo> <mrow> <mo>|</mo> <mi>a</mi> <mo>|</mo> </mrow> </mrow> </mfrac> </mrow> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mtext> </mtext> <mi>g</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mo>′</mo> </msup> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo>|</mo> <mi>a</mi> <mo>|</mo> </mrow> </mfrac> </mrow> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mtext> </mtext> <mi>g</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mo>′</mo> </msup> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo>|</mo> <mi>a</mi> <mo>|</mo> </mrow> </mfrac> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \limits _{-\infty }^{\infty }dx\ g(x)\delta (ax)={\frac {1}{-\left\vert a\right\vert }}\int \limits _{\infty }^{-\infty }dx'\ g\left({\frac {x'}{a}}\right)\delta (x')={\frac {1}{\left\vert a\right\vert }}\int \limits _{-\infty }^{\infty }dx'\ g\left({\frac {x'}{a}}\right)\delta (x')={\frac {1}{\left\vert a\right\vert }}g(0).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c4cd3eeba378f5ddca010a88235df9b78eed96a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; margin-left: -0.447ex; width:84.42ex; height:9.009ex;" alt="{\displaystyle \int \limits _{-\infty }^{\infty }dx\ g(x)\delta (ax)={\frac {1}{-\left\vert a\right\vert }}\int \limits _{\infty }^{-\infty }dx'\ g\left({\frac {x'}{a}}\right)\delta (x')={\frac {1}{\left\vert a\right\vert }}\int \limits _{-\infty }^{\infty }dx'\ g\left({\frac {x'}{a}}\right)\delta (x')={\frac {1}{\left\vert a\right\vert }}g(0).}"> Thus, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (ax)={\frac {1}{\left\vert a\right\vert }}\delta (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo>|</mo> <mi>a</mi> <mo>|</mo> </mrow> </mfrac> </mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (ax)={\frac {1}{\left\vert a\right\vert }}\delta (x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e395eba4c27b0eff920f28a102bc3913aa246fe1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.063ex; height:6.009ex;" alt="{\displaystyle \delta (ax)={\frac {1}{\left\vert a\right\vert }}\delta (x)}">. In particular, the delta function is an <a href="/wiki/Even_function" class="mw-redirect" title="Even function">even</a> distribution (symmetry), in the sense that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (-x)=\delta (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (-x)=\delta (x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e752769739457a7e50d324333e4a00910897939" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.282ex; height:2.843ex;" alt="{\displaystyle \delta (-x)=\delta (x)}"> which is <a href="/wiki/Homogeneous_function" title="Homogeneous function">homogeneous</a> of degree −1. <div class="mw-heading mw-heading3"><h3 id="Algebraic_properties">Algebraic properties</h3>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=9" title="Edit section: Algebraic properties">edit</a>]</div> The <a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">distributional product</a> of δ with x is equal to zero: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\,\delta (x)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mspace width="thinmathspace" /> <mi>δ</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 x\,\delta (x)=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c287bd4ab765b64000d550f2964b1c604995e665" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.812ex; height:2.843ex;" alt="{\displaystyle x\,\delta (x)=0.}"> More generally, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x-a)^{n}\delta (x-a)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-a)^{n}\delta (x-a)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64ffa5f7a6558522cbab7558d901ac24f4d74e1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.946ex; height:2.843ex;" alt="{\displaystyle (x-a)^{n}\delta (x-a)=0}"> for all positive integers <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><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}">. Conversely, if xf(x) = xg(x), where f and g are distributions, then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=g(x)+c\delta (x)}"> <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> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>c</mi> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=g(x)+c\delta (x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67fc4393093fe1c79bb617199052e16b8d52a143" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.806ex; height:2.843ex;" alt="{\displaystyle f(x)=g(x)+c\delta (x)}"> for some constant c.<a href="#cite_note-FOOTNOTEVladimirov1971Chapter_2,_Example_3(d)-35">[35]</a> <div class="mw-heading mw-heading3"><h3 id="Translation">Translation</h3>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=10" title="Edit section: Translation">edit</a>]</div> The integral of any function multiplied by the time-delayed Dirac delta <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{T}(t){=}\delta (t{-}T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>=</mo> </mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{T}(t){=}\delta (t{-}T)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05146f5397723a6f0b4ecf6d19b9e84f54bff989" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.021ex; height:2.843ex;" alt="{\displaystyle \delta _{T}(t){=}\delta (t{-}T)}"> is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }f(t)\,\delta (t-T)\,dt=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"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <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)\,dt=f(T).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84e51894f16a15b62a4098ddee9ffc3f09499433" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.234ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }f(t)\,\delta (t-T)\,dt=f(T).}"> This is sometimes referred to as the sifting property<a href="#cite_note-36">[36]</a> or the sampling property.<a href="#cite_note-37">[37]</a> The delta function is said to "sift out" the value of f(t) at t = T.<a href="#cite_note-38">[38]</a> It follows that the effect of <a href="/wiki/Convolution" title="Convolution">convolving</a> a function f(t) with the time-delayed Dirac delta is to time-delay f(t) by the same amount:<a href="#cite_note-39">[39]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(f*\delta _{T})(t)\ &{\stackrel {\mathrm {def} }{=}}\ \int _{-\infty }^{\infty }f(\tau )\,\delta (t-T-\tau )\,d\tau \\&=\int _{-\infty }^{\infty }f(\tau )\,\delta (\tau -(t-T))\,d\tau \qquad {\text{since}}~\delta (-x)=\delta (x)~~{\text{by (4)}}\\&=f(t-T).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <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" /> <mi>d</mi> <mi>τ</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <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" /> <mi>d</mi> <mi>τ</mi> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>since</mtext> </mrow> <mtext> </mtext> <mi>δ</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>by (4)</mtext> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(f*\delta _{T})(t)\ &{\stackrel {\mathrm {def} }{=}}\ \int _{-\infty }^{\infty }f(\tau )\,\delta (t-T-\tau )\,d\tau \\&=\int _{-\infty }^{\infty }f(\tau )\,\delta (\tau -(t-T))\,d\tau \qquad {\text{since}}~\delta (-x)=\delta (x)~~{\text{by (4)}}\\&=f(t-T).\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b307f70df02c313ac7b4b07c85a49976ab92b93f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.816ex; margin-bottom: -0.188ex; width:71.787ex; height:15.176ex;" alt="{\displaystyle {\begin{aligned}(f*\delta _{T})(t)\ &{\stackrel {\mathrm {def} }{=}}\ \int _{-\infty }^{\infty }f(\tau )\,\delta (t-T-\tau )\,d\tau \\&=\int _{-\infty }^{\infty }f(\tau )\,\delta (\tau -(t-T))\,d\tau \qquad {\text{since}}~\delta (-x)=\delta (x)~~{\text{by (4)}}\\&=f(t-T).\end{aligned}}}"> The sifting property holds under the precise condition that f be a <a href="/wiki/Distribution_(mathematics)#Tempered_distributions_and_Fourier_transform" title="Distribution (mathematics)">tempered distribution</a> (see the discussion of the Fourier transform <a href="#Fourier_transform">below</a>). As a special case, for instance, we have the identity (understood in the distribution sense) <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }\delta (\xi -x)\delta (x-\eta )\,dx=\delta (\eta -\xi ).}"> <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"> <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" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>η</mi> <mo>−</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }\delta (\xi -x)\delta (x-\eta )\,dx=\delta (\eta -\xi ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc8005bef6fb0e68024199c59f725f4ec9aae542" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.665ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }\delta (\xi -x)\delta (x-\eta )\,dx=\delta (\eta -\xi ).}"> <div class="mw-heading mw-heading3"><h3 id="Composition_with_a_function">Composition with a function</h3>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=11" title="Edit section: Composition with a function">edit</a>]</div> More generally, the delta distribution may be <a href="/wiki/Distribution_(mathematics)#Composition_with_a_smooth_function" title="Distribution (mathematics)">composed</a> with a smooth function g(x) in such a way that the familiar change of variables formula holds (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u=g(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54c52880c0cf28150ff774cb45cddcd2a027674f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.683ex; height:2.843ex;" alt="{\displaystyle u=g(x)}">), that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\mathbb {R} }\delta {\bigl (}g(x){\bigr )}f{\bigl (}g(x){\bigr )}\left|g'(x)\right|dx=\int _{g(\mathbb {R} )}\delta (u)\,f(u)\,du}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow> <mo>|</mo> <mrow> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </msub> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\mathbb {R} }\delta {\bigl (}g(x){\bigr )}f{\bigl (}g(x){\bigr )}\left|g'(x)\right|dx=\int _{g(\mathbb {R} )}\delta (u)\,f(u)\,du}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74e6840f652b0f33f41c7a02d53bb7064c87c8d7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:47.941ex; height:6.009ex;" alt="{\displaystyle \int _{\mathbb {R} }\delta {\bigl (}g(x){\bigr )}f{\bigl (}g(x){\bigr )}\left|g'(x)\right|dx=\int _{g(\mathbb {R} )}\delta (u)\,f(u)\,du}"> provided that g is a <a href="/wiki/Continuously_differentiable" class="mw-redirect" title="Continuously differentiable">continuously differentiable</a> function with g′ nowhere zero.<a href="#cite_note-FOOTNOTEGelfandShilov1966–1968Vol._1,_§II.2.5-40">[40]</a> That is, there is a unique way to assign meaning to the distribution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta \circ g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo>∘</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta \circ g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829e8b07791706c3efa4cd9a1cfaf72218b374a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.359ex; height:2.676ex;" alt="{\displaystyle \delta \circ g}"> so that this identity holds for all compactly supported test functions f. Therefore, the domain must be broken up to exclude the g′ = 0 point. This distribution satisfies δ(g(x)) = 0 if g is nowhere zero, and otherwise if g has a real <a href="/wiki/Root_of_a_function" class="mw-redirect" title="Root of a function">root</a> at x0, then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (g(x))={\frac {\delta (x-x_{0})}{|g'(x_{0})|}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <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"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </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 (g(x))={\frac {\delta (x-x_{0})}{|g'(x_{0})|}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/399637e6ed9ab64795c248fad3113272473072f1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.106ex; height:6.509ex;" alt="{\displaystyle \delta (g(x))={\frac {\delta (x-x_{0})}{|g'(x_{0})|}}.}"> It is natural therefore to define the composition δ(g(x)) for continuously differentiable functions g by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (g(x))=\sum _{i}{\frac {\delta (x-x_{i})}{|g'(x_{i})|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>g</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>g</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (g(x))=\sum _{i}{\frac {\delta (x-x_{i})}{|g'(x_{i})|}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4709614d6a0e04b2750e81ada233882b67b55c96" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.947ex; height:6.843ex;" alt="{\displaystyle \delta (g(x))=\sum _{i}{\frac {\delta (x-x_{i})}{|g'(x_{i})|}}}"> where the sum extends over all roots of g(x), which are assumed to be <a href="/wiki/Simple_root" class="mw-redirect" title="Simple root">simple</a>. Thus, for example <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta \left(x^{2}-\alpha ^{2}\right)={\frac {1}{2|\alpha |}}{\Big [}\delta \left(x+\alpha \right)+\delta \left(x-\alpha \right){\Big ]}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mi>δ</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mi>α</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>δ</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>α</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta \left(x^{2}-\alpha ^{2}\right)={\frac {1}{2|\alpha |}}{\Big [}\delta \left(x+\alpha \right)+\delta \left(x-\alpha \right){\Big ]}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6570a1d3ae92889f1cb92ad03121ad1ef10acd1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:43.085ex; height:6.009ex;" alt="{\displaystyle \delta \left(x^{2}-\alpha ^{2}\right)={\frac {1}{2|\alpha |}}{\Big [}\delta \left(x+\alpha \right)+\delta \left(x-\alpha \right){\Big ]}.}"> In the integral form, the generalized scaling property may be written as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }f(x)\,\delta (g(x))\,dx=\sum _{i}{\frac {f(x_{i})}{|g'(x_{i})|}}.}"> <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"> <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 stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</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>g</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 \int _{-\infty }^{\infty }f(x)\,\delta (g(x))\,dx=\sum _{i}{\frac {f(x_{i})}{|g'(x_{i})|}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77cade6c64311d29dcb37fada19d5dd2ca101ee8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.043ex; height:6.843ex;" alt="{\displaystyle \int _{-\infty }^{\infty }f(x)\,\delta (g(x))\,dx=\sum _{i}{\frac {f(x_{i})}{|g'(x_{i})|}}.}"> <div class="mw-heading mw-heading3"><h3 id="Indefinite_integral">Indefinite integral</h3>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=12" title="Edit section: Indefinite integral">edit</a>]</div> For a constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b044c60e01b54c7116ee355431f37ed846badc53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle a\in \mathbb {R} }"> and a "well-behaved" arbitrary real-valued function y(x), <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \displaystyle {\int }y(x)\delta (x-a)dx=y(a)H(x-a)+c,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo>∫</mo> </mrow> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>c</mi> <mo>,</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle {\int }y(x)\delta (x-a)dx=y(a)H(x-a)+c,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/241892b8bae88ecaf266ca884269b825ff2edee1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.351ex; height:5.676ex;" alt="{\displaystyle \displaystyle {\int }y(x)\delta (x-a)dx=y(a)H(x-a)+c,}"> where H(x) is the <a href="/wiki/Heaviside_step_function" title="Heaviside step function">Heaviside step function</a> and c is an integration constant. <div class="mw-heading mw-heading3"><h3 id="Properties_in_n_dimensions">Properties in n dimensions</h3>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=13" title="Edit section: Properties in n dimensions">edit</a>]</div> The delta distribution in an n-dimensional space satisfies the following scaling property instead, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (\alpha {\boldsymbol {x}})=|\alpha |^{-n}\delta ({\boldsymbol {x}})~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mi>δ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (\alpha {\boldsymbol {x}})=|\alpha |^{-n}\delta ({\boldsymbol {x}})~,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee82d520c682d697e00e4a9434169d1124d8eef7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.871ex; height:3.176ex;" alt="{\displaystyle \delta (\alpha {\boldsymbol {x}})=|\alpha |^{-n}\delta ({\boldsymbol {x}})~,}"> so that δ is a <a href="/wiki/Homogeneous_function" title="Homogeneous function">homogeneous</a> distribution of degree −n. Under any <a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">reflection</a> or <a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">rotation</a> ρ, the delta function is invariant, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (\rho {\boldsymbol {x}})=\delta ({\boldsymbol {x}})~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (\rho {\boldsymbol {x}})=\delta ({\boldsymbol {x}})~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/372d91c7d8c67e39b7c5095bedcb7f04ab721e43" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.307ex; height:2.843ex;" alt="{\displaystyle \delta (\rho {\boldsymbol {x}})=\delta ({\boldsymbol {x}})~.}"> As in the one-variable case, it is possible to define the composition of δ with a <a href="/wiki/Lipschitz_function" class="mw-redirect" title="Lipschitz function">bi-Lipschitz function</a><a href="#cite_note-41">[41]</a> g: Rn → Rn uniquely so that the following holds <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\mathbb {R} ^{n}}\delta (g({\boldsymbol {x}}))\,f(g({\boldsymbol {x}}))\left|\det g'({\boldsymbol {x}})\right|d{\boldsymbol {x}}=\int _{g(\mathbb {R} ^{n})}\delta ({\boldsymbol {u}})f({\boldsymbol {u}})\,d{\boldsymbol {u}}}"> <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>δ</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mrow> <mo>|</mo> <mrow> <mo movablelimits="true" form="prefix">det</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>g</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> </mrow> </msub> <mi>δ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\mathbb {R} ^{n}}\delta (g({\boldsymbol {x}}))\,f(g({\boldsymbol {x}}))\left|\det g'({\boldsymbol {x}})\right|d{\boldsymbol {x}}=\int _{g(\mathbb {R} ^{n})}\delta ({\boldsymbol {u}})f({\boldsymbol {u}})\,d{\boldsymbol {u}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c717f0462bb512f8a0384bb13dc1a552cbdd3cf1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:54.414ex; height:6.009ex;" alt="{\displaystyle \int _{\mathbb {R} ^{n}}\delta (g({\boldsymbol {x}}))\,f(g({\boldsymbol {x}}))\left|\det g'({\boldsymbol {x}})\right|d{\boldsymbol {x}}=\int _{g(\mathbb {R} ^{n})}\delta ({\boldsymbol {u}})f({\boldsymbol {u}})\,d{\boldsymbol {u}}}"> for all compactly supported functions f. Using the <a href="/wiki/Coarea_formula" title="Coarea formula">coarea formula</a> from <a href="/wiki/Geometric_measure_theory" title="Geometric measure theory">geometric measure theory</a>, one can also define the composition of the delta function with a <a href="/wiki/Submersion_(mathematics)" title="Submersion (mathematics)">submersion</a> from one Euclidean space to another one of different dimension; the result is a type of <a href="/wiki/Current_(mathematics)" title="Current (mathematics)">current</a>. In the special case of a continuously differentiable function g : Rn → R such that the <a href="/wiki/Gradient" title="Gradient">gradient</a> of g is nowhere zero, the following identity holds<a href="#cite_note-FOOTNOTEHörmander1983§6.1-42">[42]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\mathbb {R} ^{n}}f({\boldsymbol {x}})\,\delta (g({\boldsymbol {x}}))\,d{\boldsymbol {x}}=\int _{g^{-1}(0)}{\frac {f({\boldsymbol {x}})}{|{\boldsymbol {\nabla }}g|}}\,d\sigma ({\boldsymbol {x}})}"> <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-italic">x</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>σ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\mathbb {R} ^{n}}f({\boldsymbol {x}})\,\delta (g({\boldsymbol {x}}))\,d{\boldsymbol {x}}=\int _{g^{-1}(0)}{\frac {f({\boldsymbol {x}})}{|{\boldsymbol {\nabla }}g|}}\,d\sigma ({\boldsymbol {x}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ffc41e0f3fc237ea448743b157dd21a2e78b312" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:41.037ex; height:6.509ex;" alt="{\displaystyle \int _{\mathbb {R} ^{n}}f({\boldsymbol {x}})\,\delta (g({\boldsymbol {x}}))\,d{\boldsymbol {x}}=\int _{g^{-1}(0)}{\frac {f({\boldsymbol {x}})}{|{\boldsymbol {\nabla }}g|}}\,d\sigma ({\boldsymbol {x}})}"> where the integral on the right is over g−1(0), the (n − 1)-dimensional surface defined by g(x) = 0 with respect to the <a href="/wiki/Minkowski_content" title="Minkowski content">Minkowski content</a> measure. This is known as a simple layer integral. More generally, if S is a smooth hypersurface of Rn, then we can associate to S the distribution that integrates any compactly supported smooth function g over S: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{S}[g]=\int _{S}g({\boldsymbol {s}})\,d\sigma ({\boldsymbol {s}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>g</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mi>g</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">s</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>σ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">s</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{S}[g]=\int _{S}g({\boldsymbol {s}})\,d\sigma ({\boldsymbol {s}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d0759b7b4d698e108b006b83385cb19a2048002" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.941ex; height:5.676ex;" alt="{\displaystyle \delta _{S}[g]=\int _{S}g({\boldsymbol {s}})\,d\sigma ({\boldsymbol {s}})}"> where σ is the hypersurface measure associated to S. This generalization is associated with the <a href="/wiki/Potential_theory" title="Potential theory">potential theory</a> of <a href="/wiki/Simple_layer_potential" class="mw-redirect" title="Simple layer potential">simple layer potentials</a> on S. If D is a <a href="/wiki/Domain_(mathematical_analysis)" title="Domain (mathematical analysis)">domain</a> in Rn with smooth boundary S, then δS is equal to the <a href="/wiki/Normal_derivative" class="mw-redirect" title="Normal derivative">normal derivative</a> of the <a href="/wiki/Indicator_function" title="Indicator function">indicator function</a> of D in the distribution sense, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\int _{\mathbb {R} ^{n}}g({\boldsymbol {x}})\,{\frac {\partial 1_{D}({\boldsymbol {x}})}{\partial n}}\,d{\boldsymbol {x}}=\int _{S}\,g({\boldsymbol {s}})\,d\sigma ({\boldsymbol {s}}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <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>g</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>n</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>g</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">s</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>σ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">s</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\int _{\mathbb {R} ^{n}}g({\boldsymbol {x}})\,{\frac {\partial 1_{D}({\boldsymbol {x}})}{\partial n}}\,d{\boldsymbol {x}}=\int _{S}\,g({\boldsymbol {s}})\,d\sigma ({\boldsymbol {s}}),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83c918c2f93baf4d616cc9672aa163e1f0fda1cd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:39.729ex; height:6.176ex;" alt="{\displaystyle -\int _{\mathbb {R} ^{n}}g({\boldsymbol {x}})\,{\frac {\partial 1_{D}({\boldsymbol {x}})}{\partial n}}\,d{\boldsymbol {x}}=\int _{S}\,g({\boldsymbol {s}})\,d\sigma ({\boldsymbol {s}}),}"> where n is the outward normal.<a href="#cite_note-FOOTNOTELange2012pp.29–30-43">[43]</a><a href="#cite_note-FOOTNOTEGelfandShilov1966–1968212-44">[44]</a> For a proof, see e.g. the article on the <a href="/wiki/Laplacian_of_the_indicator#Dirac_surface_delta_function" title="Laplacian of the indicator">surface delta function</a>. In three dimensions, the delta function is represented in spherical coordinates by: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta ({\boldsymbol {r}}-{\boldsymbol {r}}_{0})={\begin{cases}\displaystyle {\frac {1}{r^{2}\sin \theta }}\delta (r-r_{0})\delta (\theta -\theta _{0})\delta (\phi -\phi _{0})&x_{0},y_{0},z_{0}\neq 0\\\displaystyle {\frac {1}{2\pi r^{2}\sin \theta }}\delta (r-r_{0})\delta (\theta -\theta _{0})&x_{0}=y_{0}=0,\ z_{0}\neq 0\\\displaystyle {\frac {1}{4\pi r^{2}}}\delta (r-r_{0})&x_{0}=y_{0}=z_{0}=0\end{cases}}}"> <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-italic">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <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> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>−</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>−</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>−</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <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 \delta ({\boldsymbol {r}}-{\boldsymbol {r}}_{0})={\begin{cases}\displaystyle {\frac {1}{r^{2}\sin \theta }}\delta (r-r_{0})\delta (\theta -\theta _{0})\delta (\phi -\phi _{0})&x_{0},y_{0},z_{0}\neq 0\\\displaystyle {\frac {1}{2\pi r^{2}\sin \theta }}\delta (r-r_{0})\delta (\theta -\theta _{0})&x_{0}=y_{0}=0,\ z_{0}\neq 0\\\displaystyle {\frac {1}{4\pi r^{2}}}\delta (r-r_{0})&x_{0}=y_{0}=z_{0}=0\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a6694f7e920490a87d787c81ae222e313a8e587" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.505ex; width:72.364ex; height:16.176ex;" alt="{\displaystyle \delta ({\boldsymbol {r}}-{\boldsymbol {r}}_{0})={\begin{cases}\displaystyle {\frac {1}{r^{2}\sin \theta }}\delta (r-r_{0})\delta (\theta -\theta _{0})\delta (\phi -\phi _{0})&x_{0},y_{0},z_{0}\neq 0\\\displaystyle {\frac {1}{2\pi r^{2}\sin \theta }}\delta (r-r_{0})\delta (\theta -\theta _{0})&x_{0}=y_{0}=0,\ z_{0}\neq 0\\\displaystyle {\frac {1}{4\pi r^{2}}}\delta (r-r_{0})&x_{0}=y_{0}=z_{0}=0\end{cases}}}"> <div class="mw-heading mw-heading2"><h2 id="Fourier_transform">Fourier transform</h2>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=14" title="Edit section: Fourier transform">edit</a>]</div> The delta function is a <a href="/wiki/Distribution_(mathematics)#Tempered_distributions_and_Fourier_transform" title="Distribution (mathematics)">tempered distribution</a>, and therefore it has a well-defined <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a>. Formally, one finds<a href="#cite_note-45">[45]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\delta }}(\xi )=\int _{-\infty }^{\infty }e^{-2\pi ix\xi }\,\delta (x)dx=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>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</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"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi>x</mi> <mi>ξ</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\delta }}(\xi )=\int _{-\infty }^{\infty }e^{-2\pi ix\xi }\,\delta (x)dx=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dfbad139241ac1c73461dfcd7b4b1bf731f0838" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.696ex; height:6.009ex;" alt="{\displaystyle {\widehat {\delta }}(\xi )=\int _{-\infty }^{\infty }e^{-2\pi ix\xi }\,\delta (x)dx=1.