Formule d'Euler — Wikipédia

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class="vector-toc-link" href="#Démonstrations"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Démonstrations</span> </div> </a> <button aria-controls="toc-Démonstrations-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>Afficher / masquer la sous-section Démonstrations</span> </button> <ul id="toc-Démonstrations-sublist" class="vector-toc-list"> <li id="toc-Par_les_séries_de_Taylor" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Par_les_séries_de_Taylor"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Par les séries de Taylor</span> </div> </a> <ul id="toc-Par_les_séries_de_Taylor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Par_le_calcul_différentiel" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Par_le_calcul_différentiel"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Par le calcul différentiel</span> </div> </a> <ul id="toc-Par_le_calcul_différentiel-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Historique" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Historique"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Historique</span> </div> </a> <ul id="toc-Historique-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voir_aussi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voir_aussi"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Voir aussi</span> </div> </a> <button aria-controls="toc-Voir_aussi-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>Afficher / masquer la sous-section Voir aussi</span> </button> <ul id="toc-Voir_aussi-sublist" class="vector-toc-list"> <li id="toc-Articles_connexes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Articles_connexes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Articles connexes</span> </div> </a> <ul id="toc-Articles_connexes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Références" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Références"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Références</span> </div> </a> <ul id="toc-Références-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="Sommaire" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table des matières" > <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="Basculer la table des matières" > <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">Basculer la table des matières</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">Formule d'Euler</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="Aller à un article dans une autre langue. Disponible en 66 langues." > <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-66" 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">66 langues</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/Euler_se_formule" title="Euler se formule – afrikaans" lang="af" hreflang="af" data-title="Euler se formule" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B5%D9%8A%D8%BA%D8%A9_%D8%A3%D9%88%D9%8A%D9%84%D8%B1" title="صيغة أويلر – arabe" lang="ar" hreflang="ar" data-title="صيغة أويلر" data-language-autonym="العربية" data-language-local-name="arabe" 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/F%C3%B3rmula_d%27Euler" title="Fórmula d'Euler – asturien" lang="ast" hreflang="ast" data-title="Fórmula d'Euler" data-language-autonym="Asturianu" data-language-local-name="asturien" 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/Eyler_d%C3%BCsturu" title="Eyler düsturu – azerbaïdjanais" lang="az" hreflang="az" data-title="Eyler düsturu" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaïdjanais" 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%AD%D0%B9%D0%BB%D0%B5%D1%80_%D1%84%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0%D2%BB%D1%8B" title="Эйлер формулаһы – bachkir" lang="ba" hreflang="ba" data-title="Эйлер формулаһы" data-language-autonym="Башҡортса" data-language-local-name="bachkir" 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%A4%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0_%D0%AD%D0%B9%D0%BB%D0%B5%D1%80%D0%B0" title="Формула Эйлера – biélorusse" lang="be" hreflang="be" data-title="Формула Эйлера" data-language-autonym="Беларуская" data-language-local-name="biélorusse" 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%A4%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0_%D0%BD%D0%B0_%D0%9E%D0%B9%D0%BB%D0%B5%D1%80" title="Формула на Ойлер – bulgare" lang="bg" hreflang="bg" data-title="Формула на Ойлер" data-language-autonym="Български" data-language-local-name="bulgare" 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%85%E0%A6%AF%E0%A6%BC%E0%A6%B2%E0%A6%BE%E0%A6%B0%E0%A7%87%E0%A6%B0_%E0%A6%B8%E0%A7%82%E0%A6%A4%E0%A7%8D%E0%A6%B0" title="অয়লারের সূত্র – bengali" lang="bn" hreflang="bn" data-title="অয়লারের সূত্র" data-language-autonym="বাংলা" data-language-local-name="bengali" 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/Eulerova_formula" title="Eulerova formula – bosniaque" lang="bs" hreflang="bs" data-title="Eulerova formula" data-language-autonym="Bosanski" data-language-local-name="bosniaque" 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/F%C3%B3rmula_d%27Euler" title="Fórmula d'Euler – catalan" lang="ca" hreflang="ca" data-title="Fórmula d'Euler" data-language-autonym="Català" data-language-local-name="catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%81%D9%88%D8%B1%D9%85%D9%88%D9%88%DA%B5%DB%8C_%D8%A6%DB%86%DB%8C%D9%84%DB%95%D8%B1" title="فورمووڵی ئۆیلەر – sorani" lang="ckb" hreflang="ckb" data-title="فورمووڵی ئۆیلەر" data-language-autonym="کوردی" data-language-local-name="sorani" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Euler%C5%AFv_vzorec" title="Eulerův vzorec – tchèque" lang="cs" hreflang="cs" data-title="Eulerův vzorec" data-language-autonym="Čeština" data-language-local-name="tchèque" 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%AD%D0%B9%D0%BB%D0%B5%D1%80_%D1%84%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B8" title="Эйлер формули – tchouvache" lang="cv" hreflang="cv" data-title="Эйлер формули" data-language-autonym="Чӑвашла" data-language-local-name="tchouvache" 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/Fformiwla_Euler" title="Fformiwla Euler – gallois" lang="cy" hreflang="cy" data-title="Fformiwla Euler" data-language-autonym="Cymraeg" data-language-local-name="gallois" 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/Eulers_formel" title="Eulers formel – danois" lang="da" hreflang="da" data-title="Eulers formel" data-language-autonym="Dansk" data-language-local-name="danois" 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/Eulersche_Formel" title="Eulersche Formel – allemand" lang="de" hreflang="de" data-title="Eulersche Formel" data-language-autonym="Deutsch" data-language-local-name="allemand" 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%A4%CF%8D%CF%80%CE%BF%CF%82_%CF%84%CE%BF%CF%85_%CE%8C%CE%B9%CE%BB%CE%B5%CF%81" title="Τύπος του Όιλερ – grec" lang="el" hreflang="el" data-title="Τύπος του Όιλερ" data-language-autonym="Ελληνικά" data-language-local-name="grec" 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/Euler%27s_formula" title="Euler's formula – anglais" lang="en" hreflang="en" data-title="Euler's formula" data-language-autonym="English" data-language-local-name="anglais" 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/E%C5%ADlera_formulo" title="Eŭlera formulo – espéranto" lang="eo" hreflang="eo" data-title="Eŭlera formulo" data-language-autonym="Esperanto" data-language-local-name="espéranto" 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/F%C3%B3rmula_de_Euler" title="Fórmula