Harmonic function - Wikipedia

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data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Etymology_of_the_term_"harmonic"" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Etymology_of_the_term_"harmonic""> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Etymology of the term "harmonic"</span> </div> </a> <ul id="toc-Etymology_of_the_term_"harmonic"-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Connections_with_complex_function_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Connections_with_complex_function_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Connections with complex function theory</span> </div> </a> <ul id="toc-Connections_with_complex_function_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties_of_harmonic_functions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties_of_harmonic_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Properties of harmonic functions</span> </div> </a> <button aria-controls="toc-Properties_of_harmonic_functions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Properties of harmonic functions subsection</span> </button> <ul id="toc-Properties_of_harmonic_functions-sublist" class="vector-toc-list"> <li id="toc-Regularity_theorem_for_harmonic_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Regularity_theorem_for_harmonic_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Regularity theorem for harmonic functions</span> </div> </a> <ul id="toc-Regularity_theorem_for_harmonic_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Maximum_principle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Maximum_principle"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Maximum principle</span> </div> </a> <ul id="toc-Maximum_principle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_mean_value_property" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_mean_value_property"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>The mean value property</span> </div> </a> <ul id="toc-The_mean_value_property-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Harnack's_inequality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Harnack's_inequality"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Harnack's inequality</span> </div> </a> <ul id="toc-Harnack's_inequality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Removal_of_singularities" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Removal_of_singularities"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Removal of singularities</span> </div> </a> <ul id="toc-Removal_of_singularities-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Liouville's_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Liouville's_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.6</span> <span>Liouville's theorem</span> </div> </a> <ul id="toc-Liouville's_theorem-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Generalizations</span> </div> </a> <button aria-controls="toc-Generalizations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Generalizations subsection</span> </button> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> <li id="toc-Weakly_harmonic_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Weakly_harmonic_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Weakly harmonic function</span> </div> </a> <ul id="toc-Weakly_harmonic_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Harmonic_functions_on_manifolds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Harmonic_functions_on_manifolds"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Harmonic functions on manifolds</span> </div> </a> <ul id="toc-Harmonic_functions_on_manifolds-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Subharmonic_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Subharmonic_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Subharmonic functions</span> </div> </a> <ul id="toc-Subharmonic_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Harmonic_forms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Harmonic_forms"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Harmonic forms</span> </div> </a> <ul id="toc-Harmonic_forms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Harmonic_maps_between_manifolds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Harmonic_maps_between_manifolds"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>Harmonic maps between manifolds</span> </div> </a> <ul id="toc-Harmonic_maps_between_manifolds-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Notes</span> </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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </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" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" 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mw-list-item"><a href="https://ca.wikipedia.org/wiki/Funci%C3%B3_harm%C3%B2nica" title="Funció harmònica – Catalan" lang="ca" hreflang="ca" data-title="Funció harmònica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Harmonick%C3%A1_funkce" title="Harmonická funkce – Czech" lang="cs" hreflang="cs" data-title="Harmonická funkce" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Harmonische_Funktion" title="Harmonische Funktion – German" lang="de" hreflang="de" data-title="Harmonische Funktion" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CF%81%CE%BC%CE%BF%CE%BD%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="Αρμονική συνάρτηση – Greek" lang="el" hreflang="el" data-title="Αρμονική συνάρτηση" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Funci%C3%B3n_arm%C3%B3nica" title="Función armónica – Spanish" lang="es" hreflang="es" data-title="Función armónica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Harmonia_funkcio" title="Harmonia funkcio – Esperanto" lang="eo" hreflang="eo" data-title="Harmonia funkcio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Funtzio_harmoniko" title="Funtzio harmoniko – Basque" lang="eu" hreflang="eu" data-title="Funtzio harmoniko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%A7%D8%A8%D8%B9_%D9%87%D8%A7%D8%B1%D9%85%D9%88%D9%86%DB%8C%DA%A9" title="تابع هارمونیک – Persian" lang="fa" hreflang="fa" data-title="تابع هارمونیک" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Fonction_harmonique" title="Fonction harmonique – French" lang="fr" hreflang="fr" data-title="Fonction harmonique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A1%B0%ED%99%94_%ED%95%A8%EC%88%98" title="조화 함수 – Korean" lang="ko" hreflang="ko" data-title="조화 함수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Funzione_armonica" title="Funzione armonica – Italian" lang="it" hreflang="it" data-title="Funzione armonica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%94%D7%A8%D7%9E%D7%95%D7%A0%D7%99%D7%AA" title="פונקציה הרמונית – Hebrew" lang="he" hreflang="he" data-title="פונקציה הרמונית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%93%D0%B0%D1%80%D0%BC%D0%BE%D0%BD%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D0%BA_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Гармоникалык функция – Kyrgyz" lang="ky" hreflang="ky" data-title="Гармоникалык функция" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Harmonikus_f%C3%BCggv%C3%A9ny" title="Harmonikus függvény – Hungarian" lang="hu" hreflang="hu" data-title="Harmonikus függvény" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Harmonische_functie" title="Harmonische functie – Dutch" lang="nl" hreflang="nl" data-title="Harmonische functie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%AA%BF%E5%92%8C%E9%96%A2%E6%95%B0" title="調和関数 – Japanese" lang="ja" hreflang="ja" data-title="調和関数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Harmonisk_funksjon" title="Harmonisk funksjon – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Harmonisk funksjon" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Harmonisk_funksjon" title="Harmonisk funksjon – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Harmonisk funksjon" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Garmonik_funksiyalar" title="Garmonik funksiyalar – Uzbek" lang="uz" hreflang="uz" data-title="Garmonik funksiyalar" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Funkcja_harmoniczna" title="Funkcja harmoniczna – Polish" lang="pl" hreflang="pl" data-title="Funkcja harmoniczna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Fun%C3%A7%C3%A3o_harm%C3%B4nica" title="Função harmônica – Portuguese" lang="pt" hreflang="pt" data-title="Função harmônica" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Func%C8%9Bie_armonic%C4%83" title="Funcție armonică – Romanian" lang="ro" hreflang="ro" data-title="Funcție armonică" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D0%B0%D1%80%D0%BC%D0%BE%D0%BD%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Гармоническая функция – Russian" lang="ru" hreflang="ru" data-title="Гармоническая функция" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Harmoni%C4%8Dna_funkcija" title="Harmonična funkcija – Slovenian" lang="sl" hreflang="sl" data-title="Harmonična funkcija" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Harmoninen_funktio" title="Harmoninen funktio – Finnish" lang="fi" hreflang="fi" data-title="Harmoninen funktio" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Harmonisk_funktion" title="Harmonisk funktion – Swedish" lang="sv" hreflang="sv" data-title="Harmonisk funktion" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Harmonik_fonksiyon" title="Harmonik fonksiyon – Turkish" lang="tr" hreflang="tr" data-title="Harmonik fonksiyon" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D0%B0%D1%80%D0%BC%D0%BE%D0%BD%D1%96%D1%87%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F" title="Гармонічна функція – Ukrainian" lang="uk" hreflang="uk" data-title="Гармонічна функція" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%B0%83%E5%92%8C%E5%87%BD%E6%95%B0" title="调和函数 – Chinese" lang="zh" hreflang="zh" data-title="调和函数" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q599027#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div 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.hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about harmonic functions in mathematics. For harmonic function in music, see <a href="/wiki/Diatonic_functionality" class="mw-redirect" title="Diatonic functionality">diatonic functionality</a>.</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Laplace%27s_equation_on_an_annulus.