}"> Properly speaking, the Fourier transform of a distribution is defined by imposing <a href="/wiki/Self-adjoint" title="Self-adjoint">self-adjointness</a> of the Fourier transform under the duality pairing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \cdot ,\cdot \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mo>⋅</mo> <mo>,</mo> <mo>⋅</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \cdot ,\cdot \rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a50080b735975d8001c9552ac2134b49ad534c0" 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 \langle \cdot ,\cdot \rangle }"> of tempered distributions with <a href="/wiki/Schwartz_functions" class="mw-redirect" title="Schwartz functions">Schwartz functions</a>. Thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\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>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\delta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe290e363518844fff77a2cae1caf27e69dea68c" 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 {\widehat {\delta }}}"> is defined as the unique tempered distribution satisfying <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle {\widehat {\delta }},\varphi \rangle =\langle \delta ,{\widehat {\varphi }}\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>^</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>^</mo> </mover> </mrow> </mrow> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle {\widehat {\delta }},\varphi \rangle =\langle \delta ,{\widehat {\varphi }}\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4ce5a2a69cc27a322c3b95775d6dbc6362b18d7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.256ex; height:3.343ex;" alt="{\displaystyle \langle {\widehat {\delta }},\varphi \rangle =\langle \delta ,{\widehat {\varphi }}\rangle }"> for all Schwartz functions φ. And indeed it follows from this that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\delta }}=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>^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\delta }}=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26a00e5358ae06e196278c0e8abc3420220a0315" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.21ex; height:2.843ex;" alt="{\displaystyle {\widehat {\delta }}=1.}"> As a result of this identity, the <a href="/wiki/Convolution" title="Convolution">convolution</a> of the delta function with any other tempered distribution S is simply S: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S*\delta =S.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>∗</mo> <mi>δ</mi> <mo>=</mo> <mi>S</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S*\delta =S.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6154a7cf81ad1158072fb3a7667efe234369dc28" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.987ex; height:2.343ex;" alt="{\displaystyle S*\delta =S.}"> That is to say that δ is an <a href="/wiki/Identity_element" title="Identity element">identity element</a> for the convolution on tempered distributions, and in fact, the space of compactly supported distributions under convolution is an <a href="/wiki/Associative_algebra" title="Associative algebra">associative algebra</a> with identity the delta function. This property is fundamental in <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a>, as convolution with a tempered distribution is a <a href="/wiki/Linear_time-invariant_system" title="Linear time-invariant system">linear time-invariant system</a>, and applying the linear time-invariant system measures its <a href="/wiki/Impulse_response" title="Impulse response">impulse response</a>. The impulse response can be computed to any desired degree of accuracy by choosing a suitable approximation for δ, and once it is known, it characterizes the system completely. See <a href="/wiki/LTI_system_theory#Impulse_response_and_convolution" class="mw-redirect" title="LTI system theory">LTI system theory § Impulse response and convolution</a>. The inverse Fourier transform of the tempered distribution f(ξ) = 1 is the delta function. Formally, this is expressed as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }1\cdot e^{2\pi ix\xi }\,d\xi =\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"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mn>1</mn> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi>x</mi> <mi>ξ</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ξ</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^{2\pi ix\xi }\,d\xi =\delta (x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7843266b55738b0876e4a6fd663d40245c7c39b8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.91ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }1\cdot e^{2\pi ix\xi }\,d\xi =\delta (x)}"> and more rigorously, it follows since <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle 1,{\widehat {f}}\rangle =f(0)=\langle \delta ,f\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mn>1</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>δ</mi> <mo>,</mo> <mi>f</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle 1,{\widehat {f}}\rangle =f(0)=\langle \delta ,f\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec113c338091a28e4216c677858bf588021e556c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.394ex; height:3.509ex;" alt="{\displaystyle \langle 1,{\widehat {f}}\rangle =f(0)=\langle \delta ,f\rangle }"> for all Schwartz functions f. In these terms, the delta function provides a suggestive statement of the orthogonality property of the Fourier kernel on R. Formally, one has <math display="block" 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]^{*}\,dt=\int _{-\infty }^{\infty }e^{-i2\pi (\xi _{2}-\xi _{1})t}\,dt=\delta (\xi _{2}-\xi _{1}).}"> <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"> <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" /> <mi>d</mi> <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"> <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" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mi>δ</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> <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]^{*}\,dt=\int _{-\infty }^{\infty }e^{-i2\pi (\xi _{2}-\xi _{1})t}\,dt=\delta (\xi _{2}-\xi _{1}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8007cb6fa52a213211396ce15fa16d836540ab0c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:56.999ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }e^{i2\pi \xi _{1}t}\left[e^{i2\pi \xi _{2}t}\right]^{*}\,dt=\int _{-\infty }^{\infty }e^{-i2\pi (\xi _{2}-\xi _{1})t}\,dt=\delta (\xi _{2}-\xi _{1}).}"> This is, of course, shorthand for the assertion that the Fourier transform of the tempered distribution <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)=e^{i2\pi \xi _{1}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> <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>1</mn> </mrow> </msub> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)=e^{i2\pi \xi _{1}t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bc510eb081b63ba18e57bca2cc67ca42d92de1e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.818ex; height:3.176ex;" alt="{\displaystyle f(t)=e^{i2\pi \xi _{1}t}}"> is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}(\xi _{2})=\delta (\xi _{1}-\xi _{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>ξ</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>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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}(\xi _{2})=\delta (\xi _{1}-\xi _{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21343f7b814a9771832d1b64f89708e453516470" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.595ex; height:3.509ex;" alt="{\displaystyle {\widehat {f}}(\xi _{2})=\delta (\xi _{1}-\xi _{2})}"> which again follows by imposing self-adjointness of the Fourier transform. By <a href="/wiki/Analytic_continuation" title="Analytic continuation">analytic continuation</a> of the Fourier transform, the <a href="/wiki/Laplace_transform" title="Laplace transform">Laplace transform</a> of the delta function is found to be<a href="#cite_note-FOOTNOTEBracewell1986-46">[46]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{\infty }\delta (t-a)\,e^{-st}\,dt=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"> <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> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <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}\,dt=e^{-sa}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95682e8e05a00ef8f98b574538d46a7275359522" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.263ex; height:5.843ex;" alt="{\displaystyle \int _{0}^{\infty }\delta (t-a)\,e^{-st}\,dt=e^{-sa}.}"> <div class="mw-heading mw-heading2"><h2 id="Derivatives">Derivatives</h2>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=15" title="Edit section: Derivatives">edit</a>]</div> The derivative of the Dirac delta distribution, denoted δ′ and also called the Dirac delta prime or Dirac delta derivative as described in <a href="/wiki/Laplacian_of_the_indicator" title="Laplacian of the indicator">Laplacian of the indicator</a>, is defined on compactly supported smooth test functions φ by<a href="#cite_note-FOOTNOTEGelfandShilov1966–196826-47">[47]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta '[\varphi ]=-\delta [\varphi ']=-\varphi '(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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta '[\varphi ]=-\delta [\varphi ']=-\varphi '(0).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dcbd7eb8726c02cb50a897f5d3ae2fdc88ad3d0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.736ex; height:3.009ex;" alt="{\displaystyle \delta '[\varphi ]=-\delta [\varphi ']=-\varphi '(0).}"> The first equality here is a kind of <a href="/wiki/Integration_by_parts" title="Integration by parts">integration by parts</a>, for if δ were a true function then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }\delta '(x)\varphi (x)\,dx=\delta (x)\varphi (x)|_{-\infty }^{\infty }-\int _{-\infty }^{\infty }\delta (x)\varphi '(x)\,dx=-\int _{-\infty }^{\infty }\delta (x)\varphi '(x)\,dx=-\varphi '(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"> <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" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>δ</mi> <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> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <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"> <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" /> <mi>d</mi> <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"> <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" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mo>−</mo> <msup> <mi>φ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }\delta '(x)\varphi (x)\,dx=\delta (x)\varphi (x)|_{-\infty }^{\infty }-\int _{-\infty }^{\infty }\delta (x)\varphi '(x)\,dx=-\int _{-\infty }^{\infty }\delta (x)\varphi '(x)\,dx=-\varphi '(0).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/214057d1d2496656b0ff40f2af4490dadd898e6c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:86.508ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }\delta '(x)\varphi (x)\,dx=\delta (x)\varphi (x)|_{-\infty }^{\infty }-\int _{-\infty }^{\infty }\delta (x)\varphi '(x)\,dx=-\int _{-\infty }^{\infty }\delta (x)\varphi '(x)\,dx=-\varphi '(0).}"> By <a href="/wiki/Mathematical_induction" title="Mathematical induction">mathematical induction</a>, the k-th derivative of δ is defined similarly as the distribution given on test functions by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta ^{(k)}[\varphi ]=(-1)^{k}\varphi ^{(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)}[\varphi ]=(-1)^{k}\varphi ^{(k)}(0).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92b8ca88b17041c8f71580ab9779320dfddadb28" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.709ex; height:3.343ex;" alt="{\displaystyle \delta ^{(k)}[\varphi ]=(-1)^{k}\varphi ^{(k)}(0).}"> In particular, δ is an infinitely differentiable distribution. The first derivative of the delta function is the distributional limit of the difference quotients:<a href="#cite_note-FOOTNOTEGelfandShilov1966–1968§2.1-48">[48]</a> <math display="block" 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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2feea26296fae6af229e60653729135b5dcfba3" class="mwe-math-fallback-image-display 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}}.}"> More properly, one has <math display="block" 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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta '=\lim _{h\to 0}{\frac {1}{h}}(\tau _{h}\delta -\delta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/633b74fd7a6feaaa89038160c091901eece9938b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.753ex; height:5.343ex;" alt="{\displaystyle \delta '=\lim _{h\to 0}{\frac {1}{h}}(\tau _{h}\delta -\delta )}"> where τh is the translation operator, defined on functions by τhφ(x) = φ(x + h), and on a distribution S by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\tau _{h}S)[\varphi ]=S[\tau _{-h}\varphi ].}"> <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)[\varphi ]=S[\tau _{-h}\varphi ].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da7641428f61e4c1de41526c105208eccffd83eb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.85ex; height:2.843ex;" alt="{\displaystyle (\tau _{h}S)[\varphi ]=S[\tau _{-h}\varphi ].}"> In the theory of <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetism</a>, the first derivative of the delta function represents a point magnetic <a href="/wiki/Dipole" title="Dipole">dipole</a> situated at the origin. Accordingly, it is referred to as a dipole or the <a href="/wiki/Unit_doublet" title="Unit doublet">doublet function</a>.<a href="#cite_note-49">[49]</a> The derivative of the delta function satisfies a number of basic properties, including:<a href="#cite_note-FOOTNOTEBracewell200086-50">[50]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\delta '(-x)&=-\delta '(x)\\x\delta '(x)&=-\delta (x)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>δ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msup> <mi>δ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> <msup> <mi>δ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\delta '(-x)&=-\delta '(x)\\x\delta '(x)&=-\delta (x)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aa01b29401817e34c4431c7d3b121d10616a63a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.221ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\delta '(-x)&=-\delta '(x)\\x\delta '(x)&=-\delta (x)\end{aligned}}}"> which can be shown by applying a test function and integrating by parts. The latter of these properties can also be demonstrated by applying distributional derivative definition, Leibniz 's theorem and linearity of inner product:<a href="#cite_note-51">[51]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\langle x\delta ',\varphi \rangle \,&=\,\langle \delta ',x\varphi \rangle \,=\,-\langle \delta ,(x\varphi )'\rangle \,=\,-\langle \delta ,x'\varphi +x\varphi '\rangle \,=\,-\langle \delta ,x'\varphi \rangle -\langle \delta ,x\varphi '\rangle \,=\,-x'(0)\varphi (0)-x(0)\varphi '(0)\\&=\,-x'(0)\langle \delta ,\varphi \rangle -x(0)\langle \delta ,\varphi '\rangle \,=\,-x'(0)\langle \delta ,\varphi \rangle +x(0)\langle \delta ',\varphi \rangle \,=\,\langle x(0)\delta '-x'(0)\delta ,\varphi \rangle \\\Longrightarrow x(t)\delta '(t)&=x(0)\delta '(t)-x'(0)\delta (t)=-x'(0)\delta (t)=-\delta (t)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <msup> <mi>δ</mi> <mo>′</mo> </msup> <mo>,</mo> <mi>φ</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> </mtd> <mtd> <mi></mi> <mo>=</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">⟨</mo> <msup> <mi>δ</mi> <mo>′</mo> </msup> <mo>,</mo> <mi>x</mi> <mi>φ</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mo>−</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>δ</mi> <mo>,</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>φ</mi> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mo>−</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>δ</mi> <mo>,</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mi>φ</mi> <mo>+</mo> <mi>x</mi> <msup> <mi>φ</mi> <mo>′</mo> </msup> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mo>−</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>δ</mi> <mo>,</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mi>φ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>−</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>δ</mi> <mo>,</mo> <mi>x</mi> <msup> <mi>φ</mi> <mo>′</mo> </msup> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mo>−</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <msup> <mi>φ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mspace width="thinmathspace" /> <mo>−</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>δ</mi> <mo>,</mo> <mi>φ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>δ</mi> <mo>,</mo> <msup> <mi>φ</mi> <mo>′</mo> </msup> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mo>−</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>δ</mi> <mo>,</mo> <mi>φ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟨</mo> <msup> <mi>δ</mi> <mo>′</mo> </msup> <mo>,</mo> <mi>φ</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <msup> <mi>δ</mi> <mo>′</mo> </msup> <mo>−</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mi>δ</mi> <mo>,</mo> <mi>φ</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">⟹</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>δ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <msup> <mi>δ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\langle x\delta ',\varphi \rangle \,&=\,\langle \delta ',x\varphi \rangle \,=\,-\langle \delta ,(x\varphi )'\rangle \,=\,-\langle \delta ,x'\varphi +x\varphi '\rangle \,=\,-\langle \delta ,x'\varphi \rangle -\langle \delta ,x\varphi '\rangle \,=\,-x'(0)\varphi (0)-x(0)\varphi '(0)\\&=\,-x'(0)\langle \delta ,\varphi \rangle -x(0)\langle \delta ,\varphi '\rangle \,=\,-x'(0)\langle \delta ,\varphi \rangle +x(0)\langle \delta ',\varphi \rangle \,=\,\langle x(0)\delta '-x'(0)\delta ,\varphi \rangle \\\Longrightarrow x(t)\delta '(t)&=x(0)\delta '(t)-x'(0)\delta (t)=-x'(0)\delta (t)=-\delta (t)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae4ee2f2fd560e4e4e83d426a090e49198f04faf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:109.735ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}\langle x\delta ',\varphi \rangle \,&=\,\langle \delta ',x\varphi \rangle \,=\,-\langle \delta ,(x\varphi )'\rangle \,=\,-\langle \delta ,x'\varphi +x\varphi '\rangle \,=\,-\langle \delta ,x'\varphi \rangle -\langle \delta ,x\varphi '\rangle \,=\,-x'(0)\varphi (0)-x(0)\varphi '(0)\\&=\,-x'(0)\langle \delta ,\varphi \rangle -x(0)\langle \delta ,\varphi '\rangle \,=\,-x'(0)\langle \delta ,\varphi \rangle +x(0)\langle \delta ',\varphi \rangle \,=\,\langle x(0)\delta '-x'(0)\delta ,\varphi \rangle \\\Longrightarrow x(t)\delta '(t)&=x(0)\delta '(t)-x'(0)\delta (t)=-x'(0)\delta (t)=-\delta (t)\end{aligned}}}"> Furthermore, the convolution of δ′ with a compactly-supported, smooth function f is <math display="block" 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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/144c0e65bbe33d5d01a3e16c247d03cd8522ada5" class="mwe-math-fallback-image-display 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',}"> which follows from the properties of the distributional derivative of a convolution. <div class="mw-heading mw-heading3"><h3 id="Higher_dimensions">Higher dimensions</h3>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=16" title="Edit section: Higher dimensions">edit</a>]</div> More generally, on an <a href="/wiki/Open_set" title="Open set">open set</a> U in the n-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> <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><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}}">, the Dirac delta distribution centered at a point a ∈ U is defined by<a href="#cite_note-FOOTNOTEHörmander198356-52">[52]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{a}[\varphi ]=\varphi (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}[\varphi ]=\varphi (a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae820065c43b667f4247c78672dcd2f83fa79316" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.606ex; height:2.843ex;" alt="{\displaystyle \delta _{a}[\varphi ]=\varphi (a)}"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \in C_{c}^{\infty }(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>∈</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \in C_{c}^{\infty }(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/569727d93f2c193d42deb512ccf7e1e3d95da6fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.626ex; height:2.843ex;" alt="{\displaystyle \varphi \in C_{c}^{\infty }(U)}">, the space of all smooth functions with compact support on U. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =(\alpha _{1},\ldots ,\alpha _{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =(\alpha _{1},\ldots ,\alpha _{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a94e85af78356a1f61786ae37197b9db2a163b85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.822ex; height:2.843ex;" alt="{\displaystyle \alpha =(\alpha _{1},\ldots ,\alpha _{n})}"> is any <a href="/wiki/Multi-index" class="mw-redirect" title="Multi-index">multi-index</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\alpha |=\alpha _{1}+\cdots +\alpha _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\alpha |=\alpha _{1}+\cdots +\alpha _{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/759fe9b27dc3715026baf55cde1ecd4395edbab1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.532ex; height:2.843ex;" alt="{\displaystyle |\alpha |=\alpha _{1}+\cdots +\alpha _{n}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial ^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial ^{\alpha }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/456c1354f357e40cdb9f5ee48e919b408d3171bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.343ex;" alt="{\displaystyle \partial ^{\alpha }}"> denotes the associated mixed <a href="/wiki/Partial_derivative" title="Partial derivative">partial derivative</a> operator, then the α-th derivative ∂αδa of δa is given by<a href="#cite_note-FOOTNOTEHörmander198356-52">[52]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle \partial ^{\alpha }\delta _{a},\,\varphi \right\rangle =(-1)^{|\alpha |}\left\langle \delta _{a},\partial ^{\alpha }\varphi \right\rangle =(-1)^{|\alpha |}\partial ^{\alpha }\varphi (x){\Big |}_{x=a}\quad {\text{ for all }}\varphi \in C_{c}^{\infty }(U).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <mi>φ</mi> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> <mrow> <mo>⟨</mo> <mrow> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>,</mo> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mi>φ</mi> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mi>a</mi> </mrow> </msub> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mi>φ</mi> <mo>∈</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle \partial ^{\alpha }\delta _{a},\,\varphi \right\rangle =(-1)^{|\alpha |}\left\langle \delta _{a},\partial ^{\alpha }\varphi \right\rangle =(-1)^{|\alpha |}\partial ^{\alpha }\varphi (x){\Big |}_{x=a}\quad {\text{ for all }}\varphi \in C_{c}^{\infty }(U).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9bf2328c5d9ef8c00251245ffd77c682378cfb5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:72.012ex; height:4.343ex;" alt="{\displaystyle \left\langle \partial ^{\alpha }\delta _{a},\,\varphi \right\rangle =(-1)^{|\alpha |}\left\langle \delta _{a},\partial ^{\alpha }\varphi \right\rangle =(-1)^{|\alpha |}\partial ^{\alpha }\varphi (x){\Big |}_{x=a}\quad {\text{ for all }}\varphi \in C_{c}^{\infty }(U).}"> That is, the α-th derivative of δa is the distribution whose value on any test function φ is the α-th derivative of φ at a (with the appropriate positive or negative sign). The first partial derivatives of the delta function are thought of as <a href="/wiki/Double_layer_potential" title="Double layer potential">double layers</a> along the coordinate planes. More generally, the <a href="/wiki/Normal_derivative" class="mw-redirect" title="Normal derivative">normal derivative</a> of a simple layer supported on a surface is a double layer supported on that surface and represents a laminar magnetic monopole. Higher derivatives of the delta function are known in physics as <a href="/wiki/Multipole" class="mw-redirect" title="Multipole">multipoles</a>. Higher derivatives enter into mathematics naturally as the building blocks for the complete structure of distributions with point support. If S is any distribution on U supported on the set {a} consisting of a single point, then there is an integer m and coefficients cα such that<a href="#cite_note-FOOTNOTEHörmander198356-52">[52]</a><a href="#cite_note-FOOTNOTERudin1991Theorem_6.25-53">[53]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\sum _{|\alpha |\leq m}c_{\alpha }\partial ^{\alpha }\delta _{a}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>m</mi> </mrow> </munder> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\sum _{|\alpha |\leq m}c_{\alpha }\partial ^{\alpha }\delta _{a}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b83a092f07ddf8d7de3c96752264e6a7c6f5466b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:17.372ex; height:6.009ex;" alt="{\displaystyle S=\sum _{|\alpha |\leq m}c_{\alpha }\partial ^{\alpha }\delta _{a}.}"> <div class="mw-heading mw-heading2"><h2 id="Representations_of_the_delta_function">Representations of the delta function</h2>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=17" title="Edit section: Representations of the delta function">edit</a>]</div> The delta function can be viewed as the limit of a sequence of functions <math display="block" 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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ebbbabf87bff1978d2d29f9cdf844b7453551ac" class="mwe-math-fallback-image-display 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),}"> where ηε(x) is sometimes called a nascent delta function. This limit is meant in a weak sense: either that <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{\varepsilon \to 0^{+}}\int _{-\infty }^{\infty }\eta _{\varepsilon }(x)f(x)\,dx=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"> <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" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{\varepsilon \to 0^{+}}\int _{-\infty }^{\infty }\eta _{\varepsilon }(x)f(x)\,dx=f(0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae9b54c787d074467e9daffc7931e3ef5ec29959" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.646ex; height:6.009ex;" alt="{\displaystyle \lim _{\varepsilon \to 0^{+}}\int _{-\infty }^{\infty }\eta _{\varepsilon }(x)f(x)\,dx=f(0)}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(5)</td></tr></tbody></table> for all <a href="/wiki/Continuous_function" title="Continuous function">continuous</a> functions f having <a href="/wiki/Compact_support" class="mw-redirect" title="Compact support">compact support</a>, or that this limit holds for all <a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">smooth</a> functions f with compact support. The difference between these two slightly different modes of weak convergence is often subtle: the former is convergence in the <a href="/wiki/Vague_topology" title="Vague topology">vague topology</a> of measures, and the latter is convergence in the sense of <a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">distributions</a>. <div class="mw-heading mw-heading3"><h3 id="Approximations_to_the_identity">Approximations to the identity</h3>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=18" title="Edit section: Approximations to the identity">edit</a>]</div> Typically a nascent delta function ηε can be constructed in the following manner. Let η be an absolutely integrable function on R of total integral 1, and define <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{\varepsilon }(x)=\varepsilon ^{-1}\eta \left({\frac {x}{\varepsilon }}\right).}"> <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> <mo>=</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>η</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>ε</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{\varepsilon }(x)=\varepsilon ^{-1}\eta \left({\frac {x}{\varepsilon }}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae344b05b1fa5b018b8884edc8356cb0f3504233" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.339ex; height:4.843ex;" alt="{\displaystyle \eta _{\varepsilon }(x)=\varepsilon ^{-1}\eta \left({\frac {x}{\varepsilon }}\right).}"> In n dimensions, one uses instead the scaling <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{\varepsilon }(x)=\varepsilon ^{-n}\eta \left({\frac {x}{\varepsilon }}\right).}"> <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> <mo>=</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mi>η</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>ε</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{\varepsilon }(x)=\varepsilon ^{-n}\eta \left({\frac {x}{\varepsilon }}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3032ed8f57dfe1e8f8f1b2b1f7732bf02553f1d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.504ex; height:4.843ex;" alt="{\displaystyle \eta _{\varepsilon }(x)=\varepsilon ^{-n}\eta \left({\frac {x}{\varepsilon }}\right).