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class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contidos</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mover á barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">agochar</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">Inicio</div> </a> </li> <li id="toc-Converxencia_a_unha_función_periódica" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Converxencia_a_unha_función_periódica"> <div 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mw-ui-icon-wikimedia-expand"></span> <span>Mostrar ou agochar a subsección "Aplicacións"</span> </button> <ul id="toc-Aplicacións-sublist" class="vector-toc-list"> <li id="toc-Solución_de_ecuacións_diferenciais" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Solución_de_ecuacións_diferenciais"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Solución de ecuacións diferenciais</span> </div> </a> <ul id="toc-Solución_de_ecuacións_diferenciais-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Enxeñaría" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Enxeñaría"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Enxeñaría</span> </div> </a> <ul id="toc-Enxeñaría-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Algunhas_consecuencias_positivas_das_propiedades_de_homomorfismo_de_exp" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Algunhas_consecuencias_positivas_das_propiedades_de_homomorfismo_de_exp"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Algunhas consecuencias positivas das propiedades de homomorfismo de exp</span> </div> </a> <ul id="toc-Algunhas_consecuencias_positivas_das_propiedades_de_homomorfismo_de_exp-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formulación_xeral" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Formulación_xeral"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Formulación xeral</span> </div> </a> <ul id="toc-Formulación_xeral-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Véxase_tamén" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Véxase_tamén"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Véxase tamén</span> </div> </a> <button aria-controls="toc-Véxase_tamén-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>Mostrar ou agochar a subsección "Véxase tamén"</span> </button> <ul id="toc-Véxase_tamén-sublist" class="vector-toc-list"> <li id="toc-Outros_artigos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Outros_artigos"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Outros artigos</span> </div> </a> <ul id="toc-Outros_artigos-sublist" class="vector-toc-list"> </ul> </li> </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="Contidos" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Mostrar ou agochar a táboa de contidos" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Mostrar ou agochar a táboa de contidos</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Serie de Fourier</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Ir a un artigo noutra lingua. Dispoñible en 65 linguas" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-65" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">65 linguas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Fourier-reeks" title="Fourier-reeks – afrikaans" lang="af" hreflang="af" data-title="Fourier-reeks" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8B%A8%E1%8D%8E%E1%88%AA%E1%8B%A8%E1%88%AD_%E1%8B%9D%E1%88%AD%E1%8B%9D%E1%88%AD" title="የፎሪየር ዝርዝር – amhárico" lang="am" hreflang="am" data-title="የፎሪየር ዝርዝር" data-language-autonym="አማርኛ" data-language-local-name="amhárico" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AA%D8%B3%D9%84%D8%B3%D9%84%D8%A9_%D9%81%D9%88%D8%B1%D9%8A%D9%8A%D9%87" title="متسلسلة فورييه – árabe" lang="ar" hreflang="ar" data-title="متسلسلة فورييه" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Serie_de_Fourier" title="Serie de Fourier – asturiano" lang="ast" hreflang="ast" data-title="Serie de Fourier" data-language-autonym="Asturianu" data-language-local-name="asturiano" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Furye_s%C4%B1ralar%C4%B1" title="Furye sıraları – acerbaixano" lang="az" hreflang="az" data-title="Furye sıraları" data-language-autonym="Azərbaycanca" data-language-local-name="acerbaixano" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A4%D1%83%D1%80%D1%8C%D0%B5_%D1%80%D3%99%D1%82%D0%B5" title="Фурье