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Laplace%27s_equation_on_an_annulus.svg/300px-Laplace%27s_equation_on_an_annulus.svg.png" decoding="async" width="300" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Laplace%27s_equation_on_an_annulus.svg/450px-Laplace%27s_equation_on_an_annulus.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Laplace%27s_equation_on_an_annulus.svg/600px-Laplace%27s_equation_on_an_annulus.svg.png 2x" data-file-width="2268" data-file-height="1134" /></a><figcaption>A harmonic function defined on an <a href="/wiki/Annulus_(mathematics)" title="Annulus (mathematics)">annulus</a>.</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 li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist 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navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Complex_analysis_sidebar" title="Template:Complex analysis sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Complex_analysis_sidebar" title="Template talk:Complex analysis sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Complex_analysis_sidebar" title="Special:EditPage/Template:Complex analysis sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <a href="/wiki/Mathematical_physics" title="Mathematical physics">mathematical physics</a> and the theory of <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic processes</a>, a <b>harmonic function</b> is a twice <a href="/wiki/Continuously_differentiable" class="mw-redirect" title="Continuously differentiable">continuously differentiable</a> <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon U\to \mathbb {R} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>U</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon U\to \mathbb {R} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4fd08e6ae92b74490847ec8ca243b70f2e5e2e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.034ex; height:2.509ex;" alt="{\displaystyle f\colon U\to \mathbb {R} ,}"></span> where <span class="texhtml mvar" style="font-style:italic;">U</span> is an <a href="/wiki/Open_set" title="Open set">open subset</a> of <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7035fcb9fe3ebecc6bc9f372f82d0352202c8bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{n},}"></span>⁠</span> that satisfies <a href="/wiki/Laplace%27s_equation" title="Laplace's equation">Laplace's equation</a>, that is, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}f}{\partial x_{1}^{2}}}+{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}+\cdots +{\frac {\partial ^{2}f}{\partial x_{n}^{2}}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}f}{\partial x_{1}^{2}}}+{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}+\cdots +{\frac {\partial ^{2}f}{\partial x_{n}^{2}}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25157cd4c4b88b223f36a2885d56d0d10b753327" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:29.284ex; height:6.676ex;" alt="{\displaystyle {\frac {\partial ^{2}f}{\partial x_{1}^{2}}}+{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}+\cdots +{\frac {\partial ^{2}f}{\partial x_{n}^{2}}}=0}"></span> everywhere on <span class="texhtml mvar" style="font-style:italic;">U</span>. This is usually written as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla ^{2}f=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>f</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}f=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44d96e48b4ac582542a70e54bf299226aaa0e812" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.53ex; height:3.009ex;" alt="{\displaystyle \nabla ^{2}f=0}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta f=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>f</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta f=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/524f4acc990d8c061ca122776ebc3b0a48f8acbb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.475ex; height:2.509ex;" alt="{\displaystyle \Delta f=0}"></span> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Etymology_of_the_term_"harmonic""><span id="Etymology_of_the_term_.22harmonic.22"></span>Etymology of the term "harmonic"</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_function&action=edit&section=1" title="Edit section: Etymology of the term "harmonic""><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The descriptor "harmonic" in the name harmonic function originates from a point on a taut string which is undergoing <a href="/wiki/Simple_harmonic_motion" title="Simple harmonic motion">harmonic motion</a>. The solution to the differential equation for this type of motion can be written in terms of sines and cosines, functions which are thus referred to as <i>harmonics</i>. <a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a> involves expanding functions on the unit circle in terms of a series of these harmonics. Considering higher dimensional analogues of the harmonics on the unit <a href="/wiki/N-sphere" title="N-sphere"><i>n</i>-sphere</a>, one arrives at the <a href="/wiki/Spherical_harmonics" title="Spherical harmonics">spherical harmonics</a>. These functions satisfy Laplace's equation and over time "harmonic" was <a href="/wiki/Synecdoche" title="Synecdoche">used to refer to all</a> functions satisfying Laplace's equation.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_function&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Examples of harmonic functions of two variables are: </p> <ul><li>The real or imaginary part of any <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic function</a>.</li> <li>The function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\!f(x,y)=e^{x}\sin y;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mi>y</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\!f(x,y)=e^{x}\sin y;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3d010032fff10057dc7f070c1b3cfafa48a8ff7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.394ex; height:2.843ex;" alt="{\displaystyle \,\!f(x,y)=e^{x}\sin y;}"></span> this is a special case of the example above, as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y)=\operatorname {Im} \left(e^{x+iy}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Im</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> </mrow> </msup> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)=\operatorname {Im} \left(e^{x+iy}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/248121cc87029bcb8260886c4e38a666b4de3ee9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.177ex; height:3.343ex;" alt="{\displaystyle f(x,y)=\operatorname {Im} \left(e^{x+iy}\right),}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x+iy}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{x+iy}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e99519a420cad88b5468c901d6bc56acc53d4fe4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.919ex; height:2.676ex;" alt="{\displaystyle e^{x+iy}}"></span> is a <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic function</a>. The second derivative with respect to <i>x</i> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\!e^{x}\sin y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\!e^{x}\sin y,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2e6645fea88e3d7732f4991f498cdda8379a9e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.688ex; height:2.676ex;" alt="{\displaystyle \,\!e^{x}\sin y,}"></span> while the second derivative with respect to <i>y</i> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\!-e^{x}\sin y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\!-e^{x}\sin y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01f94fbf08c7edfe38ddada31ea9b2c4bc5f786e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.496ex; height:2.676ex;" alt="{\displaystyle \,\!-e^{x}\sin y.}"></span></li> <li>The function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\!f(x,y)=\ln \left(x^{2}+y^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\!f(x,y)=\ln \left(x^{2}+y^{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e90079ad30602d510017c682aaaf6be41103b9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.214ex; height:3.343ex;" alt="{\displaystyle \,\!f(x,y)=\ln \left(x^{2}+y^{2}\right)}"></span> defined on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}\smallsetminus \lbrace 0\rbrace .}"> <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"> <mn>2</mn> </mrow> </msup> <mo>∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}\smallsetminus \lbrace 0\rbrace .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5353577b48c004715cdb8d33ea06d28b8e51ecb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.707ex; height:3.176ex;" alt="{\displaystyle \mathbb {R} ^{2}\smallsetminus \lbrace 0\rbrace .}"></span> This can describe the electric potential due to a line charge or the gravity potential due to a long cylindrical mass.