}"> Then a simple change of variables shows that ηε also has integral 1. One may show that (<a href="#math_5">5</a>) holds for all continuous compactly supported functions f,<a href="#cite_note-FOOTNOTESteinWeiss1971Theorem_1.18-54">[54]</a> and so ηε converges weakly to δ in the sense of measures. The ηε constructed in this way are known as an approximation to the identity.<a href="#cite_note-FOOTNOTERudin1991§II.6.31-55">[55]</a> This terminology is because the space L1(R) of absolutely integrable functions is closed under the operation of <a href="/wiki/Convolution" title="Convolution">convolution</a> of functions: f ∗ g ∈ L1(R) whenever f and g are in L1(R). However, there is no identity in L1(R) for the convolution product: no element h such that f ∗ h = f for all f. Nevertheless, the sequence ηε does approximate such an identity in the sense that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f*\eta _{\varepsilon }\to f\quad {\text{as }}\varepsilon \to 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∗</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mi>f</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>as </mtext> </mrow> <mi>ε</mi> <mo stretchy="false">→</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f*\eta _{\varepsilon }\to f\quad {\text{as }}\varepsilon \to 0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2244dfb0ca15bdedacc07a679fa469f0664378a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.009ex; height:2.676ex;" alt="{\displaystyle f*\eta _{\varepsilon }\to f\quad {\text{as }}\varepsilon \to 0.}"> This limit holds in the sense of <a href="/wiki/Mean_convergence" class="mw-redirect" title="Mean convergence">mean convergence</a> (convergence in L1). Further conditions on the ηε, for instance that it be a mollifier associated to a compactly supported function,<a href="#cite_note-56">[56]</a> are needed to ensure pointwise convergence <a href="/wiki/Almost_everywhere" title="Almost everywhere">almost everywhere</a>. If the initial η = η1 is itself smooth and compactly supported then the sequence is called a <a href="/wiki/Mollifier" title="Mollifier">mollifier</a>. The standard mollifier is obtained by choosing η to be a suitably normalized <a href="/wiki/Bump_function" title="Bump function">bump function</a>, for instance <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta (x)={\begin{cases}{\frac {1}{I_{n}}}\exp {\Big (}-{\frac {1}{1-|x|^{2}}}{\Big )}&{\text{if }}|x|<1\\0&{\text{if }}|x|\geq 1.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≥</mo> <mn>1.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta (x)={\begin{cases}{\frac {1}{I_{n}}}\exp {\Big (}-{\frac {1}{1-|x|^{2}}}{\Big )}&{\text{if }}|x|<1\\0&{\text{if }}|x|\geq 1.\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88e467429bae56f6ad7dddb8b13117cddf3e6a8d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.152ex; margin-bottom: -0.186ex; width:39.915ex; height:7.843ex;" alt="{\displaystyle \eta (x)={\begin{cases}{\frac {1}{I_{n}}}\exp {\Big (}-{\frac {1}{1-|x|^{2}}}{\Big )}&{\text{if }}|x|<1\\0&{\text{if }}|x|\geq 1.\end{cases}}}"> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aba34f081d776e30204f3458e4f50b403b09e5c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.242ex; height:2.509ex;" alt="{\displaystyle I_{n}}"> ensuring that the total integral is 1). In some situations such as <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerical analysis</a>, a <a href="/wiki/Piecewise_linear_function" title="Piecewise linear function">piecewise linear</a> approximation to the identity is desirable. This can be obtained by taking η1 to be a <a href="/wiki/Hat_function" class="mw-redirect" title="Hat function">hat function</a>. With this choice of η1, one has <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{\varepsilon }(x)=\varepsilon ^{-1}\max \left(1-\left|{\frac {x}{\varepsilon }}\right|,0\right)}"> <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> <mo>=</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo movablelimits="true" form="prefix">max</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>ε</mi> </mfrac> </mrow> <mo>|</mo> </mrow> <mo>,</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{\varepsilon }(x)=\varepsilon ^{-1}\max \left(1-\left|{\frac {x}{\varepsilon }}\right|,0\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fb1127306c60bcda672ad8ccc1032e1b1d5c7e6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.729ex; height:4.843ex;" alt="{\displaystyle \eta _{\varepsilon }(x)=\varepsilon ^{-1}\max \left(1-\left|{\frac {x}{\varepsilon }}\right|,0\right)}"> which are all continuous and compactly supported, although not smooth and so not a mollifier. <div class="mw-heading mw-heading3"><h3 id="Probabilistic_considerations">Probabilistic considerations</h3>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=19" title="Edit section: Probabilistic considerations">edit</a>]</div> In the context of <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, it is natural to impose the additional condition that the initial η1 in an approximation to the identity should be positive, as such a function then represents a <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a>. Convolution with a probability distribution is sometimes favorable because it does not result in <a href="/wiki/Overshoot_(signal)" title="Overshoot (signal)">overshoot</a> or undershoot, as the output is a <a href="/wiki/Convex_combination" title="Convex combination">convex combination</a> of the input values, and thus falls between the maximum and minimum of the input function. Taking η1 to be any probability distribution at all, and letting ηε(x) = η1(x/ε)/ε as above will give rise to an approximation to the identity. In general this converges more rapidly to a delta function if, in addition, η has mean 0 and has small higher moments. For instance, if η1 is the <a href="/wiki/Uniform_distribution_(continuous)" class="mw-redirect" title="Uniform distribution (continuous)">uniform distribution</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left[-{\frac {1}{2}},{\frac {1}{2}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \left[-{\frac {1}{2}},{\frac {1}{2}}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19b64dd6fad3e022e8b3d9f606a2a42292c6181e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.098ex; height:3.509ex;" alt="{\textstyle \left[-{\frac {1}{2}},{\frac {1}{2}}\right]}">, also known as the <a href="/wiki/Rectangular_function" title="Rectangular function">rectangular function</a>, then:<a href="#cite_note-FOOTNOTESaichevWoyczyński1997§1.1_The_"delta_function"_as_viewed_by_a_physicist_and_an_engineer,_p._3-57">[57]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{\varepsilon }(x)={\frac {1}{\varepsilon }}\operatorname {rect} \left({\frac {x}{\varepsilon }}\right)={\begin{cases}{\frac {1}{\varepsilon }},&-{\frac {\varepsilon }{2}}<x<{\frac {\varepsilon }{2}},\\0,&{\text{otherwise}}.\end{cases}}}"> <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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ε</mi> </mfrac> </mrow> <mi>rect</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>ε</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ε</mi> </mfrac> </mrow> <mo>,</mo> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ε</mi> <mn>2</mn> </mfrac> </mrow> <mo><</mo> <mi>x</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ε</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise</mtext> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{\varepsilon }(x)={\frac {1}{\varepsilon }}\operatorname {rect} \left({\frac {x}{\varepsilon }}\right)={\begin{cases}{\frac {1}{\varepsilon }},&-{\frac {\varepsilon }{2}}<x<{\frac {\varepsilon }{2}},\\0,&{\text{otherwise}}.\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1d2d3f6535136448cbf26861b5594b25e46fe34" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:43.118ex; height:6.509ex;" alt="{\displaystyle \eta _{\varepsilon }(x)={\frac {1}{\varepsilon }}\operatorname {rect} \left({\frac {x}{\varepsilon }}\right)={\begin{cases}{\frac {1}{\varepsilon }},&-{\frac {\varepsilon }{2}}<x<{\frac {\varepsilon }{2}},\\0,&{\text{otherwise}}.\end{cases}}}"> Another example is with the <a href="/wiki/Wigner_semicircle_distribution" title="Wigner semicircle distribution">Wigner semicircle distribution</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{\varepsilon }(x)={\begin{cases}{\frac {2}{\pi \varepsilon ^{2}}}{\sqrt {\varepsilon ^{2}-x^{2}}},&-\varepsilon <x<\varepsilon ,\\0,&{\text{otherwise}}.\end{cases}}}"> <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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mi>π</mi> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>ε</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> </msqrt> </mrow> <mo>,</mo> </mtd> <mtd> <mo>−</mo> <mi>ε</mi> <mo><</mo> <mi>x</mi> <mo><</mo> <mi>ε</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise</mtext> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{\varepsilon }(x)={\begin{cases}{\frac {2}{\pi \varepsilon ^{2}}}{\sqrt {\varepsilon ^{2}-x^{2}}},&-\varepsilon <x<\varepsilon ,\\0,&{\text{otherwise}}.\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bb57d399440e570d8ad80967e0081cc9abe592d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.195ex; height:7.509ex;" alt="{\displaystyle \eta _{\varepsilon }(x)={\begin{cases}{\frac {2}{\pi \varepsilon ^{2}}}{\sqrt {\varepsilon ^{2}-x^{2}}},&-\varepsilon <x<\varepsilon ,\\0,&{\text{otherwise}}.\end{cases}}}"> This is continuous and compactly supported, but not a mollifier because it is not smooth. <div class="mw-heading mw-heading3"><h3 id="Semigroups">Semigroups</h3>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=20" title="Edit section: Semigroups">edit</a>]</div> Nascent delta functions often arise as convolution <a href="/wiki/Semigroup" title="Semigroup">semigroups</a>.<a href="#cite_note-58">[58]</a> This amounts to the further constraint that the convolution of ηε with ηδ must satisfy <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{\varepsilon }*\eta _{\delta }=\eta _{\varepsilon +\delta }}"> <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>∗</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> <mo>+</mo> <mi>δ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{\varepsilon }*\eta _{\delta }=\eta _{\varepsilon +\delta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69552683fb85c78cf470cf055fa060b6cd788101" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.75ex; height:2.176ex;" alt="{\displaystyle \eta _{\varepsilon }*\eta _{\delta }=\eta _{\varepsilon +\delta }}"> for all ε, δ > 0. Convolution semigroups in L1 that form a nascent delta function are always an approximation to the identity in the above sense, however the semigroup condition is quite a strong restriction. In practice, semigroups approximating the delta function arise as <a href="/wiki/Fundamental_solution" title="Fundamental solution">fundamental solutions</a> or <a href="/wiki/Green%27s_function" title="Green's function">Green's functions</a> to physically motivated <a href="/wiki/Elliptic_partial_differential_equation" title="Elliptic partial differential equation">elliptic</a> or <a href="/wiki/Parabolic_partial_differential_equation" title="Parabolic partial differential equation">parabolic</a> <a href="/wiki/Partial_differential_equations" class="mw-redirect" title="Partial differential equations">partial differential equations</a>. In the context of <a href="/wiki/Applied_mathematics" title="Applied mathematics">applied mathematics</a>, semigroups arise as the output of a <a href="/wiki/Linear_time-invariant_system" title="Linear time-invariant system">linear time-invariant system</a>. Abstractly, if A is a linear operator acting on functions of x, then a convolution semigroup arises by solving the <a href="/wiki/Initial_value_problem" title="Initial value problem">initial value problem</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}{\dfrac {\partial }{\partial t}}\eta (t,x)=A\eta (t,x),\quad t>0\\[5pt]\displaystyle \lim _{t\to 0^{+}}\eta (t,x)=\delta (x)\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing="0.7em 0.2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> </mrow> <mi>η</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <mi>η</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>t</mi> <mo>></mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <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> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> </munder> <mi>η</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}{\dfrac {\partial }{\partial t}}\eta (t,x)=A\eta (t,x),\quad t>0\\[5pt]\displaystyle \lim _{t\to 0^{+}}\eta (t,x)=\delta (x)\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dcd8145e7871cf48f317c1c4d935ac7eb468604" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.676ex; margin-bottom: -0.328ex; width:31.474ex; height:11.176ex;" alt="{\displaystyle {\begin{cases}{\dfrac {\partial }{\partial t}}\eta (t,x)=A\eta (t,x),\quad t>0\\[5pt]\displaystyle \lim _{t\to 0^{+}}\eta (t,x)=\delta (x)\end{cases}}}"> in which the limit is as usual understood in the weak sense. Setting ηε(x) = η(ε, x) gives the associated nascent delta function. Some examples of physically important convolution semigroups arising from such a fundamental solution include the following. <div class="mw-heading mw-heading4"><h4 id="The_heat_kernel">The heat kernel</h4>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=21" title="Edit section: The heat kernel">edit</a>]</div> The <a href="/wiki/Heat_kernel" title="Heat kernel">heat kernel</a>, defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{\varepsilon }(x)={\frac {1}{\sqrt {2\pi \varepsilon }}}\mathrm {e} ^{-{\frac {x^{2}}{2\varepsilon }}}}"> <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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> <mi>ε</mi> </msqrt> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>ε</mi> </mrow> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{\varepsilon }(x)={\frac {1}{\sqrt {2\pi \varepsilon }}}\mathrm {e} ^{-{\frac {x^{2}}{2\varepsilon }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7e0dbd6b31c6cac6967775642aa8b58a572415d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.685ex; height:6.509ex;" alt="{\displaystyle \eta _{\varepsilon }(x)={\frac {1}{\sqrt {2\pi \varepsilon }}}\mathrm {e} ^{-{\frac {x^{2}}{2\varepsilon }}}}"> represents the temperature in an infinite wire at time t > 0, if a unit of heat energy is stored at the origin of the wire at time t = 0. This semigroup evolves according to the one-dimensional <a href="/wiki/Heat_equation" title="Heat equation">heat equation</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial u}{\partial t}}={\frac {1}{2}}{\frac {\partial ^{2}u}{\partial x^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial u}{\partial t}}={\frac {1}{2}}{\frac {\partial ^{2}u}{\partial x^{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcf708a9c790df7d55a4630dee915d14f2394066" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.791ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial u}{\partial t}}={\frac {1}{2}}{\frac {\partial ^{2}u}{\partial x^{2}}}.}"> In <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, ηε(x) is a <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a> of <a href="/wiki/Variance" title="Variance">variance</a> ε and mean 0. It represents the <a href="/wiki/Probability_density_function" title="Probability density function">probability density</a> at time t = ε of the position of a particle starting at the origin following a standard <a href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a>. In this context, the semigroup condition is then an expression of the <a href="/wiki/Markov_property" title="Markov property">Markov property</a> of Brownian motion. In higher-dimensional Euclidean space Rn, the heat kernel is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{\varepsilon }={\frac {1}{(2\pi \varepsilon )^{n/2}}}\mathrm {e} ^{-{\frac {x\cdot x}{2\varepsilon }}},}"> <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>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>ε</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>⋅</mo> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>ε</mi> </mrow> </mfrac> </mrow> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{\varepsilon }={\frac {1}{(2\pi \varepsilon )^{n/2}}}\mathrm {e} ^{-{\frac {x\cdot x}{2\varepsilon }}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93224832e69b68d6f99e74af2089e5f27324de43" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:20.262ex; height:6.176ex;" alt="{\displaystyle \eta _{\varepsilon }={\frac {1}{(2\pi \varepsilon )^{n/2}}}\mathrm {e} ^{-{\frac {x\cdot x}{2\varepsilon }}},}"> and has the same physical interpretation, <a href="/wiki/Mutatis_mutandis" title="Mutatis mutandis">mutatis mutandis</a>. It also represents a nascent delta function in the sense that ηε → δ in the distribution sense as ε → 0. <div class="mw-heading mw-heading4"><h4 id="The_Poisson_kernel">The Poisson kernel</h4>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=22" title="Edit section: The Poisson kernel">edit</a>]</div> The <a href="/wiki/Poisson_kernel" title="Poisson kernel">Poisson kernel</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{\varepsilon }(x)={\frac {1}{\pi }}\mathrm {Im} \left\{{\frac {1}{x-\mathrm {i} \varepsilon }}\right\}={\frac {1}{\pi }}{\frac {\varepsilon }{\varepsilon ^{2}+x^{2}}}={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\mathrm {e} ^{\mathrm {i} \xi x-|\varepsilon \xi |}\,d\xi }"> <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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>ε</mi> </mrow> </mfrac> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ε</mi> <mrow> <msup> <mi>ε</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"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>ξ</mi> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ε</mi> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{\varepsilon }(x)={\frac {1}{\pi }}\mathrm {Im} \left\{{\frac {1}{x-\mathrm {i} \varepsilon }}\right\}={\frac {1}{\pi }}{\frac {\varepsilon }{\varepsilon ^{2}+x^{2}}}={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\mathrm {e} ^{\mathrm {i} \xi x-|\varepsilon \xi |}\,d\xi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180a474804453cfbeb782988310e99eed313eaac" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:58.771ex; height:6.176ex;" alt="{\displaystyle \eta _{\varepsilon }(x)={\frac {1}{\pi }}\mathrm {Im} \left\{{\frac {1}{x-\mathrm {i} \varepsilon }}\right\}={\frac {1}{\pi }}{\frac {\varepsilon }{\varepsilon ^{2}+x^{2}}}={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\mathrm {e} ^{\mathrm {i} \xi x-|\varepsilon \xi |}\,d\xi }"> is the fundamental solution of the <a href="/wiki/Laplace_equation" class="mw-redirect" title="Laplace equation">Laplace equation</a> in the upper half-plane.<a href="#cite_note-FOOTNOTESteinWeiss1971§I.1-59">[59]</a> It represents the <a href="/wiki/Electrostatic_potential" class="mw-redirect" title="Electrostatic potential">electrostatic potential</a> in a semi-infinite plate whose potential along the edge is held at fixed at the delta function. The Poisson kernel is also closely related to the <a href="/wiki/Cauchy_distribution" title="Cauchy distribution">Cauchy distribution</a> and <a href="/wiki/Kernel_(statistics)#Kernel_functions_in_common_use" title="Kernel (statistics)">Epanechnikov and Gaussian kernel</a> functions.<a href="#cite_note-60">[60]</a> This semigroup evolves according to the equation <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial u}{\partial t}}=-\left(-{\frac {\partial ^{2}}{\partial x^{2}}}\right)^{\frac {1}{2}}u(t,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial u}{\partial t}}=-\left(-{\frac {\partial ^{2}}{\partial x^{2}}}\right)^{\frac {1}{2}}u(t,x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb389d04daadd99ee5d7d9c82ea56ad56afb9c25" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.236ex; height:7.343ex;" alt="{\displaystyle {\frac {\partial u}{\partial t}}=-\left(-{\frac {\partial ^{2}}{\partial x^{2}}}\right)^{\frac {1}{2}}u(t,x)}"> where the operator is rigorously defined as the <a href="/wiki/Fourier_multiplier" class="mw-redirect" title="Fourier multiplier">Fourier multiplier</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}\left[\left(-{\frac {\partial ^{2}}{\partial x^{2}}}\right)^{\frac {1}{2}}f\right](\xi )=|2\pi \xi |{\mathcal {F}}f(\xi ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow> <mo>[</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mi>f</mi> </mrow> <mo>]</mo> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\left[\left(-{\frac {\partial ^{2}}{\partial x^{2}}}\right)^{\frac {1}{2}}f\right](\xi )=|2\pi \xi |{\mathcal {F}}f(\xi ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9dba5209bfdb79796ee7978fbc154e029212b62" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:36.031ex; height:8.509ex;" alt="{\displaystyle {\mathcal {F}}\left[\left(-{\frac {\partial ^{2}}{\partial x^{2}}}\right)^{\frac {1}{2}}f\right](\xi )=|2\pi \xi |{\mathcal {F}}f(\xi ).}"> <div class="mw-heading mw-heading3"><h3 id="Oscillatory_integrals">Oscillatory integrals</h3>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=23" title="Edit section: Oscillatory integrals">edit</a>]</div> In areas of physics such as <a href="/wiki/Wave_propagation" class="mw-redirect" title="Wave propagation">wave propagation</a> and <a href="/wiki/Wave" title="Wave">wave mechanics</a>, the equations involved are <a href="/wiki/Hyperbolic_partial_differential_equations" class="mw-redirect" title="Hyperbolic partial differential equations">hyperbolic</a> and so may have more singular solutions. As a result, the nascent delta functions that arise as fundamental solutions of the associated <a href="/wiki/Cauchy_problem" title="Cauchy problem">Cauchy problems</a> are generally <a href="/wiki/Oscillatory_integral" title="Oscillatory integral">oscillatory integrals</a>. An example, which comes from a solution of the <a href="/wiki/Euler%E2%80%93Tricomi_equation" title="Euler–Tricomi equation">Euler–Tricomi equation</a> of <a href="/wiki/Transonic" title="Transonic">transonic</a> <a href="/wiki/Gas_dynamics" class="mw-redirect" title="Gas dynamics">gas dynamics</a>,<a href="#cite_note-FOOTNOTEValléeSoares2004§7.2-61">[61]</a> is the rescaled <a href="/wiki/Airy_function" title="Airy function">Airy function</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon ^{-1/3}\operatorname {Ai} \left(x\varepsilon ^{-1/3}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> <mi>Ai</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon ^{-1/3}\operatorname {Ai} \left(x\varepsilon ^{-1/3}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d01bd3b4864e2fee5976929a5c915567befa862e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.65ex; height:4.843ex;" alt="{\displaystyle \varepsilon ^{-1/3}\operatorname {Ai} \left(x\varepsilon ^{-1/3}\right).}"> Although using the Fourier transform, it is easy to see that this generates a semigroup in some sense—it is not absolutely integrable and so cannot define a semigroup in the above strong sense. Many nascent delta functions constructed as oscillatory integrals only converge in the sense of distributions (an example is the <a href="/wiki/Dirichlet_kernel" title="Dirichlet kernel">Dirichlet kernel</a> below), rather than in the sense of measures. Another example is the Cauchy problem for the <a href="/wiki/Wave_equation" title="Wave equation">wave equation</a> in R1+1:<a href="#cite_note-FOOTNOTEHörmander1983§7.8-62">[62]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}c^{-2}{\frac {\partial ^{2}u}{\partial t^{2}}}-\Delta u&=0\\u=0,\quad {\frac {\partial u}{\partial t}}=\delta &\qquad {\text{for }}t=0.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>u</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>δ</mi> </mtd> <mtd> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>t</mi> <mo>=</mo> <mn>0.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}c^{-2}{\frac {\partial ^{2}u}{\partial t^{2}}}-\Delta u&=0\\u=0,\quad {\frac {\partial u}{\partial t}}=\delta &\qquad {\text{for }}t=0.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd989ee1cf984a61102136b8bf0d397356d8d57" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:31.089ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}c^{-2}{\frac {\partial ^{2}u}{\partial t^{2}}}-\Delta u&=0\\u=0,\quad {\frac {\partial u}{\partial t}}=\delta &\qquad {\text{for }}t=0.\end{aligned}}}"> The solution u represents the displacement from equilibrium of an infinite elastic string, with an initial disturbance at the origin. Other approximations to the identity of this kind include the <a href="/wiki/Sinc_function" title="Sinc function">sinc function</a> (used widely in electronics and telecommunications) <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{\varepsilon }(x)={\frac {1}{\pi x}}\sin \left({\frac {x}{\varepsilon }}\right)={\frac {1}{2\pi }}\int _{-{\frac {1}{\varepsilon }}}^{\frac {1}{\varepsilon }}\cos(kx)\,dk}"> <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> <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>ε</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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ε</mi> </mfrac> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ε</mi> </mfrac> </mrow> </msubsup> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{\varepsilon }(x)={\frac {1}{\pi x}}\sin \left({\frac {x}{\varepsilon }}\right)={\frac {1}{2\pi }}\int _{-{\frac {1}{\varepsilon }}}^{\frac {1}{\varepsilon }}\cos(kx)\,dk}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18e536afde96b55e6fa81744c880a26f8bad68a8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.859ex; height:7.509ex;" alt="{\displaystyle \eta _{\varepsilon }(x)={\frac {1}{\pi x}}\sin \left({\frac {x}{\varepsilon }}\right)={\frac {1}{2\pi }}\int _{-{\frac {1}{\varepsilon }}}^{\frac {1}{\varepsilon }}\cos(kx)\,dk}"> and the <a href="/wiki/Bessel_function" title="Bessel function">Bessel function</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{\varepsilon }(x)={\frac {1}{\varepsilon }}J_{\frac {1}{\varepsilon }}\left({\frac {x+1}{\varepsilon }}\right).}"> <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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ε</mi> </mfrac> </mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ε</mi> </mfrac> </mrow> </msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>ε</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{\varepsilon }(x)={\frac {1}{\varepsilon }}J_{\frac {1}{\varepsilon }}\left({\frac {x+1}{\varepsilon }}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/314c280ad789361a72ca6cd9df0fd865196bae8c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.427ex; height:6.176ex;" alt="{\displaystyle \eta _{\varepsilon }(x)={\frac {1}{\varepsilon }}J_{\frac {1}{\varepsilon }}\left({\frac {x+1}{\varepsilon }}\right).}"> <div class="mw-heading mw-heading3"><h3 id="Plane_wave_decomposition">Plane wave decomposition</h3>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=24" title="Edit section: Plane wave decomposition">edit</a>]</div> One approach to the study of a linear partial differential equation <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L[u]=f,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">[</mo> <mi>u</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>f</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L[u]=f,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/074535ba6bdba1d73b1f0bdb8cde6be6f06d1d6c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.23ex; height:2.843ex;" alt="{\displaystyle L[u]=f,}"> where L is a <a href="/wiki/Differential_operator" title="Differential operator">differential operator</a> on Rn, is to seek first a fundamental solution, which is a solution of the equation <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L[u]=\delta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">[</mo> <mi>u</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>δ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L[u]=\delta .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6306e66d605d9cde6ddf5b7f85302c2aa915dcd7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9ex; height:2.843ex;" alt="{\displaystyle L[u]=\delta .