de Euler – espagnol" lang="es" hreflang="es" data-title="Fórmula de Euler" data-language-autonym="Español" data-language-local-name="espagnol" 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/Euleri_valem" title="Euleri valem – estonien" lang="et" hreflang="et" data-title="Euleri valem" data-language-autonym="Eesti" data-language-local-name="estonien" 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/Eulerren_formula" title="Eulerren formula – basque" lang="eu" hreflang="eu" data-title="Eulerren formula" 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/%D9%81%D8%B1%D9%85%D9%88%D9%84_%D8%A7%D9%88%DB%8C%D9%84%D8%B1" title="فرمول اویلر – persan" lang="fa" hreflang="fa" data-title="فرمول اویلر" data-language-autonym="فارسی" data-language-local-name="persan" 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/Eulerin_lause_(funktioteoria)" title="Eulerin lause (funktioteoria) – finnois" lang="fi" hreflang="fi" data-title="Eulerin lause (funktioteoria)" data-language-autonym="Suomi" data-language-local-name="finnois" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/F%C3%B3rmula_de_Euler" title="Fórmula de Euler – galicien" lang="gl" hreflang="gl" data-title="Fórmula de Euler" data-language-autonym="Galego" data-language-local-name="galicien" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A0%D7%95%D7%A1%D7%97%D7%AA_%D7%90%D7%95%D7%99%D7%9C%D7%A8_(%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%9E%D7%A8%D7%95%D7%9B%D7%91%D7%AA)" title="נוסחת אוילר (אנליזה מרוכבת) – hébreu" lang="he" hreflang="he" data-title="נוסחת אוילר (אנליזה מרוכבת)" data-language-autonym="עברית" data-language-local-name="hébreu" 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%91%E0%A4%AF%E0%A4%B2%E0%A4%B0_%E0%A4%95%E0%A4%BE_%E0%A4%B8%E0%A5%82%E0%A4%A4%E0%A5%8D%E0%A4%B0" title="ऑयलर का सूत्र – hindi" lang="hi" hreflang="hi" data-title="ऑयलर का सूत्र" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Eulerova_formula" title="Eulerova formula – croate" lang="hr" hreflang="hr" data-title="Eulerova formula" data-language-autonym="Hrvatski" data-language-local-name="croate" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Euler-k%C3%A9plet" title="Euler-képlet – hongrois" lang="hu" hreflang="hu" data-title="Euler-képlet" data-language-autonym="Magyar" data-language-local-name="hongrois" 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/%D4%B7%D5%B5%D5%AC%D5%A5%D6%80%D5%AB_%D5%A2%D5%A1%D5%B6%D5%A1%D5%B1%D6%87" title="Էյլերի բանաձև – arménien" lang="hy" hreflang="hy" data-title="Էյլերի բանաձև" data-language-autonym="Հայերեն" data-language-local-name="arménien" 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/Rumus_Euler" title="Rumus Euler – indonésien" lang="id" hreflang="id" data-title="Rumus Euler" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésien" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Jafna_Eulers" title="Jafna Eulers – islandais" lang="is" hreflang="is" data-title="Jafna Eulers" data-language-autonym="Íslenska" data-language-local-name="islandais" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Formula_di_Eulero" title="Formula di Eulero – italien" lang="it" hreflang="it" data-title="Formula di Eulero" data-language-autonym="Italiano" data-language-local-name="italien" 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%82%AA%E3%82%A4%E3%83%A9%E3%83%BC%E3%81%AE%E5%85%AC%E5%BC%8F" title="オイラーの公式 – japonais" lang="ja" hreflang="ja" data-title="オイラーの公式" data-language-autonym="日本語" data-language-local-name="japonais" 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%AD%D0%B9%D0%BB%D0%B5%D1%80_%D1%84%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0%D1%81%D1%8B" title="Эйлер формуласы – kazakh" lang="kk" hreflang="kk" data-title="Эйлер формуласы" data-language-autonym="Қазақша" data-language-local-name="kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%9A%E1%9E%BC%E1%9E%94%E1%9E%98%E1%9E%93%E1%9F%92%E1%9E%8F%E1%9E%A2%E1%9E%99%E1%9E%9B%E1%9F%90%E1%9E%9A" title="រូបមន្តអយល័រ – khmer" lang="km" hreflang="km" data-title="រូបមន្តអយល័រ" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%98%A4%EC%9D%BC%EB%9F%AC_%EA%B3%B5%EC%8B%9D" title="오일러 공식 – coréen" lang="ko" hreflang="ko" data-title="오일러 공식" data-language-autonym="한국어" data-language-local-name="coréen" 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/Euleri_formula" title="Euleri formula – latin" lang="la" hreflang="la" data-title="Euleri formula" data-language-autonym="Latina" data-language-local-name="latin" 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/Oilerio_formul%C4%97" title="Oilerio formulė – lituanien" lang="lt" hreflang="lt" data-title="Oilerio formulė" data-language-autonym="Lietuvių" data-language-local-name="lituanien" 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/Eilera_formula" title="Eilera formula – letton" lang="lv" hreflang="lv" data-title="Eilera formula" data-language-autonym="Latviešu" data-language-local-name="letton" 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%9E%D1%98%D0%BB%D0%B5%D1%80%D0%BE%D0%B2%D0%B0_%D1%84%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0" title="Ојлерова формула – macédonien" lang="mk" hreflang="mk" data-title="Ојлерова формула" data-language-autonym="Македонски" data-language-local-name="macédonien" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Formula_Euler" title="Formula Euler – malais" lang="ms" hreflang="ms" data-title="Formula Euler" data-language-autonym="Bahasa Melayu" data-language-local-name="malais" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Formule_van_Euler" title="Formule van Euler – néerlandais" lang="nl" hreflang="nl" data-title="Formule van Euler" data-language-autonym="Nederlands" data-language-local-name="néerlandais" 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/Eulerformelen" title="Eulerformelen – norvégien nynorsk" lang="nn" hreflang="nn" data-title="Eulerformelen" data-language-autonym="Norsk nynorsk" data-language-local-name="norvégien nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Eulers_formel" title="Eulers formel – norvégien bokmål" lang="nb" hreflang="nb" data-title="Eulers formel" data-language-autonym="Norsk bokmål" data-language-local-name="norvégien bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Formula_d%27Euler" title="Formula d'Euler – occitan" lang="oc" hreflang="oc" data-title="Formula d'Euler" data-language-autonym="Occitan" data-language-local-name="occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wz%C3%B3r_Eulera" title="Wzór Eulera – polonais" lang="pl" hreflang="pl" data-title="Wzór Eulera" data-language-autonym="Polski" data-language-local-name="polonais" 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/F%C3%B3rmula_de_Euler" title="Fórmula de Euler – portugais" lang="pt" hreflang="pt" data-title="Fórmula de Euler" data-language-autonym="Português" data-language-local-name="portugais" 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/Formula_lui_Euler" title="Formula lui Euler – roumain" lang="ro" hreflang="ro" data-title="Formula lui Euler" data-language-autonym="Română" data-language-local-name="roumain" 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%A4%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0_%D0%AD%D0%B9%D0%BB%D0%B5%D1%80%D0%B0" title="Формула Эйлера – russe" lang="ru" hreflang="ru" data-title="Формула Эйлера" data-language-autonym="Русский" data-language-local-name="russe" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Euler%27s_formula" title="Euler's formula – Simple English" lang="en-simple" hreflang="en-simple" data-title="Euler's