рәте – baxkir" lang="ba" hreflang="ba" data-title="Фурье рәте" data-language-autonym="Башҡортса" data-language-local-name="baxkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B4_%D0%A4%D1%83%D1%80%E2%80%99%D0%B5" title="Рад Фур’е – belaruso" lang="be" hreflang="be" data-title="Рад Фур’е" data-language-autonym="Беларуская" data-language-local-name="belaruso" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A0%D0%B5%D0%B4_%D0%BD%D0%B0_%D0%A4%D1%83%D1%80%D0%B8%D0%B5" title="Ред на Фурие – búlgaro" lang="bg" hreflang="bg" data-title="Ред на Фурие" data-language-autonym="Български" data-language-local-name="búlgaro" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AB%E0%A7%81%E0%A6%B0%E0%A6%BF%E0%A6%AF%E0%A6%BC%E0%A6%BE%E0%A6%B0_%E0%A6%A7%E0%A6%BE%E0%A6%B0%E0%A6%BE" title="ফুরিয়ার ধারা – bengalí" lang="bn" hreflang="bn" data-title="ফুরিয়ার ধারা" data-language-autonym="বাংলা" data-language-local-name="bengalí" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Fourierov_red" title="Fourierov red – bosníaco" lang="bs" hreflang="bs" data-title="Fourierov red" data-language-autonym="Bosanski" data-language-local-name="bosníaco" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/S%C3%A8rie_de_Fourier" title="Sèrie de Fourier – catalán" lang="ca" hreflang="ca" data-title="Sèrie de Fourier" data-language-autonym="Català" data-language-local-name="catalán" 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/Fourierova_%C5%99ada" title="Fourierova řada – checo" lang="cs" hreflang="cs" data-title="Fourierova řada" data-language-autonym="Čeština" data-language-local-name="checo" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A4%D1%83%D1%80%D1%8C%D0%B5_%D1%80%D0%B5%D1%87%C4%95" title="Фурье речĕ – chuvaxo" lang="cv" hreflang="cv" data-title="Фурье речĕ" data-language-autonym="Чӑвашла" data-language-local-name="chuvaxo" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Cyfres_Fourier" title="Cyfres Fourier – galés" lang="cy" hreflang="cy" data-title="Cyfres Fourier" data-language-autonym="Cymraeg" data-language-local-name="galés" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Fourierr%C3%A6kke" title="Fourierrække – dinamarqués" lang="da" hreflang="da" data-title="Fourierrække" data-language-autonym="Dansk" data-language-local-name="dinamarqués" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Fourierreihe" title="Fourierreihe – alemán" lang="de" hreflang="de" data-title="Fourierreihe" data-language-autonym="Deutsch" data-language-local-name="alemán" 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%A3%CE%B5%CE%B9%CF%81%CE%AD%CF%82_%CE%A6%CE%BF%CF%85%CF%81%CE%B9%CE%AD" title="Σειρές Φουριέ – grego" lang="el" hreflang="el" data-title="Σειρές Φουριέ" data-language-autonym="Ελληνικά" data-language-local-name="grego" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Fourier_series" title="Fourier series – inglés" lang="en" hreflang="en" data-title="Fourier series" data-language-autonym="English" data-language-local-name="inglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Vico_de_Fourier" title="Vico de Fourier – esperanto" lang="eo" hreflang="eo" data-title="Vico de Fourier" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Serie_de_Fourier" title="Serie de Fourier – español" lang="es" hreflang="es" data-title="Serie de Fourier" data-language-autonym="Español" data-language-local-name="español" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Fourier%27_rida" title="Fourier' rida – estoniano" lang="et" hreflang="et" data-title="Fourier' rida" data-language-autonym="Eesti" data-language-local-name="estoniano" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Fourierren_serie" title="Fourierren serie – éuscaro" lang="eu" hreflang="eu" data-title="Fourierren serie" data-language-autonym="Euskara" data-language-local-name="éuscaro" 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%B3%D8%B1%DB%8C_%D9%81%D9%88%D8%B1%DB%8C%D9%87" title="سری فوریه – persa" lang="fa" hreflang="fa" data-title="سری فوریه" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Fourier%E2%80%99n_sarja" title="Fourier’n sarja – finés" lang="fi" hreflang="fi" data-title="Fourier’n sarja" data-language-autonym="Suomi" data-language-local-name="finés" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/S%C3%A9rie_de_Fourier" title="Série de Fourier – francés" lang="fr" hreflang="fr" data-title="Série de Fourier" data-language-autonym="Français" data-language-local-name="francés" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%98%D7%95%D7%A8_%D7%A4%D7%95%D7%A8%D7%99%D7%99%D7%94" title="טור פורייה – hebreo" lang="he" hreflang="he" data-title="טור פורייה" data-language-autonym="עברית" data-language-local-name="hebreo" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AB%E0%A4%BC%E0%A5%82%E0%A4%B0%E0%A5%8D%E0%A4%AF%E0%A5%87_%E0%A4%B6%E0%A5%8D%E0%A4%B0%E0%A5%87%E0%A4%A3%E0%A5%80" title="फ़ूर्ये श्रेणी – hindi" lang="hi" hreflang="hi" data-title="फ़ूर्ये श्रेणी" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Fourier-sor" title="Fourier-sor – húngaro" lang="hu" hreflang="hu" data-title="Fourier-sor" data-language-autonym="Magyar" data-language-local-name="húngaro" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%96%D5%B8%D6%82%D6%80%D5%AB%D5%A5%D5%AB_%D5%B7%D5%A1%D6%80%D6%84" title="Ֆուրիեի շարք – armenio" lang="hy" hreflang="hy" data-title="Ֆուրիեի շարք" data-language-autonym="Հայերեն" data-language-local-name="armenio" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Deret_Fourier" title="Deret Fourier – indonesio" lang="id" hreflang="id" data-title="Deret Fourier" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesio" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Serie_di_Fourier" title="Serie di Fourier – italiano" lang="it" hreflang="it" data-title="Serie di Fourier" data-language-autonym="Italiano" data-language-local-name="italiano" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%95%E3%83%BC%E3%83%AA%E3%82%A8%E7%B4%9A%E6%95%B0" title="フーリエ級数 – xaponés" lang="ja" hreflang="ja" data-title="フーリエ級数" data-language-autonym="日本語" data-language-local-name="xaponés" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A4%D1%83%D1%80%D1%8C%D0%B5_%D2%9B%D0%B0%D1%82%D0%B0%D1%80%D1%8B" title="Фурье қатары – kazako" lang="kk" hreflang="kk" data-title="Фурье қатары" data-language-autonym="Қазақша" data-language-local-name="kazako" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AB%E0%B3%8B%E0%B2%B0%E0%B3%8D%E0%B2%AF%E0%B3%87_%E0%B2%B6%E0%B3%8D%E0%B2%B0%E0%B3%87%E0%B2%A3%E0%B2%BF%E0%B2%97%E0%B2%B3%E0%B3%81" title="ಫೋರ್ಯೇ ಶ್ರೇಣಿಗಳು – kannará" lang="kn" hreflang="kn" data-title="ಫೋರ್ಯೇ ಶ್ರೇಣಿಗಳು" data-language-autonym="ಕನ್ನಡ" data-language-local-name="kannará" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%91%B8%EB%A6%AC%EC%97%90_%EA%B8%89%EC%88%98" title="푸리에 급수 – coreano" lang="ko" hreflang="ko" data-title="푸리에 급수" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Series_Fourieriana" title="Series Fourieriana – latín" lang="la" hreflang="la" data-title="Series Fourieriana" data-language-autonym="Latina" data-language-local-name="latín" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Furj%C4%97_eilut%C4%97" title="Furjė eilutė – lituano" lang="lt" hreflang="lt" data-title="Furjė eilutė" data-language-autonym="Lietuvių" data-language-local-name="lituano" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Furj%C4%93_rinda" title="Furjē rinda – letón" lang="lv" hreflang="lv" data-title="Furjē rinda" data-language-autonym="Latviešu" data-language-local-name="letón" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A4%D1%83%D1%80%D0%B8%D0%B5%D0%BE%D0%B2_%D1%80%D0%B5%D0%B4" title="Фуриеов ред – macedonio" lang="mk" hreflang="mk" data-title="Фуриеов ред" data-language-autonym="Македонски" data-language-local-name="macedonio" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Serje_ta%27_Fourier" title="Serje ta' Fourier – maltés" lang="mt" hreflang="mt" data-title="Serje ta' Fourier" data-language-autonym="Malti" data-language-local-name="maltés" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Fourierreeks" title="Fourierreeks – neerlandés" lang="nl" hreflang="nl" data-title="Fourierreeks" data-language-autonym="Nederlands" data-language-local-name="neerlandés" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Fourierrekkje" title="Fourierrekkje – noruegués nynorsk" lang="nn" hreflang="nn" data-title="Fourierrekkje" data-language-autonym="Norsk nynorsk" data-language-local-name="noruegués nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Szereg_Fouriera" title="Szereg Fouriera – polaco" lang="pl" hreflang="pl" data-title="Szereg Fouriera" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Serie_%C3%ABd_Fourier" title="Serie