Formule d'Euler-Maclaurin

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id="toc-Introduction_:_comparaison_entre_série_et_intégrale-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Énoncé" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Énoncé"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Énoncé</span> </div> </a> <button aria-controls="toc-Énoncé-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 Énoncé</span> </button> <ul id="toc-Énoncé-sublist" class="vector-toc-list"> <li id="toc-Formule_d'Euler-Maclaurin" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Formule_d'Euler-Maclaurin"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Formule d'Euler-Maclaurin</span> </div> </a> <ul id="toc-Formule_d'Euler-Maclaurin-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formules_sommatoires_d'Euler-Maclaurin" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Formules_sommatoires_d'Euler-Maclaurin"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Formules sommatoires d'Euler-Maclaurin</span> </div> </a> <ul id="toc-Formules_sommatoires_d'Euler-Maclaurin-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Expressions_du_reste" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Expressions_du_reste"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Expressions du reste</span> </div> </a> <ul id="toc-Expressions_du_reste-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Majorations_du_reste" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Majorations_du_reste"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Majorations du reste</span> </div> </a> <ul id="toc-Majorations_du_reste-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Signe_du_reste" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Signe_du_reste"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Signe du reste</span> </div> </a> <ul id="toc-Signe_du_reste-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Démonstration" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Démonstration"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Démonstration</span> </div> </a> <ul id="toc-Démonstration-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Application_à_l'intégration_numérique" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Application_à_l'intégration_numérique"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Application à l'intégration numérique</span> </div> </a> <button aria-controls="toc-Application_à_l'intégration_numérique-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 Application à l'intégration numérique</span> </button> <ul id="toc-Application_à_l'intégration_numérique-sublist" class="vector-toc-list"> <li id="toc-Intégration_numérique" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Intégration_numérique"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Intégration numérique</span> </div> </a> <ul id="toc-Intégration_numérique-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formules_d'intégration_d'Euler-Maclaurin" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Formules_d'intégration_d'Euler-Maclaurin"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Formules d'intégration d'Euler-Maclaurin</span> </div> </a> <ul id="toc-Formules_d'intégration_d'Euler-Maclaurin-sublist" class="vector-toc-list"> <li id="toc-Intégration_entre_deux_entiers" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Intégration_entre_deux_entiers"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2.1</span> <span>Intégration entre deux entiers</span> </div> </a> <ul id="toc-Intégration_entre_deux_entiers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Intégration_sur_un_intervalle_quelconque" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Intégration_sur_un_intervalle_quelconque"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2.2</span> <span>Intégration sur un intervalle quelconque</span> </div> </a> <ul id="toc-Intégration_sur_un_intervalle_quelconque-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Expressions_du_reste_pour_k_=_0_et_pour_k_=_1" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Expressions_du_reste_pour_k_=_0_et_pour_k_=_1"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Expressions du reste pour <i>k</i> = 0 et pour <i>k</i> = 1</span> </div> </a> <button aria-controls="toc-Expressions_du_reste_pour_k_=_0_et_pour_k_=_1-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 Expressions du reste pour <i>k</i> = 0 et pour <i>k</i> = 1</span> </button> <ul id="toc-Expressions_du_reste_pour_k_=_0_et_pour_k_=_1-sublist" class="vector-toc-list"> <li id="toc-Expressions_de_R0_et_erreur_de_la_méthode_des_trapèzes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Expressions_de_R0_et_erreur_de_la_méthode_des_trapèzes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Expressions de <i>R</i><sub>0</sub> et erreur de la méthode des trapèzes</span> </div> </a> <ul id="toc-Expressions_de_R0_et_erreur_de_la_méthode_des_trapèzes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formule_de_quadrature_et_terme_d'erreur_pour_k_=_1" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Formule_de_quadrature_et_terme_d'erreur_pour_k_=_1"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Formule de quadrature et terme d'erreur pour <i>k</i> = 1</span> </div> </a> <ul id="toc-Formule_de_quadrature_et_terme_d'erreur_pour_k_=_1-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Autres_applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Autres_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Autres applications</span> </div> </a> <button aria-controls="toc-Autres_applications-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 Autres applications</span> </button> <ul id="toc-Autres_applications-sublist" class="vector-toc-list"> <li id="toc-Le_problème_de_Bâle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Le_problème_de_Bâle"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Le problème de Bâle</span> </div> </a> <ul id="toc-Le_problème_de_Bâle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sommes_polynomiales" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sommes_polynomiales"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Sommes polynomiales</span> </div> </a> <ul id="toc-Sommes_polynomiales-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Développements_asymptotiques_de_fonctions_définies_par_une_série" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Développements_asymptotiques_de_fonctions_définies_par_une_série"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Développements asymptotiques de fonctions définies par une série</span> </div> </a> <ul id="toc-Développements_asymptotiques_de_fonctions_définies_par_une_série-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-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">8</span> <span>Voir aussi</span> </div> </a> <ul id="toc-Voir_aussi-sublist" class="vector-toc-list"> </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">9</span> <span>Références</span> </div> </a> <ul id="toc-Références-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliographie" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliographie"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Bibliographie</span> </div> </a> <ul id="toc-Bibliographie-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" > <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-Maclaurin</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 22 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-22" 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">22 langues</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B5%D9%8A%D8%BA%D8%A9_%D8%A3%D9%88%D9%8A%D9%84%D8%B1-%D9%85%D8%A7%D9%83%D9%84%D9%88%D8%B1%D9%8A%D9%86" title="صيغة أويلر-ماكلورين – arabe" lang="ar" hreflang="ar" data-title="صيغة أويلر-ماكلورين" data-language-autonym="العربية" data-language-local-name="arabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/F%C3%B3rmula_d%27Euler-Maclaurin" title="Fórmula d'Euler-Maclaurin – catalan" lang="ca" hreflang="ca" data-title="Fórmula d'Euler-Maclaurin" data-language-autonym="Català" data-language-local-name="catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Euler%C5%AFv%E2%80%93Maclaurin%C5%AFv_vzorec" title="Eulerův–Maclaurinův vzorec – tchèque" lang="cs" hreflang="cs" data-title="Eulerův–Maclaurinův vzorec" data-language-autonym="Čeština" data-language-local-name="tchèque" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Euler-Maclaurin-Formel" title="Euler-Maclaurin-Formel – allemand" lang="de" hreflang="de" data-title="Euler-Maclaurin-Formel" data-language-autonym="Deutsch" data-language-local-name="allemand" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula" title="Euler–Maclaurin formula – anglais" lang="en" hreflang="en" data-title="Euler–Maclaurin formula" data-language-autonym="English" data-language-local-name="anglais" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/F%C3%B3rmula_de_Euler-Maclaurin" title="Fórmula de Euler-Maclaurin – espagnol" lang="es" hreflang="es" data-title="Fórmula de Euler-Maclaurin" data-language-autonym="Español" data-language-local-name="espagnol" class="interlanguage-link-target"><span>Español</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%9E%D7%A7%D7%9C%D7%95%D7%A8%D7%9F" title="נוסחת אוילר-מקלורן – hébreu" lang="he" hreflang="he" data-title="נוסחת אוילר-מקלורן" data-language-autonym="עברית" data-language-local-name="hébreu" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Euler%E2%80%93Maclaurin-k%C3%A9plet" title="Euler–Maclaurin-képlet – hongrois" lang="hu" hreflang="hu" data-title="Euler–Maclaurin-képlet" data-language-autonym="Magyar" data-language-local-name="hongrois" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Formula_di_Eulero-Maclaurin" title="Formula di Eulero-Maclaurin – italien" lang="it" hreflang="it" data-title="Formula di Eulero-Maclaurin" 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%92%8C%E5%85%AC%E5%BC%8F" title="オイラーの和公式 – japonais" lang="ja" hreflang="ja" data-title="オイラーの和公式" 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-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-%E1%9E%98%E1%9F%89%E1%9E%B6%E1%9E%80%E1%9F%92%E1%9E%9B%E1%9E%BC%E1%9E%9A%E1%9E%B8%E1%9E%93" title="រូបមន្តអយល័រ-ម៉ាក្លូរីន – khmer" lang="km" hreflang="km" data-title="រូបមន្តអយល័រ-ម៉ាក្លូរីន" 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-%EB%A7%A4%ED%81%B4%EB%A1%9C%EB%A6%B0_%EA%B3%B5%EC%8B%9D" title="오일러-매클로린 공식 – coréen" lang="ko" hreflang="ko" data-title="오일러-매클로린 공식" data-language-autonym="한국어" data-language-local-name="coréen" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Formule_van_Euler-Maclaurin" title="Formule van Euler-Maclaurin – néerlandais" lang="nl" hreflang="nl" data-title="Formule van Euler-Maclaurin" data-language-autonym="Nederlands" data-language-local-name="néerlandais" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wz%C3%B3r_Eulera-Maclaurina" title="Wzór Eulera-Maclaurina – polonais" lang="pl" hreflang="pl" data-title="Wzór Eulera-Maclaurina" 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_Euler%E2%80%93Maclaurin" title="Fórmula Euler–Maclaurin – portugais" lang="pt" hreflang="pt" data-title="Fórmula Euler–Maclaurin" data-language-autonym="Português" data-language-local-name="portugais" class="interlanguage-link-target"><span>Português</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_%E2%80%94_%D0%9C%D0%B0%D0%BA%D0%BB%D0%BE%D1%80%D0%B5%D0%BD%D0%B0" title="Формула Эйлера — Маклорена – russe" lang="ru" hreflang="ru" data-title="Формула Эйлера — Маклорена" data-language-autonym="Русский" data-language-local-name="russe" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Euler-Maclaurinova_formula" title="Euler-Maclaurinova formula – slovène" lang="sl" hreflang="sl" data-title="Euler-Maclaurinova formula" data-language-autonym="Slovenščina" data-language-local-name="slovène" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Euler-Maclaurins_formel" title="Euler-Maclaurins formel – suédois" lang="sv" hreflang="sv" data-title="Euler-Maclaurins 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 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metadata homonymie hatnote"><div class="bandeau-cell bandeau-icone" style="display:table-cell;padding-right:0.5em"><span class="noviewer" typeof="mw:File"><a href="/wiki/Aide:Homonymie" title="Aide:Homonymie"><img alt="Page d’aide sur l’homonymie" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Logo_disambig.svg/20px-Logo_disambig.svg.png" decoding="async" width="20" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Logo_disambig.svg/30px-Logo_disambig.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Logo_disambig.svg/40px-Logo_disambig.svg.png 2x" data-file-width="512" data-file-height="375" /></a></span></div><div class="bandeau-cell" style="display:table-cell;padding-right:0.5em"> <p>Pour les articles homonymes, voir <a href="/wiki/Maclaurin" class="mw-disambig" title="Maclaurin">Maclaurin</a>. </p> </div></div> <div class="bandeau-container metadata homonymie hatnote"><div class="bandeau-cell bandeau-icone" style="display:table-cell;padding-right:0.5em"><span class="noviewer" typeof="mw:File"><a href="/wiki/Aide:Homonymie" title="Aide:Homonymie"><img alt="Page d’aide sur l’homonymie" 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/Formule_d%27Euler" title="Formule d'Euler">celle définissant l'exponentielle complexe</a>. </p> </div></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier:Colin_Maclaurin_color.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Colin_Maclaurin_color.jpg/220px-Colin_Maclaurin_color.jpg" decoding="async" width="220" height="270" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Colin_Maclaurin_color.jpg/330px-Colin_Maclaurin_color.