formula" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Eulerjeva_formula" title="Eulerjeva formula – slovène" lang="sl" hreflang="sl" data-title="Eulerjeva formula" data-language-autonym="Slovenščina" data-language-local-name="slovène" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9E%D1%98%D0%BB%D0%B5%D1%80%D0%BE%D0%B2%D0%B0_%D1%84%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0" title="Ојлерова формула – serbe" lang="sr" hreflang="sr" data-title="Ојлерова формула" data-language-autonym="Српски / srpski" data-language-local-name="serbe" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Eulers_formel" title="Eulers formel – suédois" lang="sv" hreflang="sv" data-title="Eulers formel" data-language-autonym="Svenska" data-language-local-name="suédois" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%86%E0%AE%AF%E0%AF%8D%E0%AE%B2%E0%AE%B0%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%B5%E0%AE%BE%E0%AE%AF%E0%AF%8D%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AE%BE%E0%AE%9F%E0%AF%81" title="ஆய்லரின் வாய்ப்பாடு – tamoul" lang="ta" hreflang="ta" data-title="ஆய்லரின் வாய்ப்பாடு" data-language-autonym="தமிழ்" data-language-local-name="tamoul" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0%D2%B3%D0%BE%D0%B8_%D0%AD%D0%B9%D0%BB%D0%B5%D1%80" title="Формулаҳои Эйлер – tadjik" lang="tg" hreflang="tg" data-title="Формулаҳои Эйлер" data-language-autonym="Тоҷикӣ" data-language-local-name="tadjik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AA%E0%B8%B9%E0%B8%95%E0%B8%A3%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B8%AD%E0%B9%87%E0%B8%AD%E0%B8%A2%E0%B9%80%E0%B8%A5%E0%B8%AD%E0%B8%A3%E0%B9%8C" title="สูตรของอ็อยเลอร์ – thaï" lang="th" hreflang="th" data-title="สูตรของอ็อยเลอร์" data-language-autonym="ไทย" data-language-local-name="thaï" 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/Euler_form%C3%BCl%C3%BC" title="Euler formülü – turc" lang="tr" hreflang="tr" data-title="Euler formülü" data-language-autonym="Türkçe" data-language-local-name="turc" 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%A4%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0_%D0%95%D0%B9%D0%BB%D0%B5%D1%80%D0%B0" title="Формула Ейлера – ukrainien" lang="uk" hreflang="uk" data-title="Формула Ейлера" data-language-autonym="Українська" data-language-local-name="ukrainien" 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/%D8%B9%D8%A7%D8%A6%D9%84%D8%B1%DB%8C_%DA%A9%D9%84%DB%8C%DB%81" title="عائلری کلیہ – ourdou" lang="ur" hreflang="ur" data-title="عائلری کلیہ" data-language-autonym="اردو" data-language-local-name="ourdou" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Euler_formulalari" title="Euler formulalari – ouzbek" lang="uz" hreflang="uz" data-title="Euler formulalari" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="ouzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/C%C3%B4ng_th%E1%BB%A9c_Euler" title="Công thức Euler – vietnamien" lang="vi" hreflang="vi" data-title="Công thức Euler" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamien" 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/%E6%AC%A7%E6%8B%89%E5%85%AC%E5%BC%8F" title="欧拉公式 – wu" lang="wuu" hreflang="wuu" data-title="欧拉公式" data-language-autonym="吴语" data-language-local-name="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/%E6%AC%A7%E6%8B%89%E5%85%AC%E5%BC%8F" title="欧拉公式 – chinois" lang="zh" hreflang="zh" data-title="欧拉公式" data-language-autonym="中文" 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src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Confusion_colour.svg/20px-Confusion_colour.svg.png" decoding="async" width="20" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Confusion_colour.svg/30px-Confusion_colour.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Confusion_colour.svg/40px-Confusion_colour.svg.png 2x" data-file-width="260" data-file-height="200" /></a></span></div><div class="bandeau-cell" style="display:table-cell;padding-right:0.5em"> <p>Ne doit pas être confondue avec <a href="/wiki/Liste_de_sujets_nomm%C3%A9s_d%27apr%C3%A8s_Leonhard_Euler#Formules" class="mw-redirect" title="Liste de sujets nommés d'après Leonhard Euler">d'autres formules dues à Euler</a>, comme <a href="/wiki/Graphe_planaire#Formule_d'Euler_et_conséquences" title="Graphe planaire">celle concernant les graphes planaires</a>. </p> </div></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Fichier:Euler%27s_formula.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/220px-Euler%27s_formula.svg.png" decoding="async" width="220" height="226" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/330px-Euler%27s_formula.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/440px-Euler%27s_formula.svg.png 2x" data-file-width="760" data-file-height="782" /></a><figcaption><center>Formule d'Euler</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 \mathrm {e} ^{\mathrm {i} \varphi }=\cos(\varphi )+\mathrm {i} \sin(\varphi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>φ</mi> </mrow> </msup> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {e} ^{\mathrm {i} \varphi }=\cos(\varphi )+\mathrm {i} \sin(\varphi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee2b77ea02b107f41a2262edc9ff1b0c5727f414" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.395ex; height:3.176ex;" alt="{\displaystyle \mathrm {e} ^{\mathrm {i} \varphi }=\cos(\varphi )+\mathrm {i} \sin(\varphi )}"></span></figcaption></figure> <p>La <b>formule d'Euler</b> est une <a href="/wiki/Identit%C3%A9_remarquable" title="Identité remarquable">égalité</a> <a href="/wiki/Math%C3%A9matiques" title="Mathématiques">mathématique</a>, attribuée au <a href="/wiki/Math%C3%A9maticien" title="Mathématicien">mathématicien</a> <a href="/wiki/Suisse" title="Suisse">suisse</a> <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>. Elle s'écrit, pour tout <a href="/wiki/Nombre_r%C3%A9el" title="Nombre réel">nombre réel</a> <span class="texhtml"><i>x</i></span>, </p> <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 \mathrm {e} ^{\mathrm {i} \,x}=\cos x+\mathrm {i} \,\sin x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {e} ^{\mathrm {i} \,x}=\cos x+\mathrm {i} \,\sin x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1329706416e696b8f2438128c5cdf8b74e0f57f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.81ex; height:2.843ex;" alt="{\displaystyle \mathrm {e} ^{\mathrm {i} \,x}=\cos x+\mathrm {i} \,\sin x}"></span></center> <p>et se généralise aux <span class="texhtml mvar" style="font-style:italic;">x</span> <a href="/wiki/Nombre_complexe" title="Nombre complexe">complexes</a>. </p><p>Ici, le <a href="/wiki/E_(nombre)" title="E (nombre)">nombre <span class="texhtml">e</span></a> est la <a href="/wiki/Logarithme" title="Logarithme">base</a> des <a href="/wiki/Logarithme_naturel" class="mw-redirect" title="Logarithme naturel">logarithmes naturels</a>, <span class="texhtml">i</span> est l'<a href="/wiki/Unit%C3%A9_imaginaire" title="Unité imaginaire">unité imaginaire</a>, <span class="texhtml">sin</span> et <span class="texhtml">cos</span> sont des <a href="/wiki/Fonction_trigonom%C3%A9trique" title="Fonction trigonométrique">fonctions trigonométriques</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Description">Description</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler&veaction=edit&section=1" title="Modifier la section : Description" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler&action=edit&section=1" title="Modifier le code source de la section : Description"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Cette formule peut être interprétée en disant que la fonction <span class="texhtml"><i>x </i>↦ e<sup>i<i>x</i></sup></span>, appelée <b>fonction cis</b><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite_crochet">[</span>1<span class="cite_crochet">]</span></a></sup>, <a href="/wiki/Param%C3%A9trage" class="mw-redirect" title="Paramétrage">décrit</a> le <a href="/wiki/Cercle" title="Cercle">cercle</a> unité dans le <a href="/wiki/Plan_complexe" title="Plan complexe">plan complexe</a> lorsque <span class="texhtml"><i>x</i></span> varie dans l'ensemble des nombres réels.