ëd Fourier – Piedmontese" lang="pms" hreflang="pms" data-title="Serie ëd Fourier" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/S%C3%A9rie_de_Fourier" title="Série de Fourier – portugués" lang="pt" hreflang="pt" data-title="Série de Fourier" data-language-autonym="Português" data-language-local-name="portugués" 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/Serie_Fourier" title="Serie Fourier – romanés" lang="ro" hreflang="ro" data-title="Serie Fourier" data-language-autonym="Română" data-language-local-name="romanés" 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%A0%D1%8F%D0%B4_%D0%A4%D1%83%D1%80%D1%8C%D0%B5" title="Ряд Фурье – ruso" lang="ru" hreflang="ru" data-title="Ряд Фурье" data-language-autonym="Русский" data-language-local-name="ruso" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Furijeov_red" title="Furijeov red – serbocroata" lang="sh" hreflang="sh" data-title="Furijeov red" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbocroata" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B7%86%E0%B7%96%E0%B6%BB%E0%B7%92%E0%B6%BA%E0%B6%BB%E0%B7%8A_%E0%B7%81%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B7%9A%E0%B6%AB%E0%B7%92%E0%B6%BA" title="ෆූරියර් ශ්‍රේණිය – cingalés" lang="si" hreflang="si" data-title="ෆූරියර් ශ්‍රේණිය" data-language-autonym="සිංහල" data-language-local-name="cingalés" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Fourier_series" title="Fourier series – Simple English" lang="en-simple" hreflang="en-simple" data-title="Fourier series" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Fourierov_rad" title="Fourierov rad – eslovaco" lang="sk" hreflang="sk" data-title="Fourierov rad" data-language-autonym="Slovenčina" data-language-local-name="eslovaco" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Fourierova_vrsta" title="Fourierova vrsta – esloveno" lang="sl" hreflang="sl" data-title="Fourierova vrsta" data-language-autonym="Slovenščina" data-language-local-name="esloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Serit%C3%AB_e_Furierit" title="Seritë e Furierit – albanés" lang="sq" hreflang="sq" data-title="Seritë e Furierit" data-language-autonym="Shqip" data-language-local-name="albanés" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A4%D1%83%D1%80%D0%B8%D1%98%D0%B5%D0%BE%D0%B2_%D1%80%D0%B5%D0%B4" title="Фуријеов ред – serbio" lang="sr" hreflang="sr" data-title="Фуријеов ред" data-language-autonym="Српски / srpski" data-language-local-name="serbio" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/D%C3%A9r%C3%A9t_Fourier" title="Dérét Fourier – sundanés" lang="su" hreflang="su" data-title="Dérét Fourier" data-language-autonym="Sunda" data-language-local-name="sundanés" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Fourierserie" title="Fourierserie – sueco" lang="sv" hreflang="sv" data-title="Fourierserie" data-language-autonym="Svenska" data-language-local-name="sueco" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AD%E0%B8%99%E0%B8%B8%E0%B8%81%E0%B8%A3%E0%B8%A1%E0%B8%9F%E0%B8%B9%E0%B8%A3%E0%B8%B5%E0%B9%80%E0%B8%A2" title="อนุกรมฟูรีเย – tailandés" lang="th" hreflang="th" data-title="อนุกรมฟูรีเย" data-language-autonym="ไทย" data-language-local-name="tailandés" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Fourier_serisi" title="Fourier serisi – turco" lang="tr" hreflang="tr" data-title="Fourier serisi" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%A4%D1%83%D1%80%D1%8C%D0%B5_%D1%80%D3%99%D1%82%D0%B5" title="Фурье рәте – tártaro" lang="tt" hreflang="tt" data-title="Фурье рәте" data-language-autonym="Татарча / tatarça" data-language-local-name="tártaro" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D1%8F%D0%B4_%D0%A4%D1%83%D1%80%27%D1%94" title="Ряд Фур'є – ucraíno" lang="uk" hreflang="uk" data-title="Ряд Фур'є" data-language-autonym="Українська" data-language-local-name="ucraíno" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%81%D9%88%D8%B1%D8%A6%DB%8C%D8%B1_%D8%B3%DB%8C%D8%B1%DB%8C%D8%B2" title="فورئیر سیریز – urdú" lang="ur" hreflang="ur" data-title="فورئیر سیریز" data-language-autonym="اردو" data-language-local-name="urdú" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Chu%E1%BB%97i_Fourier" title="Chuỗi Fourier – vietnamita" lang="vi" hreflang="vi" data-title="Chuỗi Fourier" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%82%85%E9%87%8C%E5%8F%B6%E7%BA%A7%E6%95%B0" title="傅里叶级数 – chinés wu" lang="wuu" hreflang="wuu" data-title="傅里叶级数" data-language-autonym="吴语" data-language-local-name="chinés wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%82%85%E9%87%8C%E5%8F%B6%E7%BA%A7%E6%95%B0" title="傅里叶级数 – chinés" lang="zh" hreflang="zh" data-title="傅里叶级数" data-language-autonym="中文" data-language-local-name="chinés" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%82%85%E5%88%A9%E8%91%89%E7%B4%9A%E6%95%B8" title="傅利葉級數 – cantonés" lang="yue" hreflang="yue" data-title="傅利葉級數" data-language-autonym="粵語" data-language-local-name="cantonés" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a 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Fourier</a>»)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="gl" dir="ltr"><p>En <a href="/wiki/Matem%C3%A1ticas" title="Matemáticas">matemáticas</a>, chámase <b><a href="/wiki/Serie_(matem%C3%A1ticas)" title="Serie (matemáticas)">serie</a> de <a href="/wiki/Joseph_Fourier" class="mw-redirect" title="Joseph Fourier">Fourier</a></b>, a aquela da forma: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(x)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }\left[a_{n}\cos(nx)+b_{n}\operatorname {sen} (nx)\right],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow> <mo>[</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>sen</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(x)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }\left[a_{n}\cos(nx)+b_{n}\operatorname {sen} (nx)\right],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9132861b5517baf06b889f7a48c75fe59644acea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.122ex; height:6.843ex;" alt="{\displaystyle y(x)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }\left[a_{n}\cos(nx)+b_{n}\operatorname {sen} (nx)\right],}"></span> onde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28e2d72f6dd9375c8f1f59f1effd9b4e5492ac97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.509ex;" alt="{\displaystyle b_{n}}"></span> denomínanse <b>coeficientes de Fourier</b> da serie de Fourier da función y(x).</dd></dl> <p>Fourier foi o primeiro que estudou tales series sistematicamente, aplicándoas á solución da <a href="/w/index.php?title=Ecuaci%C3%B3n_da_calor&action=edit&redlink=1" class="new" title="Ecuación da calor (a páxina aínda non existe)">ecuación da calor</a> e publicando os seus resultados iniciais en <a href="/wiki/1807" title="1807">1807</a> e <a href="/wiki/1811" title="1811">1811</a>. Esta área de investigación chámase algunhas veces <a href="/wiki/An%C3%A1lise_harm%C3%B3nica" title="Análise harmónica">Análise harmónica</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Converxencia_a_unha_función_periódica"><span id="Converxencia_a_unha_funci.C3.B3n_peri.C3.B3dica"></span>Converxencia a unha <a href="/wiki/Funci%C3%B3n_peri%C3%B3dica" title="Función periódica">función periódica</a></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Serie_de_Fourier&veaction=edit&section=1" title="Editar a sección: «Converxencia a unha función periódica»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Serie_de_Fourier&action=edit&section=1" title="Editar o código fonte da sección: Converxencia a unha función periódica"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Se f(x) é unha función periódica de período 2π e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(x)\cos nxdx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mi>n</mi> <mi>x</mi> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(x)\cos nxdx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7acf4abc96755940441177243ed01bfef97280df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.807ex; height:6.009ex;" alt="{\displaystyle a_{n}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(x)\cos nxdx}"></span>, e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{n}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(x)\operatorname {sen} nxdx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>sen</mi> <mo>⁡</mo> <mi>n</mi> <mi>x</mi> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(x)\operatorname {sen} nxdx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2b3917b62e522af53b8f2edc43ff4b3ba6ec0bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.705ex; height:6.009ex;" alt="{\displaystyle