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Colin_Maclaurin_color.jpg/440px-Colin_Maclaurin_color.jpg 2x" data-file-width="700" data-file-height="860" /></a><figcaption>Portrait de Colin Maclaurin</figcaption></figure> <p>En <a href="/wiki/Math%C3%A9matiques" title="Mathématiques">mathématiques</a>, la <b>formule d'Euler-Maclaurin</b> (appelée parfois <b>formule sommatoire d'Euler</b>) est une relation entre <a href="/wiki/Somme_(arithm%C3%A9tique)" title="Somme (arithmétique)">sommes discrètes</a> et <a href="/wiki/Int%C3%A9gration_(math%C3%A9matiques)" title="Intégration (mathématiques)">intégrales</a>. Elle fut découverte indépendamment, aux alentours de <a href="/wiki/1735" title="1735">1735</a>, par le <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> (pour accélérer le calcul de limites de séries lentement convergentes) et par l'<a href="/wiki/%C3%89cosse" title="Écosse">Écossais</a> <a href="/wiki/Colin_Maclaurin" title="Colin Maclaurin">Colin Maclaurin</a> (pour calculer des valeurs approchées d'intégrales)<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>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduction_:_comparaison_entre_série_et_intégrale"><span id="Introduction_:_comparaison_entre_s.C3.A9rie_et_int.C3.A9grale"></span>Introduction : comparaison entre série et intégrale</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&veaction=edit&section=1" title="Modifier la section : Introduction : comparaison entre série et intégrale" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&action=edit&section=1" title="Modifier le code source de la section : Introduction : comparaison entre série et intégrale"><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/Comparaison_s%C3%A9rie-int%C3%A9grale" title="Comparaison série-intégrale">Comparaison série-intégrale</a>.</div></div> <p>Soit <span class="texhtml mvar" style="font-style:italic;">f</span> une fonction infiniment dérivable sur <span class="texhtml">[1, +∞[</span> et <i>n</i> un <a href="/wiki/Entier_naturel" title="Entier naturel">entier naturel</a> non nul. </p><p>On veut obtenir un développement asymptotique de la somme <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}f(i)=f(1)+f(2)+\cdots +f(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}f(i)=f(1)+f(2)+\cdots +f(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c24fde77edcfae2b865b18237f355e7724d3a17e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.959ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}f(i)=f(1)+f(2)+\cdots +f(n)}"></span> en la comparant à l'intégrale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{1}^{n}f(x)~{\rm {d}}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{1}^{n}f(x)~{\rm {d}}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8aeb62f7830b0968ac7059a4b79ddb57a0c0f9c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.69ex; height:5.843ex;" alt="{\displaystyle \int _{1}^{n}f(x)~{\rm {d}}x}"></span>. </p><p>La formule d'Euler-Maclaurin donne une expression de la différence <span class="mwe-math-element"><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(1)+f(2)+\cdots +f(n)-\int _{1}^{n}f(x)~{\rm {d}}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(1)+f(2)+\cdots +f(n)-\int _{1}^{n}f(x)~{\rm {d}}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85b33e2f4bbf0c016ff2036116d214105682159a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.758ex; height:5.843ex;" alt="{\displaystyle f(1)+f(2)+\cdots +f(n)-\int _{1}^{n}f(x)~{\rm {d}}x}"></span> en fonction des valeurs de la fonction et de ses dérivées aux extrémités 1 et <i>n</i> et d'un reste : </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 \sum _{i=1}^{n}f(i)-\int _{1}^{n}f(x)~{\rm {d}}x={\frac {f(1)+f(n)}{2}}+{\frac {1}{6}}{\frac {f'(n)-f'(1)}{2!}}-{\frac {1}{30}}{\frac {f'''(n)-f'''(1)}{4!}}+\cdots +b_{2k}{\frac {f^{(2k-1)}(n)-f^{(2k-1)}(1)}{(2k)!}}+R_{k,n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>30</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>f</mi> <mo>‴</mo> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mo>‴</mo> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}f(i)-\int _{1}^{n}f(x)~{\rm {d}}x={\frac {f(1)+f(n)}{2}}+{\frac {1}{6}}{\frac {f'(n)-f'(1)}{2!}}-{\frac {1}{30}}{\frac {f'''(n)-f'''(1)}{4!}}+\cdots +b_{2k}{\frac {f^{(2k-1)}(n)-f^{(2k-1)}(1)}{(2k)!}}+R_{k,n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bff1c6a51dfb098a094d99d9054820fb0558ee03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:120.217ex; height:7.176ex;" alt="{\displaystyle \sum _{i=1}^{n}f(i)-\int _{1}^{n}f(x)~{\rm {d}}x={\frac {f(1)+f(n)}{2}}+{\frac {1}{6}}{\frac {f'(n)-f'(1)}{2!}}-{\frac {1}{30}}{\frac {f'''(n)-f'''(1)}{4!}}+\cdots +b_{2k}{\frac {f^{(2k-1)}(n)-f^{(2k-1)}(1)}{(2k)!}}+R_{k,n}.}"></span></center> <p>Les nombres <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{2}={\frac {1}{6}},\quad b_{4}=-{\frac {1}{30}},\quad b_{6}={\frac {1}{42}},\quad b_{8}=-{\frac {1}{30}},\quad \ldots b_{2k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>30</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>42</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>30</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mo>…</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{2}={\frac {1}{6}},\quad b_{4}=-{\frac {1}{30}},\quad b_{6}={\frac {1}{42}},\quad b_{8}=-{\frac {1}{30}},\quad \ldots b_{2k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b29a5601b20652a5a637d5967fdc87f92f292f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:55.144ex; height:5.176ex;" alt="{\displaystyle b_{2}={\frac {1}{6}},\quad b_{4}=-{\frac {1}{30}},\quad b_{6}={\frac {1}{42}},\quad b_{8}=-{\frac {1}{30}},\quad \ldots b_{2k}}"></span> qui apparaissent dans la formule sont les <a href="/wiki/Nombre_de_Bernoulli" title="Nombre de Bernoulli">nombres de Bernoulli</a>. </p><p>La série obtenue n'est en général pas convergente mais on connaît plusieurs expressions du reste <span class="texhtml"><i>R</i><sub><i>k</i>,<i>n</i></sub></span> de la formule qui permettent de majorer l'erreur ainsi faite. </p> <div class="mw-heading mw-heading2"><h2 id="Énoncé"><span id=".C3.89nonc.C3.A9"></span>Énoncé</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&veaction=edit&section=2" title="Modifier la section : Énoncé" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&action=edit&section=2" title="Modifier le code source de la section : Énoncé"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Formule_d'Euler-Maclaurin"><span id="Formule_d.27Euler-Maclaurin"></span>Formule d'Euler-Maclaurin</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&veaction=edit&section=3" title="Modifier la section : Formule d'Euler-Maclaurin" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&action=edit&section=3" title="Modifier le code source de la section : Formule d'Euler-Maclaurin"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Soient <i>p</i> et <i>q</i> deux <a href="/wiki/Entier_relatif" title="Entier relatif">entiers relatifs</a> (<i>p</i> < <i>q</i>), <span class="texhtml mvar" style="font-style:italic;">f</span> une <a href="/wiki/Continuit%C3%A9_(math%C3%A9matiques)" title="Continuité (mathématiques)">fonction continue</a> complexe définie sur [<i>p</i>, <i>q</i>]. L'énoncé qui suit exprime la somme <span class="mwe-math-element"><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(p)}+f(p+1)+\cdots +{f(q)}=\sum _{i=p}^{q}f\left(i\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </munderover> <mi>f</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {f(p)}+f(p+1)+\cdots +{f(q)}=\sum _{i=p}^{q}f\left(i\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6325690dc9475e4f6f6781e1a020f8e7822caef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:39.037ex; height:7.343ex;" alt="{\displaystyle {f(p)}+f(p+1)+\cdots +{f(q)}=\sum _{i=p}^{q}f\left(i\right)}"></span> avec l'intégrale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{p}^{q}f(x)~{\rm {d}}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{p}^{q}f(x)~{\rm {d}}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6507bae106a5d9f333383f369a09ede12c3b3179" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:11.46ex; height:6.176ex;" alt="{\displaystyle \int _{p}^{q}f(x)~{\rm {d}}x}"></span>, les valeurs de <span class="texhtml mvar" style="font-style:italic;">f</span> (ainsi que de ses dérivées) aux extrémités <span class="texhtml"><i>f</i>(<i>p</i>)</span> et <span class="texhtml"><i>f</i>(<i>q</i>)</span> et d'un reste. </p><p>Si <span class="texhtml mvar" style="font-style:italic;">f</span> est une fonction complexe continûment dérivable une fois sur le segment [<i>p</i>, <i>q</i>], la formule d'Euler-Maclaurin s'énonce ainsi : </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 \sum _{i=p}^{q}f\left(i\right)={\frac {f(p)+f(q)}{2}}+\int _{p}^{q}f(x)~{\rm {d}}x+R_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </munderover> <mi>f</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>+</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=p}^{q}f\left(i\right)={\frac {f(p)+f(q)}{2}}+\int _{p}^{q}f(x)~{\rm {d}}x+R_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a54d4afe028f2e88e0d3b245fa93d70bc5a2281b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:43.168ex; height:7.343ex;" alt="{\displaystyle \sum _{i=p}^{q}f\left(i\right)={\frac {f(p)+f(q)}{2}}+\int _{p}^{q}f(x)~{\rm {d}}x+R_{0}}"></span> avec <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{0}=\int _{p}^{q}f'(x)\left(x-\lfloor x\rfloor -{\frac {1}{2}}\right)~{\rm {d}}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mo fence="false" stretchy="false">⌊</mo> <mi>x</mi> <mo fence="false" stretchy="false">⌋</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{0}=\int _{p}^{q}f'(x)\left(x-\lfloor x\rfloor -{\frac {1}{2}}\right)~{\rm {d}}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3d54dc69d0145e210fde0782eed2fe555d9cb6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:34.702ex; height:6.343ex;" alt="{\displaystyle R_{0}=\int _{p}^{q}f'(x)\left(x-\lfloor x\rfloor -{\frac {1}{2}}\right)~{\rm {d}}x}"></span></center> <p>où <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lfloor x\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <mi>x</mi> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor x\rfloor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738c94c88678dd08a289f90a47a609ce44eedf14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.394ex; height:2.843ex;" alt="{\displaystyle \lfloor x\rfloor }"></span> est la <a href="/wiki/Partie_enti%C3%A8re_et_partie_fractionnaire" title="Partie entière et partie fractionnaire">partie entière</a> de <i>x</i>, notée aussi E(<i>x</i>), 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 x-\lfloor x\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>−</mo> <mo fence="false" stretchy="false">⌊</mo> <mi>x</mi> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x-\lfloor x\rfloor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c7981b3e9853980abbae55827a6bbb8b02a4c7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.565ex; height:2.843ex;" alt="{\displaystyle x-\lfloor x\rfloor }"></span> est la partie fractionnaire de <i>x</i>. </p><p>Pour une fonction <span class="texhtml mvar" style="font-style:italic;">f</span> continûment dérivable 2<i>k</i> fois sur le segment [<i>p</i>, <i>q</i>] (avec <i>k </i>≥ 1), la formule d'Euler-Maclaurin s'énonce ainsi : </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 \sum _{i=p}^{q}f\left(i\right)={\frac {f\left(p\right)+f\left(q\right)}{2}}+\int _{p}^{q}f(x)~{\rm {d}}x+\sum _{j=1}^{k}{\frac {b_{2j}}{(2j)!}}\left(f^{(2j-1)}(q)-f^{(2j-1)}(p)\right)+R_{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </munderover> <mi>f</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>q</mi> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=p}^{q}f\left(i\right)={\frac {f\left(p\right)+f\left(q\right)}{2}}+\int _{p}^{q}f(x)~{\rm {d}}x+\sum _{j=1}^{k}{\frac {b_{2j}}{(2j)!}}\left(f^{(2j-1)}(q)-f^{(2j-1)}(p)\right)+R_{k}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb666cb86ff9dbf2de4898494100b0493897cdf8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:81.344ex; height:7.676ex;" alt="{\displaystyle \sum _{i=p}^{q}f\left(i\right)={\frac {f\left(p\right)+f\left(q\right)}{2}}+\int _{p}^{q}f(x)~{\rm {d}}x+\sum _{j=1}^{k}{\frac {b_{2j}}{(2j)!}}\left(f^{(2j-1)}(q)-f^{(2j-1)}(p)\right)+R_{k}.}"></span></center> <p>Les nombres <i>b</i><sub>2<i>j</i></sub> désignent les <a href="/wiki/Nombre_de_Bernoulli" title="Nombre de Bernoulli">nombres de Bernoulli</a> et le reste <i>R<sub>k</sub></i> s'exprime à l'aide du <a href="/wiki/Polyn%C3%B4me_de_Bernoulli" title="Polynôme de Bernoulli">polynôme de Bernoulli</a> <i>B</i><sub>2<i>k</i></sub> : </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 R_{k}=-{1 \over (2k)!}\int _{p}^{q}f^{(2k)}(x)B_{2k}(x-\lfloor x\rfloor )~{\rm {d}}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mo fence="false" stretchy="false">⌊</mo> <mi>x</mi> <mo fence="false" stretchy="false">⌋</mo> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{k}=-{1 \over (2k)!}\int _{p}^{q}f^{(2k)}(x)B_{2k}(x-\lfloor x\rfloor )~{\rm {d}}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f64fcb3685aad34663e7127ceeff054777cdfa3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:41.553ex; height:6.176ex;" alt="{\displaystyle R_{k}=-{1 \over (2k)!}\int _{p}^{q}f^{(2k)}(x)B_{2k}(x-\lfloor x\rfloor )~{\rm {d}}x}"></span>.