</li></ul> <p>Examples of harmonic functions of three variables are given in the table below with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r^{2}=x^{2}+y^{2}+z^{2}:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>r</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> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r^{2}=x^{2}+y^{2}+z^{2}:}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89629cb46020889bb106e6ff7f9022365a7ae8d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.917ex; height:3.009ex;" alt="{\displaystyle r^{2}=x^{2}+y^{2}+z^{2}:}"></span> </p> <dl><dd><table class="wikitable"> <tbody><tr> <th>Function</th> <th><a href="/wiki/Mathematical_singularity" class="mw-redirect" title="Mathematical singularity">Singularity</a> </th></tr> <tr> <td align="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd462218ef3cc25ed3b835b52af9b951d54edb13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{r}}}"></span> </td> <td>Unit point charge at origin </td></tr> <tr> <td align="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x}{r^{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x}{r^{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ec04e2451a575d3f25914cd2636e9e246ccfd5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:2.939ex; height:5.009ex;" alt="{\displaystyle {\frac {x}{r^{3}}}}"></span> </td> <td><i>x</i>-directed dipole at origin </td></tr> <tr> <td align="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\ln \left(r^{2}-z^{2}\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\ln \left(r^{2}-z^{2}\right)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98aade8386e8b062a58ca31dcfe5e1e45b892ee2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.739ex; height:3.343ex;" alt="{\displaystyle -\ln \left(r^{2}-z^{2}\right)\,}"></span> </td> <td>Line of unit charge density on entire z-axis </td></tr> <tr> <td align="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\ln(r+z)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\ln(r+z)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a3c33b9e281d21aa2a7e0ba9a5d38ba7b309f0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.308ex; height:2.843ex;" alt="{\displaystyle -\ln(r+z)\,}"></span> </td> <td>Line of unit charge density on negative z-axis </td></tr> <tr> <td align="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x}{r^{2}-z^{2}}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x}{r^{2}-z^{2}}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44360b53c98e0e30c8390b32bef403d1f02974a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.311ex; height:5.176ex;" alt="{\displaystyle {\frac {x}{r^{2}-z^{2}}}\,}"></span> </td> <td>Line of <i>x</i>-directed dipoles on entire <i>z</i> axis </td></tr> <tr> <td align="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x}{r(r+z)}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mi>r</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x}{r(r+z)}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee10ec00adc34a09436f21c1b97ab2eb6a2d2e4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:9.058ex; height:5.509ex;" alt="{\displaystyle {\frac {x}{r(r+z)}}\,}"></span> </td> <td>Line of <i>x</i>-directed dipoles on negative <i>z</i> axis </td></tr></tbody></table></dd></dl> <p>Harmonic functions that arise in physics are determined by their <a href="/wiki/Mathematical_singularity" class="mw-redirect" title="Mathematical singularity">singularities</a> and boundary conditions (such as <a href="/wiki/Dirichlet_boundary_conditions" class="mw-redirect" title="Dirichlet boundary conditions">Dirichlet boundary conditions</a> or <a href="/wiki/Neumann_boundary_conditions" class="mw-redirect" title="Neumann boundary conditions">Neumann boundary conditions</a>). On regions without boundaries, adding the real or imaginary part of any <a href="/wiki/Entire_function" title="Entire function">entire function</a> will produce a harmonic function with the same singularity, so in this case the harmonic function is not determined by its singularities; however, we can make the solution unique in physical situations by requiring that the solution approaches 0 as r approaches infinity. In this case, uniqueness follows by <a href="/wiki/Liouville%27s_theorem_(complex_analysis)" title="Liouville's theorem (complex analysis)">Liouville's theorem</a>. </p><p>The singular points of the harmonic functions above are expressed as "<a href="/wiki/Charge_(physics)" title="Charge (physics)">charges</a>" and "<a href="/wiki/Charge_density" title="Charge density">charge densities</a>" using the terminology of <a href="/wiki/Electrostatics" title="Electrostatics">electrostatics</a>, and so the corresponding harmonic function will be proportional to the <a href="/wiki/Electric_potential" title="Electric potential">electrostatic potential</a> due to these charge distributions. Each function above will yield another harmonic function when multiplied by a constant, rotated, and/or has a constant added. The <a href="/wiki/Method_of_inversion" class="mw-redirect" title="Method of inversion">inversion</a> of each function will yield another harmonic function which has singularities which are the images of the original singularities in a spherical "mirror". Also, the sum of any two harmonic functions will yield another harmonic function. </p><p>Finally, examples of harmonic functions of <span class="texhtml mvar" style="font-style:italic;">n</span> variables are: </p> <ul><li>The constant, linear and affine functions on all of <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>⁠</span> (for example, the <a href="/wiki/Electric_potential" title="Electric potential">electric potential</a> between the plates of a <a href="/wiki/Capacitor" title="Capacitor">capacitor</a>, and the <a href="/wiki/Gravity_potential" class="mw-redirect" title="Gravity potential">gravity potential</a> of a slab)</li> <li>The function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x_{1},\dots ,x_{n})=\left({x_{1}}^{2}+\cdots +{x_{n}}^{2}\right)^{1-n/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x_{1},\dots ,x_{n})=\left({x_{1}}^{2}+\cdots +{x_{n}}^{2}\right)^{1-n/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d778c9701834ad8e3ab9593e39c8645ad323e8df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.834ex; height:3.843ex;" alt="{\displaystyle f(x_{1},\dots ,x_{n})=\left({x_{1}}^{2}+\cdots +{x_{n}}^{2}\right)^{1-n/2}}"></span> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}\smallsetminus \lbrace 0\rbrace }"> <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> <mo>∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}\smallsetminus \lbrace 0\rbrace }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/531017b8f521f9e6db80d8be94e238eb36321be7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.224ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} ^{n}\smallsetminus \lbrace 0\rbrace }"></span> for <span class="texhtml"><i>n</i> > 2</span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_function&action=edit&section=3" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The set of harmonic functions on a given open set <span class="texhtml mvar" style="font-style:italic;">U</span> can be seen as the <a href="/wiki/Kernel_(linear_operator)" class="mw-redirect" title="Kernel (linear operator)">kernel</a> of the <a href="/wiki/Laplace_operator" title="Laplace operator">Laplace operator</a> <span class="texhtml">Δ</span> and is therefore a <a href="/wiki/Vector_space" title="Vector space">vector space</a> over <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \!:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mspace width="negativethinmathspace" /> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \!:}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f0ce7c599ad5afd27a6e1833f134c993f82ba7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.583ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} \!:}"></span>⁠</span> linear combinations of harmonic functions are again harmonic. </p><p>If <span class="texhtml mvar" style="font-style:italic;">f</span> is a harmonic function on <span class="texhtml mvar" style="font-style:italic;">U</span>, then all <a href="/wiki/Partial_derivative" title="Partial derivative">partial derivatives</a> of <span class="texhtml mvar" style="font-style:italic;">f</span> are also harmonic functions on <span class="texhtml mvar" style="font-style:italic;">U</span>. The Laplace operator <span class="texhtml">Δ</span> and the partial derivative operator will commute on this class of functions. </p><p>In several ways, the harmonic functions are real analogues to <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic functions</a>. All harmonic functions are <a href="/wiki/Analytic_function" title="Analytic function">analytic</a>, that is, they can be locally expressed as <a href="/wiki/Power_series" title="Power series">power series</a>. This is a general fact about <a href="/wiki/Elliptic_operator" title="Elliptic operator">elliptic operators</a>, of which the Laplacian is a major example. </p><p>The uniform limit of a convergent sequence of harmonic functions is still harmonic. This is true because every continuous function satisfying the mean value property is harmonic. Consider the sequence on <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\infty ,0)\times \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\infty ,0)\times \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e015bc19c4de0d428d3a63a881ee328db6184d8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.656ex; height:2.843ex;" alt="{\displaystyle (-\infty ,0)\times \mathbb {R} }"></span>⁠</span> defined by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f_{n}(x,y)={\frac {1}{n}}\exp(nx)\cos(ny);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle f_{n}(x,y)={\frac {1}{n}}\exp(nx)\cos(ny);}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/187738afbda39fedbfc55abac62da6bf9922a1ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.585ex; height:3.343ex;" alt="{\textstyle f_{n}(x,y)={\frac {1}{n}}\exp(nx)\cos(ny);}"></span> this sequence is harmonic and converges uniformly to the zero function; however note that the partial derivatives are not uniformly convergent to the zero function (the derivative of the zero function). This example shows the importance of relying on the mean value property and continuity to argue that the limit is harmonic. </p> <div class="mw-heading mw-heading2"><h2 id="Connections_with_complex_function_theory">Connections with complex function theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_function&action=edit&section=4" title="Edit section: Connections with complex function theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The real and imaginary part of any holomorphic function yield harmonic functions on <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <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"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span>⁠</span> (these are said to be a pair of <a href="/wiki/Harmonic_conjugate" title="Harmonic conjugate">harmonic conjugate</a> functions). Conversely, any harmonic function <span class="texhtml mvar" style="font-style:italic;">u</span> on an open subset <span class="texhtml">Ω</span> of <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <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"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span>⁠</span> is <i>locally</i> the real part of a holomorphic function. This is immediately seen observing that, writing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=x+iy,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=x+iy,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ee21a775fda65273b6e1a6786b7f3d51a85de65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.961ex; height:2.509ex;" alt="{\displaystyle z=x+iy,}"></span> the complex function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(z):=u_{x}-iu_{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>−</mo> <mi>i</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(z):=u_{x}-iu_{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84d3c8c6e1be2f91f28dcf7e14697e685222ad71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.283ex; height:3.009ex;" alt="{\displaystyle g(z):=u_{x}-iu_{y}}"></span> is holomorphic in <span class="texhtml">Ω</span> because it satisfies the <a href="/wiki/Cauchy%E2%80%93Riemann_equations" title="Cauchy–Riemann equations">Cauchy–Riemann equations</a>. Therefore, <span class="texhtml mvar" style="font-style:italic;">g</span> locally has a primitive <span class="texhtml mvar" style="font-style:italic;">f</span>, and <span class="texhtml mvar" style="font-style:italic;">u</span> is the real part of <span class="texhtml mvar" style="font-style:italic;">f</span> up to a constant, as <span class="texhtml mvar" style="font-style:italic;">u<sub>x</sub></span> is the real part of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'=g.