b_{n}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(x)\operatorname {sen} nxdx}"></span> daquela a serie converxe a f(x). </p> <div class="mw-heading mw-heading2"><h2 id="Forma_exponencial">Forma <a href="/wiki/Exponencial" class="mw-redirect" title="Exponencial">exponencial</a></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Serie_de_Fourier&veaction=edit&section=2" title="Editar a sección: «Forma exponencial»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Serie_de_Fourier&action=edit&section=2" title="Editar o código fonte da sección: Forma exponencial"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pola <a href="/wiki/Identidade_de_Euler" title="Identidade de Euler">identidade de Euler</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 e^{ix}=\cos(x)+i\operatorname {sen} (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>sen</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{ix}=\cos(x)+i\operatorname {sen} (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a29497716237b24f4efb0abd6562cdebd6ab129" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.582ex; height:3.176ex;" alt="{\displaystyle e^{ix}=\cos(x)+i\operatorname {sen} (x)}"></span>), e operando adecuadamente, se </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{n}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x)\,e^{-inx}\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>n</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{n}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x)\,e^{-inx}\,dx.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8f795c7a3cf975ebd64221aa510ec1e98ea1fa9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.301ex; height:6.009ex;" alt="{\displaystyle C_{n}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x)\,e^{-inx}\,dx.}"></span></dd></dl> <p>a serie de Fourier pódese expresar coma a suma de dúas series: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }C_{-n}\,e^{-inx}+\sum _{n=0}^{\infty }C_{n}\,e^{inx}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>n</mi> <mi>x</mi> </mrow> </msup> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> <mi>x</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }C_{-n}\,e^{-inx}+\sum _{n=0}^{\infty }C_{n}\,e^{inx}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4f31db4909cbdf6f09538858f89ec4207b02fc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.682ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }C_{-n}\,e^{-inx}+\sum _{n=0}^{\infty }C_{n}\,e^{inx}.}"></span></dd></dl> <p>En forma máis compacta: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=-\infty }^{\infty }C_{n}\,e^{inx}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=-\infty }^{\infty }C_{n}\,e^{inx}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e6a6011d683276c19fcb05f2c3d2bc8f12ed6dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.651ex; height:6.843ex;" alt="{\displaystyle \sum _{n=-\infty }^{\infty }C_{n}\,e^{inx}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Aplicacións"><span id="Aplicaci.C3.B3ns"></span>Aplicacións</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Serie_de_Fourier&veaction=edit&section=3" title="Editar a sección: «Aplicacións»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Serie_de_Fourier&action=edit&section=3" title="Editar o código fonte da sección: Aplicacións"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Solución_de_ecuacións_diferenciais"><span id="Soluci.C3.B3n_de_ecuaci.C3.B3ns_diferenciais"></span>Solución de ecuacións diferenciais</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Serie_de_Fourier&veaction=edit&section=4" title="Editar a sección: «Solución de ecuacións diferenciais»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Serie_de_Fourier&action=edit&section=4" title="Editar o código fonte da sección: Solución de ecuacións diferenciais"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A ecuación a resolver </p> <div class="mw-heading mw-heading3"><h3 id="Enxeñaría"><span id="Enxe.C3.B1ar.C3.ADa"></span><a href="/wiki/Enxe%C3%B1ar%C3%ADa" title="Enxeñaría">Enxeñaría</a></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Serie_de_Fourier&veaction=edit&section=5" title="Editar a sección: «Enxeñaría»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Serie_de_Fourier&action=edit&section=5" title="Editar o código fonte da sección: Enxeñaría"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>É común substituír a variábel <i>x</i> por ωt, resultando as compoñentes: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{n}={\frac {1}{T}}\int _{-{\frac {T}{2}}}^{\frac {T}{2}}f(t)\,e^{-in\omega \,t}\,dt.