</center> <p>La notation <i>B</i><sub>2<i>k</i></sub> désigne le 2<i>k</i>-ième <a href="/wiki/Polyn%C3%B4me_de_Bernoulli" title="Polynôme de Bernoulli">polynôme de Bernoulli</a> 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 B_{2k}(x-\lfloor x\rfloor )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mo fence="false" stretchy="false">⌊</mo> <mi>x</mi> <mo fence="false" stretchy="false">⌋</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{2k}(x-\lfloor x\rfloor )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef67471fbcfbe6d35de88b25df74546e703cf5a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.049ex; height:2.843ex;" alt="{\displaystyle B_{2k}(x-\lfloor x\rfloor )}"></span> en est une version <a href="/wiki/Fonction_p%C3%A9riodique" title="Fonction périodique">périodisée</a>, de période 1, égale à <i>B</i><sub>2<i>k</i></sub>(<i>x</i>) si 0 < <i>x</i> < 1. </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 B_{2}(x)=x^{2}-x+{\frac {1}{6}}\qquad B_{4}(x)=x^{4}-2x^{3}+x^{2}-{\frac {1}{30}}\qquad B_{6}(x)=x^{6}-3x^{5}+{\frac {5}{2}}x^{4}-{\frac {1}{2}}x^{2}+{\frac {1}{42}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mspace width="2em" /> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>30</mn> </mfrac> </mrow> <mspace width="2em" /> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>42</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{2}(x)=x^{2}-x+{\frac {1}{6}}\qquad B_{4}(x)=x^{4}-2x^{3}+x^{2}-{\frac {1}{30}}\qquad B_{6}(x)=x^{6}-3x^{5}+{\frac {5}{2}}x^{4}-{\frac {1}{2}}x^{2}+{\frac {1}{42}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db9737a7d89a01a3b1d1796da45edeee706a9ca0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:97.065ex; height:5.176ex;" alt="{\displaystyle B_{2}(x)=x^{2}-x+{\frac {1}{6}}\qquad B_{4}(x)=x^{4}-2x^{3}+x^{2}-{\frac {1}{30}}\qquad B_{6}(x)=x^{6}-3x^{5}+{\frac {5}{2}}x^{4}-{\frac {1}{2}}x^{2}+{\frac {1}{42}}}"></span>.</center> <p>Les nombres de Bernoulli vérifient les égalités <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{2k}=B_{2k}(0)=B_{2k}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{2k}=B_{2k}(0)=B_{2k}(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30c3f04da4add0f3015ee2f8cf07302dd52408ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.398ex; height:2.843ex;" alt="{\displaystyle b_{2k}=B_{2k}(0)=B_{2k}(1)}"></span>. </p><p>D'autres expressions du reste sont données plus loin si la fonction <span class="texhtml mvar" style="font-style:italic;">f</span> est 2<i>k</i> + 1 fois dérivable ou 2<i>k</i> + 2 fois dérivable. </p> <div class="mw-heading mw-heading3"><h3 id="Formules_sommatoires_d'Euler-Maclaurin"><span id="Formules_sommatoires_d.27Euler-Maclaurin"></span>Formules sommatoires d'Euler-Maclaurin</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&veaction=edit&section=4" title="Modifier la section : Formules sommatoires d'Euler-Maclaurin" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&action=edit&section=4" title="Modifier le code source de la section : Formules sommatoires d'Euler-Maclaurin"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Si l'on somme de <i>p</i> à <i>q</i> − 1 les nombres <span class="texhtml"><i>f</i> (<i>i</i>)</span>, on a : </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 \sum _{i=p}^{q-1}f(i)=\int _{p}^{q}f(x)~{\rm {d}}x+{\frac {f(p)-f(q)}{2}}+\sum _{j=1}^{k}{\frac {b_{2j}}{(2j)!}}\left(f^{(2j-1)}(q)-f^{(2j-1)}(p)\right)+R_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=p}^{q-1}f(i)=\int _{p}^{q}f(x)~{\rm {d}}x+{\frac {f(p)-f(q)}{2}}+\sum _{j=1}^{k}{\frac {b_{2j}}{(2j)!}}\left(f^{(2j-1)}(q)-f^{(2j-1)}(p)\right)+R_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eca2b73c4237d7b6daa6ac1d6146258f5b8375c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:79.536ex; height:7.676ex;" alt="{\displaystyle \sum _{i=p}^{q-1}f(i)=\int _{p}^{q}f(x)~{\rm {d}}x+{\frac {f(p)-f(q)}{2}}+\sum _{j=1}^{k}{\frac {b_{2j}}{(2j)!}}\left(f^{(2j-1)}(q)-f^{(2j-1)}(p)\right)+R_{k}}"></span>.</center> <p>Si l'on considère les nombres de Bernoulli d'indice impair : <span class="texhtml"><i>b</i><sub>1</sub> = –<span class="texhtml"><span style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;">1</span><span style="position:absolute;left:-10000px;top:auto;width:1px;height:1px;overflow:hidden">/</span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">2</span></span></span></span> et <span class="texhtml"><i>b</i><sub>2<i>j</i> +1</sub> = 0</span> si <i>j</i> > 1, on peut énoncer la formule d'Euler-Maclaurin de la manière suivante<sup id="cite_ref-Cohen22_2-0" class="reference"><a href="#cite_note-Cohen22-2"><span class="cite_crochet">[</span>N 1<span class="cite_crochet">]</span></a></sup> : </p><p>pour une fonction complexe <span class="texhtml mvar" style="font-style:italic;">f</span> qui est <i>r</i> fois continûment dérivable (avec <i>r</i> > 0) : </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 \sum _{i=p}^{q-1}f\left(i\right)=\int _{p}^{q}f(x)~{\rm {d}}x+\sum _{j=1}^{r}{\frac {b_{j}}{j!}}\left(f^{(j-1)}(q)-f^{(j-1)}(p)\right)+R'_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>f</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow> <mi>j</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=p}^{q-1}f\left(i\right)=\int _{p}^{q}f(x)~{\rm {d}}x+\sum _{j=1}^{r}{\frac {b_{j}}{j!}}\left(f^{(j-1)}(q)-f^{(j-1)}(p)\right)+R'_{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da549b2c94e0a661e05a4d3f4e9d60d00155a726" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:60.563ex; height:7.676ex;" alt="{\displaystyle \sum _{i=p}^{q-1}f\left(i\right)=\int _{p}^{q}f(x)~{\rm {d}}x+\sum _{j=1}^{r}{\frac {b_{j}}{j!}}\left(f^{(j-1)}(q)-f^{(j-1)}(p)\right)+R'_{r}}"></span></center> <p>On a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R'_{r}=R_{[{\frac {r}{2}}]},\qquad R_{0}=R'_{1}\qquad {\text{et}}\qquad R_{k}=R'_{2k}=R'_{2k+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">]</mo> </mrow> </msub> <mo>,</mo> <mspace width="2em" /> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>et</mtext> </mrow> <mspace width="2em" /> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R'_{r}=R_{[{\frac {r}{2}}]},\qquad R_{0}=R'_{1}\qquad {\text{et}}\qquad R_{k}=R'_{2k}=R'_{2k+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0082adad1f9bce667e0c3d82b20c92fb8390ccf5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:54.392ex; height:3.676ex;" alt="{\displaystyle R'_{r}=R_{[{\frac {r}{2}}]},\qquad R_{0}=R'_{1}\qquad {\text{et}}\qquad R_{k}=R'_{2k}=R'_{2k+1}}"></span> si <i>k</i> > 0. </p><p> Avec les notations précédentes, l'expression du reste <span class="texhtml mvar" style="font-style:italic;">R'<sub>r</sub></span> pour une fonction complexe <i>r</i> fois continûment dérivable (avec <i>r</i> > 0) est la suivante<sup id="cite_ref-Cohen22_2-1" class="reference"><a href="#cite_note-Cohen22-2"><span class="cite_crochet">[</span>N 1<span class="cite_crochet">]</span></a></sup> : </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 R'_{r}={\frac {(-1)^{r-1}}{r!}}\int _{p}^{q}f^{(r)}(x)B_{r}(x-\lfloor x\rfloor )~{\rm {d}}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <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>r</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mi>r</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mo fence="false" stretchy="false">⌊</mo> <mi>x</mi> <mo fence="false" stretchy="false">⌋</mo> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R'_{r}={\frac {(-1)^{r-1}}{r!}}\int _{p}^{q}f^{(r)}(x)B_{r}(x-\lfloor x\rfloor )~{\rm {d}}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ddbe167b1031c50432f3a5ebd9e5c9eaeda71de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.781ex; height:6.676ex;" alt="{\displaystyle R'_{r}={\frac {(-1)^{r-1}}{r!}}\int _{p}^{q}f^{(r)}(x)B_{r}(x-\lfloor x\rfloor )~{\rm {d}}x}"></span>.</center> <p>La notation <span class="texhtml mvar" style="font-style:italic;">B<sub>r</sub></span> désigne le <i>r</i>-ème <a href="/wiki/Polyn%C3%B4me_de_Bernoulli" title="Polynôme de Bernoulli">polynôme de Bernoulli</a>, 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 B_{r}(x-\lfloor x\rfloor )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mo fence="false" stretchy="false">⌊</mo> <mi>x</mi> <mo fence="false" stretchy="false">⌋</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{r}(x-\lfloor x\rfloor )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a1ac5c885cf342b919dbc4ab8738d687f3a9529" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.112ex; height:2.843ex;" alt="{\displaystyle B_{r}(x-\lfloor x\rfloor )}"></span> en est une version <a href="/wiki/Fonction_p%C3%A9riodique" title="Fonction périodique">périodisée</a>, de période 1, égale à <span class="texhtml"><i>B<sub>r</sub></i>(<i>x</i>)</span> si 0 < <i>x</i> < 1. </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 B_{0}(x)=1\qquad B_{1}(x)=x-{\frac {1}{2}}\qquad B_{2}(x)=x^{2}-x+{\frac {1}{6}}\qquad B_{3}(x)=x^{3}-{\frac {3}{2}}x^{2}+{\frac {1}{2}}x\qquad B_{4}(x)=x^{4}-2x^{3}+x^{2}-{\frac {1}{30}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mspace width="2em" /> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mspace width="2em" /> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mspace width="2em" /> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>x</mi> <mspace width="2em" /> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>30</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{0}(x)=1\qquad B_{1}(x)=x-{\frac {1}{2}}\qquad B_{2}(x)=x^{2}-x+{\frac {1}{6}}\qquad B_{3}(x)=x^{3}-{\frac {3}{2}}x^{2}+{\frac {1}{2}}x\qquad B_{4}(x)=x^{4}-2x^{3}+x^{2}-{\frac {1}{30}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d4e3845d86dbc90164bf151cdc15d49c05c6023" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:118.355ex; height:5.176ex;" alt="{\displaystyle B_{0}(x)=1\qquad B_{1}(x)=x-{\frac {1}{2}}\qquad B_{2}(x)=x^{2}-x+{\frac {1}{6}}\qquad B_{3}(x)=x^{3}-{\frac {3}{2}}x^{2}+{\frac {1}{2}}x\qquad B_{4}(x)=x^{4}-2x^{3}+x^{2}-{\frac {1}{30}}}"></span>.</center> <p>(Les polynômes et les nombres de Bernoulli sont reliés par les égalités <span class="texhtml"><i>b<sub>r</sub></i> = <i>B<sub>r</sub></i>(0) = <i>B<sub>r</sub></i>(1)</span> si <i>r</i> > 1.). </p><p>Une autre formulation équivalente, où l'on somme de <i>p</i> + 1 à <i>q</i>, est donnée par Tenenbaum<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup> : pour une fonction complexe <span class="texhtml mvar" style="font-style:italic;">f</span> qui est <i>r</i> + 1 fois continûment dérivable (avec <i>r</i> ≥ 0) : </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 \sum _{p<n\leqslant q}f\left(n\right)=\int _{p}^{q}f(x)~{\rm {d}}x+\sum _{j=0}^{r}{\frac {(-1)^{j+1}b_{j+1}}{(j+1)!}}\left(f^{(j)}(q)-f^{(j)}(p)\right)+R'_{r+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo><</mo> <mi>n</mi> <mo>⩽</mo> <mi>q</mi> </mrow> </munder> <mi>f</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</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>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{p<n\leqslant q}f\left(n\right)=\int _{p}^{q}f(x)~{\rm {d}}x+\sum _{j=0}^{r}{\frac {(-1)^{j+1}b_{j+1}}{(j+1)!}}\left(f^{(j)}(q)-f^{(j)}(p)\right)+R'_{r+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e3d203cc5918dbd8c49319fa3e1327318089e5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; margin-left: -0.063ex; width:70.78ex; height:7.343ex;" alt="{\displaystyle \sum _{p<n\leqslant q}f\left(n\right)=\int _{p}^{q}f(x)~{\rm {d}}x+\sum _{j=0}^{r}{\frac {(-1)^{j+1}b_{j+1}}{(j+1)!}}\left(f^{(j)}(q)-f^{(j)}(p)\right)+R'_{r+1}}"></span>.</center> <p>Le coefficient <span class="texhtml">(–1)<sup><i>j</i>+1</sup></span> n'intervient dans la formule que pour <i>j</i> = 0. Son rôle est de remplacer le nombre de Bernoulli d'indice 1, <span class="texhtml"><i>b</i><sub>1</sub> = –<span class="texhtml"><span style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;">1</span><span style="position:absolute;left:-10000px;top:auto;width:1px;height:1px;overflow:hidden">/</span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">2</span></span></span></span>, par <span class="texhtml"><i>b'</i><sub>1</sub> = +<span class="texhtml"><span style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;">1</span><span style="position:absolute;left:-10000px;top:auto;width:1px;height:1px;overflow:hidden">/</span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">2</span></span></span></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Expressions_du_reste">Expressions du reste</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&veaction=edit&section=5" title="Modifier la section : Expressions du reste" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&action=edit&section=5" title="Modifier le code source de la section : Expressions du reste"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Si <span class="texhtml mvar" style="font-style:italic;">f</span> est 2<i>k</i> fois continûment dérivable sur le segment [<i>p</i>, <i>q</i>] (avec <i>k </i>≥ 1), le reste <span class="texhtml mvar" style="font-style:italic;">R<sub>k</sub></span> s'exprime de la manière suivante : </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 R_{k}=-{1 \over (2k)!}\int _{p}^{q}f^{(2k)}(x)B_{2k}(x-\lfloor x\rfloor )~{\rm {d}}x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mo fence="false" stretchy="false">⌊</mo> <mi>x</mi> <mo fence="false" stretchy="false">⌋</mo> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{k}=-{1 \over (2k)!}\int _{p}^{q}f^{(2k)}(x)B_{2k}(x-\lfloor x\rfloor )~{\rm {d}}x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f66365341fa8171c5a2d0ab9aece667ae08bc6db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:42.2ex; height:6.176ex;" alt="{\displaystyle R_{k}=-{1 \over (2k)!}\int _{p}^{q}f^{(2k)}(x)B_{2k}(x-\lfloor x\rfloor )~{\rm {d}}x.}"></span></center> <p>Si <span class="texhtml mvar" style="font-style:italic;">f</span> est 2<i>k </i>+ 1 fois continûment dérivable sur le segment [<i>p</i>, <i>q</i>] (avec <i>k </i>≥ 0), le reste <span class="texhtml mvar" style="font-style:italic;">R<sub>k</sub></span> s'exprime comme suit<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite_crochet">[</span>N 2<span class="cite_crochet">]</span></a></sup> : </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 R_{k}={1 \over (2k+1)!