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>g</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'=g.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a87b9a9e564df679c8404530e7ba33a787acdaac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.867ex; height:2.843ex;" alt="{\displaystyle f'=g.}"></span> </p><p>Although the above correspondence with holomorphic functions only holds for functions of two real variables, harmonic functions in <span class="texhtml mvar" style="font-style:italic;">n</span> variables still enjoy a number of properties typical of holomorphic functions. They are (real) analytic; they have a maximum principle and a mean-value principle; a theorem of removal of singularities as well as a Liouville theorem holds for them in analogy to the corresponding theorems in complex functions theory. </p> <div class="mw-heading mw-heading2"><h2 id="Properties_of_harmonic_functions">Properties of harmonic functions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_function&action=edit&section=5" title="Edit section: Properties of harmonic functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some important properties of harmonic functions can be deduced from Laplace's equation. </p> <div class="mw-heading mw-heading3"><h3 id="Regularity_theorem_for_harmonic_functions">Regularity theorem for harmonic functions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_function&action=edit&section=6" title="Edit section: Regularity theorem for harmonic functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Harmonic functions are infinitely differentiable in open sets. In fact, harmonic functions are <a href="/wiki/Analytic_function" title="Analytic function">real analytic</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Maximum_principle">Maximum principle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_function&action=edit&section=7" title="Edit section: Maximum principle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Harmonic functions satisfy the following <i><a href="/wiki/Maximum_modulus_principle" title="Maximum modulus principle">maximum principle</a></i>: if <span class="texhtml mvar" style="font-style:italic;">K</span> is a nonempty <a href="/wiki/Compact_space" title="Compact space">compact subset</a> of <span class="texhtml mvar" style="font-style:italic;">U</span>, then <span class="texhtml mvar" style="font-style:italic;">f </span> restricted to <span class="texhtml mvar" style="font-style:italic;">K</span> attains its <a href="/wiki/Maxima_and_minima" class="mw-redirect" title="Maxima and minima">maximum and minimum</a> on the <a href="/wiki/Boundary_(topology)" title="Boundary (topology)">boundary</a> of <span class="texhtml mvar" style="font-style:italic;">K</span>. If <span class="texhtml mvar" style="font-style:italic;">U</span> is <a href="/wiki/Connected_space" title="Connected space">connected</a>, this means that <span class="texhtml mvar" style="font-style:italic;">f </span> cannot have local maxima or minima, other than the exceptional case where <span class="texhtml mvar" style="font-style:italic;">f </span> is <a href="/wiki/Constant_function" title="Constant function">constant</a>. Similar properties can be shown for <a href="/wiki/Subharmonic_function" title="Subharmonic function">subharmonic functions</a>. </p> <div class="mw-heading mw-heading3"><h3 id="The_mean_value_property">The mean value property</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_function&action=edit&section=8" title="Edit section: The mean value property"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <span class="texhtml"><i>B</i>(<i>x</i>, <i>r</i>)</span> is a <a href="/wiki/Ball_(mathematics)" title="Ball (mathematics)">ball</a> with center <span class="texhtml mvar" style="font-style:italic;">x</span> and radius <span class="texhtml mvar" style="font-style:italic;">r</span> which is completely contained in the open set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega \subset \mathbb {R} ^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>⊂</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>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega \subset \mathbb {R} ^{n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/290ac8a4dd03b6a2cd6b56cc55c2f3830e7bbc40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.32ex; height:2.676ex;" alt="{\displaystyle \Omega \subset \mathbb {R} ^{n},}"></span> then the value <span class="texhtml"><i>u</i>(<i>x</i>)</span> of a harmonic function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u:\Omega \to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>:</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u:\Omega \to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a624e1a3ee1bbe11d92c7e61e33500f7bf0e489a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.237ex; height:2.176ex;" alt="{\displaystyle u:\Omega \to \mathbb {R} }"></span> at the center of the ball is given by the average value of <span class="texhtml mvar" style="font-style:italic;">u</span> on the surface of the ball; this average value is also equal to the average value of <span class="texhtml mvar" style="font-style:italic;">u</span> in the interior of the ball. In other words, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(x)={\frac {1}{n\omega _{n}r^{n-1}}}\int _{\partial B(x,r)}u\,d\sigma ={\frac {1}{\omega _{n}r^{n}}}\int _{B(x,r)}u\,dV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi>B</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mi>u</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>σ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mi>u</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(x)={\frac {1}{n\omega _{n}r^{n-1}}}\int _{\partial B(x,r)}u\,d\sigma ={\frac {1}{\omega _{n}r^{n}}}\int _{B(x,r)}u\,dV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0adc1b631e3a9f8689cadd9fe4efe4ee4b456b43" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:49.54ex; height:6.009ex;" alt="{\displaystyle u(x)={\frac {1}{n\omega _{n}r^{n-1}}}\int _{\partial B(x,r)}u\,d\sigma ={\frac {1}{\omega _{n}r^{n}}}\int _{B(x,r)}u\,dV}"></span> where <span class="texhtml mvar" style="font-style:italic;">ω<sub>n</sub></span> is the volume of the unit ball in <span class="texhtml mvar" style="font-style:italic;">n</span> dimensions and <span class="texhtml mvar" style="font-style:italic;">σ</span> is the <span class="texhtml">(<i>n</i> − 1)</span>-dimensional surface measure. </p><p>Conversely, all locally integrable functions satisfying the (volume) mean-value property are both infinitely differentiable and harmonic. </p><p>In terms of <a href="/wiki/Convolution" title="Convolution">convolutions</a>, if <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{r}:={\frac {1}{|B(0,r)|}}\chi _{B(0,r)}={\frac {n}{\omega _{n}r^{n}}}\chi _{B(0,r)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{r}:={\frac {1}{|B(0,r)|}}\chi _{B(0,r)}={\frac {n}{\omega _{n}r^{n}}}\chi _{B(0,r)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a932e8f6938bf515fc28c5d17906ae9eab8ac07" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.458ex; height:6.009ex;" alt="{\displaystyle \chi _{r}:={\frac {1}{|B(0,r)|}}\chi _{B(0,r)}={\frac {n}{\omega _{n}r^{n}}}\chi _{B(0,r)}}"></span> denotes the <a href="/wiki/Indicator_function" title="Indicator function">characteristic function</a> of the ball with radius <span class="texhtml mvar" style="font-style:italic;">r</span> about the origin, normalized so that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \int _{\mathbb {R} ^{n}}\chi _{r}\,dx=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" 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> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \int _{\mathbb {R} ^{n}}\chi _{r}\,dx=1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5100cc2b28c3d32eb623f221b65fa0ae0a035682" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.138ex; height:3.176ex;" alt="{\textstyle \int _{\mathbb {R} ^{n}}\chi _{r}\,dx=1,}"></span> the function <span class="texhtml mvar" style="font-style:italic;">u</span> is harmonic on <span class="texhtml">Ω</span> if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(x)=u*\chi _{r}(x)\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>u</mi> <mo>∗</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(x)=u*\chi _{r}(x)\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c54f5ad82a1d04db2f67d153b7322c4637bb174" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.305ex; height:2.843ex;" alt="{\displaystyle u(x)=u*\chi _{r}(x)\;}"></span> as soon as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(x,r)\subset \Omega .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>⊂</mo> <mi mathvariant="normal">Ω</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B(x,r)\subset \Omega .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70efc98ae1eea1af8896ef306de15501342afdbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.409ex; height:2.843ex;" alt="{\displaystyle B(x,r)\subset \Omega .}"></span> </p><p><b>Sketch of the proof.