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>T</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>T</mi> <mn>2</mn> </mfrac> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>n</mi> <mi>ω</mi> <mspace width="thinmathspace" /> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{n}={\frac {1}{T}}\int _{-{\frac {T}{2}}}^{\frac {T}{2}}f(t)\,e^{-in\omega \,t}\,dt.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df8ca0ed6adc4d8f47b4173cc8cdc857fc70db9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:27.359ex; height:7.843ex;" alt="{\displaystyle C_{n}={\frac {1}{T}}\int _{-{\frac {T}{2}}}^{\frac {T}{2}}f(t)\,e^{-in\omega \,t}\,dt.}"></span></dd></dl> <p>Polo tanto: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)=\sum _{n=-\infty }^{\infty }C_{n}\,e^{in\omega \,t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> <mi>ω</mi> <mspace width="thinmathspace" /> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)=\sum _{n=-\infty }^{\infty }C_{n}\,e^{in\omega \,t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71e09bf9a91520af335f6010bc2c2de527e12825" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.739ex; height:6.843ex;" alt="{\displaystyle f(t)=\sum _{n=-\infty }^{\infty }C_{n}\,e^{in\omega \,t}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Algunhas_consecuencias_positivas_das_propiedades_de_homomorfismo_de_exp">Algunhas consecuencias positivas das propiedades de homomorfismo de exp</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Serie_de_Fourier&veaction=edit&section=6" title="Editar a sección: «Algunhas consecuencias positivas das propiedades de homomorfismo de exp»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Serie_de_Fourier&action=edit&section=6" title="Editar o código fonte da sección: Algunhas consecuencias positivas das propiedades de homomorfismo de exp"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Debido a que as "funcións base" <i>e</i><sup><i>ikx</i></sup> son <a href="/wiki/Homomorfismo" title="Homomorfismo">homomorfismos</a> da liña real (máis concretamente, do "grupo do círculo") temos certas identidades útiles: </p> <ol><li>Se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=f(x-y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)=f(x-y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af205d7efe7ddb13121d940bfd1ba2aa69f38655" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.767ex; height:2.843ex;" alt="{\displaystyle g(x)=f(x-y)}"></span> daquela <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {g}}(k)=e^{-iky}{\hat {f}}(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>k</mi> <mi>y</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {g}}(k)=e^{-iky}{\hat {f}}(k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a430f6168d6a074f8437b076d78bd3f4be57f56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.905ex; height:3.343ex;" alt="{\displaystyle {\hat {g}}(k)=e^{-iky}{\hat {f}}(k)}"></span></li> <li>A transformada de Fourier é un <a href="/wiki/Morfismo" title="Morfismo">morfismo</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*g){\hat {}}(k)={\hat {f}}(k){\hat {g}}(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow /> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f*g){\hat {}}(k)={\hat {f}}(k){\hat {g}}(k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/699f5d9f00dbdea7e49fa9632d5846b75b30d5f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.652ex; height:3.343ex;" alt="{\displaystyle (f*g){\hat {}}(k)={\hat {f}}(k){\hat {g}}(k)}"></span>—isto é, a transformada de Fourier dunha <a href="/w/index.php?title=Convoluci%C3%B3n&action=edit&redlink=1" class="new" title="Convolución (a páxina aínda non existe)">convolución</a> é o produto das transformadas de Fourier.