}\int _{p}^{q}f^{(2k+1)}(x){B_{2k+1}(x-\lfloor x\rfloor )}~{\rm {d}}x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mo fence="false" stretchy="false">⌊</mo> <mi>x</mi> <mo fence="false" stretchy="false">⌋</mo> <mo stretchy="false">)</mo> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{k}={1 \over (2k+1)!}\int _{p}^{q}f^{(2k+1)}(x){B_{2k+1}(x-\lfloor x\rfloor )}~{\rm {d}}x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c60ef7b171c12beb87a94a7f78fceae4ace459d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:48.596ex; height:6.176ex;" alt="{\displaystyle R_{k}={1 \over (2k+1)!}\int _{p}^{q}f^{(2k+1)}(x){B_{2k+1}(x-\lfloor x\rfloor )}~{\rm {d}}x.}"></span></center> <p>Si <span class="texhtml mvar" style="font-style:italic;">f</span> est une fonction réelle 2<i>k </i>+ 2 fois continûment dérivable sur le segment [<i>p</i>, <i>q</i>] (avec <i>k </i>≥ 0), le reste peut s'écrire des manières suivantes<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite_crochet">[</span>3<span class="cite_crochet">]</span></a></sup> : </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 {\begin{aligned}R_{k}&=-{1 \over (2k+2)!}\int _{p}^{q}f^{(2k+2)}(x)(B_{2k+2}(x-\lfloor x\rfloor )-b_{2k+2})~{\rm {d}}x\\&={q-p \over (2k+2)!}b_{2k+2}f^{(2k+2)}(\xi ),\qquad {\text{ avec }}\quad \xi \in \left]p,q\right[.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mo fence="false" stretchy="false">⌊</mo> <mi>x</mi> <mo fence="false" stretchy="false">⌋</mo> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <mo>−</mo> <mi>p</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> avec </mtext> </mrow> <mspace width="1em" /> <mi>ξ</mi> <mo>∈</mo> <mrow> <mo>]</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>[</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}R_{k}&=-{1 \over (2k+2)!}\int _{p}^{q}f^{(2k+2)}(x)(B_{2k+2}(x-\lfloor x\rfloor )-b_{2k+2})~{\rm {d}}x\\&={q-p \over (2k+2)!}b_{2k+2}f^{(2k+2)}(\xi ),\qquad {\text{ avec }}\quad \xi \in \left]p,q\right[.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba09f9dcefce5bf6f8ef930c42823eea914acb1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:60.167ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}R_{k}&=-{1 \over (2k+2)!}\int _{p}^{q}f^{(2k+2)}(x)(B_{2k+2}(x-\lfloor x\rfloor )-b_{2k+2})~{\rm {d}}x\\&={q-p \over (2k+2)!}b_{2k+2}f^{(2k+2)}(\xi ),\qquad {\text{ avec }}\quad \xi \in \left]p,q\right[.\end{aligned}}}"></span></center> <p>Si l'on considère les nombres de Bernoulli sans leur signe (on a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{2k}=(-1)^{k-1}|b_{2k}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{2k}=(-1)^{k-1}|b_{2k}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4076a30441a7fb1df708dd47c088468f2ce8698a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.178ex; height:3.176ex;" alt="{\displaystyle b_{2k}=(-1)^{k-1}|b_{2k}|}"></span>), la dernière formule s'écrit<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite_crochet">[</span>4<span class="cite_crochet">]</span></a></sup> : </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 R_{k}=(-1)^{k}{q-p \over (2k+2)!}|b_{2k+2}|f^{(2k+2)}(\xi ),\qquad {\text{ avec }}\quad \xi \in \left]p,q\right[.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <mo>−</mo> <mi>p</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> avec </mtext> </mrow> <mspace width="1em" /> <mi>ξ</mi> <mo>∈</mo> <mrow> <mo>]</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>[</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{k}=(-1)^{k}{q-p \over (2k+2)!}|b_{2k+2}|f^{(2k+2)}(\xi ),\qquad {\text{ avec }}\quad \xi \in \left]p,q\right[.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/540e911d61881b03195d883982eeb6ad1510afef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:60.331ex; height:6.009ex;" alt="{\displaystyle R_{k}=(-1)^{k}{q-p \over (2k+2)!}|b_{2k+2}|f^{(2k+2)}(\xi ),\qquad {\text{ avec }}\quad \xi \in \left]p,q\right[.}"></span></center> <p>Remarque : le reste <span class="texhtml mvar" style="font-style:italic;">R<sub>k</sub></span> est nul pour tout polynôme de degré au plus 2<i>k</i> + 1. </p> <div class="mw-heading mw-heading3"><h3 id="Majorations_du_reste">Majorations du reste</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&veaction=edit&section=6" title="Modifier la section : Majorations du reste" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&action=edit&section=6" title="Modifier le code source de la section : Majorations du reste"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Si <span class="texhtml mvar" style="font-style:italic;">f</span> est une fonction complexe 1 fois continûment dérivable sur le segment [<i>p</i>, <i>q</i>], le maximum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sup _{t\in [0\,;\,1]}|B_{1}(t)|={\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo>;</mo> <mspace width="thinmathspace" /> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sup _{t\in [0\,;\,1]}|B_{1}(t)|={\frac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c78651ad526d01f8dbbc3150002a27dd96651e90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.725ex; height:6.343ex;" alt="{\displaystyle \sup _{t\in [0\,;\,1]}|B_{1}(t)|={\frac {1}{2}}}"></span> du polynôme de Bernoulli <span class="texhtml"><i>B</i><sub>1</sub>(<i>t</i>)</span> permet d'obtenir la majoration  : </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 |R_{0}|\leqslant {\frac {1}{2}}\int _{p}^{q}\left|f'(t)\right|\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⩽</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <mrow> <mo>|</mo> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |R_{0}|\leqslant {\frac {1}{2}}\int _{p}^{q}\left|f'(t)\right|\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6d3025222c2266805fdd8c14c1c72310b384e24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.903ex; height:6.176ex;" alt="{\displaystyle |R_{0}|\leqslant {\frac {1}{2}}\int _{p}^{q}\left|f'(t)\right|\mathrm {d} t}"></span>.</center> <p>Si <span class="texhtml mvar" style="font-style:italic;">f</span> est une fonction complexe 2<i>k </i> fois continûment dérivable sur le segment [<i>p</i>, <i>q</i>] (avec <i>k </i>≥ 1), on peut majorer le reste (ou « terme d'erreur ») de la formule d'Euler-Maclaurin en utilisant la majoration des polynômes de Bernoulli d'indice pair<sup id="cite_ref-D301_7-0" class="reference"><a href="#cite_note-D301-7"><span class="cite_crochet">[</span>N 3<span class="cite_crochet">]</span></a></sup> : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sup _{t\in [0\,;\,1]}|B_{2k}(t)|=|b_{2k}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo>;</mo> <mspace width="thinmathspace" /> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sup _{t\in [0\,;\,1]}|B_{2k}(t)|=|b_{2k}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d135b3b348884476494c910828c393798901b619" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.785ex; height:5.009ex;" alt="{\displaystyle \sup _{t\in [0\,;\,1]}|B_{2k}(t)|=|b_{2k}|}"></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 |R_{k}|\leqslant {\frac {|b_{2k}|}{(2k)!}}\int _{p}^{q}\left|f^{(2k)}(x)\right|\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⩽</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <mrow> <mo>|</mo> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |R_{k}|\leqslant {\frac {|b_{2k}|}{(2k)!}}\int _{p}^{q}\left|f^{(2k)}(x)\right|\mathrm {d} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1756a24f87c6af18fb87365fe2228f5111937ff4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.09ex; height:6.509ex;" alt="{\displaystyle |R_{k}|\leqslant {\frac {|b_{2k}|}{(2k)!}}\int _{p}^{q}\left|f^{(2k)}(x)\right|\mathrm {d} x}"></span>.</center> <p>Par exemple, avec <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{2}={\frac {1}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{2}={\frac {1}{6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f4c6c070f9d1707a65ac8dee521339d8ed0059f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.149ex; height:5.176ex;" alt="{\displaystyle b_{2}={\frac {1}{6}}}"></span>, on a : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |R_{1}|\leqslant {\frac {1}{12}}\int _{p}^{q}\left|f''(x)\right|\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⩽</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <mrow> <mo>|</mo> <mrow> <msup> <mi>f</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |R_{1}|\leqslant {\frac {1}{12}}\int _{p}^{q}\left|f''(x)\right|\mathrm {d} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/135ed44f13b0d0cb336b48971034932ec3c0e01a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.498ex; height:6.176ex;" alt="{\displaystyle |R_{1}|\leqslant {\frac {1}{12}}\int _{p}^{q}\left|f''(x)\right|\mathrm {d} x}"></span>. </p><p>L'inégalité peut être réécrite en utilisant la formule due à Euler (pour <i>k </i>≥ 1) : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {|b_{2k}|}{(2k)!}}={\frac {2}{(2\pi )^{2k}}}\sum _{i=1}^{\infty }{\frac {1}{i^{2k}}}={\frac {2}{(2\pi )^{2k}}}\zeta (2k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {|b_{2k}|}{(2k)!}}={\frac {2}{(2\pi )^{2k}}}\sum _{i=1}^{\infty }{\frac {1}{i^{2k}}}={\frac {2}{(2\pi )^{2k}}}\zeta (2k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44cc9958880deb2e8f09273229553d4dd058ec05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.921ex; height:6.843ex;" alt="{\displaystyle {\frac {|b_{2k}|}{(2k)!}}={\frac {2}{(2\pi )^{2k}}}\sum _{i=1}^{\infty }{\frac {1}{i^{2k}}}={\frac {2}{(2\pi )^{2k}}}\zeta (2k)}"></span>. On déduit 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 {\frac {|b_{2k}|}{(2k)!}}\leqslant {\frac {2\zeta (2)}{(2\pi )^{2k}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>⩽</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {|b_{2k}|}{(2k)!}}\leqslant {\frac {2\zeta (2)}{(2\pi )^{2k}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a79607ac834cd779bd816a34de69623b66f45bda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.815ex; height:6.509ex;" alt="{\displaystyle {\frac {|b_{2k}|}{(2k)!}}\leqslant {\frac {2\zeta (2)}{(2\pi )^{2k}}}}"></span> (on a l'équivalent : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {|b_{2k}|}{(2k)!}}\sim {\frac {2}{(2\pi )^{2k}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {|b_{2k}|}{(2k)!}}\sim {\frac {2}{(2\pi )^{2k}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3275930b59a3e7c0ccc98fc5444d42ba1efb9a44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.815ex; height:6.509ex;" alt="{\displaystyle {\frac {|b_{2k}|}{(2k)!}}\sim {\frac {2}{(2\pi )^{2k}}}}"></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 |R_{k}|\leqslant {\frac {2\zeta (2)}{(2\pi )^{2k}}}\int _{p}^{q}\left|f^{(2k)}(t)\right|\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⩽</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <mrow> <mo>|</mo> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |R_{k}|\leqslant {\frac {2\zeta (2)}{(2\pi )^{2k}}}\int _{p}^{q}\left|f^{(2k)}(t)\right|\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe1b2cf6f3722af5c67f1055b3bdaebbd12c8db2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.495ex; height:6.509ex;" alt="{\displaystyle |R_{k}|\leqslant {\frac {2\zeta (2)}{(2\pi )^{2k}}}\int _{p}^{q}\left|f^{(2k)}(t)\right|\mathrm {d} t}"></span>.</center><p> Si <span class="texhtml mvar" style="font-style:italic;">f</span> est une fonction complexe 2<i>k </i>+ 1 fois continûment dérivable sur le segment [<i>p</i>, <i>q</i>] (avec <i>k </i>≥ 1), en utilisant la majoration des polynômes de Bernoulli d'indice impair : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |B_{2k+1}(t)|\leqslant {\frac {2(2k+1)!}{(2\pi )^{2k+1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⩽</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |B_{2k+1}(t)|\leqslant {\frac {2(2k+1)!}{(2\pi )^{2k+1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4be3a37518a946f60516597f4dbd9b73ced97fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.648ex; height:6.509ex;" alt="{\displaystyle |B_{2k+1}(t)|\leqslant {\frac {2(2k+1)!}{(2\pi )^{2k+1}}}}"></span> (si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in [0\,;\,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo>;</mo> <mspace width="thinmathspace" /> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in [0\,;\,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27cd225ceaf6a3ac33211c5a73bf0affbd9cc36b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.107ex; height:2.843ex;" alt="{\displaystyle t\in [0\,;\,1]}"></span>) démontrée par <a href="/wiki/Derrick_Lehmer" title="Derrick Lehmer">Derrick Lehmer</a>, on obtient l'inégalité<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite_crochet">[</span>N 4<span class="cite_crochet">]</span></a></sup> : </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 |R_{k}|\leqslant {\frac {2}{(2\pi )^{2k+1}}}\int _{p}^{q}\left|f^{(2k+1)}(x)\right|\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⩽</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <mrow> <mo>|</mo> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |R_{k}|\leqslant {\frac {2}{(2\pi )^{2k+1}}}\int _{p}^{q}\left|f^{(2k+1)}(x)\right|\mathrm {d} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8c9c3bfdf4678b5aec953bcf93631760e007d74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:34.676ex; height:6.176ex;" alt="{\displaystyle |R_{k}|\leqslant {\frac {2}{(2\pi )^{2k+1}}}\int _{p}^{q}\left|f^{(2k+1)}(x)\right|\mathrm {d} x}"></span></center> <p>Pour le polynôme de Bernoulli <span class="texhtml"><i>B</i><sub>3</sub>(<i>t</i>)</span>, on a le maximum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sup _{t\in [0\,;\,1]}|B_{3}(t)|={\frac {\sqrt {3}}{36}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo>;</mo> <mspace width="thinmathspace" /> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>36</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sup _{t\in [0\,;\,1]}|B_{3}(t)|={\frac {\sqrt {3}}{36}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd2f2113dddd83ee9fafab30f6c91b6e97b98c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.661ex; height:7.009ex;" alt="{\displaystyle \sup _{t\in [0\,;\,1]}|B_{3}(t)|={\frac {\sqrt {3}}{36}}}"></span> qui permet d'obtenir la majoration : </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 |R_{1}|\leqslant {\frac {\sqrt {3}}{216}}\int _{p}^{q}\left|f^{(3)}(t)\right|\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⩽</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>216</mn> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <mrow> <mo>|</mo> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |R_{1}|\leqslant {\frac {\sqrt {3}}{216}}\int _{p}^{q}\left|f^{(3)}(t)\right|\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c90af7e558ca0aae9fb04f66728716f7fbd4820f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.877ex; height:6.676ex;" alt="{\displaystyle |R_{1}|\leqslant {\frac {\sqrt {3}}{216}}\int _{p}^{q}\left|f^{(3)}(t)\right|\mathrm {d} t}"></span>.