</b> The proof of the mean-value property of the harmonic functions and its converse follows immediately observing that the non-homogeneous equation, for any <span class="texhtml">0 < <i>s</i> < <i>r</i></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta w=\chi _{r}-\chi _{s}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>w</mi> <mo>=</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta w=\chi _{r}-\chi _{s}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/095cf34af2509f493e8a4521d64d98a4593330df" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.071ex; height:2.509ex;" alt="{\displaystyle \Delta w=\chi _{r}-\chi _{s}\;}"></span> admits an easy explicit solution <span class="texhtml mvar" style="font-style:italic;">w<sub>r,s</sub></span> of class <span class="texhtml"><i>C</i><sup>1,1</sup></span> with compact support in <span class="texhtml"><i>B</i>(0, <i>r</i>)</span>. Thus, if <span class="texhtml mvar" style="font-style:italic;">u</span> is harmonic in <span class="texhtml">Ω</span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=\Delta u*w_{r,s}=u*\Delta w_{r,s}=u*\chi _{r}-u*\chi _{s}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>∗</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mi>u</mi> <mo>∗</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mi>u</mi> <mo>∗</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>−</mo> <mi>u</mi> <mo>∗</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=\Delta u*w_{r,s}=u*\Delta w_{r,s}=u*\chi _{r}-u*\chi _{s}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1f6e591e9b6cf8f8f5e02dd15cb7b46bf02c968" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:44.533ex; height:2.843ex;" alt="{\displaystyle 0=\Delta u*w_{r,s}=u*\Delta w_{r,s}=u*\chi _{r}-u*\chi _{s}\;}"></span> holds in the set <span class="texhtml">Ω<sub><i>r</i></sub></span> of all points <span class="texhtml mvar" style="font-style:italic;">x</span> in <span class="texhtml">Ω</span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {dist} (x,\partial \Omega )>r.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dist</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> <mo>></mo> <mi>r</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {dist} (x,\partial \Omega )>r.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f0a0324f14f141b3afb449a7317c621897a4297" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.723ex; height:2.843ex;" alt="{\displaystyle \operatorname {dist} (x,\partial \Omega )>r.}"></span> </p><p>Since <span class="texhtml mvar" style="font-style:italic;">u</span> is continuous in <span class="texhtml">Ω</span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u*\chi _{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>∗</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u*\chi _{s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a149b211df5e3fc6008384e707a04241f87e1868" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.983ex; height:2.009ex;" alt="{\displaystyle u*\chi _{s}}"></span> converges to <span class="texhtml mvar" style="font-style:italic;">u</span> as <span class="texhtml"><i>s</i> → 0</span> showing the mean value property for <span class="texhtml mvar" style="font-style:italic;">u</span> in <span class="texhtml">Ω</span>. Conversely, if <span class="texhtml mvar" style="font-style:italic;">u</span> is any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{\mathrm {loc} }^{1}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">c</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{\mathrm {loc} }^{1}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e3eb086b0395a2a88d0db0ca1eb3ddf30f0ee47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.47ex; height:3.176ex;" alt="{\displaystyle L_{\mathrm {loc} }^{1}\;}"></span> function satisfying the mean-value property in <span class="texhtml">Ω</span>, that is, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u*\chi _{r}=u*\chi _{s}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>∗</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mi>u</mi> <mo>∗</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u*\chi _{r}=u*\chi _{s}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8743bfe477815f2f8e064adb4291dcdbf06e8036" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.68ex; height:2.009ex;" alt="{\displaystyle u*\chi _{r}=u*\chi _{s}\;}"></span> holds in <span class="texhtml">Ω<sub><i>r</i></sub></span> for all <span class="texhtml">0 < <i>s</i> < <i>r</i></span> then, iterating <span class="texhtml mvar" style="font-style:italic;">m</span> times the convolution with <span class="texhtml">χ<sub><i>r</i></sub></span> one has: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=u*\chi _{r}=u*\chi _{r}*\cdots *\chi _{r}\,,\qquad x\in \Omega _{mr},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mi>u</mi> <mo>∗</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mi>u</mi> <mo>∗</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>∗</mo> <mo>⋯</mo> <mo>∗</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="2em" /> <mi>x</mi> <mo>∈</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>r</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u=u*\chi _{r}=u*\chi _{r}*\cdots *\chi _{r}\,,\qquad x\in \Omega _{mr},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8be56c1bc941bb96e91f78cae0fe17bdec8544a0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:43.953ex; height:2.509ex;" alt="{\displaystyle u=u*\chi _{r}=u*\chi _{r}*\cdots *\chi _{r}\,,\qquad x\in \Omega _{mr},}"></span> so that <span class="texhtml mvar" style="font-style:italic;">u</span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{m-1}(\Omega _{mr})\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>r</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{m-1}(\Omega _{mr})\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6cb74d047548a9b3df09ca231eae964143373bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.122ex; height:3.176ex;" alt="{\displaystyle C^{m-1}(\Omega _{mr})\;}"></span> because the <span class="texhtml mvar" style="font-style:italic;">m</span>-fold iterated convolution of <span class="texhtml">χ<sub><i>r</i></sub></span> is of class <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{m-1}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{m-1}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cb7a2d01fc60864ceab46cbf0f3ca640c335576" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.219ex; height:2.676ex;" alt="{\displaystyle C^{m-1}\;}"></span> with support <span class="texhtml"><i>B</i>(0, <i>mr</i>)</span>. Since <span class="texhtml mvar" style="font-style:italic;">r</span> and <span class="texhtml mvar" style="font-style:italic;">m</span> are arbitrary, <span class="texhtml mvar" style="font-style:italic;">u</span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }(\Omega )\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{\infty }(\Omega )\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a1fe8e46c2692826af248bbe160425991b5af16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.806ex; height:2.843ex;" alt="{\displaystyle C^{\infty }(\Omega )\;}"></span> too. Moreover, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta u*w_{r,s}=u*\Delta w_{r,s}=u*\chi _{r}-u*\chi _{s}=0\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>∗</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mi>u</mi> <mo>∗</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mi>u</mi> <mo>∗</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>−</mo> <mi>u</mi> <mo>∗</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta u*w_{r,s}=u*\Delta w_{r,s}=u*\chi _{r}-u*\chi _{s}=0\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b224130a1ef4876f1d210d6d226fb9b555039f96" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:44.533ex; height:2.843ex;" alt="{\displaystyle \Delta u*w_{r,s}=u*\Delta w_{r,s}=u*\chi _{r}-u*\chi _{s}=0\;}"></span> for all <span class="texhtml">0 < <i>s</i> < <i>r</i></span> so that <span class="texhtml">Δ<i>u</i> = 0</span> in <span class="texhtml">Ω</span> by the fundamental theorem of the calculus of variations, proving the equivalence between harmonicity and mean-value property. </p><p>This statement of the mean value property can be generalized as follows: If <span class="texhtml mvar" style="font-style:italic;">h</span> is any spherically symmetric function <a href="/wiki/Support_(mathematics)" title="Support (mathematics)">supported</a> in <span class="texhtml"><i>B</i>(<i>x</i>, <i>r</i>)</span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \int h=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∫</mo> <mi>h</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \int h=1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a90bd6488351a52b477d623f537c905e4c284d1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.052ex; height:3.176ex;" alt="{\textstyle \int h=1,}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(x)=h*u(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>h</mi> <mo>∗</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(x)=h*u(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5209fe6d440b06f9cb8976d381ff061aa153e942" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.216ex; height:2.843ex;" alt="{\displaystyle u(x)=h*u(x).}"></span> In other words, we can take the weighted average of <span class="texhtml mvar" style="font-style:italic;">u</span> about a point and recover <span class="texhtml"><i>u</i>(<i>x</i>)</span>. In particular, by taking <span class="texhtml mvar" style="font-style:italic;">h</span> to be a <span class="texhtml"><i>C</i><sup>∞</sup></span> function, we can recover the value of <span class="texhtml mvar" style="font-style:italic;">u</span> at any point even if we only know how <span class="texhtml mvar" style="font-style:italic;">u</span> acts as a <a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">distribution</a>. See <a href="/wiki/Weyl%27s_lemma_(Laplace_equation)" title="Weyl's lemma (Laplace equation)">Weyl's lemma</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Harnack's_inequality"><span id="Harnack.27s_inequality"></span>Harnack's inequality</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_function&action=edit&section=9" title="Edit section: Harnack's inequality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\subset {\overline {V}}\subset \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>⊂</mo> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\subset {\overline {V}}\subset \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c3d768083748a09445f143127eff062f8a97f88" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.845ex; height:3.009ex;" alt="{\displaystyle V\subset {\overline {V}}\subset \Omega }"></span> be a connected set in a bounded domain <span class="texhtml">Ω</span>. Then for every non-negative harmonic