<br /> <span class="texhtml"><i>x</i></span> représente la mesure (en radians) de l'<a href="/wiki/Angle#Angles_orientés_dans_le_plan" title="Angle">angle orienté</a> que fait la <a href="/wiki/Demi-droite" title="Demi-droite">demi-droite</a> d'extrémité l'origine et passant par un point du cercle unité avec la demi-droite des réels positifs. La formule n'est valable que si <span class="texhtml">sin</span> et <span class="texhtml">cos</span> ont des arguments exprimés en radians plutôt qu'en degrés. </p><p>La démonstration est fondée sur les développements en <a href="/wiki/S%C3%A9rie_enti%C3%A8re" title="Série entière">série entière</a> de la fonction <a href="/wiki/Exponentielle_complexe" title="Exponentielle complexe">exponentielle</a> <span class="texhtml"><i>z </i>↦ e<sup><i>z</i></sup></span> de la variable complexe <span class="texhtml"><i>z</i></span> et des fonctions <span class="texhtml">sin</span> et <span class="texhtml">cos</span> considérées à variables réelles.<br /> En fait, la même démonstration montre que la formule d'<a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a> est encore valable pour tous les nombres complexes <span class="texhtml"><i>x</i></span>. </p><p>La formule établit un puissant lien entre l'<a href="/wiki/Analyse_(math%C3%A9matiques)" title="Analyse (mathématiques)">analyse</a> et la <a href="/wiki/Trigonom%C3%A9trie" title="Trigonométrie">trigonométrie</a>. Selon <a href="/wiki/Richard_Feynman" title="Richard Feynman">Richard Feynman</a>, c'est <span class="citation">« l'une des formules les plus remarquables […] de toutes les mathématiques<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup>. »</span> Elle est utilisée pour représenter les nombres complexes sous forme trigonométrique et permet la définition du <a href="/wiki/Logarithme" title="Logarithme">logarithme</a> pour les arguments complexes. En utilisant les propriétés de l'exponentielle </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 {\rm {e}}^{a+b}={\rm {e}}^{a}{\rm {e}}^{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {e}}^{a+b}={\rm {e}}^{a}{\rm {e}}^{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29e75866f94da818d4baa64b47beb2831fa88dce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.321ex; height:2.676ex;" alt="{\displaystyle {\rm {e}}^{a+b}={\rm {e}}^{a}{\rm {e}}^{b}}"></span></dd></dl> <p>et </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 ({\rm {e}}^{a})^{k}={\rm {e}}^{ak}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\rm {e}}^{a})^{k}={\rm {e}}^{ak}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a3f0c64de4aef0f201722a07ba583d07b42ddce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.121ex; height:3.176ex;" alt="{\displaystyle ({\rm {e}}^{a})^{k}={\rm {e}}^{ak}}"></span></dd></dl> <p>(qui sont aussi valables pour tous les nombres complexes <span class="texhtml"><i>a</i>, <i>b</i></span> et pour tout entier <span class="texhtml"><i>k</i></span>), il devient facile de dériver plusieurs <a href="/wiki/Identit%C3%A9_trigonom%C3%A9trique" class="mw-redirect" title="Identité trigonométrique">identités trigonométriques</a> ou d'en déduire la <a href="/wiki/Formule_de_Moivre" title="Formule de Moivre">formule de Moivre</a>. La formule d'Euler permet une interprétation des fonctions cosinus et sinus comme <a href="/wiki/Combinaison_lin%C3%A9aire" title="Combinaison linéaire">combinaisons linéaires</a> de fonctions exponentielles : </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 \cos(x)=\displaystyle {\frac {\mathrm {e} ^{\mathrm {i} \,x}+\mathrm {e} ^{-\mathrm {i} \,x}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> <mi>x</mi> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> <mi>x</mi> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(x)=\displaystyle {\frac {\mathrm {e} ^{\mathrm {i} \,x}+\mathrm {e} ^{-\mathrm {i} \,x}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7751dc49f8b90d34a2694d3c6d73abf70a9d866d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.402ex; height:5.676ex;" alt="{\displaystyle \cos(x)=\displaystyle {\frac {\mathrm {e} ^{\mathrm {i} \,x}+\mathrm {e} ^{-\mathrm {i} \,x}}{2}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(x)=\displaystyle {\frac {\mathrm {e} ^{\mathrm {i} \,x}-\mathrm {e} ^{-\mathrm {i} \,x}}{2\mathrm {i} \,}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> <mi>x</mi> </mrow> </msup> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> <mi>x</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(x)=\displaystyle {\frac {\mathrm {e} ^{\mathrm {i} \,x}-\mathrm {e} ^{-\mathrm {i} \,x}}{2\mathrm {i} \,}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3c72a9ae92acfd8dbe9fec28ec67234def91466" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.147ex; height:5.676ex;" alt="{\displaystyle \sin(x)=\displaystyle {\frac {\mathrm {e} ^{\mathrm {i} \,x}-\mathrm {e} ^{-\mathrm {i} \,x}}{2\mathrm {i} \,}}}"></span></dd></dl> <p>Ces formules (aussi appelées <b>formules d'Euler</b>) constituent la définition moderne des fonctions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e473a3de151d75296f141f9f482fe59d582a7509" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.111ex; height:1.676ex;" alt="{\displaystyle \cos }"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee55beec18afd710e7ab767964b915b020c65093" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.856ex; height:2.176ex;" alt="{\displaystyle \sin }"></span> (y compris <a href="/wiki/Trigonom%C3%A9trie_complexe" title="Trigonométrie complexe">lorsque <span class="texhtml"><i>x</i></span> est une variable complexe</a>)<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite_crochet">[</span>3<span class="cite_crochet">]</span></a></sup> et <a href="/wiki/Syst%C3%A8me_d%27%C3%A9quations_lin%C3%A9aires" title="Système d'équations linéaires">sont équivalentes</a><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite_crochet">[</span>4<span class="cite_crochet">]</span></a></sup> à la formule d'Euler (appliquée à <span class="texhtml"><i>x</i></span> et à <span class="texhtml">–<i>x</i></span>), qui devient alors une <a href="/wiki/Tautologie_(logique)" title="Tautologie (logique)">tautologie</a>. </p><p>Dans les équations différentielles, la fonction <span class="texhtml"><i>x </i>↦ e<sup>i<i>x</i></sup></span>, est souvent utilisée pour simplifier les dérivations, même si le problème est de déterminer les solutions réelles exprimées à l'aide de sinus et cosinus. L'<a href="/wiki/Identit%C3%A9_d%27Euler" title="Identité d'Euler">identité d'Euler</a> est une conséquence immédiate de la formule d'Euler. </p><p>En <a href="/wiki/%C3%89lectrotechnique" title="Électrotechnique">électrotechnique</a> et dans d'autres domaines, les signaux qui varient périodiquement en fonction du temps sont souvent décrits par des combinaisons linéaires des fonctions sinus et cosinus (voir <a href="/wiki/Analyse_de_Fourier" class="mw-redirect" title="Analyse de Fourier">analyse de Fourier</a>), et ces dernières sont plus commodément exprimées comme parties réelles de fonctions exponentielles avec des exposants imaginaires, en utilisant la formule d'Euler. </p> <div class="mw-heading mw-heading2"><h2 id="Démonstrations"><span id="D.C3.A9monstrations"></span>Démonstrations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler&veaction=edit&section=2" title="Modifier la section : Démonstrations" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler&action=edit&section=2" title="Modifier le code source de la section : Démonstrations"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Par_les_séries_de_Taylor"><span id="Par_les_s.C3.A9ries_de_Taylor"></span>Par les séries de Taylor</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler&veaction=edit&section=3" title="Modifier la section : Par les séries de Taylor" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler&action=edit&section=3" title="Modifier le code source de la section : Par les séries de Taylor"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier:Euler%27s_formula_proof.