</center> <div class="mw-heading mw-heading3"><h3 id="Signe_du_reste">Signe du reste</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&veaction=edit&section=7" title="Modifier la section : Signe du reste" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&action=edit&section=7" title="Modifier le code source de la section : Signe du reste"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div><p> Si <span class="texhtml mvar" style="font-style:italic;">f</span> est une fonction réelle 2<i>k </i>+ 2 fois continûment dérivable sur le segment [<i>p</i>, <i>q</i>] (avec <i>k </i>≥ 0), dont la dérivée d'ordre 2<i>k</i> + 2 est de signe constant<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite_crochet">[</span>N 5<span class="cite_crochet">]</span></a></sup>, le reste <span class="texhtml mvar" style="font-style:italic;">R<sub>k</sub></span> a le même signe que le « premier terme négligé<sup id="cite_ref-Cohen25_10-0" class="reference"><a href="#cite_note-Cohen25-10"><span class="cite_crochet">[</span>5<span class="cite_crochet">]</span></a></sup> » : </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 R_{k}\approx {\frac {b_{2k+2}}{(2k+2)!}}\left(f^{(2k+1)}(q)-f^{(2k+1)}(p)\right)\approx (-1)^{k}\left(f^{(2k+1)}(q)-f^{(2k+1)}(p)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{k}\approx {\frac {b_{2k+2}}{(2k+2)!}}\left(f^{(2k+1)}(q)-f^{(2k+1)}(p)\right)\approx (-1)^{k}\left(f^{(2k+1)}(q)-f^{(2k+1)}(p)\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/277e1ae64f7be18d0e61e056239a922a4d933f3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:74.753ex; height:6.176ex;" alt="{\displaystyle R_{k}\approx {\frac {b_{2k+2}}{(2k+2)!}}\left(f^{(2k+1)}(q)-f^{(2k+1)}(p)\right)\approx (-1)^{k}\left(f^{(2k+1)}(q)-f^{(2k+1)}(p)\right)}"></span>.</center> <p>De plus on a les majorations suivantes<sup id="cite_ref-Cohen25_10-1" class="reference"><a href="#cite_note-Cohen25-10"><span class="cite_crochet">[</span>5<span class="cite_crochet">]</span></a></sup> : </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 |R_{k}|\leqslant 2\left(1-2^{-(2k+2)}\right){\frac {|b_{2k+2}|}{(2k+2)!}}\left|f^{(2k+1)}(q)-f^{(2k+1)}(p)\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⩽</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow> <mo>|</mo> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |R_{k}|\leqslant 2\left(1-2^{-(2k+2)}\right){\frac {|b_{2k+2}|}{(2k+2)!}}\left|f^{(2k+1)}(q)-f^{(2k+1)}(p)\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9742ff73f536ff43f65f1d8474cd5abef56d8448" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:56.961ex; height:6.509ex;" alt="{\displaystyle |R_{k}|\leqslant 2\left(1-2^{-(2k+2)}\right){\frac {|b_{2k+2}|}{(2k+2)!}}\left|f^{(2k+1)}(q)-f^{(2k+1)}(p)\right|}"></span>.</center><p> et </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 |R_{k+1}|\leqslant {\frac {|b_{2k+2}|}{(2k+2)!}}\left|f^{(2k+1)}(q)-f^{(2k+1)}(p)\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⩽</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow> <mo>|</mo> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |R_{k+1}|\leqslant {\frac {|b_{2k+2}|}{(2k+2)!}}\left|f^{(2k+1)}(q)-f^{(2k+1)}(p)\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f55b0a467aa835249848ae0ddc11313fb96626f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:42.615ex; height:6.509ex;" alt="{\displaystyle |R_{k+1}|\leqslant {\frac {|b_{2k+2}|}{(2k+2)!}}\left|f^{(2k+1)}(q)-f^{(2k+1)}(p)\right|}"></span>.</center><p> (le reste suivant <span class="texhtml"><i>R</i><sub><i>k</i> + 1</sub></span> n'excède pas (en valeur absolue) le « dernier terme retenu »<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite_crochet">[</span>6<span class="cite_crochet">]</span></a></sup>). </p><p>Si <span class="texhtml mvar" style="font-style:italic;">f</span> est une fonction réelle 2<i>k </i>+ 4 fois continûment dérivable sur le segment [<i>p</i>, <i>q</i>] (avec <i>k </i>≥ 0), dont les dérivées d'ordre 2<i>k</i> + 2 et 2<i>k</i> + 4 sont de signe constant et de même signe, alors les restes <span class="texhtml mvar" style="font-style:italic;">R<sub>k</sub></span> et <span class="texhtml"><i>R</i><sub><i>k</i> + 1</sub></span> sont de signes opposés et le reste <span class="texhtml mvar" style="font-style:italic;">R<sub>k</sub></span> est majoré (en valeur absolue) par le premier terme négligé<sup id="cite_ref-Cohen25_10-2" class="reference"><a href="#cite_note-Cohen25-10"><span class="cite_crochet">[</span>5<span class="cite_crochet">]</span></a></sup> : </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 |R_{k}|\leqslant {\frac {|b_{2k+2}|}{(2k+2)!}}\left|f^{(2k+1)}(q)-f^{(2k+1)}(p)\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⩽</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow> <mo>|</mo> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |R_{k}|\leqslant {\frac {|b_{2k+2}|}{(2k+2)!}}\left|f^{(2k+1)}(q)-f^{(2k+1)}(p)\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/150796c142fd12407f28e95e72b994cc1ce2219f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.515ex; height:6.509ex;" alt="{\displaystyle |R_{k}|\leqslant {\frac {|b_{2k+2}|}{(2k+2)!}}\left|f^{(2k+1)}(q)-f^{(2k+1)}(p)\right|}"></span>.</center> <div class="mw-heading mw-heading2"><h2 id="Démonstration"><span id="D.C3.A9monstration"></span>Démonstration</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&veaction=edit&section=8" title="Modifier la section : Démonstration" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&action=edit&section=8" title="Modifier le code source de la section : Démonstration"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>On démontre la formule </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 R_{k}=-{1 \over (2k)!}\int _{p}^{q}f^{(2k)}(x)B_{2k}(x-\lfloor x\rfloor )~{\rm {d}}x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mo fence="false" stretchy="false">⌊</mo> <mi>x</mi> <mo fence="false" stretchy="false">⌋</mo> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{k}=-{1 \over (2k)!}\int _{p}^{q}f^{(2k)}(x)B_{2k}(x-\lfloor x\rfloor )~{\rm {d}}x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f66365341fa8171c5a2d0ab9aece667ae08bc6db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:42.2ex; height:6.176ex;" alt="{\displaystyle R_{k}=-{1 \over (2k)!}\int _{p}^{q}f^{(2k)}(x)B_{2k}(x-\lfloor x\rfloor )~{\rm {d}}x.}"></span></center> <p>sur l'intervalle [<i>n</i>, <i>n </i>+ 1], avec <i>n </i>∈ ℤ, puis on déduit la formule précédente par sommation sur <i>n </i>∈ ℤ (<i>p </i>≤ <i>n </i>≤ <i>q </i>– 1). </p> <div class="NavFrame" style="border: thin solid #aaaaaa; margin:1em 2em; padding: 0 1em; font-size:100%; text-align:justify; overflow:hidden;"> <div class="NavHead" style="background-color:transparent; color:inherit; padding:0;">Démonstration</div><div class="NavContent" style="padding-bottom:0.4em"> <p>Soit <i>g</i> une fonction continûment dérivable sur [<i>n</i>, <i>n </i>+ 1]. En utilisant la propriété des polynômes de Bernoulli : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall m\in \mathbb {N} \quad B_{m+1}'=\left(m+1\right)B_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>m</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mspace width="1em" /> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall m\in \mathbb {N} \quad B_{m+1}'=\left(m+1\right)B_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58f6b45537400c880996723fc15ad8ef6194c97c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:30.491ex; height:3.176ex;" alt="{\displaystyle \forall m\in \mathbb {N} \quad B_{m+1}'=\left(m+1\right)B_{m}}"></span>, on trouve en faisant une intégration par parties : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{n}^{n+1}g(t)B_{m}(t-n)~{\rm {d}}t=\left[g(t){\frac {B_{m+1}(t-n)}{m+1}}\right]_{n}^{n+1}-\int _{n}^{n+1}g'(t){\frac {B_{m+1}(t-n)}{m+1}}~{\rm {d}}t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>t</mi> <mo>=</mo> <msubsup> <mrow> <mo>[</mo> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{n}^{n+1}g(t)B_{m}(t-n)~{\rm {d}}t=\left[g(t){\frac {B_{m+1}(t-n)}{m+1}}\right]_{n}^{n+1}-\int _{n}^{n+1}g'(t){\frac {B_{m+1}(t-n)}{m+1}}~{\rm {d}}t.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7eb4bf8f4cf06aa4570648180ee92dd7a16d303" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:78.95ex; height:6.676ex;" alt="{\displaystyle \int _{n}^{n+1}g(t)B_{m}(t-n)~{\rm {d}}t=\left[g(t){\frac {B_{m+1}(t-n)}{m+1}}\right]_{n}^{n+1}-\int _{n}^{n+1}g'(t){\frac {B_{m+1}(t-n)}{m+1}}~{\rm {d}}t.}"></span></dd></dl> <p>Or, sachant que pour <i>m </i>≥ 1, on a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{m+1}(1)=B_{m+1}(0)=b_{m+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{m+1}(1)=B_{m+1}(0)=b_{m+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a695637f7450dc82d17c2d0750224c75cee397c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.993ex; height:2.843ex;" alt="{\displaystyle B_{m+1}(1)=B_{m+1}(0)=b_{m+1}}"></span>, on en déduit : </p> <dl><dd>pour <i>m </i>≥ 1 : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{n}^{n+1}g(t)B_{m}(t-n)~{\rm {d}}t={\frac {b_{m+1}}{m+1}}\left(g(n+1)-g(n)\right)-{\frac {1}{m+1}}\int _{n}^{n+1}g'(t)B_{m+1}(t-n)~{\rm {d}}t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{n}^{n+1}g(t)B_{m}(t-n)~{\rm {d}}t={\frac {b_{m+1}}{m+1}}\left(g(n+1)-g(n)\right)-{\frac {1}{m+1}}\int _{n}^{n+1}g'(t)B_{m+1}(t-n)~{\rm {d}}t.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2915baab4fd7d27bc743e21e2dcd57950ebf97f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:87.141ex; height:6.176ex;" alt="{\displaystyle \int _{n}^{n+1}g(t)B_{m}(t-n)~{\rm {d}}t={\frac {b_{m+1}}{m+1}}\left(g(n+1)-g(n)\right)-{\frac {1}{m+1}}\int _{n}^{n+1}g'(t)B_{m+1}(t-n)~{\rm {d}}t.}"></span></dd></dl> Sachant que (pour <i>m</i> = 0) on a<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 B_{0}(t)=1{\text{ et }}B_{1}(1)={\frac {1}{2}}{\text{ et }}B_{1}(0)=-{\frac {1}{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> et </mtext> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> et </mtext> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{0}(t)=1{\text{ et }}B_{1}(1)={\frac {1}{2}}{\text{ et }}B_{1}(0)=-{\frac {1}{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/450a0d20b880d25b1c5c8341e85f90385300e231" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:40.154ex; height:5.176ex;" alt="{\displaystyle B_{0}(t)=1{\text{ et }}B_{1}(1)={\frac {1}{2}}{\text{ et }}B_{1}(0)=-{\frac {1}{2}},}"></span></center>on en déduit si <i>g</i> est dérivable continûment deux fois : <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 {\begin{aligned}\int _{n}^{n+1}g(t)~{\rm {d}}t&=\int _{n}^{n+1}g(t)B_{0}(t-n)~{\rm {d}}t\\&={\frac {1}{2}}\left(g(n+1)+g(n)\right)-\int _{n}^{n+1}g'(t)B_{1}(t-n)~{\rm {d}}t\\&={\frac {g(n+1)+g(n)}{2}}-{\frac {b_{2}}{2}}\left(g'(n+1)-g'(n)\right)+{\frac {1}{2}}\int _{n}^{n+1}g''(t){B_{2}(t-n)}~{\rm {d}}t.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>t</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <msup> <mi>g</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>t</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int _{n}^{n+1}g(t)~{\rm {d}}t&=\int _{n}^{n+1}g(t)B_{0}(t-n)~{\rm {d}}t\\&={\frac {1}{2}}\left(g(n+1)+g(n)\right)-\int _{n}^{n+1}g'(t)B_{1}(t-n)~{\rm {d}}t\\&={\frac {g(n+1)+g(n)}{2}}-{\frac {b_{2}}{2}}\left(g'(n+1)-g'(n)\right)+{\frac {1}{2}}\int _{n}^{n+1}g''(t){B_{2}(t-n)}~{\rm {d}}t.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a442cee7253b292348c151cec2b83bc599973552" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.712ex; margin-bottom: -0.292ex; width:86.962ex; height:19.176ex;" alt="{\displaystyle {\begin{aligned}\int _{n}^{n+1}g(t)~{\rm {d}}t&=\int _{n}^{n+1}g(t)B_{0}(t-n)~{\rm {d}}t\\&={\frac {1}{2}}\left(g(n+1)+g(n)\right)-\int _{n}^{n+1}g'(t)B_{1}(t-n)~{\rm {d}}t\\&={\frac {g(n+1)+g(n)}{2}}-{\frac {b_{2}}{2}}\left(g'(n+1)-g'(n)\right)+{\frac {1}{2}}\int _{n}^{n+1}g''(t){B_{2}(t-n)}~{\rm {d}}t.\end{aligned}}}"></span> </p><p>Soit <span class="texhtml mvar" style="font-style:italic;">f</span> une fonction continûment dérivable 2<i>k</i> fois (<i>k</i> > 0). Par récurrence sur <i>k</i>, on montre : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{n}^{n+1}f(t)~{\rm {d}}t={\frac {f(n)+f(n+1)}{2}}+\sum _{i=2}^{2k}{\frac {\left(-1\right)^{i-1}b_{i}}{i!}}\left(f^{(i-1)}(n+1)-f^{(i-1)}(n)\right)+{\frac {1}{(2k)!}}\int _{n}^{n+1}f^{(2k)}(t)B_{2k}(t-n)~{\rm {d}}t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>i</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>i</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>i</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{n}^{n+1}f(t)~{\rm {d}}t={\frac {f(n)+f(n+1)}{2}}+\sum _{i=2}^{2k}{\frac {\left(-1\right)^{i-1}b_{i}}{i!