function <span class="texhtml mvar" style="font-style:italic;">u</span>, <a href="/wiki/Harnack%27s_inequality" title="Harnack's inequality">Harnack's inequality</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sup _{V}u\leq C\inf _{V}u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </munder> <mi>u</mi> <mo>≤</mo> <mi>C</mi> <munder> <mo movablelimits="true" form="prefix">inf</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </munder> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sup _{V}u\leq C\inf _{V}u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1f4c8cb9b1da2f8d45c01b5c5e074eb11c981e5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.838ex; height:4.343ex;" alt="{\displaystyle \sup _{V}u\leq C\inf _{V}u}"></span> holds for some constant <span class="texhtml mvar" style="font-style:italic;">C</span> that depends only on <span class="texhtml mvar" style="font-style:italic;">V</span> and <span class="texhtml">Ω</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Removal_of_singularities">Removal of singularities</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_function&action=edit&section=10" title="Edit section: Removal of singularities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following principle of removal of singularities holds for harmonic functions. If <span class="texhtml mvar" style="font-style:italic;">f</span> is a harmonic function defined on a dotted open subset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega \smallsetminus \{x_{0}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>∖</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega \smallsetminus \{x_{0}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5f170ef6641002a35ab73a6f734ee10d63c886f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.227ex; height:2.843ex;" alt="{\displaystyle \Omega \smallsetminus \{x_{0}\}}"></span> of <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>⁠</span>, which is less singular at <span class="texhtml"><i>x</i><sub>0</sub></span> than the fundamental solution (for <span class="texhtml"><i>n</i> > 2</span>), that is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=o\left(\vert x-x_{0}\vert ^{2-n}\right),\qquad {\text{as }}x\to x_{0},}"> <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>o</mi> <mrow> <mo>(</mo> <mrow> <mo fence="false" stretchy="false">|</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo fence="false" stretchy="false">|</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−</mo> <mi>n</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>as </mtext> </mrow> <mi>x</mi> <mo stretchy="false">→</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=o\left(\vert x-x_{0}\vert ^{2-n}\right),\qquad {\text{as }}x\to x_{0},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f4650abeaf7dc108c1f6d2b448dbf825457a0a4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.027ex; height:3.343ex;" alt="{\displaystyle f(x)=o\left(\vert x-x_{0}\vert ^{2-n}\right),\qquad {\text{as }}x\to x_{0},}"></span> then <span class="texhtml mvar" style="font-style:italic;">f</span> extends to a harmonic function on <span class="texhtml">Ω</span> (compare <a href="/wiki/Removable_singularity#Riemann's_theorem" title="Removable singularity">Riemann's theorem</a> for functions of a complex variable). </p> <div class="mw-heading mw-heading3"><h3 id="Liouville's_theorem"><span id="Liouville.27s_theorem"></span>Liouville's theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_function&action=edit&section=11" title="Edit section: Liouville's theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Theorem</b>: If <span class="texhtml mvar" style="font-style:italic;">f</span> is a harmonic function defined on all of <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>⁠</span> which is bounded above or bounded below, then <span class="texhtml mvar" style="font-style:italic;">f</span> is constant. </p><p>(Compare <a href="/wiki/Liouville%27s_theorem_(complex_analysis)" title="Liouville's theorem (complex analysis)">Liouville's theorem for functions of a complex variable</a>). </p><p><a href="/wiki/Edward_Nelson" title="Edward Nelson">Edward Nelson</a> gave a particularly short proof of this theorem for the case of bounded functions,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> using the mean value property mentioned above: </p> <blockquote><p>Given two points, choose two balls with the given points as centers and of equal radius. If the radius is large enough, the two balls will coincide except for an arbitrarily small proportion of their volume. Since <span class="texhtml mvar" style="font-style:italic;">f </span> is bounded, the averages of it over the two balls are arbitrarily close, and so <span class="texhtml mvar" style="font-style:italic;">f </span> assumes the same value at any two points. </p></blockquote> <p>The proof can be adapted to the case where the harmonic function <span class="texhtml mvar" style="font-style:italic;">f </span> is merely bounded above or below. By adding a constant and possibly multiplying by –1, we may assume that <span class="texhtml mvar" style="font-style:italic;">f </span> is non-negative. Then for any two points <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span>, and any positive number <span class="texhtml mvar" style="font-style:italic;">R</span>, we let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=R+d(x,y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>R</mi> <mo>+</mo> <mi>d</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 r=R+d(x,y).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d01ed8cb1fc3bd40b87731dc2bad6cdeeb935a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.943ex; height:2.843ex;" alt="{\displaystyle r=R+d(x,y).}"></span> We then consider the balls <span class="texhtml"><i>B<sub>R</sub></i>(<i>x</i>)</span> and <span class="texhtml"><i>B<sub>r</sub></i>(<i>y</i>)</span> where by the triangle inequality, the first ball is contained in the second. </p><p>By the averaging property and the monotonicity of the integral, we have <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {1}{\operatorname {vol} (B_{R})}}\int _{B_{R}(x)}f(z)\,dz\leq {\frac {1}{\operatorname {vol} (B_{R})}}\int _{B_{r}(y)}f(z)\,dz.}"> <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> <mi>vol</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>vol</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {1}{\operatorname {vol} (B_{R})}}\int _{B_{R}(x)}f(z)\,dz\leq {\frac {1}{\operatorname {vol} (B_{R})}}\int _{B_{r}(y)}f(z)\,dz.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4c9a570205b630f6f72f5b3ecc4964ddb1ffec4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:56.199ex; height:6.009ex;" alt="{\displaystyle f(x)={\frac {1}{\operatorname {vol} (B_{R})}}\int _{B_{R}(x)}f(z)\,dz\leq {\frac {1}{\operatorname {vol} (B_{R})}}\int _{B_{r}(y)}f(z)\,dz.}"></span> (Note that since <span class="texhtml">vol <i>B<sub>R</sub></i>(<i>x</i>)</span> is independent of <span class="texhtml mvar" style="font-style:italic;">x</span>, we denote it merely as <span class="texhtml">vol <i>B<sub>R</sub></i></span>.) In the last expression, we may multiply and divide by <span class="texhtml">vol <i>B<sub>r</sub></i></span> and use the averaging property again, to obtain <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\leq {\frac {\operatorname {vol} (B_{r})}{\operatorname {vol} (B_{R})}}f(y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>vol</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>vol</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\leq {\frac {\operatorname {vol} (B_{r})}{\operatorname {vol} (B_{R})}}f(y).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5cc4769dd46ed72ba3ded276bc71e88d9060a76" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.332ex; height:6.509ex;" alt="{\displaystyle f(x)\leq {\frac {\operatorname {vol} (B_{r})}{\operatorname {vol} (B_{R})}}f(y).}"></span> But as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\rightarrow \infty ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\rightarrow \infty ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2b5097b37a31bb5a64b9278f726ee34c873624c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.349ex; height:2.509ex;" alt="{\displaystyle R\rightarrow \infty ,}"></span> the quantity <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\operatorname {vol} (B_{r})}{\operatorname {vol} (B_{R})}}={\frac {\left(R+d(x,y)\right)^{n}}{R^{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>vol</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>vol</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mi>R</mi> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {vol} (B_{r})}{\operatorname {vol} (B_{R})}}={\frac {\left(R+d(x,y)\right)^{n}}{R^{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c6f6798bf9482aad95e38a9f324611ec594223c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.037ex; height:6.509ex;" alt="{\displaystyle {\frac {\operatorname {vol} (B_{r})}{\operatorname {vol} (B_{R})}}={\frac {\left(R+d(x,y)\right)^{n}}{R^{n}}}}"></span> tends to 1. Thus, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\leq f(y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\leq f(y).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8d015e6c1f3e6af0bcd329f462ccd9267f72737" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.406ex; height:2.843ex;" alt="{\displaystyle f(x)\leq f(y).}"></span> The same argument with the roles of <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span> reversed shows that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(y)\leq f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(y)\leq f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0123e88fe6f496eb9cf546248bad5322d1c65c90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.759ex; height:2.843ex;" alt="{\displaystyle f(y)\leq f(x)}"></span>, so that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=f(y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=f(y).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1779b3b82f8ea31cb8753ba439d40d53a41df60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.406ex; height:2.843ex;" alt="{\displaystyle f(x)=f(y).