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Euler%27s_formula_proof.gif/350px-Euler%27s_formula_proof.gif" decoding="async" width="350" height="234" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Euler%27s_formula_proof.gif/525px-Euler%27s_formula_proof.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/54/Euler%27s_formula_proof.gif/700px-Euler%27s_formula_proof.gif 2x" data-file-width="901" data-file-height="603" /></a><figcaption>Animation de la démonstration par les séries de Taylor.</figcaption></figure> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/S%C3%A9rie_de_Taylor" title="Série de Taylor">Série de Taylor</a>.</div></div> <p>Le développement en série de la fonction <span class="texhtml">exp</span> de la <a href="/wiki/Variable_(math%C3%A9matiques)" title="Variable (mathématiques)">variable</a> réelle <span class="texhtml"><i>t</i></span> peut s'écrire : </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 {\mathrm {e} }^{t}={\frac {t^{0}}{0\,!}}+{\frac {t^{1}}{1\,!}}+{\frac {t^{2}}{2\,!}}+{\frac {t^{3}}{3\,!}}+{\frac {t^{4}}{4\,!}}+\cdots =\sum _{n=0}^{\infty }{\frac {t^{n}}{n\,!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mrow> <mn>0</mn> <mspace width="thinmathspace" /> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mrow> <mn>1</mn> <mspace width="thinmathspace" /> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mspace width="thinmathspace" /> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mn>3</mn> <mspace width="thinmathspace" /> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow> <mn>4</mn> <mspace width="thinmathspace" /> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mspace width="thinmathspace" /> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathrm {e} }^{t}={\frac {t^{0}}{0\,!}}+{\frac {t^{1}}{1\,!}}+{\frac {t^{2}}{2\,!}}+{\frac {t^{3}}{3\,!}}+{\frac {t^{4}}{4\,!}}+\cdots =\sum _{n=0}^{\infty }{\frac {t^{n}}{n\,!}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/444394b6d00ebbcef713446525d4dfeb7fdb3e89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:47.15ex; height:6.843ex;" alt="{\displaystyle {\mathrm {e} }^{t}={\frac {t^{0}}{0\,!}}+{\frac {t^{1}}{1\,!}}+{\frac {t^{2}}{2\,!}}+{\frac {t^{3}}{3\,!}}+{\frac {t^{4}}{4\,!}}+\cdots =\sum _{n=0}^{\infty }{\frac {t^{n}}{n\,!}}}"></span></dd></dl> <p>et s'étend à tout nombre complexe <span class="texhtml"><i>t</i></span> : le développement en <a href="/wiki/S%C3%A9rie_de_Taylor" title="Série de Taylor">série de Taylor</a> reste <a href="/wiki/Convergence_absolue" title="Convergence absolue">absolument convergent</a> et définit l'exponentielle complexe. </p><p>En particulier pour <span class="texhtml"><i>t </i>= i<i>x</i></span> avec <span class="texhtml"><i>x</i></span> réel : </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 {\mathrm {e} }^{{\mathrm {i} \,}x}=\sum _{n=0}^{\infty }{\frac {{({\mathrm {i} \,}x)}^{n}}{n\,!}}=\sum _{n=0}^{\infty }{\frac {{\mathrm {i} \,}^{n}x^{n}}{n\,!}}\cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> </mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mspace width="thinmathspace" /> <mo>!</mo> </mrow> </mfrac> </mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mi>n</mi> <mspace width="thinmathspace" /> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathrm {e} }^{{\mathrm {i} \,}x}=\sum _{n=0}^{\infty }{\frac {{({\mathrm {i} \,}x)}^{n}}{n\,!}}=\sum _{n=0}^{\infty }{\frac {{\mathrm {i} \,}^{n}x^{n}}{n\,!}}\cdot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbe5e396f2aedbb9e99254de8f6266e2f371041a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.242ex; height:6.843ex;" alt="{\displaystyle {\mathrm {e} }^{{\mathrm {i} \,}x}=\sum _{n=0}^{\infty }{\frac {{({\mathrm {i} \,}x)}^{n}}{n\,!}}=\sum _{n=0}^{\infty }{\frac {{\mathrm {i} \,}^{n}x^{n}}{n\,!}}\cdot }"></span></dd></dl> <p>Cette série peut être séparée en deux en regroupant les termes pairs et impairs. En effet, un réarrangement de l'ordre des termes de la série est possible ici, car il s'agit d'une série <a href="/wiki/Convergence_absolue" title="Convergence absolue">absolument convergente</a>, autrement dit d'une <a href="/wiki/Famille_sommable" title="Famille sommable">famille sommable</a>. On obtient alors, en utilisant le fait que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {i} \,^{2k}=(\mathrm {i} \,^{2})^{k}=(-1)^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {i} \,^{2k}=(\mathrm {i} \,^{2})^{k}=(-1)^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcebb369bc73e482cb41aef0365492a1e9bbcb96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.996ex; height:3.176ex;" alt="{\displaystyle \mathrm {i} \,^{2k}=(\mathrm {i} \,^{2})^{k}=(-1)^{k}}"></span> : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {e} ^{\mathrm {i} \,x}=\sum _{k=0}^{\infty }{\frac {\mathrm {i} \,^{2k}x^{2k}}{(2k)\,!}}+\sum _{k=0}^{\infty }{\frac {\mathrm {i} \,^{2k+1}x^{2k+1}}{(2k+1)\,!}}=\sum _{k=0}^{\infty }{\frac {(-1)^{k}x^{2k}}{(2k)\,!}}+\mathrm {i} \,\sum _{k=0}^{\infty }{\frac {(-1)^{k}x^{2k+1}}{(2k+1)\,!}}\cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> <mi>x</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {e} ^{\mathrm {i} \,x}=\sum _{k=0}^{\infty }{\frac {\mathrm {i} \,^{2k}x^{2k}}{(2k)\,!}}+\sum _{k=0}^{\infty }{\frac {\mathrm {i} \,^{2k+1}x^{2k+1}}{(2k+1)\,!}}=\sum _{k=0}^{\infty }{\frac {(-1)^{k}x^{2k}}{(2k)\,!}}+\mathrm {i} \,\sum _{k=0}^{\infty }{\frac {(-1)^{k}x^{2k+1}}{(2k+1)\,!}}\cdot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/653ea313874a84c147e581e2b08c3a5971208f5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:72.197ex; height:7.176ex;" alt="{\displaystyle \mathrm {e} ^{\mathrm {i} \,x}=\sum _{k=0}^{\infty }{\frac {\mathrm {i} \,^{2k}x^{2k}}{(2k)\,!}}+\sum _{k=0}^{\infty }{\frac {\mathrm {i} \,^{2k+1}x^{2k+1}}{(2k+1)\,!}}=\sum _{k=0}^{\infty }{\frac {(-1)^{k}x^{2k}}{(2k)\,!}}+\mathrm {i} \,\sum _{k=0}^{\infty }{\frac {(-1)^{k}x^{2k+1}}{(2k+1)\,!}}\cdot }"></span></dd></dl> <p>On voit <i>ainsi</i> apparaître les développements en série de Taylor des fonctions cosinus et sinus<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite_crochet">[</span>5<span class="cite_crochet">]</span></a></sup> : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(x)=1-{\frac {x^{2}}{2\,!}}+{\frac {x^{4}}{4\,!}}-{\frac {x^{6}}{6\,!}}+\cdots =\sum _{k=0}^{\infty }{\frac {(-1)^{k}x^{2k}}{(2k)\,!