}}\left(f^{(i-1)}(n+1)-f^{(i-1)}(n)\right)+{\frac {1}{(2k)!}}\int _{n}^{n+1}f^{(2k)}(t)B_{2k}(t-n)~{\rm {d}}t.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4e994a5d2c79b18afb08731aa30eccd9e93a002" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:113.005ex; height:7.343ex;" alt="{\displaystyle \int _{n}^{n+1}f(t)~{\rm {d}}t={\frac {f(n)+f(n+1)}{2}}+\sum _{i=2}^{2k}{\frac {\left(-1\right)^{i-1}b_{i}}{i!}}\left(f^{(i-1)}(n+1)-f^{(i-1)}(n)\right)+{\frac {1}{(2k)!}}\int _{n}^{n+1}f^{(2k)}(t)B_{2k}(t-n)~{\rm {d}}t.}"></span></dd></dl> <p>Enfin, avec la propriété : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall j\geq 1,b_{2j+1}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>j</mi> <mo>≥</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall j\geq 1,b_{2j+1}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a2fd29a79e627e16380874abca0e4a6e1855d0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.636ex; height:2.843ex;" alt="{\displaystyle \forall j\geq 1,b_{2j+1}=0}"></span>, on en déduit (avec <span class="texhtml"><i>i</i> = 2<i>j</i></span>) : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{n}^{n+1}f(t)~{\rm {d}}t={\frac {f(n)+f(n+1)}{2}}-\sum _{j=1}^{k}{\frac {b_{2j}}{(2j)!}}\left(f^{(2j-1)}(n+1)-f^{(2j-1)}(n)\right)+{\frac {1}{(2k)!}}\int _{n}^{n+1}f^{(2k)}(t)B_{2k}(t-n)~{\rm {d}}t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{n}^{n+1}f(t)~{\rm {d}}t={\frac {f(n)+f(n+1)}{2}}-\sum _{j=1}^{k}{\frac {b_{2j}}{(2j)!}}\left(f^{(2j-1)}(n+1)-f^{(2j-1)}(n)\right)+{\frac {1}{(2k)!}}\int _{n}^{n+1}f^{(2k)}(t)B_{2k}(t-n)~{\rm {d}}t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5c19ef487181aec0bfa09cb75ed9db55924775" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:109.322ex; height:7.676ex;" alt="{\displaystyle \int _{n}^{n+1}f(t)~{\rm {d}}t={\frac {f(n)+f(n+1)}{2}}-\sum _{j=1}^{k}{\frac {b_{2j}}{(2j)!}}\left(f^{(2j-1)}(n+1)-f^{(2j-1)}(n)\right)+{\frac {1}{(2k)!}}\int _{n}^{n+1}f^{(2k)}(t)B_{2k}(t-n)~{\rm {d}}t}"></span></dd></dl> <p>donc, avec <span class="texhtml"><i>n </i>= ⌊<i>t</i>⌋</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 {\frac {f(n)+f(n+1)}{2}}=\int _{n}^{n+1}f(t)~{\rm {d}}t+\sum _{j=1}^{k}{\frac {b_{2j}}{(2j)!}}\left(f^{(2j-1)}(n+1)-f^{(2j-1)}(n)\right)-{\frac {1}{(2k)!}}\int _{n}^{n+1}f^{(2k)}(t)B_{2k}\left(t-\lfloor t\rfloor \right)~{\rm {d}}t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>t</mi> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <mo fence="false" stretchy="false">⌊</mo> <mi>t</mi> <mo fence="false" stretchy="false">⌋</mo> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {f(n)+f(n+1)}{2}}=\int _{n}^{n+1}f(t)~{\rm {d}}t+\sum _{j=1}^{k}{\frac {b_{2j}}{(2j)!}}\left(f^{(2j-1)}(n+1)-f^{(2j-1)}(n)\right)-{\frac {1}{(2k)!}}\int _{n}^{n+1}f^{(2k)}(t)B_{2k}\left(t-\lfloor t\rfloor \right)~{\rm {d}}t.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2422948897e2cf92c9ac47d9c33fb016776ccb53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:112.252ex; height:7.676ex;" alt="{\displaystyle {\frac {f(n)+f(n+1)}{2}}=\int _{n}^{n+1}f(t)~{\rm {d}}t+\sum _{j=1}^{k}{\frac {b_{2j}}{(2j)!}}\left(f^{(2j-1)}(n+1)-f^{(2j-1)}(n)\right)-{\frac {1}{(2k)!}}\int _{n}^{n+1}f^{(2k)}(t)B_{2k}\left(t-\lfloor t\rfloor \right)~{\rm {d}}t.}"></span></dd></dl> <p>D'où par sommation sur sommation sur <i>n </i>∈ ℤ (<i>p </i>≤ <i>n </i>≤ <i>q </i>– 1) : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=p}^{q-1}{\frac {f(n)+f(n+1)}{2}}=\int _{p}^{q}f(t)~{\rm {d}}t+\sum _{j=1}^{k}{\frac {b_{2j}}{(2j)!}}\left(f^{(2j-1)}(q)-f^{(2j-1)}(p)\right)-{\frac {1}{(2k)!}}\int _{p}^{q}f^{(2k)}(t)B_{2k}\left(t-\lfloor t\rfloor \right)~{\rm {d}}t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>t</mi> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <mo fence="false" stretchy="false">⌊</mo> <mi>t</mi> <mo fence="false" stretchy="false">⌋</mo> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=p}^{q-1}{\frac {f(n)+f(n+1)}{2}}=\int _{p}^{q}f(t)~{\rm {d}}t+\sum _{j=1}^{k}{\frac {b_{2j}}{(2j)!}}\left(f^{(2j-1)}(q)-f^{(2j-1)}(p)\right)-{\frac {1}{(2k)!}}\int _{p}^{q}f^{(2k)}(t)B_{2k}\left(t-\lfloor t\rfloor \right)~{\rm {d}}t.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d473347db54231dba1a0efa55af16b8e7d005bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:106.78ex; height:7.676ex;" alt="{\displaystyle \sum _{n=p}^{q-1}{\frac {f(n)+f(n+1)}{2}}=\int _{p}^{q}f(t)~{\rm {d}}t+\sum _{j=1}^{k}{\frac {b_{2j}}{(2j)!}}\left(f^{(2j-1)}(q)-f^{(2j-1)}(p)\right)-{\frac {1}{(2k)!}}\int _{p}^{q}f^{(2k)}(t)B_{2k}\left(t-\lfloor t\rfloor \right)~{\rm {d}}t.}"></span></dd></dl> </div><div class="clear" style="clear:both;"></div> </div> <div class="mw-heading mw-heading2"><h2 id="Application_à_l'intégration_numérique"><span id="Application_.C3.A0_l.27int.C3.A9gration_num.C3.A9rique"></span>Application à l'intégration numérique</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&veaction=edit&section=9" title="Modifier la section : Application à l'intégration numérique" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&action=edit&section=9" title="Modifier le code source de la section : Application à l'intégration numérique"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Intégration_numérique"><span id="Int.C3.A9gration_num.C3.A9rique"></span>Intégration numérique</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&veaction=edit&section=10" title="Modifier la section : Intégration numérique" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&action=edit&section=10" title="Modifier le code source de la section : Intégration numérique"><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/Calcul_num%C3%A9rique_d%27une_int%C3%A9grale" title="Calcul numérique d'une intégrale">Calcul numérique d'une intégrale</a>.</div></div> <p>La formule sommatoire peut être utilisée pour approcher des intégrales par un procédé discret, par exemple dans la <a href="/wiki/M%C3%A9thode_des_trap%C3%A8zes" title="Méthode des trapèzes">méthode des trapèzes</a> ou celle de <a href="/wiki/M%C3%A9thode_de_Romberg" title="Méthode de Romberg">Romberg</a>, ou à l'inverse pour transformer une somme discrète (finie ou non) et lui appliquer les techniques du <a href="/wiki/Calcul_infinit%C3%A9simal" title="Calcul infinitésimal">calcul infinitésimal</a>. </p><p>La formule d'Euler-Maclaurin peut aussi être utilisée pour une estimation précise de l'erreur commise dans le <a href="/wiki/Calcul_num%C3%A9rique_d%27une_int%C3%A9grale" title="Calcul numérique d'une intégrale">calcul numérique d'une intégrale</a> ; en particulier, c'est sur elle que reposent les méthodes d'extrapolation. La <a href="/wiki/M%C3%A9thode_de_quadrature_de_Clenshaw-Curtis" title="Méthode de quadrature de Clenshaw-Curtis">méthode de quadrature de Clenshaw-Curtis</a> est essentiellement un changement de variables ramenant une intégrale arbitraire à l'intégration de fonctions périodiques, pour lesquelles la formule sommatoire est très précise (dans ce cas, elle prend la forme d'une <a href="/wiki/Transform%C3%A9e_en_cosinus_discr%C3%A8te" title="Transformée en cosinus discrète">transformée en cosinus discrète</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Formules_d'intégration_d'Euler-Maclaurin"><span id="Formules_d.27int.C3.A9gration_d.27Euler-Maclaurin"></span>Formules d'intégration d'Euler-Maclaurin</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&veaction=edit&section=11" title="Modifier la section : Formules d'intégration d'Euler-Maclaurin" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&action=edit&section=11" title="Modifier le code source de la section : Formules d'intégration d'Euler-Maclaurin"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Intégration_entre_deux_entiers"><span id="Int.C3.A9gration_entre_deux_entiers"></span>Intégration entre deux entiers</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&veaction=edit&section=12" title="Modifier la section : Intégration entre deux entiers" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&action=edit&section=12" title="Modifier le code source de la section : Intégration entre deux entiers"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dans la <a href="/wiki/M%C3%A9thode_des_trap%C3%A8zes" title="Méthode des trapèzes">méthode des trapèzes</a>, on approxime l'intégrale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{n}^{n+1}f(x)~{\rm {d}}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{n}^{n+1}f(x)~{\rm {d}}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2ecf3abe7a1f0d75d9d3de402db042f2be14ebd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.791ex; height:6.176ex;" alt="{\displaystyle \int _{n}^{n+1}f(x)~{\rm {d}}x}"></span> par <a href="/wiki/Interpolation_lin%C3%A9aire" title="Interpolation linéaire">interpolation linéaire</a> sur chaque intervalle [<i>n</i>, <i>n</i> + 1] : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{n}^{n+1}f(x)~{\rm {d}}x\approx {\frac {f(n)+f(n+1)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{n}^{n+1}f(x)~{\rm {d}}x\approx {\frac {f(n)+f(n+1)}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fefb82542aecbde557f025b0237aee0a91aaf7e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.534ex; height:6.176ex;" alt="{\displaystyle \int _{n}^{n+1}f(x)~{\rm {d}}x\approx {\frac {f(n)+f(n+1)}{2}}}"></span>. </p><p>En sommant sur tous les intervalles de longueur 1, on approxime l'intégrale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{p}^{q}f(x)~{\rm {d}}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{p}^{q}f(x)~{\rm {d}}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6507bae106a5d9f333383f369a09ede12c3b3179" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:11.46ex; height:6.176ex;" alt="{\displaystyle \int _{p}^{q}f(x)~{\rm {d}}x}"></span> par la somme </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 {\frac {f(p)}{2}}+f(p+1)+f(p+2)+\ldots +f(q-1)+{\frac {f(q)}{2}}={\frac {f\left(p\right)+f\left(q\right)}{2}}+\sum _{i=p+1}^{q-1}f\left(i\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>q</mi> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>f</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {f(p)}{2}}+f(p+1)+f(p+2)+\ldots +f(q-1)+{\frac {f(q)}{2}}={\frac {f\left(p\right)+f\left(q\right)}{2}}+\sum _{i=p+1}^{q-1}f\left(i\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/976c65328a576925ea2d1754467546540e532aea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:79.935ex; height:7.676ex;" alt="{\displaystyle {\frac {f(p)}{2}}+f(p+1)+f(p+2)+\ldots +f(q-1)+{\frac {f(q)}{2}}={\frac {f\left(p\right)+f\left(q\right)}{2}}+\sum _{i=p+1}^{q-1}f\left(i\right)}"></span></center> <p>La formule d'Euler-Maclaurin peut s'écrire : </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 \int _{p}^{q}f(x)~{\rm {d}}x={\frac {f\left(p\right)+f\left(q\right)}{2}}+\sum _{i=p+1}^{q-1}f\left(i\right)-\sum _{j=1}^{k}{\frac {b_{2j}}{(2j)!}}\left(f^{(2j-1)}(q)-f^{(2j-1)}(p)\right)-R_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>q</mi> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>f</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{p}^{q}f(x)~{\rm {d}}x={\frac {f\left(p\right)+f\left(q\right)}{2}}+\sum _{i=p+1}^{q-1}f\left(i\right)-\sum _{j=1}^{k}{\frac {b_{2j}}{(2j)!}}\left(f^{(2j-1)}(q)-f^{(2j-1)}(p)\right)-R_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/683f5254c7204fb02b031201be44ec318c229030" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:82.116ex; height:7.676ex;" alt="{\displaystyle \int _{p}^{q}f(x)~{\rm {d}}x={\frac {f\left(p\right)+f\left(q\right)}{2}}+\sum _{i=p+1}^{q-1}f\left(i\right)-\sum _{j=1}^{k}{\frac {b_{2j}}{(2j)!}}\left(f^{(2j-1)}(q)-f^{(2j-1)}(p)\right)-R_{k}}"></span></center> <div class="mw-heading mw-heading4"><h4 id="Intégration_sur_un_intervalle_quelconque"><span id="Int.C3.A9gration_sur_un_intervalle_quelconque"></span>Intégration sur un intervalle quelconque</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&veaction=edit&section=13" title="Modifier la section : Intégration sur un intervalle quelconque" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&action=edit&section=13" title="Modifier le code source de la section : Intégration sur un intervalle quelconque"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Un simple <a href="/wiki/Int%C3%A9gration_par_changement_de_variable" title="Intégration par changement de variable">changement de variable</a> permet d'obtenir une formule analogue pour une fonction définie sur un segment à bornes non entières. Les restes sont donnés avec le « point moyen » <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\xi \in ~]a,b[}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>ξ</mi> <mo>∈</mo> <mtext> </mtext> <mo stretchy="false">]</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">[</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\xi \in ~]a,b[}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b33fc3848b6a8607c6399b31fefb7455dd1c56f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.652ex; height:2.843ex;" alt="{\displaystyle \;\xi \in ~]a,b[}"></span> pour une fonction dérivable <span class="texhtml">2<i>k</i> + 2</span> fois. </p><p>En posant <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h={\frac {b-a}{N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mi>N</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h={\frac {b-a}{N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40884140b7e265bf43da90fef7dcf90a340b96eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.341ex; height:5.343ex;" alt="{\displaystyle h={\frac {b-a}{N}}}"></span>, on a<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite_crochet">[</span>7<span class="cite_crochet">]</span></a></sup> : </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 \int _{a}^{b}f(x)~{\rm {d}}x=h{\frac {f\left(a\right)+f\left(b\right)}{2}}+h\sum _{m=1}^{N-1}f\left(a+mh\right)-\sum _{j=1}^{k}{\frac {b_{2j}}{(2j)!