}"> When L is particularly simple, this problem can often be resolved using the Fourier transform directly (as in the case of the Poisson kernel and heat kernel already mentioned). For more complicated operators, it is sometimes easier first to consider an equation of the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L[u]=h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">[</mo> <mi>u</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L[u]=h}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0a532cf7d2fd63a65144c98a7bbf8456eceaf91" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.644ex; height:2.843ex;" alt="{\displaystyle L[u]=h}"> where h is a <a href="/wiki/Plane_wave" title="Plane wave">plane wave</a> function, meaning that it has the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=h(x\cdot \xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⋅</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h=h(x\cdot \xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/146f8ba7d25f89f5d88cbc236194c68c68a79eac" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.625ex; height:2.843ex;" alt="{\displaystyle h=h(x\cdot \xi )}"> for some vector ξ. Such an equation can be resolved (if the coefficients of L are <a href="/wiki/Analytic_function" title="Analytic function">analytic functions</a>) by the <a href="/wiki/Cauchy%E2%80%93Kovalevskaya_theorem" title="Cauchy–Kovalevskaya theorem">Cauchy–Kovalevskaya theorem</a> or (if the coefficients of L are constant) by quadrature. So, if the delta function can be decomposed into plane waves, then one can in principle solve linear partial differential equations. Such a decomposition of the delta function into plane waves was part of a general technique first introduced essentially by <a href="/wiki/Johann_Radon" title="Johann Radon">Johann Radon</a>, and then developed in this form by <a href="/wiki/Fritz_John" title="Fritz John">Fritz John</a> (<a href="#CITEREFJohn1955">1955</a>).<a href="#cite_note-FOOTNOTECourantHilbert1962§14-63">[63]</a> Choose k so that n + k is an even integer, and for a real number s, put <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(s)=\operatorname {Re} \left[{\frac {-s^{k}\log(-is)}{k!(2\pi i)^{n}}}\right]={\begin{cases}{\frac {|s|^{k}}{4k!(2\pi i)^{n-1}}}&n{\text{ odd}}\\[5pt]-{\frac {|s|^{k}\log |s|}{k!(2\pi i)^{n}}}&n{\text{ even.}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Re</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>k</mi> <mo>!</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing="0.7em 0.2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>s</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <mi>k</mi> <mo>!</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> odd</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>s</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mi>k</mi> <mo>!</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> even.</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(s)=\operatorname {Re} \left[{\frac {-s^{k}\log(-is)}{k!(2\pi i)^{n}}}\right]={\begin{cases}{\frac {|s|^{k}}{4k!(2\pi i)^{n-1}}}&n{\text{ odd}}\\[5pt]-{\frac {|s|^{k}\log |s|}{k!(2\pi i)^{n}}}&n{\text{ even.}}\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb36840bedf496b8822ad586804f91ae10fcba1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:51.457ex; height:11.843ex;" alt="{\displaystyle g(s)=\operatorname {Re} \left[{\frac {-s^{k}\log(-is)}{k!(2\pi i)^{n}}}\right]={\begin{cases}{\frac {|s|^{k}}{4k!(2\pi i)^{n-1}}}&n{\text{ odd}}\\[5pt]-{\frac {|s|^{k}\log |s|}{k!(2\pi i)^{n}}}&n{\text{ even.}}\end{cases}}}"> Then δ is obtained by applying a power of the <a href="/wiki/Laplacian" class="mw-redirect" title="Laplacian">Laplacian</a> to the integral with respect to the unit <a href="/wiki/Sphere_measure" class="mw-redirect" title="Sphere measure">sphere measure</a> dω of g(x · ξ) for ξ in the <a href="/wiki/Unit_sphere" title="Unit sphere">unit sphere</a> Sn−1: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (x)=\Delta _{x}^{(n+k)/2}\int _{S^{n-1}}g(x\cdot \xi )\,d\omega _{\xi }.}"> <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> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msub> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⋅</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ξ</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (x)=\Delta _{x}^{(n+k)/2}\int _{S^{n-1}}g(x\cdot \xi )\,d\omega _{\xi }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdc46867ccbcc15a2c6296cbad99579ecdbc65de" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.164ex; height:5.676ex;" alt="{\displaystyle \delta (x)=\Delta _{x}^{(n+k)/2}\int _{S^{n-1}}g(x\cdot \xi )\,d\omega _{\xi }.}"> The Laplacian here is interpreted as a weak derivative, so that this equation is taken to mean that, for any test function φ, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (x)=\int _{\mathbf {R} ^{n}}\varphi (y)\,dy\,\Delta _{x}^{\frac {n+k}{2}}\int _{S^{n-1}}g((x-y)\cdot \xi )\,d\omega _{\xi }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> <mspace width="thinmathspace" /> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </msubsup> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msub> <mi>g</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ξ</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (x)=\int _{\mathbf {R} ^{n}}\varphi (y)\,dy\,\Delta _{x}^{\frac {n+k}{2}}\int _{S^{n-1}}g((x-y)\cdot \xi )\,d\omega _{\xi }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e625e0b8145c4f397e81491209230712d1dd883c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:48.69ex; height:6.176ex;" alt="{\displaystyle \varphi (x)=\int _{\mathbf {R} ^{n}}\varphi (y)\,dy\,\Delta _{x}^{\frac {n+k}{2}}\int _{S^{n-1}}g((x-y)\cdot \xi )\,d\omega _{\xi }.}"> The result follows from the formula for the <a href="/wiki/Newtonian_potential" title="Newtonian potential">Newtonian potential</a> (the fundamental solution of Poisson's equation). This is essentially a form of the inversion formula for the <a href="/wiki/Radon_transform" title="Radon transform">Radon transform</a> because it recovers the value of φ(x) from its integrals over hyperplanes. For instance, if n is odd and k = 1, then the integral on the right hand side is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&c_{n}\Delta _{x}^{\frac {n+1}{2}}\iint _{S^{n-1}}\varphi (y)|(y-x)\cdot \xi |\,d\omega _{\xi }\,dy\\[5pt]&\qquad =c_{n}\Delta _{x}^{(n+1)/2}\int _{S^{n-1}}\,d\omega _{\xi }\int _{-\infty }^{\infty }|p|R\varphi (\xi ,p+x\cdot \xi )\,dp\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.8em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msubsup> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msub> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ξ</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mspace width="2em" /> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ξ</mi> </mrow> </msub> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>R</mi> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo>,</mo> <mi>p</mi> <mo>+</mo> <mi>x</mi> <mo>⋅</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>p</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&c_{n}\Delta _{x}^{\frac {n+1}{2}}\iint _{S^{n-1}}\varphi (y)|(y-x)\cdot \xi |\,d\omega _{\xi }\,dy\\[5pt]&\qquad =c_{n}\Delta _{x}^{(n+1)/2}\int _{S^{n-1}}\,d\omega _{\xi }\int _{-\infty }^{\infty }|p|R\varphi (\xi ,p+x\cdot \xi )\,dp\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7bb9fbfb3720307465949837a5c76ae4a649295" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.947ex; margin-bottom: -0.224ex; width:54.616ex; height:13.509ex;" alt="{\displaystyle {\begin{aligned}&c_{n}\Delta _{x}^{\frac {n+1}{2}}\iint _{S^{n-1}}\varphi (y)|(y-x)\cdot \xi |\,d\omega _{\xi }\,dy\\[5pt]&\qquad =c_{n}\Delta _{x}^{(n+1)/2}\int _{S^{n-1}}\,d\omega _{\xi }\int _{-\infty }^{\infty }|p|R\varphi (\xi ,p+x\cdot \xi )\,dp\end{aligned}}}"> where Rφ(ξ, p) is the Radon transform of φ: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\varphi (\xi ,p)=\int _{x\cdot \xi =p}f(x)\,d^{n-1}x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>⋅</mo> <mi>ξ</mi> <mo>=</mo> <mi>p</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\varphi (\xi ,p)=\int _{x\cdot \xi =p}f(x)\,d^{n-1}x.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c90603a79a6038922477ab48abed31e39c7a7460" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.887ex; height:6.009ex;" alt="{\displaystyle R\varphi (\xi ,p)=\int _{x\cdot \xi =p}f(x)\,d^{n-1}x.}"> An alternative equivalent expression of the plane wave decomposition is:<a href="#cite_note-FOOTNOTEGelfandShilov1966–1968I,_§3.10-64">[64]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (x)={\begin{cases}{\frac {(n-1)!}{(2\pi i)^{n}}}\displaystyle \int _{S^{n-1}}(x\cdot \xi )^{-n}\,d\omega _{\xi }&n{\text{ even}}\\{\frac {1}{2(2\pi i)^{n-1}}}\displaystyle \int _{S^{n-1}}\delta ^{(n-1)}(x\cdot \xi )\,d\omega _{\xi }&n{\text{ odd}}.\end{cases}}}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⋅</mo> <mi>ξ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ξ</mi> </mrow> </msub> </mstyle> </mtd> <mtd> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> even</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msub> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⋅</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ξ</mi> </mrow> </msub> </mstyle> </mtd> <mtd> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> odd</mtext> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (x)={\begin{cases}{\frac {(n-1)!}{(2\pi i)^{n}}}\displaystyle \int _{S^{n-1}}(x\cdot \xi )^{-n}\,d\omega _{\xi }&n{\text{ even}}\\{\frac {1}{2(2\pi i)^{n-1}}}\displaystyle \int _{S^{n-1}}\delta ^{(n-1)}(x\cdot \xi )\,d\omega _{\xi }&n{\text{ odd}}.\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baa0a1404f8c491e30ace43d98c23142c3722e22" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:48.48ex; height:11.509ex;" alt="{\displaystyle \delta (x)={\begin{cases}{\frac {(n-1)!}{(2\pi i)^{n}}}\displaystyle \int _{S^{n-1}}(x\cdot \xi )^{-n}\,d\omega _{\xi }&n{\text{ even}}\\{\frac {1}{2(2\pi i)^{n-1}}}\displaystyle \int _{S^{n-1}}\delta ^{(n-1)}(x\cdot \xi )\,d\omega _{\xi }&n{\text{ odd}}.\end{cases}}}"> <div class="mw-heading mw-heading3"><h3 id="Fourier_kernels">Fourier kernels</h3>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=25" title="Edit section: Fourier kernels">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Convergence_of_Fourier_series" title="Convergence of Fourier series">Convergence of Fourier series</a></div> In the study of <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a>, a major question consists of determining whether and in what sense the Fourier series associated with a <a href="/wiki/Periodic_function" title="Periodic function">periodic function</a> converges to the function. The n-th partial sum of the Fourier series of a function f of period 2π is defined by convolution (on the interval [−π,π]) with the <a href="/wiki/Dirichlet_kernel" title="Dirichlet kernel">Dirichlet kernel</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{N}(x)=\sum _{n=-N}^{N}e^{inx}={\frac {\sin \left(\left(N+{\frac {1}{2}}\right)x\right)}{\sin(x/2)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mi>N</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{N}(x)=\sum _{n=-N}^{N}e^{inx}={\frac {\sin \left(\left(N+{\frac {1}{2}}\right)x\right)}{\sin(x/2)}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd0acecb02108faf03064ce48ba289949a3d48e7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.028ex; height:7.676ex;" alt="{\displaystyle D_{N}(x)=\sum _{n=-N}^{N}e^{inx}={\frac {\sin \left(\left(N+{\frac {1}{2}}\right)x\right)}{\sin(x/2)}}.}"> Thus, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{N}(f)(x)=D_{N}*f(x)=\sum _{n=-N}^{N}a_{n}e^{inx}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>∗</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{N}(f)(x)=D_{N}*f(x)=\sum _{n=-N}^{N}a_{n}e^{inx}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6704591e4b27edda19fc806b498b50e13c532234" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:37.081ex; height:7.509ex;" alt="{\displaystyle s_{N}(f)(x)=D_{N}*f(x)=\sum _{n=-N}^{N}a_{n}e^{inx}}"> where <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(y)e^{-iny}\,dy.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <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>π</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>n</mi> <mi>y</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(y)e^{-iny}\,dy.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d66fc2d1b35f208e01d3925df300c52cd1a8fed5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.01ex; height:6.009ex;" alt="{\displaystyle a_{n}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(y)e^{-iny}\,dy.}"> A fundamental result of elementary Fourier series states that the Dirichlet kernel restricted to the interval [−π,π] tends to a multiple of the delta function as N → ∞. This is interpreted in the distribution sense, that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{N}(f)(0)=\int _{-\pi }^{\pi }D_{N}(x)f(x)\,dx\to 2\pi f(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</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" /> <mi>d</mi> <mi>x</mi> <mo stretchy="false">→</mo> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{N}(f)(0)=\int _{-\pi }^{\pi }D_{N}(x)f(x)\,dx\to 2\pi f(0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/685e0989da170fa02a67fce0f7d67c6ddc0e0290" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.536ex; height:6.009ex;" alt="{\displaystyle s_{N}(f)(0)=\int _{-\pi }^{\pi }D_{N}(x)f(x)\,dx\to 2\pi f(0)}"> for every compactly supported smooth function f. Thus, formally one has <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (x)={\frac {1}{2\pi }}\sum _{n=-\infty }^{\infty }e^{inx}}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (x)={\frac {1}{2\pi }}\sum _{n=-\infty }^{\infty }e^{inx}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c1589f14927f4c3cd4bf079b1f6e443a86fe649" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.387ex; height:6.843ex;" alt="{\displaystyle \delta (x)={\frac {1}{2\pi }}\sum _{n=-\infty }^{\infty }e^{inx}}"> on the interval [−π,π]. Despite this, the result does not hold for all compactly supported continuous functions: that is DN does not converge weakly in the sense of measures. The lack of convergence of the Fourier series has led to the introduction of a variety of <a href="/wiki/Summability_methods" class="mw-redirect" title="Summability methods">summability methods</a> to produce convergence. The method of <a href="/wiki/Ces%C3%A0ro_summation" title="Cesàro summation">Cesàro summation</a> leads to the <a href="/wiki/Fej%C3%A9r_kernel" title="Fejér kernel">Fejér kernel</a><a href="#cite_note-FOOTNOTELang1997312-65">[65]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{N}(x)={\frac {1}{N}}\sum _{n=0}^{N-1}D_{n}(x)={\frac {1}{N}}\left({\frac {\sin {\frac {Nx}{2}}}{\sin {\frac {x}{2}}}}\right)^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</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> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>D</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> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>N</mi> <mi>x</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{N}(x)={\frac {1}{N}}\sum _{n=0}^{N-1}D_{n}(x)={\frac {1}{N}}\left({\frac {\sin {\frac {Nx}{2}}}{\sin {\frac {x}{2}}}}\right)^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89c4ac7d6be64ffa987ce54aa64cd0bc14bcf4d2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:41.634ex; height:8.176ex;" alt="{\displaystyle F_{N}(x)={\frac {1}{N}}\sum _{n=0}^{N-1}D_{n}(x)={\frac {1}{N}}\left({\frac {\sin {\frac {Nx}{2}}}{\sin {\frac {x}{2}}}}\right)^{2}.}"> The <a href="/wiki/Fej%C3%A9r_kernel" title="Fejér kernel">Fejér kernels</a> tend to the delta function in a stronger sense that<a href="#cite_note-66">[66]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\pi }^{\pi }F_{N}(x)f(x)\,dx\to 2\pi f(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</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" /> <mi>d</mi> <mi>x</mi> <mo stretchy="false">→</mo> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\pi }^{\pi }F_{N}(x)f(x)\,dx\to 2\pi f(0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc7f4850da6a6e786ac553e45262a7ea345aa66c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.166ex; height:6.009ex;" alt="{\displaystyle \int _{-\pi }^{\pi }F_{N}(x)f(x)\,dx\to 2\pi f(0)}"> for every compactly supported continuous function f. The implication is that the Fourier series of any continuous function is Cesàro summable to the value of the function at every point. <div class="mw-heading mw-heading3"><h3 id="Hilbert_space_theory">Hilbert space theory</h3>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=26" title="Edit section: Hilbert space theory">edit</a>]</div> The Dirac delta distribution is a <a href="/wiki/Densely_defined" class="mw-redirect" title="Densely defined">densely defined</a> <a href="/wiki/Unbounded_operator" title="Unbounded operator">unbounded</a> <a href="/wiki/Linear_functional" class="mw-redirect" title="Linear functional">linear functional</a> on the <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> <a href="/wiki/Lp_space" title="Lp space">L2</a> of <a href="/wiki/Square-integrable_function" title="Square-integrable function">square-integrable functions</a>. Indeed, smooth compactly supported functions are <a href="/wiki/Dense_set" title="Dense set">dense</a> in L2, and the action of the delta distribution on such functions is well-defined. In many applications, it is possible to identify subspaces of L2 and to give a stronger <a href="/wiki/Topology" title="Topology">topology</a> on which the delta function defines a <a href="/wiki/Bounded_linear_functional" class="mw-redirect" title="Bounded linear functional">bounded linear functional</a>. <div class="mw-heading mw-heading4"><h4 id="Sobolev_spaces">Sobolev spaces</h4>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=27" title="Edit section: Sobolev spaces">edit</a>]</div> The <a href="/wiki/Sobolev_embedding_theorem" class="mw-redirect" title="Sobolev embedding theorem">Sobolev embedding theorem</a> for <a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev spaces</a> on the real line R implies that any square-integrable function f such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{H^{1}}^{2}=\int _{-\infty }^{\infty }|{\widehat {f}}(\xi )|^{2}(1+|\xi |^{2})\,d\xi <\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msubsup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ξ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ξ</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{H^{1}}^{2}=\int _{-\infty }^{\infty }|{\widehat {f}}(\xi )|^{2}(1+|\xi |^{2})\,d\xi <\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dabe7b3ee46741e2a55f4b0ecd9c3046e61af29" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.29ex; height:6.009ex;" alt="{\displaystyle \|f\|_{H^{1}}^{2}=\int _{-\infty }^{\infty }|{\widehat {f}}(\xi )|^{2}(1+|\xi |^{2})\,d\xi <\infty }"> is automatically continuous, and satisfies in particular <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta [f]=|f(0)|<C\|f\|_{H^{1}}.}"> <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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>C</mi> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta [f]=|f(0)|<C\|f\|_{H^{1}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d546155e53e1d2a324b2639790b8845db523fca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.93ex; height:3.009ex;" alt="{\displaystyle \delta [f]=|f(0)|<C\|f\|_{H^{1}}.}"> Thus δ is a bounded linear functional on the Sobolev space H1. Equivalently δ is an element of the <a href="/wiki/Continuous_dual_space" class="mw-redirect" title="Continuous dual space">continuous dual space</a> H−1 of H1. More generally, in n dimensions, one has δ ∈ H−s(Rn) provided s > <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠n/2⁠. <div class="mw-heading mw-heading4"><h4 id="Spaces_of_holomorphic_functions">Spaces of holomorphic functions</h4>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=28" title="Edit section: Spaces of holomorphic functions">edit</a>]</div> In <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a>, the delta function enters via <a href="/wiki/Cauchy%27s_integral_formula" title="Cauchy's integral formula">Cauchy's integral formula</a>, which asserts that if D is a domain in the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a> with smooth boundary, then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)={\frac {1}{2\pi i}}\oint _{\partial D}{\frac {f(\zeta )\,d\zeta }{\zeta -z}},\quad z\in D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> <mi>i</mi> </mrow> </mfrac> </mrow> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi>D</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>ζ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ζ</mi> </mrow> <mrow> <mi>ζ</mi> <mo>−</mo> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>z</mi> <mo>∈</mo> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z)={\frac {1}{2\pi i}}\oint _{\partial D}{\frac {f(\zeta )\,d\zeta }{\zeta -z}},\quad z\in D}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28bc45a2dfeda82ebea4bce4e25df5bf0efaa14f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.926ex; height:6.176ex;" alt="{\displaystyle f(z)={\frac {1}{2\pi i}}\oint _{\partial D}{\frac {f(\zeta )\,d\zeta }{\zeta -z}},\quad z\in D}"> for all <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic functions</a> f in D that are continuous on the closure of D. As a result, the delta function δz is represented in this class of holomorphic functions by the Cauchy integral: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{z}[f]=f(z)={\frac {1}{2\pi i}}\oint _{\partial D}{\frac {f(\zeta )\,d\zeta }{\zeta -z}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> <mi>i</mi> </mrow> </mfrac> </mrow> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi>D</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>ζ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ζ</mi> </mrow> <mrow> <mi>ζ</mi> <mo>−</mo> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{z}[f]=f(z)={\frac {1}{2\pi i}}\oint _{\partial D}{\frac {f(\zeta )\,d\zeta }{\zeta -z}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35176ad52679bd943137c4ab9d002ad92a8c4dea" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.068ex; height:6.176ex;" alt="{\displaystyle \delta _{z}[f]=f(z)={\frac {1}{2\pi i}}\oint _{\partial D}{\frac {f(\zeta )\,d\zeta }{\zeta -z}}.}"> Moreover, let H2(∂D) be the <a href="/wiki/Hardy_space" title="Hardy space">Hardy space</a> consisting of the closure in L2(∂D) of all holomorphic functions in D continuous up to the boundary of D. Then functions in H2(∂D) uniquely extend to holomorphic functions in D, and the Cauchy integral formula continues to hold. In particular for z ∈ D, the delta function δz is a continuous linear functional on H2(∂D). This is a special case of the situation in <a href="/wiki/Several_complex_variables" class="mw-redirect" title="Several complex variables">several complex variables</a> in which, for smooth domains D, the <a href="/wiki/Szeg%C5%91_kernel" title="Szegő kernel">Szegő kernel</a> plays the role of the Cauchy integral.<a href="#cite_note-FOOTNOTEHazewinkel1995[httpsbooksgooglecombooksidPE1a-EIG22kCpgPA357_357]-67">[67]</a> Another representation of the delta function in a space of holomorphic functions is on the space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(D)\cap L^{2}(D)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>∩</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(D)\cap L^{2}(D)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/badb776f85fd3d09048ed5681c7636b5653a8695" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.75ex; height:3.176ex;" alt="{\displaystyle H(D)\cap L^{2}(D)}"> of square-integrable holomorphic functions in an open set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D\subset \mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>⊂</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D\subset \mathbb {C} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d340ebde44f48772693969cb72e0da51eaa51a0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.919ex; height:2.343ex;" alt="{\displaystyle D\subset \mathbb {C} ^{n}}">. This is a closed subspace of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(D)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}(D)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22d37c2a8854bd3c82f3791b039ca9b30c1192ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.371ex; height:3.176ex;" alt="{\displaystyle L^{2}(D)}">, and therefore is a Hilbert space. On the other hand, the functional that evaluates a holomorphic function in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(D)\cap L^{2}(D)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>∩</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(D)\cap L^{2}(D)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/badb776f85fd3d09048ed5681c7636b5653a8695" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.75ex; height:3.176ex;" alt="{\displaystyle H(D)\cap L^{2}(D)}"> at a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"> is a continuous functional, and so by the Riesz representation theorem, is represented by integration against a kernel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{z}(\zeta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ζ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{z}(\zeta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac8fa06d3e4f50a748aed118e684f248749a0f3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.879ex; height:2.843ex;" alt="{\displaystyle K_{z}(\zeta )}">, the <a href="/wiki/Bergman_kernel" title="Bergman kernel">Bergman kernel</a>. This kernel is the analog of the delta function in this Hilbert space. A Hilbert space having such a kernel is called a <a href="/wiki/Reproducing_kernel_Hilbert_space" title="Reproducing kernel Hilbert space">reproducing kernel Hilbert space</a>. In the special case of the unit disc, one has <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{w}[f]=f(w)={\frac {1}{\pi }}\iint _{|z|<1}{\frac {f(z)\,dx\,dy}{(1-{\bar {z}}w)^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mi>w</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{w}[f]=f(w)={\frac {1}{\pi }}\iint _{|z|<1}{\frac {f(z)\,dx\,dy}{(1-{\bar {z}}w)^{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4107a6750747d8aad60c24262e3c1bc56d9f9e4b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.791ex; height:6.509ex;" alt="{\displaystyle \delta _{w}[f]=f(w)={\frac {1}{\pi }}\iint _{|z|<1}{\frac {f(z)\,dx\,dy}{(1-{\bar {z}}w)^{2}}}.}"> <div class="mw-heading mw-heading4"><h4 id="Resolutions_of_the_identity">Resolutions of the identity</h4>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=29" title="Edit section: Resolutions of the identity">edit</a>]</div> Given a complete <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> set of functions {φn} in a separable Hilbert space, for example, the normalized <a href="/wiki/Eigenvector" class="mw-redirect" title="Eigenvector">eigenvectors</a> of a <a href="/wiki/Compact_operator_on_Hilbert_space#Spectral_theorem" title="Compact operator on Hilbert space">compact self-adjoint operator</a>, any vector f can be expressed as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=\sum _{n=1}^{\infty }\alpha _{n}\varphi _{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=\sum _{n=1}^{\infty }\alpha _{n}\varphi _{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/242aff14e02ae75d7435b67b533e3d80f4065ca9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.211ex; height:6.843ex;" alt="{\displaystyle f=\sum _{n=1}^{\infty }\alpha _{n}\varphi _{n}.}"> The coefficients {αn} are found as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{n}=\langle \varphi _{n},f\rangle ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mi>f</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{n}=\langle \varphi _{n},f\rangle ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbc66b65d0dfcd14127efd71860b4685c44db863" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.312ex; height:2.843ex;" alt="{\displaystyle \alpha _{n}=\langle \varphi _{n},f\rangle ,}"> which may be represented by the notation: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{n}=\varphi _{n}^{\dagger }f,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> <mi>f</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{n}=\varphi _{n}^{\dagger }f,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/631e1a8067274f1fb77f31f485fcfc2f22b9796e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.469ex; height:3.343ex;" alt="{\displaystyle \alpha _{n}=\varphi _{n}^{\dagger }f,}"> a form of the <a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">bra–ket notation</a> of Dirac.