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mspace width="thinmathspace" /> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow> <mn>4</mn> <mspace width="thinmathspace" /> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mrow> <mn>6</mn> <mspace width="thinmathspace" /> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(x)=1-{\frac {x^{2}}{2\,!}}+{\frac {x^{4}}{4\,!}}-{\frac {x^{6}}{6\,!}}+\cdots =\sum _{k=0}^{\infty }{\frac {(-1)^{k}x^{2k}}{(2k)\,!}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c54a5e9909aafd34b3581cb6a5de2923634512e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:51.042ex; height:7.176ex;" alt="{\displaystyle \cos(x)=1-{\frac {x^{2}}{2\,!}}+{\frac {x^{4}}{4\,!}}-{\frac {x^{6}}{6\,!}}+\cdots =\sum _{k=0}^{\infty }{\frac {(-1)^{k}x^{2k}}{(2k)\,!}}}"></span></dd></dl> <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 \sin(x)=x-{\frac {x^{3}}{3\,!}}+{\frac {x^{5}}{5\,!}}-{\frac {x^{7}}{7\,!}}+\cdots =\sum _{k=0}^{\infty }{\frac {(-1)^{k}x^{2k+1}}{(2k+1)\,!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mn>3</mn> <mspace width="thinmathspace" /> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mrow> <mn>5</mn> <mspace width="thinmathspace" /> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mrow> <mn>7</mn> <mspace width="thinmathspace" /> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(x)=x-{\frac {x^{3}}{3\,!}}+{\frac {x^{5}}{5\,!}}-{\frac {x^{7}}{7\,!}}+\cdots =\sum _{k=0}^{\infty }{\frac {(-1)^{k}x^{2k+1}}{(2k+1)\,!}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7220a6a47777abc45945bbb301252757a2f45c31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:53.054ex; height:7.176ex;" alt="{\displaystyle \sin(x)=x-{\frac {x^{3}}{3\,!}}+{\frac {x^{5}}{5\,!}}-{\frac {x^{7}}{7\,!}}+\cdots =\sum _{k=0}^{\infty }{\frac {(-1)^{k}x^{2k+1}}{(2k+1)\,!}}}"></span></dd></dl> <p>ce qui, en remplaçant dans l'expression précédente de <span class="texhtml">e<sup>i<i>x</i></sup></span>, donne bien : </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 \mathrm {e} ^{\mathrm {i} \,x}=\cos(x)+\mathrm {i} \,\sin(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> <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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {e} ^{\mathrm {i} \,x}=\cos(x)+\mathrm {i} \,\sin(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08363db1783f1dacbda80b05b41c5b006c34c2ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.301ex; height:3.176ex;" alt="{\displaystyle \mathrm {e} ^{\mathrm {i} \,x}=\cos(x)+\mathrm {i} \,\sin(x).}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Par_le_calcul_différentiel"><span id="Par_le_calcul_diff.C3.A9rentiel"></span>Par le <a href="/wiki/Calcul_diff%C3%A9rentiel" title="Calcul différentiel">calcul différentiel</a></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler&veaction=edit&section=4" title="Modifier la section : Par le calcul différentiel" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler&action=edit&section=4" title="Modifier le code source de la section : Par le calcul différentiel"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Exponentielle_de_base_a#Par_une_équation_différentielle" title="Exponentielle de base a">Exponentielle de base a#Par une équation différentielle</a>.</div></div> <p>Pour tout nombre complexe <span class="texhtml"><i>k</i></span>, la seule <a href="/wiki/Application_(math%C3%A9matiques)" title="Application (mathématiques)">application</a> <span class="texhtml"><i>f</i></span> : ℝ → ℂ vérifiant <i><span class="texhtml">f ' = kf</span></i> et <span class="texhtml"><i>f</i>(0) = 1</span> est l'application <span class="texhtml"><i>x</i> ↦ exp(<i>kx</i>)</span> (la démonstration est identique à celle pour <span class="texhtml"><i>k</i></span> réel, donnée dans l'article détaillé). </p><p>L'application <span class="texhtml"><i>f</i></span> définie par <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><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)=\cos x+{\rm {i}}\sin x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\cos x+{\rm {i}}\sin x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf564e693fd8cb6d09e8042d574b5593af5ecae1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.791ex; height:2.843ex;" alt="{\displaystyle f(x)=\cos x+{\rm {i}}\sin x}"></span></span> vérifie <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><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'={\rm {i}}f{\text{ et }}f(0)=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> et </mtext> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'={\rm {i}}f{\text{ et }}f(0)=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45f846db4f285b7805a9b89e9c10bca9e2c866ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.286ex; height:3.009ex;" alt="{\displaystyle f'={\rm {i}}f{\text{ et }}f(0)=1.}"></span></span> </p><p>Elle coïncide donc avec l'application <span class="texhtml"><i>x</i> ↦ exp(i<i>x</i>)</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Historique">Historique</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler&veaction=edit&section=5" title="Modifier la section : Historique" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler&action=edit&section=5" title="Modifier le code source de la section : Historique"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Histoire_des_nombres_complexes" title="Histoire des nombres complexes">Histoire des nombres complexes</a>.</div></div> <p>La formule d'Euler fut mise en évidence pour la première fois par <a href="/wiki/Roger_Cotes" title="Roger Cotes">Roger Cotes</a> en 1714 sous la forme <span class="texhtml">ln(cos <i>x</i> + i sin <i>x</i>) = i<i>x</i></span> (où <span class="texhtml">ln</span> désigne le <a href="/wiki/Logarithme_n%C3%A9p%C3%A9rien" title="Logarithme népérien">logarithme népérien</a>, c'est-à-dire le logarithme de base <span class="texhtml">e</span>)<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite_crochet">[</span>6<span class="cite_crochet">]</span></a></sup><sup class="reference cite_virgule">,</sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite_crochet">[</span>7<span class="cite_crochet">]</span></a></sup>. Ce fut Euler qui publia la formule sous sa forme actuelle en 1748, en basant sa démonstration sur la <a href="/wiki/Formule_de_Moivre" title="Formule de Moivre">formule de Moivre</a> et à l'aide d'équivalents et de passages à la limite<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite_crochet">[</span>8<span class="cite_crochet">]</span></a></sup><sup class="reference cite_virgule">,</sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite_crochet">[</span>9<span class="cite_crochet">]</span></a></sup>. Aucun des deux mathématiciens ne donna une interprétation géométrique de la formule : l'interprétation des nombres complexes comme des affixes de points d'un <a href="/wiki/Plan_complexe" title="Plan complexe">plan</a> ne fut vraiment évoquée que cinquante années plus tard (voir <a href="/wiki/Caspar_Wessel" title="Caspar Wessel">Caspar Wessel</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler&veaction=edit&section=6" title="Modifier la section : Applications" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler&action=edit&section=6" title="Modifier le code source de la section : Applications"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>La formule d'Euler permet d'affirmer que la <a href="/wiki/Logarithme_complexe#Détermination_principale" title="Logarithme complexe">détermination principale du logarithme complexe</a> de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos x+\mathrm {i} \,\sin x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos x+\mathrm {i} \,\sin x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f9ebaa9e5d4ddaf24aaaaf3fe8a7e4b3c7972ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.662ex; height:2.343ex;" alt="{\displaystyle \cos x+\mathrm {i} \,\sin x}"></span> est <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {i} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {i} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9390ca6168e08e7c276092958becdc85846dbebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.977ex; height:2.176ex;" alt="{\displaystyle \mathrm {i} x}"></span>, pour tout <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \left]-\pi ,\pi \right[}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow> <mo>]</mo> <mrow> <mo>−</mo> <mi>π</mi> <mo>,</mo> <mi>π</mi> </mrow> <mo>[</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \left]-\pi ,\pi \right[}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64bfae9d92c65381cde6e58d18e374748bfab4b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.97ex; height:2.843ex;" alt="{\displaystyle x\in \left]-\pi ,\pi \right[}"></span>.