</li></ol> <div class="mw-heading mw-heading2"><h2 id="Formulación_xeral"><span id="Formulaci.C3.B3n_xeral"></span>Formulación xeral</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Serie_de_Fourier&veaction=edit&section=7" title="Editar a sección: «Formulación xeral»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Serie_de_Fourier&action=edit&section=7" title="Editar o código fonte da sección: Formulación xeral"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As útiles propiedades das series de Fourier son debidas principalmente á <a href="/w/index.php?title=Ortogonalidade&action=edit&redlink=1" class="new" title="Ortogonalidade (a páxina aínda non existe)">ortogonalidade</a> e á <a href="/wiki/Homomorfismo" title="Homomorfismo">propiedade de homomorfismo</a> das funcións <i>e</i><sup><i>i n x</i></sup>. </p><p>Outras sucesións de <a href="/w/index.php?title=Funci%C3%B3n_ortogonal&action=edit&redlink=1" class="new" title="Función ortogonal (a páxina aínda non existe)">funcións ortogonais</a> teñen propiedades similares, aínda que algunhas identidades útiles, concernendo por exemplo ás convolucións, non seguirán cumpríndose se se perde a "propiedade de homomorfismo". </p><p>Algúns exemplos son as secuencias de <a href="/w/index.php?title=Funci%C3%B3n_de_Bessel&action=edit&redlink=1" class="new" title="Función de Bessel (a páxina aínda non existe)">funcións de Bessel</a> e os <a href="/w/index.php?title=Polinomio_ortogonal&action=edit&redlink=1" class="new" title="Polinomio ortogonal (a páxina aínda non existe)">polinomios ortogonais</a>. Tales sucesións obtéñense normalmente como solucións dunha ecuación diferencial; unha gran clase de tales sucesións útiles son solucións dos chamados <a href="/w/index.php?title=Teor%C3%ADa_de_Sturm-Liouville&action=edit&redlink=1" class="new" title="Teoría de Sturm-Liouville (a páxina aínda non existe)">problemas de Sturm-Liouville</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Véxase_tamén"><span id="V.C3.A9xase_tam.C3.A9n"></span>Véxase tamén</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Serie_de_Fourier&veaction=edit&section=8" title="Editar a sección: «Véxase tamén»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Serie_de_Fourier&action=edit&section=8" title="Editar o código fonte da sección: Véxase tamén"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Outros_artigos">Outros artigos</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Serie_de_Fourier&veaction=edit&section=9" title="Editar a sección: «Outros artigos»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Serie_de_Fourier&action=edit&section=9" title="Editar o código fonte da sección: Outros artigos"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Transformada_de_Fourier" title="Transformada de Fourier">Transformada de Fourier</a>.</li> <li><a href="/wiki/An%C3%A1lise_harm%C3%B3nica" title="Análise harmónica">Análise harmónica</a>.</li> <li><a href="/w/index.php?title=Fen%C3%B3meno_de_Gibbs&action=edit&redlink=1" class="new" title="Fenómeno de Gibbs (a páxina aínda non existe)">Fenómeno de Gibbs</a>.</li></ul> <hr /> <table> <tbody><tr> <td><span typeof="mw:File"><span title="Este artigo tan só é un bosquexo"><img alt="Este artigo tan só é un bosquexo" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Nuvola_apps_kfig.svg/30px-Nuvola_apps_kfig.svg.png" decoding="async" width="30" height="30" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Nuvola_apps_kfig.svg/45px-Nuvola_apps_kfig.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/57/Nuvola_apps_kfig.svg/60px-Nuvola_apps_kfig.svg.png 2x" data-file-width="109" data-file-height="110" /></span></span> </td> <td><div style="line-height: 110%;"> <span style="font-size:95%"><i>Este artigo sobre <a href="/wiki/Matem%C3%A1ticas" title="Matemáticas">matemáticas</a> é, polo de agora, só un bosquexo. <span class="plainlinks"><a class="external text" href="https://gl.wikipedia.org/w/index.php?title=Serie_de_Fourier&action=edit">Traballa nel</a></span> para axudar a contribuír a que a Galipedia <a href="/wiki/Wikipedia:Mantemento" title="Wikipedia:Mantemento">mellore e medre</a>.</i></span><br /> <i><span style="font-size:85%">Existen igualmente outros <a href="/wiki/Categor%C3%ADa:Wikipedia:Matem%C3%A1ticas_en_progreso" title="Categoría:Wikipedia:Matemáticas en progreso">artigos relacionados con este tema</a> nos que tamén podes contribuír.</span></i></div> </td></tr></tbody></table> <hr /> <div role="navigation" class="navbox" aria-labelledby="Control_de_autoridades" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th id="Control_de_autoridades" scope="row" class="navbox-group" style="width:1%;width: 12%; 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