<a href="#cite_note-68">[68]</a> Adopting this notation, the expansion of f takes the <a href="/wiki/Dyadic_tensor" class="mw-redirect" title="Dyadic tensor">dyadic</a> form:<a href="#cite_note-FOOTNOTEDavisThomson2000Perfect_operators,_p.344-69">[69]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=\sum _{n=1}^{\infty }\varphi _{n}\left(\varphi _{n}^{\dagger }f\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> <mi>f</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=\sum _{n=1}^{\infty }\varphi _{n}\left(\varphi _{n}^{\dagger }f\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76884d3b270e2611b34ac8f4d19310402e725459" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.071ex; height:6.843ex;" alt="{\displaystyle f=\sum _{n=1}^{\infty }\varphi _{n}\left(\varphi _{n}^{\dagger }f\right).}"> Letting I denote the <a href="/wiki/Identity_operator" class="mw-redirect" title="Identity operator">identity operator</a> on the Hilbert space, the expression <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=\sum _{n=1}^{\infty }\varphi _{n}\varphi _{n}^{\dagger },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=\sum _{n=1}^{\infty }\varphi _{n}\varphi _{n}^{\dagger },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b7ea5af73fc1ecda81e36ca57383d1893bc3388" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.136ex; height:6.843ex;" alt="{\displaystyle I=\sum _{n=1}^{\infty }\varphi _{n}\varphi _{n}^{\dagger },}"> is called a <a href="/wiki/Borel_functional_calculus#Resolution_of_the_identity" title="Borel functional calculus">resolution of the identity</a>. When the Hilbert space is the space L2(D) of square-integrable functions on a domain D, the quantity: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{n}\varphi _{n}^{\dagger },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{n}\varphi _{n}^{\dagger },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/656acc880baffbf97b7aa58c788db6b7dee767e4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.124ex; height:3.343ex;" alt="{\displaystyle \varphi _{n}\varphi _{n}^{\dagger },}"> is an integral operator, and the expression for f can be rewritten <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\sum _{n=1}^{\infty }\int _{D}\,\left(\varphi _{n}(x)\varphi _{n}^{*}(\xi )\right)f(\xi )\,d\xi .}"> <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> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ξ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\sum _{n=1}^{\infty }\int _{D}\,\left(\varphi _{n}(x)\varphi _{n}^{*}(\xi )\right)f(\xi )\,d\xi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/615cd30ab3f8592eba06de5f58e102aacb0a6542" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.968ex; height:6.843ex;" alt="{\displaystyle f(x)=\sum _{n=1}^{\infty }\int _{D}\,\left(\varphi _{n}(x)\varphi _{n}^{*}(\xi )\right)f(\xi )\,d\xi .}"> The right-hand side converges to f in the L2 sense. It need not hold in a pointwise sense, even when f is a continuous function. Nevertheless, it is common to abuse notation and write <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\int \,\delta (x-\xi )f(\xi )\,d\xi ,}"> <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> <mo>∫</mo> <mspace width="thinmathspace" /> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ξ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\int \,\delta (x-\xi )f(\xi )\,d\xi ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0456236abf9ba80b7ec235a2bcdb7c8cd262bfdd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.94ex; height:5.676ex;" alt="{\displaystyle f(x)=\int \,\delta (x-\xi )f(\xi )\,d\xi ,}"> resulting in the representation of the delta function:<a href="#cite_note-FOOTNOTEDavisThomson2000Equation_8.9.11,_p._344-70">[70]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (x-\xi )=\sum _{n=1}^{\infty }\varphi _{n}(x)\varphi _{n}^{*}(\xi ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (x-\xi )=\sum _{n=1}^{\infty }\varphi _{n}(x)\varphi _{n}^{*}(\xi ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eec7029b19f112482e0e54d1605a3560359ceefa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.001ex; height:6.843ex;" alt="{\displaystyle \delta (x-\xi )=\sum _{n=1}^{\infty }\varphi _{n}(x)\varphi _{n}^{*}(\xi ).}"> With a suitable <a href="/wiki/Rigged_Hilbert_space" title="Rigged Hilbert space">rigged Hilbert space</a> (Φ, L2(D), Φ*) where Φ ⊂ L2(D) contains all compactly supported smooth functions, this summation may converge in Φ*, depending on the properties of the basis φn. In most cases of practical interest, the orthonormal basis comes from an integral or differential operator, in which case the series converges in the <a href="/wiki/Distribution_(mathematics)#Distributions" title="Distribution (mathematics)">distribution</a> sense.<a href="#cite_note-FOOTNOTEde_la_MadridBohmGadella2002-71">[71]</a> <div class="mw-heading mw-heading3"><h3 id="Infinitesimal_delta_functions">Infinitesimal delta functions</h3>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=30" title="Edit section: Infinitesimal delta functions">edit</a>]</div> <a href="/wiki/Cauchy" class="mw-redirect" title="Cauchy">Cauchy</a> used an infinitesimal α to write down a unit impulse, infinitely tall and narrow Dirac-type delta function δα satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \int F(x)\delta _{\alpha }(x)\,dx=F(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∫</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \int F(x)\delta _{\alpha }(x)\,dx=F(0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38c6e7188142732aee4402b7732a04e82bce0508" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.884ex; height:3.176ex;" alt="{\textstyle \int F(x)\delta _{\alpha }(x)\,dx=F(0)}"> in a number of articles in 1827.<a href="#cite_note-FOOTNOTELaugwitz1989-72">[72]</a> Cauchy defined an infinitesimal in <a href="/wiki/Cours_d%27Analyse" title="Cours d'Analyse">Cours d'Analyse</a> (1827) in terms of a sequence tending to zero. Namely, such a null sequence becomes an infinitesimal in Cauchy's and <a href="/wiki/Lazare_Carnot" title="Lazare Carnot">Lazare Carnot</a>'s terminology. <a href="/wiki/Non-standard_analysis" class="mw-redirect" title="Non-standard analysis">Non-standard analysis</a> allows one to rigorously treat infinitesimals. The article by <a href="#CITEREFYamashita2007">Yamashita (2007)</a> contains a bibliography on modern Dirac delta functions in the context of an infinitesimal-enriched continuum provided by the <a href="/wiki/Hyperreal_number" title="Hyperreal number">hyperreals</a>. Here the Dirac delta can be given by an actual function, having the property that for every real function F one has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \int F(x)\delta _{\alpha }(x)\,dx=F(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∫</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \int F(x)\delta _{\alpha }(x)\,dx=F(0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38c6e7188142732aee4402b7732a04e82bce0508" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.884ex; height:3.176ex;" alt="{\textstyle \int F(x)\delta _{\alpha }(x)\,dx=F(0)}"> as anticipated by Fourier and Cauchy. <div class="mw-heading mw-heading2"><h2 id="Dirac_comb">Dirac comb</h2>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=31" title="Edit section: Dirac comb">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Dirac_comb" title="Dirac comb">Dirac comb</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Dirac_comb.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Dirac_comb.svg/220px-Dirac_comb.svg.png" decoding="async" width="220" height="139" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Dirac_comb.svg/330px-Dirac_comb.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/49/Dirac_comb.svg/440px-Dirac_comb.svg.png 2x" data-file-width="512" data-file-height="323" /></a><figcaption>A Dirac comb is an infinite series of Dirac delta functions spaced at intervals of T</figcaption></figure> A so-called uniform "pulse train" of Dirac delta measures, which is known as a <a href="/wiki/Dirac_comb" title="Dirac comb">Dirac comb</a>, or as the <a href="/wiki/Sha_(Cyrillic)" title="Sha (Cyrillic)">Sha</a> distribution, creates a <a href="/wiki/Sampling_(signal_processing)" title="Sampling (signal processing)">sampling</a> function, often used in <a href="/wiki/Digital_signal_processing" title="Digital signal processing">digital signal processing</a> (DSP) and discrete time signal analysis. The Dirac comb is given as the <a href="/wiki/Infinite_sum" class="mw-redirect" title="Infinite sum">infinite sum</a>, whose limit is understood in the distribution sense, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {\text{Ш}} (x)=\sum _{n=-\infty }^{\infty }\delta (x-n),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mtext>Ш</mtext> </mrow> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {\text{Ш}} (x)=\sum _{n=-\infty }^{\infty }\delta (x-n),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2835ee1a00bd8d0a06fafbfd1f6b89d8d3416bff" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.297ex; height:6.843ex;" alt="{\displaystyle \operatorname {\text{Ш}} (x)=\sum _{n=-\infty }^{\infty }\delta (x-n),}"> which is a sequence of point masses at each of the integers. Up to an overall normalizing constant, the Dirac comb is equal to its own Fourier transform. This is significant because if f is any <a href="/wiki/Schwartz_space" title="Schwartz space">Schwartz function</a>, then the <a href="/wiki/Wrapped_distribution" title="Wrapped distribution">periodization</a> of f is given by the convolution <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f*\operatorname {\text{Ш}} )(x)=\sum _{n=-\infty }^{\infty }f(x-n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mtext>Ш</mtext> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f*\operatorname {\text{Ш}} )(x)=\sum _{n=-\infty }^{\infty }f(x-n).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/983960c780f976eaea4bb35bb4d1d516c171c15e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.81ex; height:6.843ex;" alt="{\displaystyle (f*\operatorname {\text{Ш}} )(x)=\sum _{n=-\infty }^{\infty }f(x-n).}"> In particular, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f*\operatorname {\text{Ш}} )^{\wedge }={\widehat {f}}{\widehat {\operatorname {\text{Ш}} }}={\widehat {f}}\operatorname {\text{Ш}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mtext>Ш</mtext> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∧</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mtext>Ш</mtext> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mtext>Ш</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f*\operatorname {\text{Ш}} )^{\wedge }={\widehat {f}}{\widehat {\operatorname {\text{Ш}} }}={\widehat {f}}\operatorname {\text{Ш}} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43226de936a90d00bd4da2da78d2a107d269a63" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.987ex; height:4.176ex;" alt="{\displaystyle (f*\operatorname {\text{Ш}} )^{\wedge }={\widehat {f}}{\widehat {\operatorname {\text{Ш}} }}={\widehat {f}}\operatorname {\text{Ш}} }"> is precisely the <a href="/wiki/Poisson_summation_formula" title="Poisson summation formula">Poisson summation formula</a>.<a href="#cite_note-FOOTNOTECórdoba1988-73">[73]</a><a href="#cite_note-FOOTNOTEHörmander1983[httpsbooksgooglecombooksidaaLrCAAAQBAJpgPA177_§7.2]-74">[74]</a> More generally, this formula remains to be true if f is a tempered distribution of rapid descent or, equivalently, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/153a4a4d50ef7099fc5f8804e34dccc539f08743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.019ex; width:1.79ex; height:3.343ex;" alt="{\displaystyle {\widehat {f}}}"> is a slowly growing, ordinary function within the space of tempered distributions. <div class="mw-heading mw-heading2"><h2 id="Sokhotski–Plemelj_theorem">Sokhotski–Plemelj theorem</h2>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=32" title="Edit section: Sokhotski–Plemelj theorem">edit</a>]</div> The <a href="/wiki/Sokhotski%E2%80%93Plemelj_theorem" title="Sokhotski–Plemelj theorem">Sokhotski–Plemelj theorem</a>, important in quantum mechanics, relates the delta function to the distribution p.v. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/x⁠, the <a href="/wiki/Cauchy_principal_value" title="Cauchy principal value">Cauchy principal value</a> of the function <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/x⁠, defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle \operatorname {p.v.} {\frac {1}{x}},\varphi \right\rangle =\lim _{\varepsilon \to 0^{+}}\int _{|x|>\varepsilon }{\frac {\varphi (x)}{x}}\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mrow> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">p</mi> <mo>.</mo> <mi mathvariant="normal">v</mi> <mo>.</mo> </mrow> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>,</mo> <mi>φ</mi> </mrow> <mo>⟩</mo> </mrow> <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> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mi>ε</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>x</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle \operatorname {p.v.} {\frac {1}{x}},\varphi \right\rangle =\lim _{\varepsilon \to 0^{+}}\int _{|x|>\varepsilon }{\frac {\varphi (x)}{x}}\,dx.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e979de266c5abe799ed95a15eb9b1833cdd82c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:35.599ex; height:6.509ex;" alt="{\displaystyle \left\langle \operatorname {p.v.} {\frac {1}{x}},\varphi \right\rangle =\lim _{\varepsilon \to 0^{+}}\int _{|x|>\varepsilon }{\frac {\varphi (x)}{x}}\,dx.}"> Sokhotsky's formula states that<a href="#cite_note-FOOTNOTEVladimirov1971§5.7-75">[75]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{\varepsilon \to 0^{+}}{\frac {1}{x\pm i\varepsilon }}=\operatorname {p.v.} {\frac {1}{x}}\mp i\pi \delta (x),}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>±</mo> <mi>i</mi> <mi>ε</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">p</mi> <mo>.</mo> <mi mathvariant="normal">v</mi> <mo>.</mo> </mrow> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>∓</mo> <mi>i</mi> <mi>π</mi> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{\varepsilon \to 0^{+}}{\frac {1}{x\pm i\varepsilon }}=\operatorname {p.v.} {\frac {1}{x}}\mp i\pi \delta (x),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5d0e91d56b1d0eb984500bbdb6d4dbcbec63331" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.374ex; height:5.676ex;" alt="{\displaystyle \lim _{\varepsilon \to 0^{+}}{\frac {1}{x\pm i\varepsilon }}=\operatorname {p.v.} {\frac {1}{x}}\mp i\pi \delta (x),}"> Here the limit is understood in the distribution sense, that for all compactly supported smooth functions f, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }\lim _{\varepsilon \to 0^{+}}{\frac {f(x)}{x\pm i\varepsilon }}\,dx=\mp i\pi f(0)+\lim _{\varepsilon \to 0^{+}}\int _{|x|>\varepsilon }{\frac {f(x)}{x}}\,dx.}"> <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"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>x</mi> <mo>±</mo> <mi>i</mi> <mi>ε</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mo>∓</mo> <mi>i</mi> <mi>π</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <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> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mi>ε</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>x</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }\lim _{\varepsilon \to 0^{+}}{\frac {f(x)}{x\pm i\varepsilon }}\,dx=\mp i\pi f(0)+\lim _{\varepsilon \to 0^{+}}\int _{|x|>\varepsilon }{\frac {f(x)}{x}}\,dx.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a46a66e6ea0beb6a72312ccd41a6754bf51339d7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:53.076ex; height:6.509ex;" alt="{\displaystyle \int _{-\infty }^{\infty }\lim _{\varepsilon \to 0^{+}}{\frac {f(x)}{x\pm i\varepsilon }}\,dx=\mp i\pi f(0)+\lim _{\varepsilon \to 0^{+}}\int _{|x|>\varepsilon }{\frac {f(x)}{x}}\,dx.}"> <div class="mw-heading mw-heading2"><h2 id="Relationship_to_the_Kronecker_delta">Relationship to the Kronecker delta</h2>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=33" title="Edit section: Relationship to the Kronecker delta">edit</a>]</div> The <a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a> δij is the quantity defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{ij}={\begin{cases}1&i=j\\0&i\not =j\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <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> <mi>i</mi> <mo>=</mo> <mi>j</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>i</mi> <mo>≠</mo> <mi>j</mi> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{ij}={\begin{cases}1&i=j\\0&i\not =j\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c59930d1f15174ec9d84282cec18212b265cfb09" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.447ex; height:6.176ex;" alt="{\displaystyle \delta _{ij}={\begin{cases}1&i=j\\0&i\not =j\end{cases}}}"> for all integers i, j. This function then satisfies the following analog of the sifting property: if ai (for i in the set of all integers) is any <a href="/wiki/Infinite_sequence#Doubly-infinite_sequences" class="mw-redirect" title="Infinite sequence">doubly infinite sequence</a>, then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=-\infty }^{\infty }a_{i}\delta _{ik}=a_{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=-\infty }^{\infty }a_{i}\delta _{ik}=a_{k}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b595c7a9598cbbdf0dd7cf682dfd92733ae03aa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:15.937ex; height:7.009ex;" alt="{\displaystyle \sum _{i=-\infty }^{\infty }a_{i}\delta _{ik}=a_{k}.}"> Similarly, for any real or complex valued continuous function f on R, the Dirac delta satisfies the sifting property <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }f(x)\delta (x-x_{0})\,dx=f(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"> <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>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <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 \int _{-\infty }^{\infty }f(x)\delta (x-x_{0})\,dx=f(x_{0}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2e6c78232fd0dfe7b791e431d437fab8e861ec0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.813ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }f(x)\delta (x-x_{0})\,dx=f(x_{0}).}"> This exhibits the Kronecker delta function as a discrete analog of the Dirac delta function.<a href="#cite_note-FOOTNOTEHartmann1997pp._154–155-76">[76]</a> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=34" title="Edit section: Applications">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Probability_theory">Probability theory</h3>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=35" title="Edit section: Probability theory">edit</a>]</div> In <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a> and <a href="/wiki/Statistics" title="Statistics">statistics</a>, the Dirac delta function is often used to represent a <a href="/wiki/Discrete_distribution" class="mw-redirect" title="Discrete distribution">discrete distribution</a>, or a partially discrete, partially <a href="/wiki/Continuous_distribution" class="mw-redirect" title="Continuous distribution">continuous distribution</a>, using a <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a> (which is normally used to represent absolutely continuous distributions). For example, the probability density function f(x) of a discrete distribution consisting of points x = {x1, ..., xn}, with corresponding probabilities p1, ..., pn, can be written as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\sum _{i=1}^{n}p_{i}\delta (x-x_{i}).}"> <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> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\sum _{i=1}^{n}p_{i}\delta (x-x_{i}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df030dbfcfdbc915326e856b29fc21edd2ca9a94" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.031ex; height:6.843ex;" alt="{\displaystyle f(x)=\sum _{i=1}^{n}p_{i}\delta (x-x_{i}).}"> As another example, consider a distribution in which 6/10 of the time returns a standard <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a>, and 4/10 of the time returns exactly the value 3.5 (i.e. a partly continuous, partly discrete <a href="/wiki/Mixture_distribution" title="Mixture distribution">mixture distribution</a>). The density function of this distribution can be written as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=0.6\,{\frac {1}{\sqrt {2\pi }}}e^{-{\frac {x^{2}}{2}}}+0.4\,\delta (x-3.5).}"> <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>0.6</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mo>+</mo> <mn>0.4</mn> <mspace width="thinmathspace" /> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>3.5</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=0.6\,{\frac {1}{\sqrt {2\pi }}}e^{-{\frac {x^{2}}{2}}}+0.4\,\delta (x-3.5).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42997822d0745a22f02620d6bb8e566762b0ee97" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:37.982ex; height:6.509ex;" alt="{\displaystyle f(x)=0.6\,{\frac {1}{\sqrt {2\pi }}}e^{-{\frac {x^{2}}{2}}}+0.4\,\delta (x-3.5).}"> The delta function is also used to represent the resulting probability density function of a random variable that is transformed by continuously differentiable function. If Y = g(X) is a continuous differentiable function, then the density of Y can be written as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{Y}(y)=\int _{-\infty }^{+\infty }f_{X}(x)\delta (y-g(x))\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</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> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{Y}(y)=\int _{-\infty }^{+\infty }f_{X}(x)\delta (y-g(x))\,dx.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1859af9ba4e81ed72ae8cd7952fa91fc2b75744" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.293ex; height:6.176ex;" alt="{\displaystyle f_{Y}(y)=\int _{-\infty }^{+\infty }f_{X}(x)\delta (y-g(x))\,dx.}"> The delta function is also used in a completely different way to represent the <a href="/wiki/Local_time_(mathematics)" title="Local time (mathematics)">local time</a> of a <a href="/wiki/Diffusion_process" title="Diffusion process">diffusion process</a> (like <a href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a>). The local time of a stochastic process B(t) is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell (x,t)=\int _{0}^{t}\delta (x-B(s))\,ds}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell (x,t)=\int _{0}^{t}\delta (x-B(s))\,ds}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0200632fe8ac164dabb35f5c5ff1798dec3a7bb0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.143ex; height:6.176ex;" alt="{\displaystyle \ell (x,t)=\int _{0}^{t}\delta (x-B(s))\,ds}"> and represents the amount of time that the process spends at the point x in the range of the process. More precisely, in one dimension this integral can be written <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell (x,t)=\lim _{\varepsilon \to 0^{+}}{\frac {1}{2\varepsilon }}\int _{0}^{t}\mathbf {1} _{[x-\varepsilon ,x+\varepsilon ]}(B(s))\,ds}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ε</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mi>x</mi> <mo>−</mo> <mi>ε</mi> <mo>,</mo> <mi>x</mi> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">]</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell (x,t)=\lim _{\varepsilon \to 0^{+}}{\frac {1}{2\varepsilon }}\int _{0}^{t}\mathbf {1} _{[x-\varepsilon ,x+\varepsilon ]}(B(s))\,ds}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5559ee5dae6a003812081e6f4785fe97c5c7ce3e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:39.125ex; height:6.176ex;" alt="{\displaystyle \ell (x,t)=\lim _{\varepsilon \to 0^{+}}{\frac {1}{2\varepsilon }}\int _{0}^{t}\mathbf {1} _{[x-\varepsilon ,x+\varepsilon ]}(B(s))\,ds}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1} _{[x-\varepsilon ,x+\varepsilon ]}}"> <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> <mi>x</mi> <mo>−</mo> <mi>ε</mi> <mo>,</mo> <mi>x</mi> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">]</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} _{[x-\varepsilon ,x+\varepsilon ]}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d2638b50ca5e6ffd77a1e6e4747b7b98c5b786f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.911ex; height:3.009ex;" alt="{\displaystyle \mathbf {1} _{[x-\varepsilon ,x+\varepsilon ]}}"> is the <a href="/wiki/Indicator_function" title="Indicator function">indicator function</a> of the interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x-\varepsilon ,x+\varepsilon ].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>x</mi> <mo>−</mo> <mi>ε</mi> <mo>,</mo> <mi>x</mi> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x-\varepsilon ,x+\varepsilon ].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee3e4409bd7fe914f16bb104c11de59e5ee1a225" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.482ex; height:2.843ex;" alt="{\displaystyle [x-\varepsilon ,x+\varepsilon ].}"> <div class="mw-heading mw-heading3"><h3 id="Quantum_mechanics">Quantum mechanics</h3>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=36" title="Edit section: Quantum mechanics">edit</a>]</div> The delta function is expedient in <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>. The <a href="/wiki/Wave_function" title="Wave function">wave function</a> of a particle gives the <a href="/wiki/Probability_amplitude" title="Probability amplitude">probability amplitude</a> of finding a particle within a given region of space. Wave functions are assumed to be elements of the Hilbert space L2 of <a href="/wiki/Square-integrable_function" title="Square-integrable function">square-integrable functions</a>, and the total probability of finding a particle within a given interval is the integral of the magnitude of the wave function squared over the interval. A set {|φn⟩} of wave functions is orthonormal if <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \varphi _{n}\mid \varphi _{m}\rangle =\delta _{nm},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∣</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>m</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \varphi _{n}\mid \varphi _{m}\rangle =\delta _{nm},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b29e7171ed63cda3fe7acbb0db4db2c14a802d0c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.119ex; height:2.843ex;" alt="{\displaystyle \langle \varphi _{n}\mid \varphi _{m}\rangle =\delta _{nm},}"> where δnm is the Kronecker delta. A set of orthonormal wave functions is complete in the space of square-integrable functions if any wave function |ψ⟩ can be expressed as a linear combination of the {|φn⟩} with complex coefficients: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi =\sum c_{n}\varphi _{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <mo>∑</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi =\sum c_{n}\varphi _{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01a42296ab42c7a0687a6a983d4f0ddccc353604" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:13.964ex; height:3.843ex;" alt="{\displaystyle \psi =\sum c_{n}\varphi _{n},}"> where cn = ⟨φn|ψ⟩. Complete orthonormal systems of wave functions appear naturally as the <a href="/wiki/Eigenfunction" title="Eigenfunction">eigenfunctions</a> of the <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a> (of a <a href="/wiki/Bound_state" title="Bound state">bound system</a>) in quantum mechanics that measures the energy levels, which are called the eigenvalues. The set of eigenvalues, in this case, is known as the <a href="/wiki/Spectrum_(functional_analysis)" title="Spectrum (functional analysis)">spectrum</a> of the Hamiltonian. In <a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">bra–ket notation</a> this equality implies the <a href="/wiki/Borel_functional_calculus#Resolution_of_the_identity" title="Borel functional calculus">resolution of the identity</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=\sum |\varphi _{n}\rangle \langle \varphi _{n}|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=\sum |\varphi _{n}\rangle \langle \varphi _{n}|.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4b383161286e2ba9ce85e0a4c0495001c7b293c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:17.239ex; height:3.843ex;" alt="{\displaystyle I=\sum |\varphi _{n}\rangle \langle \varphi _{n}|.