</li> <li>Un exemple d'application en <a href="/wiki/%C3%89lectromagn%C3%A9tisme" title="Électromagnétisme">électromagnétisme</a> est le <a href="/wiki/Courant_alternatif" title="Courant alternatif">courant alternatif</a> : puisque la <a href="/wiki/Diff%C3%A9rence_de_potentiel" class="mw-redirect" title="Différence de potentiel">différence de potentiel</a> d'un tel circuit <a href="/wiki/Onde" title="Onde">oscille</a>, elle peut être représentée par un <a href="/wiki/Nombre_complexe" title="Nombre complexe">nombre complexe</a> :<span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><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=V_{0}{\rm {e}}^{{\rm {i}}\omega t}=V_{0}\left(\cos \omega t+{\rm {i}}\sin \omega t\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>ω</mi> <mi>t</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>ω</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=V_{0}{\rm {e}}^{{\rm {i}}\omega t}=V_{0}\left(\cos \omega t+{\rm {i}}\sin \omega t\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dc662e2028ceede9c034ee2eec6299b80c8fb3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.558ex; height:3.176ex;" alt="{\displaystyle V=V_{0}{\rm {e}}^{{\rm {i}}\omega t}=V_{0}\left(\cos \omega t+{\rm {i}}\sin \omega t\right).}"></span></span>Afin d'obtenir une quantité mesurable, on prend la partie réelle<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite_crochet">[</span>10<span class="cite_crochet">]</span></a></sup> :<span style="display: block; margin-left:1.6em;"></span></li></ul> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Re} (V)=\mathrm {Re} \left[V_{0}{\rm {e}}^{{\rm {i}}\omega t}\right]=V_{0}\cos \omega t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mrow> <mo>[</mo> <mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>ω</mi> <mi>t</mi> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>ω</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Re} (V)=\mathrm {Re} \left[V_{0}{\rm {e}}^{{\rm {i}}\omega t}\right]=V_{0}\cos \omega t.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e33990c0e17d5962226323ef0a8f4a4cc7c5e88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.58ex; height:3.343ex;" alt="{\displaystyle \mathrm {Re} (V)=\mathrm {Re} \left[V_{0}{\rm {e}}^{{\rm {i}}\omega t}\right]=V_{0}\cos \omega t.}"></span> </p> <ul><li>La <a href="/wiki/Identit%C3%A9_trigonom%C3%A9trique#Linéarisation" class="mw-redirect" title="Identité trigonométrique">linéarisation</a>, qui repose sur la formule d'Euler et la <a href="/wiki/Formule_du_bin%C3%B4me_de_Newton" title="Formule du binôme de Newton">formule du binôme de Newton</a>, transforme tout <a href="/wiki/Polyn%C3%B4me_en_plusieurs_ind%C3%A9termin%C3%A9es" title="Polynôme en plusieurs indéterminées">polynôme</a> en <span class="texhtml">cos(<i>x</i>)</span> et <span class="texhtml">sin(<i>x</i>)</span> en une <a href="/wiki/Combinaison_lin%C3%A9aire" title="Combinaison linéaire">combinaison linéaire</a> de divers <span class="texhtml">cos(<i>nx</i>)</span> et <span class="texhtml">sin(<i>nx</i>)</span>, ce qui rend alors immédiat le calcul de ses <a href="/wiki/Primitive" title="Primitive">primitives</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Voir_aussi">Voir aussi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler&veaction=edit&section=7" title="Modifier la section : Voir aussi" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler&action=edit&section=7" title="Modifier le code source de la section : Voir aussi"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Articles_connexes">Articles connexes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler&veaction=edit&section=8" title="Modifier la section : Articles connexes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler&action=edit&section=8" title="Modifier le code source de la section : Articles connexes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Th%C3%A9or%C3%A8me_de_Descartes-Euler" title="Théorème de Descartes-Euler">Théorème de Descartes-Euler</a></li> <li><a href="/wiki/Triangle#Droite_et_cercle_d'Euler" title="Triangle">Relation d'Euler</a> dans le triangle</li> <li><a href="/wiki/M%C3%A9thode_d%27Euler" title="Méthode d'Euler">Méthode d'Euler</a>, calcul approché d'équations différentielles et de primitives</li> <li><a href="/wiki/Identit%C3%A9_d%27Euler" title="Identité d'Euler">Identité d'Euler</a> : <span class="texhtml">e<sup>iπ</sup></span> + 1 = 0</li></ul> <div class="mw-heading mw-heading2"><h2 id="Références"><span id="R.C3.A9f.C3.A9rences"></span>Références</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler&veaction=edit&section=9" title="Modifier la section : Références" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler&action=edit&section=9" title="Modifier le code source de la section : Références"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a> </span><span class="reference-text"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Alan Sultan et Alice F. Artzt, <i>The mathematics that every secondary school math teacher needs to know</i>, Studies in Mathematical Thinking and Learning, Taylor & Francis, 2010, <a rel="nofollow" class="external text" href="https://books.google.fr/books?id=N16lz_AsU4AC&pg=PA326">p. 326</a>.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a> </span><span class="reference-text"><span class="ouvrage"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <a href="/wiki/Richard_Feynman" title="Richard Feynman">Richard P. Feynman</a>, <a href="/wiki/Robert_B._Leighton" title="Robert B. Leighton">Robert B. Leighton</a> et <a href="/w/index.php?title=Matthew_Sands&action=edit&redlink=1" class="new" title="Matthew Sands (page inexistante)">Matthew Sands</a> <a href="https://en.wikipedia.org/wiki/Matthew_Sands" class="extiw" title="en:Matthew Sands"><span class="indicateur-langue" title="Article en anglais : « Matthew Sands »">(en)</span></a>, <cite class="italique" lang="en"><a href="/wiki/Cours_de_physique_de_Feynman" class="mw-redirect" title="Cours de physique de Feynman">Feynman Lectures on Physics</a></cite> <small>[<a href="/wiki/R%C3%A9f%C3%A9rence:Le_cours_de_physique_de_Feynman" class="mw-redirect" title="Référence:Le cours de physique de Feynman">détail de l’édition</a>]</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Feynman+Lectures+on+Physics&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AFormule+d%27Euler"></span></span>, vol. 1, p. 22, d'après <abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> « <a href="https://fr.wikiquote.org/wiki/en:Euler%27s_identity" class="extiw" title="q:en:Euler's identity">Euler's identity</a> » sur <a href="/wiki/Wikiquote" title="Wikiquote">Wikiquote</a>.