}"></span> </p><p>Another proof uses the fact that given a <a href="/wiki/Wiener_process" title="Wiener process">Brownian motion</a> <span class="texhtml mvar" style="font-style:italic;">B<sub>t</sub></span> in <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7035fcb9fe3ebecc6bc9f372f82d0352202c8bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{n},}"></span>⁠</span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{0}=x_{0},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{0}=x_{0},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89cacfc247178ac5a5a6fe9b221090ee01dc5b34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.947ex; height:2.509ex;" alt="{\displaystyle B_{0}=x_{0},}"></span> we have <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E[f(B_{t})]=f(x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E[f(B_{t})]=f(x_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e4e041d925744c36c7e8761cb60df0eaa0bf22b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.317ex; height:2.843ex;" alt="{\displaystyle E[f(B_{t})]=f(x_{0})}"></span> for all <span class="texhtml"><i>t</i> ≥ 0</span>. In words, it says that a harmonic function defines a <a href="/wiki/Martingale_(probability_theory)" title="Martingale (probability theory)">martingale</a> for the Brownian motion. Then a <a href="/wiki/Coupling_(probability)" title="Coupling (probability)">probabilistic coupling</a> argument finishes the proof.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_function&action=edit&section=12" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Weakly_harmonic_function">Weakly harmonic function</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_function&action=edit&section=13" title="Edit section: Weakly harmonic function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A function (or, more generally, a <a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">distribution</a>) is <a href="/wiki/Weakly_harmonic" class="mw-redirect" title="Weakly harmonic">weakly harmonic</a> if it satisfies Laplace's equation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta f=0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>f</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta f=0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e8cb3aa9955ef5133611f45b17f68256bcfd67" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.862ex; height:2.509ex;" alt="{\displaystyle \Delta f=0\,}"></span> in a <a href="/wiki/Weak_derivative" title="Weak derivative">weak</a> sense (or, equivalently, in the sense of distributions). A weakly harmonic function coincides almost everywhere with a strongly harmonic function, and is in particular smooth. A weakly harmonic distribution is precisely the distribution associated to a strongly harmonic function, and so also is smooth. This is <a href="/wiki/Weyl%27s_lemma_(Laplace_equation)" title="Weyl's lemma (Laplace equation)">Weyl's lemma</a>. </p><p>There are other <a href="/wiki/Weak_formulation" title="Weak formulation">weak formulations</a> of Laplace's equation that are often useful. One of which is <a href="/wiki/Dirichlet%27s_principle" title="Dirichlet's principle">Dirichlet's principle</a>, representing harmonic functions in the <a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev space</a> <span class="texhtml"><i>H</i><sup>1</sup>(Ω)</span> as the minimizers of the <a href="/wiki/Dirichlet_energy" title="Dirichlet energy">Dirichlet energy</a> integral <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J(u):=\int _{\Omega }|\nabla u|^{2}\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <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> <mi mathvariant="normal">∇</mi> <mi>u</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J(u):=\int _{\Omega }|\nabla u|^{2}\,dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b3287d934a45f37f678c7092ceaee3cb206a0ba" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20ex; height:5.676ex;" alt="{\displaystyle J(u):=\int _{\Omega }|\nabla u|^{2}\,dx}"></span> with respect to local variations, that is, all functions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in H^{1}(\Omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>∈</mo> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u\in H^{1}(\Omega )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65bd83cb973fa409d513986bada2dee73d529eb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.815ex; height:3.176ex;" alt="{\displaystyle u\in H^{1}(\Omega )}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J(u)\leq J(u+v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>J</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>+</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J(u)\leq J(u+v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d05f23345f725efc2905d0c1931fc283691cc6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.287ex; height:2.843ex;" alt="{\displaystyle J(u)\leq J(u+v)}"></span> holds for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in C_{c}^{\infty }(\Omega ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</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 mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\in C_{c}^{\infty }(\Omega ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1e05cc7488b8e009391cf19f0f99c69e1c6c01b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.776ex; height:2.843ex;" alt="{\displaystyle v\in C_{c}^{\infty }(\Omega ),}"></span> or equivalently, for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in H_{0}^{1}(\Omega ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>∈</mo> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\in H_{0}^{1}(\Omega ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cae6f39f00143fc95faf324a8250c7564d181b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.26ex; height:3.176ex;" alt="{\displaystyle v\in H_{0}^{1}(\Omega ).}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Harmonic_functions_on_manifolds">Harmonic functions on manifolds</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_function&action=edit&section=14" title="Edit section: Harmonic functions on manifolds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Harmonic functions can be defined on an arbitrary <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a>, using the <a href="/wiki/Laplace%E2%80%93Beltrami_operator" title="Laplace–Beltrami operator">Laplace–Beltrami operator</a> <span class="texhtml">Δ</span>. In this context, a function is called <i>harmonic</i> if <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \Delta f=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi mathvariant="normal">Δ</mi> <mi>f</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \Delta f=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e2de6ab827f22eb955cd5f81799edd04bc679d8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.703ex; height:2.509ex;" alt="{\displaystyle \ \Delta f=0.}"></span> Many of the properties of harmonic functions on domains in Euclidean space carry over to this more general setting, including the mean value theorem (over <a href="/wiki/Geodesic" title="Geodesic">geodesic</a> balls), the maximum principle, and the Harnack inequality. With the exception of the mean value theorem, these are easy consequences of the corresponding results for general linear <a href="/wiki/Elliptic_partial_differential_equation" title="Elliptic partial differential equation">elliptic partial differential equations</a> of the second order. </p> <div class="mw-heading mw-heading3"><h3 id="Subharmonic_functions">Subharmonic functions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_function&action=edit&section=15" title="Edit section: Subharmonic functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <span class="texhtml"><i>C</i><sup>2</sup></span> function that satisfies <span class="texhtml">Δ<i>f</i> ≥ 0</span> is called subharmonic. This condition guarantees that the maximum principle will hold, although other properties of harmonic functions may fail. More generally, a function is subharmonic if and only if, in the interior of any ball in its domain, its graph lies below that of the harmonic function interpolating its boundary values on the ball. </p> <div class="mw-heading mw-heading3"><h3 id="Harmonic_forms">Harmonic forms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_function&action=edit&section=16" title="Edit section: Harmonic forms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One generalization of the study of harmonic functions is the study of <a href="/wiki/Harmonic_form" class="mw-redirect" title="Harmonic form">harmonic forms</a> on <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifolds</a>, and it is related to the study of <a href="/wiki/Cohomology" title="Cohomology">cohomology</a>. Also, it is possible to define harmonic vector-valued functions, or harmonic maps of two Riemannian manifolds, which are critical points of a generalized Dirichlet energy functional (this includes harmonic functions as a special case, a result known as <a href="/wiki/Dirichlet_principle" class="mw-redirect" title="Dirichlet principle">Dirichlet principle</a>). This kind of harmonic map appears in the theory of minimal surfaces. For example, a curve, that is, a map from an interval in <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>⁠</span> to a Riemannian manifold, is a harmonic map if and only if it is a <a href="/wiki/Geodesic" title="Geodesic">geodesic</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Harmonic_maps_between_manifolds">Harmonic maps between manifolds</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_function&action=edit&section=17" title="Edit section: Harmonic maps between manifolds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Harmonic_map" title="Harmonic map">Harmonic map</a></div> <p>If <span class="texhtml mvar" style="font-style:italic;">M</span> and <span class="texhtml mvar" style="font-style:italic;">N</span> are two Riemannian manifolds, then a harmonic map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u:M\to N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>:</mo> <mi>M</mi> <mo stretchy="false">→</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u:M\to N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f332b0d99a1aa8414eb34892bd842e033e0ad60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.387ex; height:2.176ex;" alt="{\displaystyle u:M\to N}"></span> is defined to be a critical point of the Dirichlet energy <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D[u]={\frac {1}{2}}\int _{M}\left\|du\right\|^{2}\,d\operatorname {Vol} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">[</mo> <mi>u</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <msup> <mrow> <mo symmetric="true">‖</mo> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>Vol</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D[u]={\frac {1}{2}}\int _{M}\left\|du\right\|^{2}\,d\operatorname {Vol} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbbacda9536efc8200c39ac87f74b1d65fbad3cd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.138ex; height:5.676ex;" alt="{\displaystyle D[u]={\frac {1}{2}}\int _{M}\left\|du\right\|^{2}\,d\operatorname {Vol} }"></span> in which <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle du:TM\to TN}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>u</mi> <mo>:</mo> <mi>T</mi> <mi>M</mi> <mo stretchy="false">→</mo> <mi>T</mi> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle du:TM\to TN}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f55dcfcfdc7e0e923300062bdae5ebb4c69c302" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.875ex; height:2.176ex;" alt="{\displaystyle du:TM\to TN}"></span> is the differential of <span class="texhtml mvar" style="font-style:italic;">u</span>, and the norm is that induced by the metric on <span class="texhtml mvar" style="font-style:italic;">M</span> and that on <span class="texhtml mvar" style="font-style:italic;">N</span> on the tensor product bundle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{\ast }M\otimes u^{-1}TN.