}"> Here the eigenvalues are assumed to be discrete, but the set of eigenvalues of an <a href="/wiki/Observable" title="Observable">observable</a> can also be continuous. An example is the <a href="/wiki/Position_operator" title="Position operator">position operator</a>, Qψ(x) = xψ(x). The spectrum of the position (in one dimension) is the entire real line and is called a <a href="/wiki/Spectrum_(physical_sciences)#In_quantum_mechanics" title="Spectrum (physical sciences)">continuous spectrum</a>. However, unlike the Hamiltonian, the position operator lacks proper eigenfunctions. The conventional way to overcome this shortcoming is to widen the class of available functions by allowing distributions as well, i.e., to replace the Hilbert space with a <a href="/wiki/Rigged_Hilbert_space" title="Rigged Hilbert space">rigged Hilbert space</a>.<a href="#cite_note-FOOTNOTEIsham1995§6.2-77">[77]</a> In this context, the position operator has a complete set of "generalized eigenfunctions", labeled by the points y of the real line, given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{y}(x)=\delta (x-y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{y}(x)=\delta (x-y).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee92e79b4ce4b11ae6a1f92f7aba827cf9292580" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.637ex; height:3.009ex;" alt="{\displaystyle \varphi _{y}(x)=\delta (x-y).}"> The generalized eigenfunctions of the position operator are called the eigenkets and are denoted by φy = |y⟩.<a href="#cite_note-FOOTNOTEde_la_Madrid_Modino200196,_106-78">[78]</a> Similar considerations apply to any other <a href="/wiki/Spectral_theorem#Unbounded_self-adjoint_operators" title="Spectral theorem">(unbounded) self-adjoint operator</a> with continuous spectrum and no degenerate eigenvalues, such as the <a href="/wiki/Momentum_operator" title="Momentum operator">momentum operator</a> P. In that case, there is a set Ω of real numbers (the spectrum) and a collection of distributions φy with y ∈ Ω such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\varphi _{y}=y\varphi _{y}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mi>y</mi> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\varphi _{y}=y\varphi _{y}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3853f92b0c24a3eb6a0d85fd184fd0e4f513504" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.785ex; height:2.843ex;" alt="{\displaystyle P\varphi _{y}=y\varphi _{y}.}"> That is, φy are the generalized eigenvectors of P. If they form an "orthonormal basis" in the distribution sense, that is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \varphi _{y},\varphi _{y'}\rangle =\delta (y-y'),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>y</mi> <mo>′</mo> </msup> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \varphi _{y},\varphi _{y'}\rangle =\delta (y-y'),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd9a3bae543ed91fbf83f9eba81d467c7776533" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.962ex; height:3.176ex;" alt="{\displaystyle \langle \varphi _{y},\varphi _{y'}\rangle =\delta (y-y'),}"> then for any test function ψ, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)=\int _{\Omega }c(y)\varphi _{y}(x)\,dy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mi>c</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)=\int _{\Omega }c(y)\varphi _{y}(x)\,dy}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db273ee8be939ee91d25a6071556402b81b92add" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.288ex; height:5.676ex;" alt="{\displaystyle \psi (x)=\int _{\Omega }c(y)\varphi _{y}(x)\,dy}"> where c(y) = ⟨ψ, φy⟩. That is, there is a resolution of the identity <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=\int _{\Omega }|\varphi _{y}\rangle \,\langle \varphi _{y}|\,dy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=\int _{\Omega }|\varphi _{y}\rangle \,\langle \varphi _{y}|\,dy}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71af2b6e5c50f2a9d956bfaf06919f84ff19083f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.756ex; height:5.676ex;" alt="{\displaystyle I=\int _{\Omega }|\varphi _{y}\rangle \,\langle \varphi _{y}|\,dy}"> where the operator-valued integral is again understood in the weak sense. If the spectrum of P has both continuous and discrete parts, then the resolution of the identity involves a summation over the discrete spectrum and an integral over the continuous spectrum. The delta function also has many more specialized applications in quantum mechanics, such as the <a href="/wiki/Delta_potential" title="Delta potential">delta potential</a> models for a single and double potential well. <div class="mw-heading mw-heading3"><h3 id="Structural_mechanics">Structural mechanics</h3>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=37" title="Edit section: Structural mechanics">edit</a>]</div> The delta function can be used in <a href="/wiki/Structural_mechanics" title="Structural mechanics">structural mechanics</a> to describe transient loads or point loads acting on structures. The governing equation of a simple <a href="/wiki/Harmonic_oscillator" title="Harmonic oscillator">mass–spring system</a> excited by a sudden force <a href="/wiki/Impulse_(physics)" title="Impulse (physics)">impulse</a> I at time t = 0 can be written <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m{\frac {d^{2}\xi }{dt^{2}}}+k\xi =I\delta (t),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ξ</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>k</mi> <mi>ξ</mi> <mo>=</mo> <mi>I</mi> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m{\frac {d^{2}\xi }{dt^{2}}}+k\xi =I\delta (t),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18c095dc7dfac9d0a604cf5f01c7d330b777c618" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.875ex; height:6.009ex;" alt="{\displaystyle m{\frac {d^{2}\xi }{dt^{2}}}+k\xi =I\delta (t),}"> where m is the mass, ξ is the deflection, and k is the <a href="/wiki/Spring_constant" class="mw-redirect" title="Spring constant">spring constant</a>. As another example, the equation governing the static deflection of a slender <a href="/wiki/Beam_(structure)" title="Beam (structure)">beam</a> is, according to <a href="/wiki/Euler%E2%80%93Bernoulli_beam_equation" class="mw-redirect" title="Euler–Bernoulli beam equation">Euler–Bernoulli theory</a>, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle EI{\frac {d^{4}w}{dx^{4}}}=q(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>w</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle EI{\frac {d^{4}w}{dx^{4}}}=q(x),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c79f36e1a3902ecf7531e851fc917662483bd69" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.674ex; height:6.009ex;" alt="{\displaystyle EI{\frac {d^{4}w}{dx^{4}}}=q(x),}"> where EI is the <a href="/wiki/Bending_stiffness" title="Bending stiffness">bending stiffness</a> of the beam, w is the <a href="/wiki/Deflection_(engineering)" title="Deflection (engineering)">deflection</a>, x is the spatial coordinate, and q(x) is the load distribution. If a beam is loaded by a point force F at x = x0, the load distribution is written <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(x)=F\delta (x-x_{0}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> <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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q(x)=F\delta (x-x_{0}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/339b0c1e87ddd55ace2ae43d72d4b54537ca5ee3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.107ex; height:2.843ex;" alt="{\displaystyle q(x)=F\delta (x-x_{0}).}"> As the integration of the delta function results in the <a href="/wiki/Heaviside_step_function" title="Heaviside step function">Heaviside step function</a>, it follows that the static deflection of a slender beam subject to multiple point loads is described by a set of piecewise <a href="/wiki/Polynomial" title="Polynomial">polynomials</a>. Also, a point <a href="/wiki/Bending_moment" title="Bending moment">moment</a> acting on a beam can be described by delta functions. Consider two opposing point forces F at a distance d apart. They then produce a moment M = Fd acting on the beam. Now, let the distance d approach the <a href="/wiki/Limit_of_a_function" title="Limit of a function">limit</a> zero, while M is kept constant. The load distribution, assuming a clockwise moment acting at x = 0, is written <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}q(x)&=\lim _{d\to 0}{\Big (}F\delta (x)-F\delta (x-d){\Big )}\\[4pt]&=\lim _{d\to 0}\left({\frac {M}{d}}\delta (x)-{\frac {M}{d}}\delta (x-d)\right)\\[4pt]&=M\lim _{d\to 0}{\frac {\delta (x)-\delta (x-d)}{d}}\\[4pt]&=M\delta '(x).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.7em 0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mi>F</mi> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>M</mi> <mi>d</mi> </mfrac> </mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>M</mi> <mi>d</mi> </mfrac> </mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>M</mi> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</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 stretchy="false">)</mo> <mo>−</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>M</mi> <msup> <mi>δ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}q(x)&=\lim _{d\to 0}{\Big (}F\delta (x)-F\delta (x-d){\Big )}\\[4pt]&=\lim _{d\to 0}\left({\frac {M}{d}}\delta (x)-{\frac {M}{d}}\delta (x-d)\right)\\[4pt]&=M\lim _{d\to 0}{\frac {\delta (x)-\delta (x-d)}{d}}\\[4pt]&=M\delta '(x).\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd72d95be9328aeeea207fd8a970c1bdd5304b56" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.782ex; margin-bottom: -0.223ex; width:37.02ex; height:23.176ex;" alt="{\displaystyle {\begin{aligned}q(x)&=\lim _{d\to 0}{\Big (}F\delta (x)-F\delta (x-d){\Big )}\\[4pt]&=\lim _{d\to 0}\left({\frac {M}{d}}\delta (x)-{\frac {M}{d}}\delta (x-d)\right)\\[4pt]&=M\lim _{d\to 0}{\frac {\delta (x)-\delta (x-d)}{d}}\\[4pt]&=M\delta '(x).\end{aligned}}}"> Point moments can thus be represented by the <a href="/wiki/Derivative" title="Derivative">derivative</a> of the delta function. Integration of the beam equation again results in piecewise <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> deflection. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=38" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Atom_(measure_theory)" title="Atom (measure theory)">Atom (measure theory)</a></li> <li><a href="/wiki/Laplacian_of_the_indicator" title="Laplacian of the indicator">Laplacian of the indicator</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=39" title="Edit section: Notes">edit</a>]</div> <div style="clear:both;" class=""></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-FOOTNOTEatis2013unit_impulse-1"><a href="#cite_ref-FOOTNOTEatis2013unit_impulse_1-0">^</a> <a href="#CITEREFatis2013">atis 2013</a>, unit impulse. </li> <li id="cite_note-FOOTNOTEArfkenWeber200084-2"><a href="#cite_ref-FOOTNOTEArfkenWeber200084_2-0">^</a> <a href="#CITEREFArfkenWeber2000">Arfken & Weber 2000</a>, p. 84. </li> <li id="cite_note-FOOTNOTEDirac1930§22_The_''δ''_function-3">^ <a href="#cite_ref-FOOTNOTEDirac1930§22_The_''δ''_function_3-0">a</a> <a href="#cite_ref-FOOTNOTEDirac1930§22_The_''δ''_function_3-1">b</a> <a href="#CITEREFDirac1930">Dirac 1930</a>, §22 The δ function. </li> <li id="cite_note-FOOTNOTEGelfandShilov1966–1968Volume_I,_§1.1-4"><a href="#cite_ref-FOOTNOTEGelfandShilov1966–1968Volume_I,_§1.1_4-0">^</a> <a href="#CITEREFGelfandShilov1966–1968">Gelfand & Shilov 1966–1968</a>, Volume I, §1.1. </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFZhao2011" class="citation book cs1">Zhao, Ji-Cheng (2011-05-05). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=blZYGDREpk8C&pg=PA174">Methods for Phase Diagram Determination</a>. Elsevier. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08-054996-5" title="Special:BookSources/978-0-08-054996-5"><bdi>978-0-08-054996-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Methods+for+Phase+Diagram+Determination&rft.pub=Elsevier&rft.date=2011-05-05&rft.isbn=978-0-08-054996-5&rft.aulast=Zhao&rft.aufirst=Ji-Cheng&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DblZYGDREpk8C%26pg%3DPA174&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988"> </li> <li id="cite_note-Fourier-6"><a href="#cite_ref-Fourier_6-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFourier1822" class="citation book cs1"><a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Fourier, JB</a> (1822). The Analytical Theory of Heat (English translation by Alexander Freeman, 1878 ed.). The University Press. p. <a rel="nofollow" class="external autonumber" href="https://books.google.com/books?id=-N8EAAAAYAAJ&pg=PA408">[1]</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Analytical+Theory+of+Heat&rft.pages=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D-N8EAAAAYAAJ%26pg%3DPA408&rft.edition=English+translation+by+Alexander+Freeman%2C+1878&rft.pub=The+University+Press&rft.date=1822&rft.aulast=Fourier&rft.aufirst=JB&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988">, cf. <a rel="nofollow" class="external free" href="https://books.google.com/books?id=-N8EAAAAYAAJ&pg=PA449">https://books.google.com/books?id=-N8EAAAAYAAJ&pg=PA449</a> and pp. 546–551. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TDQJAAAAIAAJ&pg=PA525">Original French text</a>. </li> <li id="cite_note-Kawai-7"><a href="#cite_ref-Kawai_7-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKomatsu2002" class="citation book cs1">Komatsu, Hikosaburo (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8GwKzEemrIcC">"Fourier's hyperfunctions and Heaviside's pseudodifferential operators"</a>. In <a href="/wiki/Takahiro_Kawai" title="Takahiro Kawai">Takahiro Kawai</a>; Keiko Fujita (eds.). Microlocal Analysis and Complex Fourier Analysis. World Scientific. p. <a rel="nofollow" class="external autonumber" href="https://books.google.com/books?id=8GwKzEemrIcC&pg=PA200">[2]</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-238-161-3" title="Special:BookSources/978-981-238-161-3"><bdi>978-981-238-161-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Fourier%27s+hyperfunctions+and+Heaviside%27s+pseudodifferential+operators&rft.btitle=Microlocal+Analysis+and+Complex+Fourier+Analysis&rft.pages=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D8GwKzEemrIcC%26pg%3DPA200&rft.pub=World+Scientific&rft.date=2002&rft.isbn=978-981-238-161-3&rft.aulast=Komatsu&rft.aufirst=Hikosaburo&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D8GwKzEemrIcC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988"> </li> <li id="cite_note-Myint-U-8"><a href="#cite_ref-Myint-U_8-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMyint-U.Debnath2007" class="citation book cs1">Myint-U., Tyn; <a href="/wiki/Lokenath_Debnath" title="Lokenath Debnath">Debnath, Lokenath</a> (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Zbz5_UvERIIC">Linear Partial Differential Equations for Scientists And Engineers</a> (4th ed.). Springer. p. <a rel="nofollow" class="external autonumber" href="https://books.google.com/books?id=Zbz5_UvERIIC&pg=PA4">[3]</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8176-4393-5" title="Special:BookSources/978-0-8176-4393-5"><bdi>978-0-8176-4393-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Partial+Differential+Equations+for+Scientists+And+Engineers&rft.pages=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DZbz5_UvERIIC%26pg%3DPA4&rft.edition=4th&rft.pub=Springer&rft.date=2007&rft.isbn=978-0-8176-4393-5&rft.aulast=Myint-U.&rft.aufirst=Tyn&rft.au=Debnath%2C+Lokenath&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DZbz5_UvERIIC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988"> </li> <li id="cite_note-Debnath-9"><a href="#cite_ref-Debnath_9-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDebnathBhatta2007" class="citation book cs1">Debnath, Lokenath; Bhatta, Dambaru (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=WbZcqdvCEfwC">Integral Transforms And Their Applications</a> (2nd ed.). <a href="/wiki/CRC_Press" title="CRC Press">CRC Press</a>. p. <a rel="nofollow" class="external autonumber" href="https://books.google.com/books?id=WbZcqdvCEfwC&pg=PA2">[4]</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-58488-575-7" title="Special:BookSources/978-1-58488-575-7"><bdi>978-1-58488-575-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Integral+Transforms+And+Their+Applications&rft.pages=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DWbZcqdvCEfwC%26pg%3DPA2&rft.edition=2nd&rft.pub=CRC+Press&rft.date=2007&rft.isbn=978-1-58488-575-7&rft.aulast=Debnath&rft.aufirst=Lokenath&rft.au=Bhatta%2C+Dambaru&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DWbZcqdvCEfwC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988"> </li> <li id="cite_note-Grattan-Guinness-10"><a href="#cite_ref-Grattan-Guinness_10-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrattan-Guinness2009" class="citation book cs1"><a href="/wiki/Ivor_Grattan-Guinness" title="Ivor Grattan-Guinness">Grattan-Guinness, Ivor</a> (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_GgioErrbW8C">Convolutions in French Mathematics, 1800–1840: From the Calculus and Mechanics to Mathematical Analysis and Mathematical Physics, Volume 2</a>. Birkhäuser. p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_GgioErrbW8C&pg=PA653">653</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-7643-2238-0" title="Special:BookSources/978-3-7643-2238-0"><bdi>978-3-7643-2238-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Convolutions+in+French+Mathematics%2C+1800%E2%80%931840%3A+From+the+Calculus+and+Mechanics+to+Mathematical+Analysis+and+Mathematical+Physics%2C+Volume+2&rft.pages=653&rft.pub=Birkh%C3%A4user&rft.date=2009&rft.isbn=978-3-7643-2238-0&rft.aulast=Grattan-Guinness&rft.aufirst=Ivor&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_GgioErrbW8C&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988"> </li> <li id="cite_note-Cauchy-11"><a href="#cite_ref-Cauchy_11-0">^</a> See, for example, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCauchy1882–1974" class="citation book cs1">Cauchy, Augustin-Louis (1789-1857) Auteur du texte (1882–1974). <a rel="nofollow" class="external text" href="http://gallica.bnf.fr/ark:/12148/bpt6k90181x/f387">"Des intégrales doubles qui se présentent sous une forme indéterminèe"</a>. <a rel="nofollow" class="external text" href="https://gallica.bnf.fr/ark:/12148/bpt6k90181x">Oeuvres complètes d'Augustin Cauchy. Série 1, tome 1 / publiées sous la direction scientifique de l'Académie des sciences et sous les auspices de M. le ministre de l'Instruction publique...</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Des+int%C3%A9grales+doubles+qui+se+pr%C3%A9sentent+sous+une+forme+ind%C3%A9termin%C3%A8e&rft.btitle=Oeuvres+compl%C3%A8tes+d%27Augustin+Cauchy.+S%C3%A9rie+1%2C+tome+1+%2F+publi%C3%A9es+sous+la+direction+scientifique+de+l%27Acad%C3%A9mie+des+sciences+et+sous+les+auspices+de+M.+le+ministre+de+l%27Instruction+publique...&rft.date=1882%2F1974&rft.aulast=Cauchy&rft.aufirst=Augustin-Louis+%281789-1857%29+Auteur+du+texte&rft_id=http%3A%2F%2Fgallica.bnf.fr%2Fark%3A%2F12148%2Fbpt6k90181x%2Ff387&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: numeric names: authors list (<a href="/wiki/Category:CS1_maint:_numeric_names:_authors_list" title="Category:CS1 maint: numeric names: authors list">link</a>) </li> <li id="cite_note-Mitrović-12"><a href="#cite_ref-Mitrović_12-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMitrovićŽubrinić1998" class="citation book cs1">Mitrović, Dragiša; Žubrinić, Darko (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Od5BxTEN0VsC">Fundamentals of Applied Functional Analysis: Distributions, Sobolev Spaces</a>. CRC Press. p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Od5BxTEN0VsC&pg=PA62">62</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-582-24694-2" title="Special:BookSources/978-0-582-24694-2"><bdi>978-0-582-24694-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Applied+Functional+Analysis%3A+Distributions%2C+Sobolev+Spaces&rft.pages=62&rft.pub=CRC+Press&rft.date=1998&rft.isbn=978-0-582-24694-2&rft.aulast=Mitrovi%C4%87&rft.aufirst=Dragi%C5%A1a&rft.au=%C5%BDubrini%C4%87%2C+Darko&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DOd5BxTEN0VsC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988"> </li> <li id="cite_note-Kracht-13"><a href="#cite_ref-Kracht_13-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKrachtKreyszig1989" class="citation book cs1">Kracht, Manfred; <a href="/wiki/Erwin_Kreyszig" title="Erwin Kreyszig">Kreyszig, Erwin</a> (1989). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xIsPrSiDlZIC">"On singular integral operators and generalizations"</a>. In Themistocles M. Rassias (ed.). Topics in Mathematical Analysis: A Volume Dedicated to the Memory of A.L. Cauchy. World Scientific. p. <a rel="nofollow" class="external free" href="https://books.google.com/books?id=xIsPrSiDlZIC&pg=PA553">https://books.google.com/books?id=xIsPrSiDlZIC&pg=PA553</a> 553]. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-9971-5-0666-7" title="Special:BookSources/978-9971-5-0666-7"><bdi>978-9971-5-0666-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=On+singular+integral+operators+and+generalizations&rft.btitle=Topics+in+Mathematical+Analysis%3A+A+Volume+Dedicated+to+the+Memory+of+A.L.+Cauchy&rft.pages=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DxIsPrSiDlZIC%26pg%3DPA553+553&rft.pub=World+Scientific&rft.date=1989&rft.isbn=978-9971-5-0666-7&rft.aulast=Kracht&rft.aufirst=Manfred&rft.au=Kreyszig%2C+Erwin&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DxIsPrSiDlZIC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988"> </li> <li id="cite_note-FOOTNOTELaugwitz1989230-14"><a href="#cite_ref-FOOTNOTELaugwitz1989230_14-0">^</a> <a href="#CITEREFLaugwitz1989">Laugwitz 1989</a>, p. 230. </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> A more complete historical account can be found in <a href="#CITEREFvan_der_PolBremmer1987">van der Pol & Bremmer 1987</a>, §V.4. </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDirac1927" class="citation journal cs1">Dirac, P. A. M. (January 1927). <a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1927.0012">"The physical interpretation of the quantum dynamics"</a>. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. 113 (765): 621–641. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1927RSPSA.113..621D">1927RSPSA.113..621D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1927.0012">10.1098/rspa.1927.0012</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0950-1207">0950-1207</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122855515">122855515</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Society+of+London.+Series+A%2C+Containing+Papers+of+a+Mathematical+and+Physical+Character&rft.atitle=The+physical+interpretation+of+the+quantum+dynamics&rft.volume=113&rft.issue=765&rft.pages=621-641&rft.date=1927-01&rft_id=info%3Adoi%2F10.1098%2Frspa.1927.0012&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122855515%23id-name%3DS2CID&rft.issn=0950-1207&rft_id=info%3Abibcode%2F1927RSPSA.113..621D&rft.aulast=Dirac&rft.aufirst=P.+A.+M.&rft_id=https%3A%2F%2Fdoi.org%2F10.1098%252Frspa.1927.0012&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988"> </li> <li id="cite_note-FOOTNOTEGelfandShilov1966–1968Volume_I,_§1.1,_p._1-17"><a href="#cite_ref-FOOTNOTEGelfandShilov1966–1968Volume_I,_§1.1,_p._1_17-0">^</a> <a href="#CITEREFGelfandShilov1966–1968">Gelfand & Shilov 1966–1968</a>, Volume I, §1.1, p. 1. </li> <li id="cite_note-FOOTNOTEDirac193063-18"><a href="#cite_ref-FOOTNOTEDirac193063_18-0">^</a> <a href="#CITEREFDirac1930">Dirac 1930</a>, p. 63. </li> <li id="cite_note-Rudin_1966_loc=§1.20-19"><a href="#cite_ref-Rudin_1966_loc=§1.20_19-0">^</a> <a href="#CITEREFRudin1966">Rudin 1966</a>, §1.20 </li> <li id="cite_note-FOOTNOTEHewittStromberg1963§19.61-20"><a href="#cite_ref-FOOTNOTEHewittStromberg1963§19.61_20-0">^</a> <a href="#CITEREFHewittStromberg1963">Hewitt & Stromberg 1963</a>, §19.61. </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> <a href="#CITEREFDriggers2003">Driggers 2003</a>, p. 2321 See also <a href="#CITEREFBracewell1986">Bracewell 1986</a>, Chapter 5 for a different interpretation. Other conventions for the assigning the value of the Heaviside function at zero exist, and some of these are not consistent with what follows. </li> <li id="cite_note-FOOTNOTEHewittStromberg1963§9.19-22"><a href="#cite_ref-FOOTNOTEHewittStromberg1963§9.19_22-0">^</a> <a href="#CITEREFHewittStromberg1963">Hewitt & Stromberg 1963</a>, §9.19. </li> <li id="cite_note-FOOTNOTEHazewinkel2011[httpsbooksgooglecombooksid_YPtCAAAQBAJpgPA41_41]-23"><a href="#cite_ref-FOOTNOTEHazewinkel2011[httpsbooksgooglecombooksid_YPtCAAAQBAJpgPA41_41]_23-0">^</a> <a href="#CITEREFHazewinkel2011">Hazewinkel 2011</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_YPtCAAAQBAJ&pg=PA41">41</a>. </li> <li id="cite_note-FOOTNOTEStrichartz1994§2.2-24"><a href="#cite_ref-FOOTNOTEStrichartz1994§2.2_24-0">^</a> <a href="#CITEREFStrichartz1994">Strichartz 1994</a>, §2.2. </li> <li id="cite_note-FOOTNOTEHörmander1983Theorem_2.1.5-25"><a href="#cite_ref-FOOTNOTEHörmander1983Theorem_2.1.5_25-0">^</a> <a href="#CITEREFHörmander1983">Hörmander 1983</a>, Theorem 2.1.5. </li> <li id="cite_note-FOOTNOTEBracewell1986Chapter_5-26"><a href="#cite_ref-FOOTNOTEBracewell1986Chapter_5_26-0">^</a> <a href="#CITEREFBracewell1986">Bracewell 1986</a>, Chapter 5. </li> <li id="cite_note-FOOTNOTEHörmander1983§3.1-27"><a href="#cite_ref-FOOTNOTEHörmander1983§3.1_27-0">^</a> <a href="#CITEREFHörmander1983">Hörmander 1983</a>, §3.1. </li> <li id="cite_note-FOOTNOTEStrichartz1994§2.3-28"><a href="#cite_ref-FOOTNOTEStrichartz1994§2.3_28-0">^</a> <a href="#CITEREFStrichartz1994">Strichartz 1994</a>, §2.3. </li> <li id="cite_note-FOOTNOTEHörmander1983§8.2-29"><a href="#cite_ref-FOOTNOTEHörmander1983§8.2_29-0">^</a> <a href="#CITEREFHörmander1983">Hörmander 1983</a>, §8.2. </li> <li id="cite_note-FOOTNOTERudin1966§1.20-30"><a href="#cite_ref-FOOTNOTERudin1966§1.20_30-0">^</a> <a href="#CITEREFRudin1966">Rudin 1966</a>, §1.20. </li> <li id="cite_note-FOOTNOTEDieudonné1972§17.3.3-31"><a href="#cite_ref-FOOTNOTEDieudonné1972§17.3.3_31-0">^</a> <a href="#CITEREFDieudonné1972">Dieudonné 1972</a>, §17.3.3. </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKrantzParks2008" class="citation book cs1">Krantz, Steven G.; Parks, Harold R. (2008-12-15). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=X_BKmVphIcsC&q">Geometric Integration Theory</a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8176-4679-0" title="Special:BookSources/978-0-8176-4679-0"><bdi>978-0-8176-4679-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometric+Integration+Theory&rft.pub=Springer+Science+%26+Business+Media&rft.date=2008-12-15&rft.isbn=978-0-8176-4679-0&rft.aulast=Krantz&rft.aufirst=Steven+G.&rft.au=Parks%2C+Harold+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DX_BKmVphIcsC%26q&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988"> </li> <li id="cite_note-FOOTNOTEFederer1969§2.5.19-33"><a href="#cite_ref-FOOTNOTEFederer1969§2.5.19_33-0">^</a> <a href="#CITEREFFederer1969">Federer 1969</a>, §2.5.19. </li> <li id="cite_note-FOOTNOTEStrichartz1994Problem_2.6.2-34"><a href="#cite_ref-FOOTNOTEStrichartz1994Problem_2.6.2_34-0">^</a> <a href="#CITEREFStrichartz1994">Strichartz 1994</a>, Problem 2.6.2. </li> <li id="cite_note-FOOTNOTEVladimirov1971Chapter_2,_Example_3(d)-35"><a href="#cite_ref-FOOTNOTEVladimirov1971Chapter_2,_Example_3(d)_35-0">^</a> <a href="#CITEREFVladimirov1971">Vladimirov 1971</a>, Chapter 2, Example 3(d). </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/SiftingProperty.html">"Sifting Property"</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Sifting+Property&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FSiftingProperty.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988"> </li> <li id="cite_note-37"><a href="#cite_ref-37">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKarris2003" class="citation book cs1">Karris, Steven T. (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=f0RdM1zv_dkC">Signals and Systems with MATLAB Applications</a>. Orchard Publications. p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=f0RdM1zv_dkC&pg=SA1-PA15">15</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-9709511-6-8" title="Special:BookSources/978-0-9709511-6-8"><bdi>978-0-9709511-6-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Signals+and+Systems+with+MATLAB+Applications&rft.pages=15&rft.pub=Orchard+Publications&rft.date=2003&rft.isbn=978-0-9709511-6-8&rft.aulast=Karris&rft.aufirst=Steven+T.