}}h^{2j}\left(f^{(2j-1)}(b)-f^{(2j-1)}(a)\right)-(b-a){\frac {b_{2k+2}}{(2k+2)!}}f^{(2k+2)}(\xi )h^{2k+2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>=</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>b</mi> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>h</mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>+</mo> <mi>m</mi> <mi>h</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)~{\rm {d}}x=h{\frac {f\left(a\right)+f\left(b\right)}{2}}+h\sum _{m=1}^{N-1}f\left(a+mh\right)-\sum _{j=1}^{k}{\frac {b_{2j}}{(2j)!}}h^{2j}\left(f^{(2j-1)}(b)-f^{(2j-1)}(a)\right)-(b-a){\frac {b_{2k+2}}{(2k+2)!}}f^{(2k+2)}(\xi )h^{2k+2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/842c8f361d7ce86fa702c2f3678d8ccfe2226882" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:122.104ex; height:7.676ex;" alt="{\displaystyle \int _{a}^{b}f(x)~{\rm {d}}x=h{\frac {f\left(a\right)+f\left(b\right)}{2}}+h\sum _{m=1}^{N-1}f\left(a+mh\right)-\sum _{j=1}^{k}{\frac {b_{2j}}{(2j)!}}h^{2j}\left(f^{(2j-1)}(b)-f^{(2j-1)}(a)\right)-(b-a){\frac {b_{2k+2}}{(2k+2)!}}f^{(2k+2)}(\xi )h^{2k+2}}"></span>.</center> <p>Le « terme d'erreur » peut également s'écrire : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -R_{k}=-(b-a){\frac {b_{2k+2}}{(2k+2)!}}f^{(2k+2)}(\xi )h^{2k+2}=-{\frac {b_{2k+2}(b-a)^{2k+3}}{(2k+2)!N^{2k+2}}}f^{(2k+2)}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>3</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>!</mo> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -R_{k}=-(b-a){\frac {b_{2k+2}}{(2k+2)!}}f^{(2k+2)}(\xi )h^{2k+2}=-{\frac {b_{2k+2}(b-a)^{2k+3}}{(2k+2)!N^{2k+2}}}f^{(2k+2)}(\xi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/158eae3c504b601bc3c3516a3800369abfed9826" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:72.004ex; height:6.676ex;" alt="{\displaystyle -R_{k}=-(b-a){\frac {b_{2k+2}}{(2k+2)!}}f^{(2k+2)}(\xi )h^{2k+2}=-{\frac {b_{2k+2}(b-a)^{2k+3}}{(2k+2)!N^{2k+2}}}f^{(2k+2)}(\xi )}"></span>. </p><p>Si <span class="texhtml"><i>N</i> = 1</span>, on a une formule où n'interviennent que les extrémités <i>a</i> et <i>b</i><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite_crochet">[</span>8<span class="cite_crochet">]</span></a></sup> : </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 \int _{a}^{b}f(x)~{\rm {d}}x=(b-a){\frac {f(a)+f(b)}{2}}-\sum _{j=1}^{k}{\frac {(b-a)^{2j}}{(2j)!}}b_{2j}\left(f^{(2j-1)}(b)-f^{(2j-1)}(a)\right)-(b-a)^{2k+3}{\frac {b_{2k+2}}{(2k+2)!}}f^{(2k+2)}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)~{\rm {d}}x=(b-a){\frac {f(a)+f(b)}{2}}-\sum _{j=1}^{k}{\frac {(b-a)^{2j}}{(2j)!}}b_{2j}\left(f^{(2j-1)}(b)-f^{(2j-1)}(a)\right)-(b-a)^{2k+3}{\frac {b_{2k+2}}{(2k+2)!}}f^{(2k+2)}(\xi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f79ffc18e58ade5c771925e227f929f52385201" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:109.782ex; height:7.676ex;" alt="{\displaystyle \int _{a}^{b}f(x)~{\rm {d}}x=(b-a){\frac {f(a)+f(b)}{2}}-\sum _{j=1}^{k}{\frac {(b-a)^{2j}}{(2j)!}}b_{2j}\left(f^{(2j-1)}(b)-f^{(2j-1)}(a)\right)-(b-a)^{2k+3}{\frac {b_{2k+2}}{(2k+2)!}}f^{(2k+2)}(\xi )}"></span></center> <div class="mw-heading mw-heading2"><h2 id="Expressions_du_reste_pour_k_=_0_et_pour_k_=_1"><span id="Expressions_du_reste_pour_k_.3D_0_et_pour_k_.3D_1"></span>Expressions du reste pour <i>k</i> = 0 et pour <i>k</i> = 1</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&veaction=edit&section=14" title="Modifier la section : Expressions du reste pour k = 0 et pour k = 1" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&action=edit&section=14" title="Modifier le code source de la section : Expressions du reste pour k = 0 et pour k = 1"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Expressions_de_R0_et_erreur_de_la_méthode_des_trapèzes"><span id="Expressions_de_R0_et_erreur_de_la_m.C3.A9thode_des_trap.C3.A8zes"></span>Expressions de <i>R</i><sub>0</sub> et erreur de la méthode des trapèzes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&veaction=edit&section=15" title="Modifier la section : Expressions de R0 et erreur de la méthode des trapèzes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&action=edit&section=15" title="Modifier le code source de la section : Expressions de R0 et erreur de la méthode des trapèzes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Les premiers polynômes de Bernoulli sont : </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 B_{0}(y)=1,\quad B_{1}(y)=y-{\frac {1}{2}},\quad B_{2}(y)=y^{2}-y+{\frac {1}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="1em" /> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>y</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{0}(y)=1,\quad B_{1}(y)=y-{\frac {1}{2}},\quad B_{2}(y)=y^{2}-y+{\frac {1}{6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0117d4b1f5d1e65a2a58c1f705b4522c695d0a41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:51.564ex; height:5.176ex;" alt="{\displaystyle B_{0}(y)=1,\quad B_{1}(y)=y-{\frac {1}{2}},\quad B_{2}(y)=y^{2}-y+{\frac {1}{6}}}"></span>.</dd></dl> <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 R_{0}={\frac {f\left(p\right)+f\left(q\right)}{2}}+\sum _{n=p+1}^{q-1}f\left(n\right)-\int _{p}^{q}f(x)~{\rm {d}}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>q</mi> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>f</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{0}={\frac {f\left(p\right)+f\left(q\right)}{2}}+\sum _{n=p+1}^{q-1}f\left(n\right)-\int _{p}^{q}f(x)~{\rm {d}}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85e753ba6ac4e18bf82de390fc64db0a6f6e27c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:46.372ex; height:7.676ex;" alt="{\displaystyle R_{0}={\frac {f\left(p\right)+f\left(q\right)}{2}}+\sum _{n=p+1}^{q-1}f\left(n\right)-\int _{p}^{q}f(x)~{\rm {d}}x}"></span>.</center> <p><span class="texhtml">(-<i>R</i><sub>0</sub>)</span> est l'erreur faite en approximant l'intégrale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{n}^{n+1}f(x)~{\rm {d}}x\approx {\frac {f(n)+f(n+1)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{n}^{n+1}f(x)~{\rm {d}}x\approx {\frac {f(n)+f(n+1)}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fefb82542aecbde557f025b0237aee0a91aaf7e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.534ex; height:6.176ex;" alt="{\displaystyle \int _{n}^{n+1}f(x)~{\rm {d}}x\approx {\frac {f(n)+f(n+1)}{2}}}"></span> par la méthode des trapèzes sur chaque intervalle [<i>n</i>, <i>n</i> + 1]. </p><p>Si <span class="texhtml mvar" style="font-style:italic;">f</span> est continûment dérivable une fois (on pose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=x-\lfloor x\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mo fence="false" stretchy="false">⌊</mo> <mi>x</mi> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=x-\lfloor x\rfloor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66610226f6e9c7e9c0ab8a8debe3cd38e941c02d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.818ex; height:2.843ex;" alt="{\displaystyle y=x-\lfloor x\rfloor }"></span><sup id="cite_ref-Hardy_14-0" class="reference"><a href="#cite_note-Hardy-14"><span class="cite_crochet">[</span>9<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 R_{0}=\int _{p}^{q}f'(x){B_{1}(y)}~{\rm {d}}x=\int _{p}^{q}f'(x)\left(y-{\tfrac {1}{2}}\right)~{\rm {d}}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mi>y</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{0}=\int _{p}^{q}f'(x){B_{1}(y)}~{\rm {d}}x=\int _{p}^{q}f'(x)\left(y-{\tfrac {1}{2}}\right)~{\rm {d}}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b92cf59edaa52e6a86173ed14731122ca9294f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:47.73ex; height:6.176ex;" alt="{\displaystyle R_{0}=\int _{p}^{q}f'(x){B_{1}(y)}~{\rm {d}}x=\int _{p}^{q}f'(x)\left(y-{\tfrac {1}{2}}\right)~{\rm {d}}x}"></span></dd></dl> <p>Si <span class="texhtml mvar" style="font-style:italic;">f</span> est une fonction réelle continûment dérivable deux fois<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite_crochet">[</span>10<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 {\begin{aligned}R_{0}=-{\frac {1}{2!}}\int _{p}^{q}f''(x)\left(B_{2}(y)-{\tfrac {1}{6}}\right)~{\rm {d}}x&=-{\frac {1}{2}}\int _{p}^{q}f''(x)(y^{2}-y)~{\rm {d}}x\\&={\frac {q-p}{12}}f''(\xi )\qquad ({\text{avec }}\;\xi \in \left]p,q\right[).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <mo>−</mo> <mi>p</mi> </mrow> <mn>12</mn> </mfrac> </mrow> <msup> <mi>f</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mspace width="2em" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>avec </mtext> </mrow> <mspace width="thickmathspace" /> <mi>ξ</mi> <mo>∈</mo> <mrow> <mo>]</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>[</mo> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}R_{0}=-{\frac {1}{2!}}\int _{p}^{q}f''(x)\left(B_{2}(y)-{\tfrac {1}{6}}\right)~{\rm {d}}x&=-{\frac {1}{2}}\int _{p}^{q}f''(x)(y^{2}-y)~{\rm {d}}x\\&={\frac {q-p}{12}}f''(\xi )\qquad ({\text{avec }}\;\xi \in \left]p,q\right[).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1d303826dfbdf125419bf36a4f182ea59e14c6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:73.51ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}R_{0}=-{\frac {1}{2!}}\int _{p}^{q}f''(x)\left(B_{2}(y)-{\tfrac {1}{6}}\right)~{\rm {d}}x&=-{\frac {1}{2}}\int _{p}^{q}f''(x)(y^{2}-y)~{\rm {d}}x\\&={\frac {q-p}{12}}f''(\xi )\qquad ({\text{avec }}\;\xi \in \left]p,q\right[).\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Formule_de_quadrature_et_terme_d'erreur_pour_k_=_1"><span id="Formule_de_quadrature_et_terme_d.27erreur_pour_k_.3D_1"></span>Formule de quadrature et terme d'erreur pour <i>k</i> = 1</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&veaction=edit&section=16" title="Modifier la section : Formule de quadrature et terme d'erreur pour k = 1" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&action=edit&section=16" title="Modifier le code source de la section : Formule de quadrature et terme d'erreur pour k = 1"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Les polynômes de Bernoulli qui interviennent sont<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite_crochet">[</span>N 6<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 B_{2}(y)=y^{2}-y+{\frac {1}{6}},\quad B_{3}(y)=y^{3}-{\frac {3}{2}}y^{2}+{\frac {1}{2}}y=y(y-1)\left(y-{\tfrac {1}{2}}\right),\quad B_{4}(y)=y^{4}-2y^{3}+y^{2}-{\frac {1}{30}}=(y^{2}-y)^{2}-{\frac {1}{30}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>y</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>y</mi> <mo>=</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mi>y</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>30</mn> </mfrac> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>30</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{2}(y)=y^{2}-y+{\frac {1}{6}},\quad B_{3}(y)=y^{3}-{\frac {3}{2}}y^{2}+{\frac {1}{2}}y=y(y-1)\left(y-{\tfrac {1}{2}}\right),\quad B_{4}(y)=y^{4}-2y^{3}+y^{2}-{\frac {1}{30}}=(y^{2}-y)^{2}-{\frac {1}{30}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13328f060f18f22bfd0a9c945b1fca0839df974a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:117.113ex; height:5.176ex;" alt="{\displaystyle B_{2}(y)=y^{2}-y+{\frac {1}{6}},\quad B_{3}(y)=y^{3}-{\frac {3}{2}}y^{2}+{\frac {1}{2}}y=y(y-1)\left(y-{\tfrac {1}{2}}\right),\quad B_{4}(y)=y^{4}-2y^{3}+y^{2}-{\frac {1}{30}}=(y^{2}-y)^{2}-{\frac {1}{30}}}"></span>.</dd></dl> <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 R_{1}={\frac {f\left(p\right)+f\left(q\right)}{2}}+\sum _{n=p+1}^{q-1}f\left(n\right)-\int _{p}^{q}f(x)~{\rm {d}}x-{\frac {1}{12}}(f'(q)-f'(p))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>q</mi> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>f</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{1}={\frac {f\left(p\right)+f\left(q\right)}{2}}+\sum _{n=p+1}^{q-1}f\left(n\right)-\int _{p}^{q}f(x)~{\rm {d}}x-{\frac {1}{12}}(f'(q)-f'(p))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7f34a9e877ce1349a01a97207f5fa967511050d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:66.891ex; height:7.676ex;" alt="{\displaystyle R_{1}={\frac {f\left(p\right)+f\left(q\right)}{2}}+\sum _{n=p+1}^{q-1}f\left(n\right)-\int _{p}^{q}f(x)~{\rm {d}}x-{\frac {1}{12}}(f'(q)-f'(p))}"></span>.</center> <p><span class="texhtml">(–<i>R</i><sub>1</sub>)</span> est le terme d'erreur correspondant à la formule de quadrature<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite_crochet">[</span>11<span class="cite_crochet">]</span></a></sup>, exacte pour les polynômes de degré inférieur ou égal à trois : </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 \int _{p}^{q}f(x)~{\rm {d}}x\approx {\frac {f\left(p\right)+f\left(q\right)}{2}}+\sum _{n=p+1}^{q-1}f\left(n\right)+{\frac {f'(p)-f'(q)}{12}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>q</mi> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>f</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> <mn>12</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{p}^{q}f(x)~{\rm {d}}x\approx {\frac {f\left(p\right)+f\left(q\right)}{2}}+\sum _{n=p+1}^{q-1}f\left(n\right)+{\frac {f'(p)-f'(q)}{12}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c9b2cac77465b58c365faf92f9dc50b489e615d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:57.098ex; height:7.676ex;" alt="{\displaystyle \int _{p}^{q}f(x)~{\rm {d}}x\approx {\frac {f\left(p\right)+f\left(q\right)}{2}}+\sum _{n=p+1}^{q-1}f\left(n\right)+{\frac {f'(p)-f'(q)}{12}}}"></span></center> <p>(dans l'article <a href="/wiki/Calcul_num%C3%A9rique_d%27une_int%C3%A9grale" title="Calcul numérique d'une intégrale">Calcul numérique d'une intégrale</a>, il s'agit de la formule de Newton-Coates généralisée NC-2-2). </p><p>Si <span class="texhtml mvar" style="font-style:italic;">f</span> est continûment dérivable deux fois (en posant <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=x-\lfloor x\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mo fence="false" stretchy="false">⌊</mo> <mi>x</mi> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=x-\lfloor x\rfloor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66610226f6e9c7e9c0ab8a8debe3cd38e941c02d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.818ex; height:2.843ex;" alt="{\displaystyle y=x-\lfloor x\rfloor }"></span><sup id="cite_ref-Hardy_14-1" class="reference"><a href="#cite_note-Hardy-14"><span class="cite_crochet">[</span>9<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 R_{1}=-{\frac {1}{2!