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>M</mi> <mo>⊗</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>T</mi> <mi>N</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{\ast }M\otimes u^{-1}TN.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d090f7424da4a7895a2598cb274f468d2d8a692b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.066ex; height:2.843ex;" alt="{\displaystyle T^{\ast }M\otimes u^{-1}TN.}"></span> </p><p>Important special cases of harmonic maps between manifolds include <a href="/wiki/Minimal_surface" title="Minimal surface">minimal surfaces</a>, which are precisely the harmonic immersions of a surface into three-dimensional Euclidean space. More generally, minimal submanifolds are harmonic immersions of one manifold in another. <a href="/wiki/Harmonic_coordinates" title="Harmonic coordinates">Harmonic coordinates</a> are a harmonic <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphism</a> from a manifold to an open subset of a Euclidean space of the same dimension. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_function&action=edit&section=18" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 20em;"> <ul><li><a href="/wiki/Balayage" title="Balayage">Balayage</a></li> <li><a href="/wiki/Biharmonic_map" title="Biharmonic map">Biharmonic map</a></li> <li><a href="/wiki/Dirichlet_problem" title="Dirichlet problem">Dirichlet problem</a></li> <li><a href="/wiki/Harmonic_morphism" title="Harmonic morphism">Harmonic morphism</a></li> <li><a href="/wiki/Harmonic_polynomial" title="Harmonic polynomial">Harmonic polynomial</a></li> <li><a href="/wiki/Heat_equation" title="Heat equation">Heat equation</a></li> <li><a href="/wiki/Laplace_equation_for_irrotational_flow" title="Laplace equation for irrotational flow">Laplace equation for irrotational flow</a></li> <li><a href="/wiki/Poisson%27s_equation" title="Poisson's equation">Poisson's equation</a></li> <li><a href="/wiki/Quadrature_domains" title="Quadrature domains">Quadrature domains</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_function&action=edit&section=19" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><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="CITEREFAxlerBourdonRamey2001" class="citation book cs1">Axler, Sheldon; Bourdon, Paul; Ramey, Wade (2001). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/harmonicfunction00axle_418"><i>Harmonic Function Theory</i></a></span>. New York: Springer. p. <a rel="nofollow" class="external text" href="https://archive.org/details/harmonicfunction00axle_418/page/n34">25</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-95218-7" title="Special:BookSources/0-387-95218-7"><bdi>0-387-95218-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Harmonic+Function+Theory&rft.place=New+York&rft.pages=25&rft.pub=Springer&rft.date=2001&rft.isbn=0-387-95218-7&rft.aulast=Axler&rft.aufirst=Sheldon&rft.au=Bourdon%2C+Paul&rft.au=Ramey%2C+Wade&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fharmonicfunction00axle_418&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic+function" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNelson1961" class="citation journal cs1">Nelson, Edward (1961). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9939-1961-0259149-4">"A proof of Liouville's theorem"</a>. <i><a href="/wiki/Proceedings_of_the_American_Mathematical_Society" title="Proceedings of the American Mathematical Society">Proceedings of the American Mathematical Society</a></i>. <b>12</b> (6): 995. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9939-1961-0259149-4">10.1090/S0002-9939-1961-0259149-4</a></span>.</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+American+Mathematical+Society&rft.atitle=A+proof+of+Liouville%27s+theorem&rft.volume=12&rft.issue=6&rft.pages=995&rft.date=1961&rft_id=info%3Adoi%2F10.1090%2FS0002-9939-1961-0259149-4&rft.aulast=Nelson&rft.aufirst=Edward&rft_id=https%3A%2F%2Fdoi.org%2F10.1090%252FS0002-9939-1961-0259149-4&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic+function" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20210508091536/https://blameitontheanalyst.wordpress.com/2012/01/24/probabilistic-coupling/">"Probabilistic Coupling"</a>. <i>Blame It On The Analyst</i>. 2012-01-24. 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Retrieved <span class="nowrap">2022-05-26</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Blame+It+On+The+Analyst&rft.atitle=Probabilistic+Coupling&rft.date=2012-01-24&rft_id=https%3A%2F%2Fblameitontheanalyst.wordpress.com%2F2012%2F01%2F24%2Fprobabilistic-coupling%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic+function" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_function&action=edit&section=20" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEvans1998" class="citation cs2"><a href="/wiki/Lawrence_C._Evans" title="Lawrence C. Evans">Evans, Lawrence C.</a> (1998), <i>Partial Differential Equations</i>, American Mathematical Society</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Partial+Differential+Equations&rft.pub=American+Mathematical+Society&rft.date=1998&rft.aulast=Evans&rft.aufirst=Lawrence+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic+function" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGilbargTrudinger2001" class="citation cs2"><a href="/wiki/David_Gilbarg" title="David Gilbarg">Gilbarg, David</a>; <a href="/wiki/Neil_Trudinger" title="Neil Trudinger">Trudinger, Neil</a> (12 January 2001), <i>Elliptic Partial Differential Equations of Second Order</i>, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-41160-7" title="Special:BookSources/3-540-41160-7"><bdi>3-540-41160-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elliptic+Partial+Differential+Equations+of+Second+Order&rft.pub=Springer&rft.date=2001-01-12&rft.isbn=3-540-41160-7&rft.aulast=Gilbarg&rft.aufirst=David&rft.au=Trudinger%2C+Neil&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic+function" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHanLin2000" class="citation cs2">Han, Q.; Lin, F. (2000), <i>Elliptic Partial Differential Equations</i>, American Mathematical Society</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elliptic+Partial+Differential+Equations&rft.pub=American+Mathematical+Society&rft.date=2000&rft.aulast=Han&rft.aufirst=Q.&rft.au=Lin%2C+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic+function" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJost2005" class="citation cs2">Jost, Jürgen (2005), <i>Riemannian Geometry and Geometric Analysis</i> (4th ed.), Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-25907-7" title="Special:BookSources/978-3-540-25907-7"><bdi>978-3-540-25907-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Riemannian+Geometry+and+Geometric+Analysis&rft.place=Berlin%2C+New+York&rft.edition=4th&rft.pub=Springer-Verlag&rft.date=2005&rft.isbn=978-3-540-25907-7&rft.aulast=Jost&rft.aufirst=J%C3%BCrgen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic+function" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAxlerBourdonRamey2001" class="citation cs2">Axler, Sheldon; Bourdon, Paul; Ramey, Wade (2001), <i>Harmonic Function Theory</i>, vol. 137 (Second ed.), New York: Springer-Verlag, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4757-8137-3">10.1007/978-1-4757-8137-3</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-95218-7" title="Special:BookSources/0-387-95218-7"><bdi>0-387-95218-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Harmonic+Function+Theory&rft.place=New+York&rft.edition=Second&rft.pub=Springer-Verlag&rft.date=2001&rft_id=info%3Adoi%2F10.1007%2F978-1-4757-8137-3&rft.isbn=0-387-95218-7&rft.aulast=Axler&rft.aufirst=Sheldon&rft.au=Bourdon%2C+Paul&rft.au=Ramey%2C+Wade&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic+function" class="Z3988"></span>.</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_function&action=edit&section=21" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><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=Harmonic_function">"Harmonic function"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <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=Harmonic+function&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DHarmonic_function&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic+function" class="Z3988"></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Harmonic_Function"><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/HarmonicFunction.html">"Harmonic Function"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</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=Harmonic+Function&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FHarmonicFunction.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic+function" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="http://www.axler.net/HFT.html">Harmonic Function Theory by S.Axler, Paul Bourdon, and Wade Ramey</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 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Harmonic function - Wikipedia