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Df0RdM1zv_dkC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988"> </li> <li id="cite_note-38"><a href="#cite_ref-38">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRoden2014" class="citation book cs1">Roden, Martin S. (2014-05-17). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YEKeBQAAQBAJ">Introduction to Communication Theory</a>. Elsevier. p. <a rel="nofollow" class="external autonumber" href="https://books.google.com/books?id=YEKeBQAAQBAJ&pg=PA40">[5]</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4831-4556-3" title="Special:BookSources/978-1-4831-4556-3"><bdi>978-1-4831-4556-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Communication+Theory&rft.pages=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DYEKeBQAAQBAJ%26pg%3DPA40&rft.pub=Elsevier&rft.date=2014-05-17&rft.isbn=978-1-4831-4556-3&rft.aulast=Roden&rft.aufirst=Martin+S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DYEKeBQAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988"> </li> <li id="cite_note-39"><a href="#cite_ref-39">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRottwittTidemand-Lichtenberg2014" class="citation book cs1">Rottwitt, Karsten; Tidemand-Lichtenberg, Peter (2014-12-11). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=G1jSBQAAQBAJ">Nonlinear Optics: Principles and Applications</a>. CRC Press. p. <a rel="nofollow" class="external autonumber" href="https://books.google.com/books?id=G1jSBQAAQBAJ&pg=PA276">[6]</a> 276. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4665-6583-8" title="Special:BookSources/978-1-4665-6583-8"><bdi>978-1-4665-6583-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Nonlinear+Optics%3A+Principles+and+Applications&rft.pages=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DG1jSBQAAQBAJ%26pg%3DPA276+276&rft.pub=CRC+Press&rft.date=2014-12-11&rft.isbn=978-1-4665-6583-8&rft.aulast=Rottwitt&rft.aufirst=Karsten&rft.au=Tidemand-Lichtenberg%2C+Peter&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DG1jSBQAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988"> </li> <li id="cite_note-FOOTNOTEGelfandShilov1966–1968Vol._1,_§II.2.5-40"><a href="#cite_ref-FOOTNOTEGelfandShilov1966–1968Vol._1,_§II.2.5_40-0">^</a> <a href="#CITEREFGelfandShilov1966–1968">Gelfand & Shilov 1966–1968</a>, Vol. 1, §II.2.5. </li> <li id="cite_note-41"><a href="#cite_ref-41">^</a> Further refinement is possible, namely to <a href="/wiki/Submersion_(mathematics)" title="Submersion (mathematics)">submersions</a>, although these require a more involved change of variables formula. </li> <li id="cite_note-FOOTNOTEHörmander1983§6.1-42"><a href="#cite_ref-FOOTNOTEHörmander1983§6.1_42-0">^</a> <a href="#CITEREFHörmander1983">Hörmander 1983</a>, §6.1. </li> <li id="cite_note-FOOTNOTELange2012pp.29–30-43"><a href="#cite_ref-FOOTNOTELange2012pp.29–30_43-0">^</a> <a href="#CITEREFLange2012">Lange 2012</a>, pp.29–30. </li> <li id="cite_note-FOOTNOTEGelfandShilov1966–1968212-44"><a href="#cite_ref-FOOTNOTEGelfandShilov1966–1968212_44-0">^</a> <a href="#CITEREFGelfandShilov1966–1968">Gelfand & Shilov 1966–1968</a>, p. 212. </li> <li id="cite_note-45"><a href="#cite_ref-45">^</a> The numerical factors depend on the <a href="/wiki/Fourier_transform#Other_conventions" title="Fourier transform">conventions</a> for the Fourier transform. </li> <li id="cite_note-FOOTNOTEBracewell1986-46"><a href="#cite_ref-FOOTNOTEBracewell1986_46-0">^</a> <a href="#CITEREFBracewell1986">Bracewell 1986</a>. </li> <li id="cite_note-FOOTNOTEGelfandShilov1966–196826-47"><a href="#cite_ref-FOOTNOTEGelfandShilov1966–196826_47-0">^</a> <a href="#CITEREFGelfandShilov1966–1968">Gelfand & Shilov 1966–1968</a>, p. 26. </li> <li id="cite_note-FOOTNOTEGelfandShilov1966–1968§2.1-48"><a href="#cite_ref-FOOTNOTEGelfandShilov1966–1968§2.1_48-0">^</a> <a href="#CITEREFGelfandShilov1966–1968">Gelfand & Shilov 1966–1968</a>, §2.1. </li> <li id="cite_note-49"><a href="#cite_ref-49">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/DoubletFunction.html">"Doublet Function"</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Doublet+Function&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FDoubletFunction.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988"> </li> <li id="cite_note-FOOTNOTEBracewell200086-50"><a href="#cite_ref-FOOTNOTEBracewell200086_50-0">^</a> <a href="#CITEREFBracewell2000">Bracewell 2000</a>, p. 86. </li> <li id="cite_note-51"><a href="#cite_ref-51">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.matematicamente.it/forum/viewtopic.php?f=36&t=62388&start=10#wrap">"Gugo82's comment on the distributional derivative of Dirac's delta"</a>. matematicamente.it. 12 September 2010.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=matematicamente.it&rft.atitle=Gugo82%27s+comment+on+the+distributional+derivative+of+Dirac%27s+delta&rft.date=2010-09-12&rft_id=https%3A%2F%2Fwww.matematicamente.it%2Fforum%2Fviewtopic.php%3Ff%3D36%26t%3D62388%26start%3D10%23wrap&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988"> </li> <li id="cite_note-FOOTNOTEHörmander198356-52">^ <a href="#cite_ref-FOOTNOTEHörmander198356_52-0">a</a> <a href="#cite_ref-FOOTNOTEHörmander198356_52-1">b</a> <a href="#cite_ref-FOOTNOTEHörmander198356_52-2">c</a> <a href="#CITEREFHörmander1983">Hörmander 1983</a>, p. 56. </li> <li id="cite_note-FOOTNOTERudin1991Theorem_6.25-53"><a href="#cite_ref-FOOTNOTERudin1991Theorem_6.25_53-0">^</a> <a href="#CITEREFRudin1991">Rudin 1991</a>, Theorem 6.25. </li> <li id="cite_note-FOOTNOTESteinWeiss1971Theorem_1.18-54"><a href="#cite_ref-FOOTNOTESteinWeiss1971Theorem_1.18_54-0">^</a> <a href="#CITEREFSteinWeiss1971">Stein & Weiss 1971</a>, Theorem 1.18. </li> <li id="cite_note-FOOTNOTERudin1991§II.6.31-55"><a href="#cite_ref-FOOTNOTERudin1991§II.6.31_55-0">^</a> <a href="#CITEREFRudin1991">Rudin 1991</a>, §II.6.31. </li> <li id="cite_note-56"><a href="#cite_ref-56">^</a> More generally, one only needs η = η1 to have an integrable radially symmetric decreasing rearrangement. </li> <li id="cite_note-FOOTNOTESaichevWoyczyński1997§1.1_The_"delta_function"_as_viewed_by_a_physicist_and_an_engineer,_p._3-57"><a href="#cite_ref-FOOTNOTESaichevWoyczyński1997§1.1_The_"delta_function"_as_viewed_by_a_physicist_and_an_engineer,_p._3_57-0">^</a> <a href="#CITEREFSaichevWoyczyński1997">Saichev & Woyczyński 1997</a>, §1.1 The "delta function" as viewed by a physicist and an engineer, p. 3. </li> <li id="cite_note-58"><a href="#cite_ref-58">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMilovanovićRassias2014" class="citation book cs1">Milovanović, Gradimir V.; Rassias, Michael Th (2014-07-08). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4U-5BQAAQBAJ">Analytic Number Theory, Approximation Theory, and Special Functions: In Honor of Hari M. Srivastava</a>. Springer. p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4U-5BQAAQBAJ&pg=PA748">748</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4939-0258-3" title="Special:BookSources/978-1-4939-0258-3"><bdi>978-1-4939-0258-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analytic+Number+Theory%2C+Approximation+Theory%2C+and+Special+Functions%3A+In+Honor+of+Hari+M.+Srivastava&rft.pages=748&rft.pub=Springer&rft.date=2014-07-08&rft.isbn=978-1-4939-0258-3&rft.aulast=Milovanovi%C4%87&rft.aufirst=Gradimir+V.&rft.au=Rassias%2C+Michael+Th&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D4U-5BQAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988"> </li> <li id="cite_note-FOOTNOTESteinWeiss1971§I.1-59"><a href="#cite_ref-FOOTNOTESteinWeiss1971§I.1_59-0">^</a> <a href="#CITEREFSteinWeiss1971">Stein & Weiss 1971</a>, §I.1. </li> <li id="cite_note-60"><a href="#cite_ref-60">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMader2006" class="citation book cs1">Mader, Heidy M. (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=e5Y_RRPxdyYC">Statistics in Volcanology</a>. Geological Society of London. p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=e5Y_RRPxdyYC&pg=PA81">81</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-86239-208-3" title="Special:BookSources/978-1-86239-208-3"><bdi>978-1-86239-208-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistics+in+Volcanology&rft.pages=81&rft.pub=Geological+Society+of+London&rft.date=2006&rft.isbn=978-1-86239-208-3&rft.aulast=Mader&rft.aufirst=Heidy+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3De5Y_RRPxdyYC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988"> </li> <li id="cite_note-FOOTNOTEValléeSoares2004§7.2-61"><a href="#cite_ref-FOOTNOTEValléeSoares2004§7.2_61-0">^</a> <a href="#CITEREFValléeSoares2004">Vallée & Soares 2004</a>, §7.2. </li> <li id="cite_note-FOOTNOTEHörmander1983§7.8-62"><a href="#cite_ref-FOOTNOTEHörmander1983§7.8_62-0">^</a> <a href="#CITEREFHörmander1983">Hörmander 1983</a>, §7.8. </li> <li id="cite_note-FOOTNOTECourantHilbert1962§14-63"><a href="#cite_ref-FOOTNOTECourantHilbert1962§14_63-0">^</a> <a href="#CITEREFCourantHilbert1962">Courant & Hilbert 1962</a>, §14. </li> <li id="cite_note-FOOTNOTEGelfandShilov1966–1968I,_§3.10-64"><a href="#cite_ref-FOOTNOTEGelfandShilov1966–1968I,_§3.10_64-0">^</a> <a href="#CITEREFGelfandShilov1966–1968">Gelfand & Shilov 1966–1968</a>, I, §3.10. </li> <li id="cite_note-FOOTNOTELang1997312-65"><a href="#cite_ref-FOOTNOTELang1997312_65-0">^</a> <a href="#CITEREFLang1997">Lang 1997</a>, p. 312. </li> <li id="cite_note-66"><a href="#cite_ref-66">^</a> In the terminology of <a href="#CITEREFLang1997">Lang (1997)</a>, the Fejér kernel is a Dirac sequence, whereas the Dirichlet kernel is not. </li> <li id="cite_note-FOOTNOTEHazewinkel1995[httpsbooksgooglecombooksidPE1a-EIG22kCpgPA357_357]-67"><a href="#cite_ref-FOOTNOTEHazewinkel1995[httpsbooksgooglecombooksidPE1a-EIG22kCpgPA357_357]_67-0">^</a> <a href="#CITEREFHazewinkel1995">Hazewinkel 1995</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PE1a-EIG22kC&pg=PA357">357</a>. </li> <li id="cite_note-68"><a href="#cite_ref-68">^</a> The development of this section in bra–ket notation is found in (<a href="#CITEREFLevin2002">Levin 2002</a>, Coordinate-space wave functions and completeness, pp.=109ff) </li> <li id="cite_note-FOOTNOTEDavisThomson2000Perfect_operators,_p.344-69"><a href="#cite_ref-FOOTNOTEDavisThomson2000Perfect_operators,_p.344_69-0">^</a> <a href="#CITEREFDavisThomson2000">Davis & Thomson 2000</a>, Perfect operators, p.344. </li> <li id="cite_note-FOOTNOTEDavisThomson2000Equation_8.9.11,_p._344-70"><a href="#cite_ref-FOOTNOTEDavisThomson2000Equation_8.9.11,_p._344_70-0">^</a> <a href="#CITEREFDavisThomson2000">Davis & Thomson 2000</a>, Equation 8.9.11, p. 344. </li> <li id="cite_note-FOOTNOTEde_la_MadridBohmGadella2002-71"><a href="#cite_ref-FOOTNOTEde_la_MadridBohmGadella2002_71-0">^</a> <a href="#CITEREFde_la_MadridBohmGadella2002">de la Madrid, Bohm & Gadella 2002</a>. </li> <li id="cite_note-FOOTNOTELaugwitz1989-72"><a href="#cite_ref-FOOTNOTELaugwitz1989_72-0">^</a> <a href="#CITEREFLaugwitz1989">Laugwitz 1989</a>. </li> <li id="cite_note-FOOTNOTECórdoba1988-73"><a href="#cite_ref-FOOTNOTECórdoba1988_73-0">^</a> <a href="#CITEREFCórdoba1988">Córdoba 1988</a>. </li> <li id="cite_note-FOOTNOTEHörmander1983[httpsbooksgooglecombooksidaaLrCAAAQBAJpgPA177_§7.2]-74"><a href="#cite_ref-FOOTNOTEHörmander1983[httpsbooksgooglecombooksidaaLrCAAAQBAJpgPA177_§7.2]_74-0">^</a> <a href="#CITEREFHörmander1983">Hörmander 1983</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aaLrCAAAQBAJ&pg=PA177">§7.2</a>. </li> <li id="cite_note-FOOTNOTEVladimirov1971§5.7-75"><a href="#cite_ref-FOOTNOTEVladimirov1971§5.7_75-0">^</a> <a href="#CITEREFVladimirov1971">Vladimirov 1971</a>, §5.7. </li> <li id="cite_note-FOOTNOTEHartmann1997pp._154–155-76"><a href="#cite_ref-FOOTNOTEHartmann1997pp._154–155_76-0">^</a> <a href="#CITEREFHartmann1997">Hartmann 1997</a>, pp. 154–155. </li> <li id="cite_note-FOOTNOTEIsham1995§6.2-77"><a href="#cite_ref-FOOTNOTEIsham1995§6.2_77-0">^</a> <a href="#CITEREFIsham1995">Isham 1995</a>, §6.2. </li> <li id="cite_note-FOOTNOTEde_la_Madrid_Modino200196,_106-78"><a href="#cite_ref-FOOTNOTEde_la_Madrid_Modino200196,_106_78-0">^</a> <a href="#CITEREFde_la_Madrid_Modino2001">de la Madrid Modino 2001</a>, pp. 96, 106. </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=40" title="Edit section: References">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAratynRasinariu2006" class="citation cs2">Aratyn, Henrik; 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Arfken">Arfken, G. B.</a>; Weber, H. J. 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New York: McGraw-Hill (published 1987). <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-100276-6" title="Special:BookSources/0-07-100276-6"><bdi>0-07-100276-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Real+and+complex+analysis&rft.place=New+York&rft.edition=3rd&rft.pub=McGraw-Hill&rft.date=1966&rft.isbn=0-07-100276-6&rft.aulast=Rudin&rft.aufirst=Walter&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRudin1991" class="citation cs2"><a href="/wiki/Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1991), <a rel="nofollow" class="external text" href="https://archive.org/details/functionalanalys00rudi">Functional Analysis</a> (2nd ed.), McGraw-Hill, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-054236-5" title="Special:BookSources/978-0-07-054236-5"><bdi>978-0-07-054236-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Functional+Analysis&rft.edition=2nd&rft.pub=McGraw-Hill&rft.date=1991&rft.isbn=978-0-07-054236-5&rft.aulast=Rudin&rft.aufirst=Walter&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffunctionalanalys00rudi&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFValléeSoares2004" class="citation cs2">Vallée, Olivier; Soares, Manuel (2004), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8M42DwAAQBAJ">Airy functions and applications to physics</a>, London: Imperial College Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781911299486" title="Special:BookSources/9781911299486"><bdi>9781911299486</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Airy+functions+and+applications+to+physics&rft.place=London&rft.pub=Imperial+College+Press&rft.date=2004&rft.isbn=9781911299486&rft.aulast=Vall%C3%A9e&rft.aufirst=Olivier&rft.au=Soares%2C+Manuel&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D8M42DwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSaichevWoyczyński1997" class="citation cs2">Saichev, A I; Woyczyński, Wojbor Andrzej (1997), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=42I7huO-hiYC&pg=PA3">"Chapter1: Basic definitions and operations"</a>, Distributions in the Physical and Engineering Sciences: Distributional and fractal calculus, integral transforms, and wavelets, Birkhäuser, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8176-3924-2" title="Special:BookSources/978-0-8176-3924-2"><bdi>978-0-8176-3924-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter1%3A+Basic+definitions+and+operations&rft.btitle=Distributions+in+the+Physical+and+Engineering+Sciences%3A+Distributional+and+fractal+calculus%2C+integral+transforms%2C+and+wavelets&rft.pub=Birkh%C3%A4user&rft.date=1997&rft.isbn=978-0-8176-3924-2&rft.aulast=Saichev&rft.aufirst=A+I&rft.au=Woyczy%C5%84ski%2C+Wojbor+Andrzej&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D42I7huO-hiYC%26pg%3DPA3&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchwartz1950" class="citation cs2"><a href="/wiki/Laurent_Schwartz" title="Laurent Schwartz">Schwartz, L.</a> (1950), Théorie des distributions, vol. 1, Hermann</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Th%C3%A9orie+des+distributions&rft.pub=Hermann&rft.date=1950&rft.aulast=Schwartz&rft.aufirst=L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchwartz1951" class="citation cs2"><a href="/wiki/Laurent_Schwartz" title="Laurent Schwartz">Schwartz, L.</a> (1951), Théorie des distributions, vol. 2, Hermann</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Th%C3%A9orie+des+distributions&rft.pub=Hermann&rft.date=1951&rft.aulast=Schwartz&rft.aufirst=L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteinWeiss1971" class="citation cs2"><a href="/wiki/Elias_Stein" class="mw-redirect" title="Elias Stein">Stein, Elias</a>; Weiss, Guido (1971), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xnIwDAAAQBAJ">Introduction to Fourier Analysis on Euclidean Spaces</a>, Princeton University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-08078-9" title="Special:BookSources/978-0-691-08078-9"><bdi>978-0-691-08078-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Fourier+Analysis+on+Euclidean+Spaces&rft.pub=Princeton+University+Press&rft.date=1971&rft.isbn=978-0-691-08078-9&rft.aulast=Stein&rft.aufirst=Elias&rft.au=Weiss%2C+Guido&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DxnIwDAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStrichartz1994" class="citation cs2"><a href="/wiki/Robert_Strichartz" title="Robert Strichartz">Strichartz, R.</a> (1994), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=T7vEOGGDCh4C">A Guide to Distribution Theory and Fourier Transforms</a>, CRC Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8493-8273-4" title="Special:BookSources/978-0-8493-8273-4"><bdi>978-0-8493-8273-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Guide+to+Distribution+Theory+and+Fourier+Transforms&rft.pub=CRC+Press&rft.date=1994&rft.isbn=978-0-8493-8273-4&rft.aulast=Strichartz&rft.aufirst=R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DT7vEOGGDCh4C&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVladimirov1971" class="citation cs2">Vladimirov, V. S. (1971), Equations of mathematical physics, Marcel Dekker, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8247-1713-1" title="Special:BookSources/978-0-8247-1713-1"><bdi>978-0-8247-1713-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Equations+of+mathematical+physics&rft.pub=Marcel+Dekker&rft.date=1971&rft.isbn=978-0-8247-1713-1&rft.aulast=Vladimirov&rft.aufirst=V.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/DeltaFunction.html">"Delta Function"</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Delta+Function&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FDeltaFunction.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYamashita2006" class="citation cs2">Yamashita, H. (2006), "Pointwise analysis of scalar fields: A nonstandard approach", <a href="/wiki/Journal_of_Mathematical_Physics" title="Journal of Mathematical Physics">Journal of Mathematical Physics</a>, 47 (9): 092301, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006JMP....47i2301Y">2006JMP....47i2301Y</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.2339017">10.1063/1.2339017</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Mathematical+Physics&rft.atitle=Pointwise+analysis+of+scalar+fields%3A+A+nonstandard+approach&rft.volume=47&rft.issue=9&rft.pages=092301&rft.date=2006&rft_id=info%3Adoi%2F10.1063%2F1.2339017&rft_id=info%3Abibcode%2F2006JMP....47i2301Y&rft.aulast=Yamashita&rft.aufirst=H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYamashita2007" class="citation cs2">Yamashita, H. (2007), "Comment on "Pointwise analysis of scalar fields: A nonstandard approach" [J. Math. Phys. 47, 092301 (2006)]", <a href="/wiki/Journal_of_Mathematical_Physics" title="Journal of Mathematical Physics">Journal of Mathematical Physics</a>, 48 (8): 084101, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007JMP....48h4101Y">2007JMP....48h4101Y</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.2771422">10.1063/1.2771422</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Mathematical+Physics&rft.atitle=Comment+on+%22Pointwise+analysis+of+scalar+fields%3A+A+nonstandard+approach%22+%5BJ.+Math.+Phys.+47%2C+092301+%282006%29%5D&rft.volume=48&rft.issue=8&rft.pages=084101&rft.date=2007&rft_id=info%3Adoi%2F10.1063%2F1.2771422&rft_id=info%3Abibcode%2F2007JMP....48h4101Y&rft.aulast=Yamashita&rft.aufirst=H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Dirac_delta_function&action=edit&section=41" title="Edit section: External links">edit</a>]</div> <ul><li><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Dirac_distribution" class="extiw" title="commons:Category:Dirac distribution">Dirac distribution</a> at Wikimedia Commons</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Delta-function">"Delta-function"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Delta-function&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DDelta-function&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirac+delta+function" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="https://www.khanacademy.org/video/dirac-delta-function">KhanAcademy.org video lesson</a></li> <li><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><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="/wiki/Arthur_Mattuck" title="Arthur Mattuck">Arthur Mattuck</a>.</li> <li><a rel="nofollow" class="external text" href="http://www.osaka-kyoiku.ac.jp/~ashino/pdf/chinaproceedings.pdf">The Dirac delta measure is a hyperfunction</a></li> <li><a rel="nofollow" class="external text" href="http://www.ing-mat.udec.cl/~rodolfo/Papers/BGR-3.pdf">We show the existence of a unique solution and analyze a finite element approximation when the source term is a Dirac delta measure</a></li> <li><a rel="nofollow" class="external text" href="http://www.mathematik.uni-muenchen.de/~lerdos/WS04/FA/content.html">Non-Lebesgue measures on R. Lebesgue-Stieltjes measure, Dirac delta measure.</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080307221128/http://www.mathematik.uni-muenchen.de/~lerdos/WS04/FA/content.html">Archived</a> 2008-03-07 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 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href="mw-data:TemplateStyles:r886047488"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"></div><div role="navigation" class="navbox" aria-labelledby="Probability_distributions_(list)" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Probability_distributions" title="Template:Probability 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style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">with finite support</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Benford%27s_law" title="Benford's law">Benford</a></li> <li><a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli</a></li> <li><a href="/wiki/Beta-binomial_distribution" title="Beta-binomial distribution">Beta-binomial</a></li> <li><a href="/wiki/Binomial_distribution" title="Binomial distribution">Binomial</a></li> <li><a href="/wiki/Categorical_distribution" title="Categorical distribution">Categorical</a></li> <li><a href="/wiki/Hypergeometric_distribution" title="Hypergeometric distribution">Hypergeometric</a> <ul><li><a href="/wiki/Negative_hypergeometric_distribution" title="Negative hypergeometric distribution">Negative</a></li></ul></li> <li><a href="/wiki/Poisson_binomial_distribution" title="Poisson binomial distribution">Poisson binomial</a></li> <li><a href="/wiki/Rademacher_distribution" title="Rademacher distribution">Rademacher</a></li> <li><a href="/wiki/Soliton_distribution" title="Soliton distribution">Soliton</a></li> <li><a href="/wiki/Discrete_uniform_distribution" title="Discrete uniform distribution">Discrete uniform</a></li> <li><a href="/wiki/Zipf%27s_law" title="Zipf's law">Zipf</a></li> <li><a href="/wiki/Zipf%E2%80%93Mandelbrot_law" title="Zipf–Mandelbrot law">Zipf–Mandelbrot</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">with infinite support</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Beta_negative_binomial_distribution" title="Beta negative binomial distribution">Beta negative binomial</a></li> <li><a href="/wiki/Borel_distribution" title="Borel distribution">Borel</a></li> <li><a href="/wiki/Conway%E2%80%93Maxwell%E2%80%93Poisson_distribution" title="Conway–Maxwell–Poisson distribution">Conway–Maxwell–Poisson</a></li> <li><a href="/wiki/Discrete_phase-type_distribution" title="Discrete phase-type distribution">Discrete phase-type</a></li> <li><a href="/wiki/Delaporte_distribution" title="Delaporte distribution">Delaporte</a></li> <li><a href="/wiki/Extended_negative_binomial_distribution" title="Extended negative binomial distribution">Extended negative binomial</a></li> <li><a href="/wiki/Flory%E2%80%93Schulz_distribution" title="Flory–Schulz distribution">Flory–Schulz</a></li> <li><a href="/wiki/Gauss%E2%80%93Kuzmin_distribution" title="Gauss–Kuzmin distribution">Gauss–Kuzmin</a></li> <li><a href="/wiki/Geometric_distribution" title="Geometric distribution">Geometric</a></li> <li><a href="/wiki/Logarithmic_distribution" title="Logarithmic distribution">Logarithmic</a></li> <li><a href="/wiki/Mixed_Poisson_distribution" title="Mixed Poisson distribution">Mixed Poisson</a></li> <li><a href="/wiki/Negative_binomial_distribution" title="Negative binomial distribution">Negative binomial</a></li> <li><a href="/wiki/(a,b,0)_class_of_distributions" title="(a,b,0) class of distributions">Panjer</a></li> <li><a href="/wiki/Parabolic_fractal_distribution" title="Parabolic fractal distribution">Parabolic fractal</a></li> <li><a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson</a></li> <li><a href="/wiki/Skellam_distribution" title="Skellam distribution">Skellam</a></li> <li><a href="/wiki/Yule%E2%80%93Simon_distribution" title="Yule–Simon distribution">Yule–Simon</a></li> <li><a href="/wiki/Zeta_distribution" title="Zeta distribution">Zeta</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Continuous univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">supported on a bounded interval</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arcsine_distribution" title="Arcsine distribution">Arcsine</a></li> <li><a href="/wiki/ARGUS_distribution" title="ARGUS distribution">ARGUS</a></li> <li><a href="/wiki/Balding%E2%80%93Nichols_model" title="Balding–Nichols model">Balding–Nichols</a></li> <li><a href="/wiki/Bates_distribution" title="Bates distribution">Bates</a></li> <li><a href="/wiki/Beta_distribution" title="Beta distribution">Beta</a> <ul><li><a href="/wiki/Generalized_beta_distribution" title="Generalized beta distribution">Generalized</a></li></ul></li> <li><a href="/wiki/Beta_rectangular_distribution" title="Beta rectangular distribution">Beta rectangular</a></li> <li><a href="/wiki/Continuous_Bernoulli_distribution" title="Continuous Bernoulli distribution">Continuous Bernoulli</a></li> <li><a href="/wiki/Irwin%E2%80%93Hall_distribution" title="Irwin–Hall distribution">Irwin–Hall</a></li> <li><a href="/wiki/Kumaraswamy_distribution" title="Kumaraswamy distribution">Kumaraswamy</a></li> <li><a href="/wiki/Logit-normal_distribution" title="Logit-normal distribution">Logit-normal</a></li> <li><a href="/wiki/Noncentral_beta_distribution" title="Noncentral beta distribution">Noncentral beta</a></li> <li><a href="/wiki/PERT_distribution" title="PERT distribution">PERT</a></li> <li><a href="/wiki/Raised_cosine_distribution" title="Raised cosine distribution">Raised cosine</a></li> <li><a href="/wiki/Reciprocal_distribution" title="Reciprocal distribution">Reciprocal</a></li> <li><a href="/wiki/Triangular_distribution" title="Triangular distribution">Triangular</a></li> <li><a href="/wiki/U-quadratic_distribution" title="U-quadratic distribution">U-quadratic</a></li> <li><a href="/wiki/Continuous_uniform_distribution" title="Continuous uniform distribution">Uniform</a></li> <li><a href="/wiki/Wigner_semicircle_distribution" title="Wigner semicircle distribution">Wigner semicircle</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">supported on a semi-infinite interval</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Benini_distribution" title="Benini distribution">Benini</a></li> <li><a href="/wiki/Benktander_type_I_distribution" title="Benktander type I distribution">Benktander 1st kind</a></li> <li><a href="/wiki/Benktander_type_II_distribution" title="Benktander type II distribution">Benktander 2nd kind</a></li> <li><a href="/wiki/Beta_prime_distribution" title="Beta prime distribution">Beta prime</a></li> <li><a href="/wiki/Burr_distribution" title="Burr distribution">Burr</a></li> <li><a href="/wiki/Chi_distribution" title="Chi distribution">Chi</a></li> <li><a href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">Chi-squared</a> <ul><li><a href="/wiki/Noncentral_chi-squared_distribution" title="Noncentral chi-squared distribution">Noncentral</a></li> <li><a href="/wiki/Inverse-chi-squared_distribution" title="Inverse-chi-squared distribution">Inverse</a> <ul><li><a href="/wiki/Scaled_inverse_chi-squared_distribution" title="Scaled inverse chi-squared distribution">Scaled</a></li></ul></li></ul></li> <li><a