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Chatterji1997"><span class="ouvrage" id="Srishti_D._Chatterji1997">Srishti D. Chatterji, <cite class="italique">Cours d'analyse</cite>, <abbr class="abbr" title="volume">vol.</abbr> 2 : <span class="italique">Analyse complexe</span>, <a href="/wiki/PPUR" class="mw-redirect" title="PPUR">PPUR</a>, <time>1997</time> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="//books.google.com/books?id=oMbaI618xmMC&pg=PA96">lire en ligne</a>)</small>, <abbr class="abbr" title="page">p.</abbr> 96<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Cours+d%27analyse&rft.pub=PPUR&rft.aulast=Chatterji&rft.aufirst=Srishti+D.&rft.date=1997&rft.volume=2&rft.pages=96&rft_id=%2F%2Fbooks.google.com%2Fbooks%3Fid%3DoMbaI618xmMC%26pg%3DPA96&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AFormule+d%27Euler"></span></span></span>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a> </span><span class="reference-text"><a href="#Chatterji1997">Chatterji 1997</a>, <abbr class="abbr" title="page(s)">p.</abbr> 97.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a> </span><span class="reference-text"><span class="ouvrage" id="HairerWanner2001"><span class="ouvrage" id="Ernst_HairerGerhard_Wanner2001">Ernst Hairer et Gerhard Wanner, <cite class="italique">L'Analyse au fil de l'histoire</cite>, Springer, <time>2001</time><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=L%27Analyse+au+fil+de+l%27histoire&rft.pub=Springer&rft.aulast=Hairer&rft.aufirst=Ernst&rft.au=Gerhard+Wanner&rft.date=2001&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AFormule+d%27Euler"></span></span></span>, p. 59</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Flament2003"><span class="ouvrage" id="Dominique_Flament2003">Dominique <span class="nom_auteur">Flament</span>, <cite class="italique">Histoire des nombres complexes : Entre algèbre et géométrie</cite>, Paris, CNRS Éditions, <time>2003</time> <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/2_271_06128_8" title="Spécial:Ouvrages de référence/2 271 06128 8"><span class="nowrap">2 271 06128 8</span></a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Histoire+des+nombres+complexes&rft.place=Paris&rft.pub=CNRS+%C3%89ditions&rft.stitle=Entre+alg%C3%A8bre+et+g%C3%A9om%C3%A9trie&rft.aulast=Flament&rft.aufirst=Dominique&rft.date=2003&rft.isbn=2+271+06128+8&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AFormule+d%27Euler"></span></span></span>, p. 80.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Stillwell"><span class="ouvrage" id="John_Stillwell"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <a href="/wiki/John_Stillwell" title="John Stillwell">John <span class="nom_auteur">Stillwell</span></a>, <cite class="italique" lang="en">Mathematics and Its History</cite> <small>[<a href="/wiki/R%C3%A9f%C3%A9rence:Stillwell" title="Référence:Stillwell">détail des éditions</a>]</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+and+Its+History&rft.aulast=Stillwell&rft.aufirst=John&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AFormule+d%27Euler"></span></span></span>, p. 294.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><a href="#cite_ref-8">↑</a> </span><span class="reference-text"> L. Euler, <i>Introduction à l'analyse infinitésimale</i>, <a rel="nofollow" class="external text" href="https://books.google.fr/books?id=XMo7AAAAcAAJ&hl=fr&pg=PA102">article 138</a>.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><a href="#cite_ref-9">↑</a> </span><span class="reference-text"><a href="#Flament2003">Flament 2003</a>, <abbr class="abbr" title="page(s)">p.</abbr> 83-84.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><a href="#cite_ref-10">↑</a> </span><span class="reference-text">Voir des exemples dans : <i>Electromagnetism</i> (<abbr class="abbr" title="Deuxième">2<sup>e</sup></abbr> édition), I.S. Grant, W.R. Phillips, Manchester Physics Series, 2008 <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/0-471-92712-0" title="Spécial:Ouvrages de référence/0-471-92712-0"><span class="nowrap">0-471-92712-0</span></a>)</small>.</span> </li> </ol></div> <div class="navbox-container" style="clear:both;"> <table class="navbox collapsible noprint autocollapse" style=""> <tbody><tr><th class="navbox-title" colspan="2" style=""><div style="float:left; width:6em; text-align:left"><div class="noprint plainlinks nowrap tnavbar" style="padding:0; font-size:xx-small; color:var(--color-emphasized, #000000);"><a href="/wiki/Mod%C3%A8le:Palette_Nombre_e" title="Modèle:Palette Nombre e"><abbr class="abbr" title="Voir ce modèle.">v</abbr></a> · <a class="external text" href="https://fr.wikipedia.org/w/index.php?title=Mod%C3%A8le:Palette_Nombre_e&action=edit"><abbr class="abbr" title="Modifier ce modèle. Merci de prévisualiser avant de sauvegarder.">m</abbr></a></div></div><div style="font-size:110%"><a href="/wiki/E_(nombre)" title="E (nombre)">Nombre e</a></div></th> </tr> <tr> <th class="navbox-group" style="">Applications</th> <td class="navbox-list" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Int%C3%A9r%C3%AAts_compos%C3%A9s" title="Intérêts composés">Intérêts composés</a></li> <li><a href="/wiki/Identit%C3%A9_d%27Euler" title="Identité d'Euler">Identité d'Euler</a></li> <li><a class="mw-selflink selflink">Formule d'Euler</a></li> <li><a href="/wiki/Demi-vie" title="Demi-vie">Demi-vie</a> <ul><li><a href="/wiki/Croissance_exponentielle" title="Croissance exponentielle">Croissance exponentielle</a></li> <li><a href="/wiki/D%C3%A9croissance_exponentielle" title="Décroissance exponentielle">Décroissance exponentielle</a></li></ul></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style="">Définitions</th> <td class="navbox-list navbox-even" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/E_(nombre)#Irrationalité" title="E (nombre)">Démonstration de l'irrationalité de e</a></li> <li><a href="/wiki/Repr%C3%A9sentations_de_e" title="Représentations de e">Représentations de e</a></li> <li><a href="/wiki/Th%C3%A9or%C3%A8me_d%27Hermite-Lindemann" title="Théorème d'Hermite-Lindemann">Théorème d'Hermite-Lindemann</a></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style="">Personnes</th> <td class="navbox-list" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/John_Napier" title="John Napier">John Napier</a></li> <li><small><a href="/wiki/John_Speidell" title="John Speidell">John Speidell</a></small></li> <li><a href="/wiki/Jacques_Bernoulli" title="Jacques Bernoulli">Jacques Bernoulli</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li></ul> </div></td> </tr> </tbody></table> </div> <ul id="bandeau-portail" class="bandeau-portail"><li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer" typeof="mw:File"><a href="/wiki/Portail:Analyse" title="Portail de l'analyse"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Nuvola_apps_kmplot.svg/24px-Nuvola_apps_kmplot.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Nuvola_apps_kmplot.svg/36px-Nuvola_apps_kmplot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Nuvola_apps_kmplot.svg/48px-Nuvola_apps_kmplot.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span></span> <span class="bandeau-portail-texte"><a href="/wiki/Portail:Analyse" title="Portail:Analyse">Portail de l'analyse</a></span> </span></li> </ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; 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Formule d'Euler — Wikipédia