href="/wiki/Dagum_distribution" title="Dagum distribution">Dagum</a></li> <li><a href="/wiki/Davis_distribution" title="Davis distribution">Davis</a></li> <li><a href="/wiki/Erlang_distribution" title="Erlang distribution">Erlang</a> <ul><li><a href="/wiki/Hyper-Erlang_distribution" title="Hyper-Erlang distribution">Hyper</a></li></ul></li> <li><a href="/wiki/Exponential_distribution" title="Exponential distribution">Exponential</a> <ul><li><a href="/wiki/Hyperexponential_distribution" title="Hyperexponential distribution">Hyperexponential</a></li> <li><a href="/wiki/Hypoexponential_distribution" title="Hypoexponential distribution">Hypoexponential</a></li> <li><a href="/wiki/Exponential-logarithmic_distribution" title="Exponential-logarithmic distribution">Logarithmic</a></li></ul></li> <li><a href="/wiki/F-distribution" title="F-distribution">F</a> <ul><li><a href="/wiki/Noncentral_F-distribution" title="Noncentral F-distribution">Noncentral</a></li></ul></li> <li><a href="/wiki/Folded_normal_distribution" title="Folded normal distribution">Folded normal</a></li> <li><a href="/wiki/Fr%C3%A9chet_distribution" title="Fréchet distribution">Fréchet</a></li> <li><a href="/wiki/Gamma_distribution" title="Gamma distribution">Gamma</a> <ul><li><a href="/wiki/Generalized_gamma_distribution" title="Generalized gamma distribution">Generalized</a></li> <li><a href="/wiki/Inverse-gamma_distribution" title="Inverse-gamma distribution">Inverse</a></li></ul></li> <li><a href="/wiki/Gamma/Gompertz_distribution" title="Gamma/Gompertz distribution">gamma/Gompertz</a></li> <li><a href="/wiki/Gompertz_distribution" title="Gompertz distribution">Gompertz</a> <ul><li><a href="/wiki/Shifted_Gompertz_distribution" title="Shifted Gompertz distribution">Shifted</a></li></ul></li> <li><a href="/wiki/Half-logistic_distribution" title="Half-logistic distribution">Half-logistic</a></li> <li><a href="/wiki/Half-normal_distribution" title="Half-normal distribution">Half-normal</a></li> <li><a href="/wiki/Hotelling%27s_T-squared_distribution" title="Hotelling's T-squared distribution">Hotelling's T-squared</a></li> <li><a href="/wiki/Inverse_Gaussian_distribution" title="Inverse Gaussian distribution">Inverse Gaussian</a> <ul><li><a href="/wiki/Generalized_inverse_Gaussian_distribution" title="Generalized inverse Gaussian distribution">Generalized</a></li></ul></li> <li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov</a></li> <li><a href="/wiki/L%C3%A9vy_distribution" title="Lévy distribution">Lévy</a></li> <li><a href="/wiki/Log-Cauchy_distribution" title="Log-Cauchy distribution">Log-Cauchy</a></li> <li><a href="/wiki/Log-Laplace_distribution" title="Log-Laplace distribution">Log-Laplace</a></li> <li><a href="/wiki/Log-logistic_distribution" title="Log-logistic distribution">Log-logistic</a></li> <li><a href="/wiki/Log-normal_distribution" title="Log-normal distribution">Log-normal</a></li> <li><a href="/wiki/Log-t_distribution" title="Log-t distribution">Log-t</a></li> <li><a href="/wiki/Lomax_distribution" title="Lomax distribution">Lomax</a></li> <li><a href="/wiki/Matrix-exponential_distribution" title="Matrix-exponential distribution">Matrix-exponential</a></li> <li><a href="/wiki/Maxwell%E2%80%93Boltzmann_distribution" title="Maxwell–Boltzmann distribution">Maxwell–Boltzmann</a></li> <li><a href="/wiki/Maxwell%E2%80%93J%C3%BCttner_distribution" title="Maxwell–Jüttner distribution">Maxwell–Jüttner</a></li> <li><a href="/wiki/Mittag-Leffler_distribution" title="Mittag-Leffler distribution">Mittag-Leffler</a></li> <li><a href="/wiki/Nakagami_distribution" title="Nakagami distribution">Nakagami</a></li> <li><a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto</a></li> <li><a href="/wiki/Phase-type_distribution" title="Phase-type distribution">Phase-type</a></li> <li><a href="/wiki/Poly-Weibull_distribution" title="Poly-Weibull distribution">Poly-Weibull</a></li> <li><a href="/wiki/Rayleigh_distribution" title="Rayleigh distribution">Rayleigh</a></li> <li><a href="/wiki/Relativistic_Breit%E2%80%93Wigner_distribution" title="Relativistic Breit–Wigner distribution">Relativistic Breit–Wigner</a></li> <li><a href="/wiki/Rice_distribution" title="Rice distribution">Rice</a></li> <li><a href="/wiki/Truncated_normal_distribution" title="Truncated normal distribution">Truncated normal</a></li> <li><a href="/wiki/Type-2_Gumbel_distribution" title="Type-2 Gumbel distribution">type-2 Gumbel</a></li> <li><a href="/wiki/Weibull_distribution" title="Weibull distribution">Weibull</a> <ul><li><a href="/wiki/Discrete_Weibull_distribution" title="Discrete Weibull distribution">Discrete</a></li></ul></li> <li><a href="/wiki/Wilks%27s_lambda_distribution" title="Wilks's lambda distribution">Wilks's lambda</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">supported on the whole real line</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cauchy_distribution" title="Cauchy distribution">Cauchy</a></li> <li><a href="/wiki/Generalized_normal_distribution#Version_1" title="Generalized normal distribution">Exponential power</a></li> <li><a href="/wiki/Fisher%27s_z-distribution" title="Fisher's z-distribution">Fisher's z</a></li> <li><a href="/wiki/Kaniadakis_Gaussian_distribution" title="Kaniadakis Gaussian distribution">Kaniadakis κ-Gaussian</a></li> <li><a href="/wiki/Gaussian_q-distribution" title="Gaussian q-distribution">Gaussian q</a></li> <li><a href="/wiki/Generalized_normal_distribution" title="Generalized normal distribution">Generalized normal</a></li> <li><a href="/wiki/Generalised_hyperbolic_distribution" title="Generalised hyperbolic distribution">Generalized hyperbolic</a></li> <li><a href="/wiki/Geometric_stable_distribution" title="Geometric stable distribution">Geometric stable</a></li> <li><a href="/wiki/Gumbel_distribution" title="Gumbel distribution">Gumbel</a></li> <li><a href="/wiki/Holtsmark_distribution" title="Holtsmark distribution">Holtsmark</a></li> <li><a href="/wiki/Hyperbolic_secant_distribution" title="Hyperbolic secant distribution">Hyperbolic secant</a></li> <li><a href="/wiki/Johnson%27s_SU-distribution" title="Johnson's SU-distribution">Johnson's SU</a></li> <li><a href="/wiki/Landau_distribution" title="Landau distribution">Landau</a></li> <li><a href="/wiki/Laplace_distribution" title="Laplace distribution">Laplace</a> <ul><li><a href="/wiki/Asymmetric_Laplace_distribution" title="Asymmetric Laplace distribution">Asymmetric</a></li></ul></li> <li><a href="/wiki/Logistic_distribution" title="Logistic distribution">Logistic</a></li> <li><a href="/wiki/Noncentral_t-distribution" title="Noncentral t-distribution">Noncentral t</a></li> <li><a href="/wiki/Normal_distribution" title="Normal distribution">Normal (Gaussian)</a></li> <li><a href="/wiki/Normal-inverse_Gaussian_distribution" title="Normal-inverse Gaussian distribution">Normal-inverse Gaussian</a></li> <li><a href="/wiki/Skew_normal_distribution" title="Skew normal distribution">Skew normal</a></li> <li><a href="/wiki/Slash_distribution" title="Slash distribution">Slash</a></li> <li><a href="/wiki/Stable_distribution" title="Stable distribution">Stable</a></li> <li><a href="/wiki/Student%27s_t-distribution" title="Student's t-distribution">Student's t</a></li> <li><a href="/wiki/Tracy%E2%80%93Widom_distribution" title="Tracy–Widom distribution">Tracy–Widom</a></li> <li><a href="/wiki/Variance-gamma_distribution" title="Variance-gamma distribution">Variance-gamma</a></li> <li><a href="/wiki/Voigt_profile" title="Voigt profile">Voigt</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">with support whose type varies</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Generalized_chi-squared_distribution" title="Generalized chi-squared distribution">Generalized chi-squared</a></li> <li><a href="/wiki/Generalized_extreme_value_distribution" title="Generalized extreme value distribution">Generalized extreme value</a></li> <li><a href="/wiki/Generalized_Pareto_distribution" title="Generalized Pareto distribution">Generalized Pareto</a></li> <li><a href="/wiki/Marchenko%E2%80%93Pastur_distribution" title="Marchenko–Pastur distribution">Marchenko–Pastur</a></li> <li><a href="/wiki/Kaniadakis_Exponential_distribution" class="mw-redirect" title="Kaniadakis Exponential distribution">Kaniadakis κ-exponential</a></li> <li><a href="/wiki/Kaniadakis_Gamma_distribution" title="Kaniadakis Gamma distribution">Kaniadakis κ-Gamma</a></li> <li><a href="/wiki/Kaniadakis_Weibull_distribution" title="Kaniadakis Weibull distribution">Kaniadakis κ-Weibull</a></li> <li><a href="/wiki/Kaniadakis_Logistic_distribution" class="mw-redirect" title="Kaniadakis Logistic distribution">Kaniadakis κ-Logistic</a></li> <li><a href="/wiki/Kaniadakis_Erlang_distribution" title="Kaniadakis Erlang distribution">Kaniadakis κ-Erlang</a></li> <li><a href="/wiki/Q-exponential_distribution" title="Q-exponential distribution">q-exponential</a></li> <li><a href="/wiki/Q-Gaussian_distribution" title="Q-Gaussian distribution">q-Gaussian</a></li> <li><a href="/wiki/Q-Weibull_distribution" title="Q-Weibull distribution">q-Weibull</a></li> <li><a href="/wiki/Shifted_log-logistic_distribution" title="Shifted log-logistic distribution">Shifted log-logistic</a></li> <li><a href="/wiki/Tukey_lambda_distribution" title="Tukey lambda distribution">Tukey lambda</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mixed univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">continuous- discrete</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Rectified_Gaussian_distribution" title="Rectified Gaussian distribution">Rectified Gaussian</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Joint_probability_distribution" title="Joint probability distribution">Multivariate (joint)</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Discrete: </li> <li><a href="/wiki/Ewens%27s_sampling_formula" title="Ewens's sampling formula">Ewens</a></li> <li><a href="/wiki/Multinomial_distribution" title="Multinomial distribution">Multinomial</a> <ul><li><a href="/wiki/Dirichlet-multinomial_distribution" title="Dirichlet-multinomial distribution">Dirichlet</a></li> <li><a href="/wiki/Negative_multinomial_distribution" title="Negative multinomial distribution">Negative</a></li></ul></li> <li>Continuous: </li> <li><a href="/wiki/Dirichlet_distribution" title="Dirichlet distribution">Dirichlet</a> <ul><li><a href="/wiki/Generalized_Dirichlet_distribution" title="Generalized Dirichlet distribution">Generalized</a></li></ul></li> <li><a href="/wiki/Multivariate_Laplace_distribution" title="Multivariate Laplace distribution">Multivariate Laplace</a></li> <li><a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">Multivariate normal</a></li> <li><a href="/wiki/Multivariate_stable_distribution" title="Multivariate stable distribution">Multivariate stable</a></li> <li><a href="/wiki/Multivariate_t-distribution" title="Multivariate t-distribution">Multivariate t</a></li> <li><a href="/wiki/Normal-gamma_distribution" title="Normal-gamma distribution">Normal-gamma</a> <ul><li><a href="/wiki/Normal-inverse-gamma_distribution" title="Normal-inverse-gamma distribution">Inverse</a></li></ul></li> <li><a href="/wiki/Random_matrix" title="Random matrix">Matrix-valued: </a></li> <li><a href="/wiki/Lewandowski-Kurowicka-Joe_distribution" title="Lewandowski-Kurowicka-Joe distribution">LKJ</a></li> <li><a href="/wiki/Matrix_normal_distribution" title="Matrix normal distribution">Matrix normal</a></li> <li><a href="/wiki/Matrix_t-distribution" title="Matrix t-distribution">Matrix t</a></li> <li><a href="/wiki/Matrix_gamma_distribution" title="Matrix gamma distribution">Matrix gamma</a> <ul><li><a href="/wiki/Inverse_matrix_gamma_distribution" title="Inverse matrix gamma distribution">Inverse</a></li></ul></li> <li><a href="/wiki/Wishart_distribution" title="Wishart distribution">Wishart</a> <ul><li><a href="/wiki/Normal-Wishart_distribution" title="Normal-Wishart distribution">Normal</a></li> <li><a href="/wiki/Inverse-Wishart_distribution" title="Inverse-Wishart distribution">Inverse</a></li> <li><a href="/wiki/Normal-inverse-Wishart_distribution" title="Normal-inverse-Wishart distribution">Normal-inverse</a></li> <li><a href="/wiki/Complex_Wishart_distribution" title="Complex Wishart distribution">Complex</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Directional_statistics" title="Directional statistics">Directional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <dl><dt>Univariate (circular) <a href="/wiki/Directional_statistics" title="Directional statistics">directional</a></dt> <dd><a href="/wiki/Circular_uniform_distribution" title="Circular uniform distribution">Circular uniform</a></dd> <dd><a href="/wiki/Von_Mises_distribution" title="Von Mises distribution">Univariate von Mises</a></dd> <dd><a href="/wiki/Wrapped_normal_distribution" title="Wrapped normal distribution">Wrapped normal</a></dd> <dd><a href="/wiki/Wrapped_Cauchy_distribution" title="Wrapped Cauchy distribution">Wrapped Cauchy</a></dd> <dd><a href="/wiki/Wrapped_exponential_distribution" title="Wrapped exponential distribution">Wrapped exponential</a></dd> <dd><a href="/wiki/Wrapped_asymmetric_Laplace_distribution" title="Wrapped asymmetric Laplace distribution">Wrapped asymmetric Laplace</a></dd> <dd><a href="/wiki/Wrapped_L%C3%A9vy_distribution" title="Wrapped Lévy distribution">Wrapped Lévy</a></dd> <dt>Bivariate (spherical)</dt> <dd><a href="/wiki/Kent_distribution" title="Kent distribution">Kent</a></dd> <dt>Bivariate (toroidal)</dt> <dd><a href="/wiki/Bivariate_von_Mises_distribution" title="Bivariate von Mises distribution">Bivariate von Mises</a></dd> <dt>Multivariate</dt> <dd><a href="/wiki/Von_Mises%E2%80%93Fisher_distribution" title="Von Mises–Fisher distribution">von Mises–Fisher</a></dd> <dd><a href="/wiki/Bingham_distribution" title="Bingham distribution">Bingham</a></dd></dl> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Degenerate_distribution" title="Degenerate distribution">Degenerate</a> and <a href="/wiki/Singular_distribution" title="Singular distribution">singular</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <dl><dt>Degenerate</dt> <dd><a class="mw-selflink selflink">Dirac delta function</a></dd> <dt>Singular</dt> <dd><a href="/wiki/Cantor_distribution" title="Cantor distribution">Cantor</a></dd></dl> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Families</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Circular_distribution" title="Circular distribution">Circular</a></li> <li><a href="/wiki/Compound_Poisson_distribution" title="Compound Poisson distribution">Compound Poisson</a></li> <li><a href="/wiki/Elliptical_distribution" title="Elliptical distribution">Elliptical</a></li> <li><a href="/wiki/Exponential_family" title="Exponential family">Exponential</a></li> <li><a href="/wiki/Natural_exponential_family" title="Natural exponential family">Natural exponential</a></li> <li><a href="/wiki/Location%E2%80%93scale_family" title="Location–scale family">Location–scale</a></li> <li><a href="/wiki/Maximum_entropy_probability_distribution" title="Maximum entropy probability distribution">Maximum entropy</a></li> <li><a href="/wiki/Mixture_distribution" title="Mixture distribution">Mixture</a></li> <li><a href="/wiki/Pearson_distribution" title="Pearson distribution">Pearson</a></li> <li><a href="/wiki/Tweedie_distribution" title="Tweedie distribution">Tweedie</a></li> <li><a href="/wiki/Wrapped_distribution" title="Wrapped distribution">Wrapped</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:Probability_distributions" title="Category:Probability distributions">Category</a></li> <li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /> <a href="https://commons.wikimedia.org/wiki/Category:Probability_distributions" class="extiw" title="commons:Category:Probability distributions">Commons</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Differential_equations" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Differential_equations_topics" title="Template:Differential equations topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Differential_equations_topics" title="Template talk:Differential equations topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Differential_equations_topics" title="Special:EditPage/Template:Differential equations topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Differential_equations" style="font-size:114%;margin:0 4em"><a href="/wiki/Differential_equation" title="Differential equation">Differential equations</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classification</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Operations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Differential_operator" title="Differential operator">Differential operator</a></li> <li><a href="/wiki/Notation_for_differentiation" title="Notation for differentiation">Notation for differentiation</a></li> <li><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary</a></li> <li><a href="/wiki/Partial_differential_equation" title="Partial differential equation">Partial</a></li> <li><a href="/wiki/Differential-algebraic_system_of_equations" title="Differential-algebraic system of equations">Differential-algebraic</a></li> <li><a href="/wiki/Integro-differential_equation" title="Integro-differential equation">Integro-differential</a></li> <li><a href="/wiki/Fractional_differential_equations" class="mw-redirect" title="Fractional differential equations">Fractional</a></li> <li><a href="/wiki/Linear_differential_equation" title="Linear differential equation">Linear</a></li> <li><a href="/wiki/Non-linear_differential_equation" class="mw-redirect" title="Non-linear differential equation">Non-linear</a></li> <li><a href="/wiki/Holonomic_function" title="Holonomic function">Holonomic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Attributes of variables</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dependent_and_independent_variables" title="Dependent and independent variables">Dependent and independent variables</a></li> <li><a href="/wiki/Homogeneous_differential_equation" title="Homogeneous differential equation">Homogeneous</a></li> <li><a href="/wiki/Non-homogeneous_differential_equation" class="mw-redirect" title="Non-homogeneous differential equation">Nonhomogeneous</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Coupled</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Decoupled</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Order</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Degree</a></li> <li><a href="/wiki/Autonomous_system_(mathematics)" title="Autonomous system (mathematics)">Autonomous</a></li> <li><a href="/wiki/Exact_differential_equation" title="Exact differential equation">Exact differential equation</a></li> <li><a href="/wiki/Jet_bundle#Partial_differential_equations" title="Jet bundle">On jet bundles</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Relation to processes</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Difference_equation" class="mw-redirect" title="Difference equation">Difference</a> (discrete analogue)</li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic</a> <ul><li><a href="/wiki/Stochastic_partial_differential_equation" title="Stochastic partial differential equation">Stochastic partial</a></li></ul></li> <li><a href="/wiki/Delay_differential_equation" title="Delay differential equation">Delay</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Solutions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Existence/uniqueness</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem" title="Picard–Lindelöf theorem">Picard–Lindelöf theorem</a></li> <li><a href="/wiki/Peano_existence_theorem" title="Peano existence theorem">Peano existence theorem</a></li> <li><a href="/wiki/Carath%C3%A9odory%27s_existence_theorem" title="Carathéodory's existence theorem">Carathéodory's existence theorem</a></li> <li><a href="/wiki/Cauchy%E2%80%93Kowalevski_theorem" class="mw-redirect" title="Cauchy–Kowalevski theorem">Cauchy–Kowalevski theorem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Solution topics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Wronskian" title="Wronskian">Wronskian</a></li> <li><a href="/wiki/Phase_portrait" title="Phase portrait">Phase portrait</a></li> <li><a href="/wiki/Phase_space" title="Phase space">Phase space</a></li> <li><a href="/wiki/Lyapunov_stability" title="Lyapunov stability">Lyapunov stability</a></li> <li><a href="/wiki/Asymptotic_stability" class="mw-redirect" title="Asymptotic stability">Asymptotic stability</a></li> <li><a href="/wiki/Exponential_stability" title="Exponential stability">Exponential stability</a></li> <li><a href="/wiki/Rate_of_convergence" title="Rate of convergence">Rate of convergence</a></li> <li><a href="/wiki/Power_series_solution_of_differential_equations" title="Power series solution of differential equations">Series solutions</a></li> <li><a href="/wiki/Integral" title="Integral">Integral</a> solutions</li> <li><a href="/wiki/Numerical_integration" title="Numerical integration">Numerical integration</a></li> <li><a class="mw-selflink selflink">Dirac delta function</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Solution methods</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_mathematical_jargon#Proof_techniques" class="mw-redirect" title="List of mathematical jargon">Inspection</a></li> <li><a href="/wiki/Integration_by_substitution" title="Integration by substitution">Substitution</a></li> <li><a href="/wiki/Separation_of_variables" title="Separation of variables">Separation of variables</a></li> <li><a href="/wiki/Method_of_undetermined_coefficients" title="Method of undetermined coefficients">Method of undetermined coefficients</a></li> <li><a href="/wiki/Variation_of_parameters" title="Variation of parameters">Variation of parameters</a></li> <li><a href="/wiki/Integrating_factor" title="Integrating factor">Integrating factor</a></li> <li><a href="/wiki/Integral_transform" title="Integral transform">Integral transforms</a></li> <li><a href="/wiki/Euler_method" title="Euler method">Euler method</a></li> <li><a href="/wiki/Finite_difference_method" title="Finite difference method">Finite difference method</a></li> <li><a href="/wiki/Crank%E2%80%93Nicolson_method" title="Crank–Nicolson method">Crank–Nicolson method</a></li> <li><a href="/wiki/Runge%E2%80%93Kutta_methods" title="Runge–Kutta methods">Runge–Kutta methods</a></li> <li><a href="/wiki/Finite_element_method" title="Finite element method">Finite element method</a></li> <li><a href="/wiki/Finite_volume_method" title="Finite volume method">Finite volume method</a></li> <li><a href="/wiki/Galerkin_method" title="Galerkin method">Galerkin method</a></li> <li><a href="/wiki/Perturbation_theory" title="Perturbation theory">Perturbation theory</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Examples</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_named_differential_equations" title="List of named differential equations">List of named differential equations</a></li> <li><a href="/wiki/List_of_linear_ordinary_differential_equations" title="List of linear ordinary differential equations">List of linear ordinary differential equations</a></li> <li><a href="/wiki/List_of_nonlinear_ordinary_differential_equations" title="List of nonlinear ordinary differential equations">List of nonlinear ordinary differential equations</a></li> <li><a href="/wiki/List_of_nonlinear_partial_differential_equations" title="List of nonlinear partial differential equations">List of nonlinear partial differential equations</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mathematicians</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li> <li><a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><a href="/wiki/Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a></li> <li><a href="/wiki/%C3%89mile_Picard" title="Émile Picard">Émile Picard</a></li> <li><a href="/wiki/J%C3%B3zef_Maria_Hoene-Wro%C5%84ski" title="Józef Maria Hoene-Wroński">Józef Maria Hoene-Wroński</a></li> <li><a href="/wiki/Ernst_Leonard_Lindel%C3%B6f" title="Ernst Leonard Lindelöf">Ernst Lindelöf</a></li> <li><a href="/wiki/Rudolf_Lipschitz" title="Rudolf Lipschitz">Rudolf Lipschitz</a></li> <li><a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li> <li><a href="/wiki/John_Crank" title="John Crank">John Crank</a></li> <li><a href="/wiki/Phyllis_Nicolson" title="Phyllis Nicolson">Phyllis Nicolson</a></li> <li><a href="/wiki/Carl_David_Tolm%C3%A9_Runge" class="mw-redirect" title="Carl David Tolmé Runge">Carl David Tolmé Runge</a></li> <li><a href="/wiki/Martin_Kutta" title="Martin Kutta">Martin Kutta</a></li> <li><a href="/wiki/Sofya_Kovalevskaya" title="Sofya Kovalevskaya">Sofya Kovalevskaya</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Dirac_delta_function&oldid=1257959343">https://en.wikipedia.org/w/index.php?title=Dirac_delta_function&oldid=1257959343</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Fourier_analysis" title="Category:Fourier analysis">Fourier analysis</a></li><li><a href="/wiki/Category:Generalized_functions" title="Category:Generalized functions">Generalized functions</a></li><li><a href="/wiki/Category:Measure_theory" title="Category:Measure 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Template:Math"," 8.87% 148.357 1 Template:Differential_equations"," 8.71% 145.815 1 Template:Sidebar_with_collapsible_lists"," 7.91% 132.311 1 Template:Short_description"," 6.77% 113.268 1 Template:Lang"]},"scribunto":{"limitreport-timeusage":{"value":"1.015","limit":"10.000"},"limitreport-memusage":{"value":16677336,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFAratynRasinariu2006\"] = 1,\n [\"CITEREFArfkenWeber2000\"] = 1,\n [\"CITEREFBracewell1986\"] = 1,\n [\"CITEREFBracewell2000\"] = 1,\n [\"CITEREFCauchy1882–1974\"] = 1,\n [\"CITEREFCourantHilbert1962\"] = 1,\n [\"CITEREFCórdoba1988\"] = 1,\n [\"CITEREFDavisThomson2000\"] = 1,\n [\"CITEREFDebnathBhatta2007\"] = 1,\n [\"CITEREFDieudonné1972\"] = 1,\n [\"CITEREFDieudonné1976\"] = 1,\n [\"CITEREFDirac1927\"] = 1,\n [\"CITEREFDirac1930\"] = 1,\n [\"CITEREFDriggers2003\"] = 1,\n [\"CITEREFDuistermaatKolk2010\"] = 1,\n [\"CITEREFFederer1969\"] = 1,\n [\"CITEREFFourier1822\"] = 1,\n [\"CITEREFGannon2008\"] = 1,\n [\"CITEREFGelfandShilov1966–1968\"] = 1,\n [\"CITEREFGrattan-Guinness2009\"] = 1,\n [\"CITEREFHartmann1997\"] = 1,\n [\"CITEREFHazewinkel1995\"] = 1,\n [\"CITEREFHazewinkel2011\"] = 1,\n [\"CITEREFHewittStromberg1963\"] = 1,\n [\"CITEREFHörmander1983\"] = 1,\n [\"CITEREFIsham1995\"] = 1,\n [\"CITEREFJohn1955\"] = 1,\n [\"CITEREFKarris2003\"] = 1,\n [\"CITEREFKomatsu2002\"] = 1,\n [\"CITEREFKrachtKreyszig1989\"] = 1,\n [\"CITEREFKrantzParks2008\"] = 1,\n [\"CITEREFLang1997\"] = 1,\n [\"CITEREFLange2012\"] = 1,\n [\"CITEREFLaugwitz1989\"] = 1,\n [\"CITEREFLevin2002\"] = 1,\n [\"CITEREFLiWong2008\"] = 1,\n [\"CITEREFMader2006\"] = 1,\n [\"CITEREFMcMahon2005\"] = 1,\n [\"CITEREFMilovanovićRassias2014\"] = 1,\n [\"CITEREFMitrovićŽubrinić1998\"] = 1,\n [\"CITEREFMyint-U.Debnath2007\"] = 1,\n [\"CITEREFRoden2014\"] = 1,\n [\"CITEREFRottwittTidemand-Lichtenberg2014\"] = 1,\n [\"CITEREFRudin1966\"] = 1,\n [\"CITEREFRudin1991\"] = 1,\n [\"CITEREFSaichevWoyczyński1997\"] = 1,\n [\"CITEREFSchwartz1950\"] = 1,\n [\"CITEREFSchwartz1951\"] = 1,\n [\"CITEREFSteinWeiss1971\"] = 1,\n [\"CITEREFStrichartz1994\"] = 1,\n [\"CITEREFValléeSoares2004\"] = 1,\n [\"CITEREFVladimirov1971\"] = 1,\n [\"CITEREFYamashita2006\"] = 1,\n [\"CITEREFYamashita2007\"] = 1,\n [\"CITEREFZhao2011\"] = 1,\n [\"CITEREFatis2013\"] = 1,\n [\"CITEREFde_la_MadridBohmGadella2002\"] = 1,\n [\"CITEREFde_la_Madrid_Modino2001\"] = 1,\n [\"CITEREFvan_der_PolBremmer1987\"] = 1,\n [\"nascent_delta_function\"] = 1,\n}\ntemplate_list = table#1 {\n [\"!\"] = 2,\n [\"Anchor\"] = 1,\n [\"Angbr\"] = 1,\n [\"Bra-ket\"] = 1,\n [\"Brace\"] = 6,\n [\"Citation\"] = 39,\n [\"Cite book\"] = 18,\n [\"Cite journal\"] = 1,\n [\"Cite thesis\"] = 1,\n [\"Cite web\"] = 1,\n [\"Clear\"] = 1,\n [\"Closed-closed\"] = 3,\n [\"Commons category-inline\"] = 1,\n [\"Differential equations\"] = 1,\n [\"Differential equations topics\"] = 1,\n [\"Em\"] = 5,\n [\"EquationNote\"] = 4,\n [\"EquationRef\"] = 5,\n [\"Good article\"] = 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projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2002-01-31T12:47:36Z","dateModified":"2024-11-17T11:39:21Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/4\/48\/Dirac_distribution_PDF.svg","headline":"pseudo-function \u03b4 such that an integral of \u03b4(x-c)f(x) always takes the value of f(c)"}</script> </body> </html>

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Dirac delta function - Wikipedia