}}\int _{p}^{q}f''(x){B_{2}(y)}~{\rm {d}}x=-{\frac {1}{2}}\int _{p}^{q}f''(x)\left(y^{2}-y+{\tfrac {1}{6}}\right)~{\rm {d}}x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>y</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{1}=-{\frac {1}{2!}}\int _{p}^{q}f''(x){B_{2}(y)}~{\rm {d}}x=-{\frac {1}{2}}\int _{p}^{q}f''(x)\left(y^{2}-y+{\tfrac {1}{6}}\right)~{\rm {d}}x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5156ba34d7edb9ada1616d51b3230a91eaf1eb30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:64.017ex; height:6.176ex;" alt="{\displaystyle R_{1}=-{\frac {1}{2!}}\int _{p}^{q}f''(x){B_{2}(y)}~{\rm {d}}x=-{\frac {1}{2}}\int _{p}^{q}f''(x)\left(y^{2}-y+{\tfrac {1}{6}}\right)~{\rm {d}}x.}"></span></dd></dl> <p>Si <span class="texhtml mvar" style="font-style:italic;">f</span> est continûment dérivable trois fois : </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 R_{1}={\frac {1}{3!}}\int _{p}^{q}f'''(x){B_{3}(y)}~{\rm {d}}x={\frac {1}{6}}\int _{p}^{q}f'''(x)\left(y^{3}-{\tfrac {3}{2}}y^{2}+{\tfrac {1}{2}}y\right)~{\rm {d}}x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mo>‴</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mo>‴</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{1}={\frac {1}{3!}}\int _{p}^{q}f'''(x){B_{3}(y)}~{\rm {d}}x={\frac {1}{6}}\int _{p}^{q}f'''(x)\left(y^{3}-{\tfrac {3}{2}}y^{2}+{\tfrac {1}{2}}y\right)~{\rm {d}}x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6817e045c5ba5bc320a21de080e9a0ffd51b34bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:65.178ex; height:6.176ex;" alt="{\displaystyle R_{1}={\frac {1}{3!}}\int _{p}^{q}f'''(x){B_{3}(y)}~{\rm {d}}x={\frac {1}{6}}\int _{p}^{q}f'''(x)\left(y^{3}-{\tfrac {3}{2}}y^{2}+{\tfrac {1}{2}}y\right)~{\rm {d}}x.}"></span></dd></dl> <p>Si <span class="texhtml mvar" style="font-style:italic;">f</span> est une fonction réelle continûment dérivable quatre fois : </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 R_{1}=-{\frac {1}{4!}}\int _{p}^{q}f^{(4)}(x)\left(B_{4}(y)+{\tfrac {1}{30}}\right)~{\rm {d}}x=-{\frac {1}{24}}\int _{p}^{q}f^{(4)}(x)(y^{2}-y)^{2}~{\rm {d}}x=-{\frac {q-p}{720}}f^{(4)}(\xi )\qquad ({\text{avec }}\;\xi \in \left]p,q\right[).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>30</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>24</mn> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <mo>−</mo> <mi>p</mi> </mrow> <mn>720</mn> </mfrac> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mspace width="2em" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>avec </mtext> </mrow> <mspace width="thickmathspace" /> <mi>ξ</mi> <mo>∈</mo> <mrow> <mo>]</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>[</mo> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{1}=-{\frac {1}{4!}}\int _{p}^{q}f^{(4)}(x)\left(B_{4}(y)+{\tfrac {1}{30}}\right)~{\rm {d}}x=-{\frac {1}{24}}\int _{p}^{q}f^{(4)}(x)(y^{2}-y)^{2}~{\rm {d}}x=-{\frac {q-p}{720}}f^{(4)}(\xi )\qquad ({\text{avec }}\;\xi \in \left]p,q\right[).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a8530806de13b4b7cf6e46612c38901984d08a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:109.146ex; height:6.176ex;" alt="{\displaystyle R_{1}=-{\frac {1}{4!}}\int _{p}^{q}f^{(4)}(x)\left(B_{4}(y)+{\tfrac {1}{30}}\right)~{\rm {d}}x=-{\frac {1}{24}}\int _{p}^{q}f^{(4)}(x)(y^{2}-y)^{2}~{\rm {d}}x=-{\frac {q-p}{720}}f^{(4)}(\xi )\qquad ({\text{avec }}\;\xi \in \left]p,q\right[).}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Autres_applications">Autres applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&veaction=edit&section=17" title="Modifier la section : Autres applications" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&action=edit&section=17" title="Modifier le code source de la section : Autres applications"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Le_problème_de_Bâle"><span id="Le_probl.C3.A8me_de_B.C3.A2le"></span>Le problème de Bâle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&veaction=edit&section=18" title="Modifier la section : Le problème de Bâle" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&action=edit&section=18" title="Modifier le code source de la section : Le problème de Bâle"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Le <a href="/wiki/Probl%C3%A8me_de_B%C3%A2le" title="Problème de Bâle">problème de Bâle</a> demandait de déterminer la somme <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+{\frac {1}{4}}+{\frac {1}{9}}+{\frac {1}{16}}+{\frac {1}{25}}+\cdots =\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>9</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>25</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+{\frac {1}{4}}+{\frac {1}{9}}+{\frac {1}{16}}+{\frac {1}{25}}+\cdots =\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09c78a1f2fb594dfe67841b595213d4f69c07394" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:39.179ex; height:6.843ex;" alt="{\displaystyle 1+{\frac {1}{4}}+{\frac {1}{9}}+{\frac {1}{16}}+{\frac {1}{25}}+\cdots =\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}.}"></span> </p><p>Euler calcula cette somme à 20 décimales en utilisant seulement quelques termes de la formule d'Euler-Maclaurin. Ce calcul le convainquit probablement qu'elle valait <span class="texhtml"><span style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;">π<sup>2</sup></span><span style="position:absolute;left:-10000px;top:auto;width:1px;height:1px;overflow:hidden">/</span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">6</span></span></span>, résultat qu'il publia en 1735 (mais avec des arguments incorrects ; il lui fallut six ans de plus pour trouver <a href="/wiki/Probl%C3%A8me_de_B%C3%A2le#La_démonstration_d'Euler" title="Problème de Bâle">une démonstration rigoureuse</a>)<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite_crochet">[</span>12<span class="cite_crochet">]</span></a></sup>. </p> <div class="mw-heading mw-heading3"><h3 id="Sommes_polynomiales">Sommes polynomiales</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&veaction=edit&section=19" title="Modifier la section : Sommes polynomiales" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&action=edit&section=19" title="Modifier le code source de la section : Sommes polynomiales"><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/Formule_de_Faulhaber" title="Formule de Faulhaber">formule de Faulhaber</a>.</div></div> <p>Si <span class="texhtml mvar" style="font-style:italic;">f</span> est un <a href="/wiki/Polyn%C3%B4me" title="Polynôme">polynôme</a> de degré <i>d</i> et si l'on applique la formule sommatoire avec <i>p</i> = 0, <i>q</i> = <i>n</i> et <i>k</i> choisi tel que <span class="texhtml"><i>d</i> ≤ 2<i>k</i> +1</span>, le reste <span class="texhtml mvar" style="font-style:italic;">R<sub>k</sub></span> disparaît. </p><p>Par exemple, si <span class="texhtml"><i>f</i>(<i>x</i>) = <i>x</i><sup>3</sup></span>, on peut prendre <i>k</i> = 1 pour obtenir </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 \sum _{i=0}^{n}i^{3}=\int _{0}^{n}x^{3}\mathrm {d} x+{\frac {0^{3}+n^{3}}{2}}+{\frac {1}{6}}{\frac {3n^{2}-0}{2!}}={\frac {n^{4}}{4}}+{\frac {n^{3}}{2}}+{\frac {n^{2}}{4}}={\frac {n^{4}+2n^{3}+n^{2}}{4}}=\left({\frac {n(n+1)}{2}}\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>0</mn> </mrow> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=0}^{n}i^{3}=\int _{0}^{n}x^{3}\mathrm {d} x+{\frac {0^{3}+n^{3}}{2}}+{\frac {1}{6}}{\frac {3n^{2}-0}{2!}}={\frac {n^{4}}{4}}+{\frac {n^{3}}{2}}+{\frac {n^{2}}{4}}={\frac {n^{4}+2n^{3}+n^{2}}{4}}=\left({\frac {n(n+1)}{2}}\right)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20daeaa1814b3013a27bf3a85ef1561183fa0fcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:96.016ex; height:7.176ex;" alt="{\displaystyle \sum _{i=0}^{n}i^{3}=\int _{0}^{n}x^{3}\mathrm {d} x+{\frac {0^{3}+n^{3}}{2}}+{\frac {1}{6}}{\frac {3n^{2}-0}{2!}}={\frac {n^{4}}{4}}+{\frac {n^{3}}{2}}+{\frac {n^{2}}{4}}={\frac {n^{4}+2n^{3}+n^{2}}{4}}=\left({\frac {n(n+1)}{2}}\right)^{2}}"></span>.</center> <div class="mw-heading mw-heading3"><h3 id="Développements_asymptotiques_de_fonctions_définies_par_une_série"><span id="D.C3.A9veloppements_asymptotiques_de_fonctions_d.C3.A9finies_par_une_s.C3.A9rie"></span>Développements asymptotiques de fonctions définies par une série</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&veaction=edit&section=20" title="Modifier la section : Développements asymptotiques de fonctions définies par une série" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&action=edit&section=20" title="Modifier le code source de la section : Développements asymptotiques de fonctions définies par une série"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pour déterminer des <a href="/wiki/D%C3%A9veloppement_asymptotique" title="Développement asymptotique">développements asymptotiques</a> de sommes et de séries, la forme la plus utile de la formule sommatoire est sans doute (pour <i>a</i> et <i>b</i> entiers) : </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 \sum _{n=a}^{b}f(n)\sim \int _{a}^{b}f(x)~{\rm {d}}x+{\frac {f(a)+f(b)}{2}}+\sum _{j=1}^{\infty }\,{\frac {B_{2j}}{(2j)!}}\left(f^{(2j-1)}(b)-f^{(2j-1)}(a)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=a}^{b}f(n)\sim \int _{a}^{b}f(x)~{\rm {d}}x+{\frac {f(a)+f(b)}{2}}+\sum _{j=1}^{\infty }\,{\frac {B_{2j}}{(2j)!}}\left(f^{(2j-1)}(b)-f^{(2j-1)}(a)\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cedce66b95f1682760cd902af167103b013983c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:74.748ex; height:7.676ex;" alt="{\displaystyle \sum _{n=a}^{b}f(n)\sim \int _{a}^{b}f(x)~{\rm {d}}x+{\frac {f(a)+f(b)}{2}}+\sum _{j=1}^{\infty }\,{\frac {B_{2j}}{(2j)!}}\left(f^{(2j-1)}(b)-f^{(2j-1)}(a)\right)}"></span></center> <p>Ce développement reste souvent valide même lorsque l'on prend les limites quand <span class="texhtml"><i>a </i>→ –∞</span> ou <span class="texhtml"><i>b </i>→ +∞</span>, ou les deux. Dans de nombreux cas, l'intégrale de droite peut être calculée de manière exacte avec des fonctions élémentaires, alors que ce n'est pas le cas de la somme. </p><p>L'écriture précédente doit être interprétée comme une <a href="/wiki/S%C3%A9rie_formelle" title="Série formelle">série formelle</a>, car, le plus souvent, cette série est <a href="/wiki/S%C3%A9rie_divergente" title="Série divergente">divergente</a> ; la formule ne peut en général pas être exploitée directement sous cette forme. Toutefois, Euler avait déjà remarqué qu'on obtenait une précision numérique remarquable en tronquant la formule au plus petit terme de la série, ce qui fut précisé et expliqué par les travaux d'<a href="/wiki/%C3%89mile_Borel" title="Émile Borel">Émile Borel</a><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite_crochet">[</span>13<span class="cite_crochet">]</span></a></sup>. </p><p>Par exemple : </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 \underbrace {\sum _{k=0}^{\infty }{\frac {1}{(z+k)^{2}}}} _{=\psi _{1}(z)}\sim \underbrace {\int _{0}^{\infty }{\frac {1}{(z+x)^{2}}}~{\rm {d}}x} _{=1/z}+{\frac {1}{2z^{2}}}+\sum _{j=1}^{\infty }{\frac {B_{2j}}{z^{2j+1}}}.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <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> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>k</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>=</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </munder> <mo>∼</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>z</mi> </mrow> </munder> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \underbrace {\sum _{k=0}^{\infty }{\frac {1}{(z+k)^{2}}}} _{=\psi _{1}(z)}\sim \underbrace {\int _{0}^{\infty }{\frac {1}{(z+x)^{2}}}~{\rm {d}}x} _{=1/z}+{\frac {1}{2z^{2}}}+\sum _{j=1}^{\infty }{\frac {B_{2j}}{z^{2j+1}}}.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9de04dfc80bb87965374180d9d100d1c4b38e829" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.005ex; width:52.926ex; height:10.843ex;" alt="{\displaystyle \underbrace {\sum _{k=0}^{\infty }{\frac {1}{(z+k)^{2}}}} _{=\psi _{1}(z)}\sim \underbrace {\int _{0}^{\infty }{\frac {1}{(z+x)^{2}}}~{\rm {d}}x} _{=1/z}+{\frac {1}{2z^{2}}}+\sum _{j=1}^{\infty }{\frac {B_{2j}}{z^{2j+1}}}.\,}"></span></center> <p>Ici, le membre de gauche est égal à <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{1}(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{1}(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68b482f10339e8d27125edcc1ab412367db38865" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.465ex; height:2.843ex;" alt="{\displaystyle \psi _{1}(z)}"></span>, c'est-à-dire à la <a href="/wiki/Fonction_polygamma" title="Fonction polygamma">fonction polygamma</a> d'ordre 1 (appelée aussi fonction trigamma) définie à partir de la <a href="/wiki/Fonction_Gamma" class="mw-redirect" title="Fonction Gamma">fonction Gamma</a> : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{1}(z)={\frac {\mathrm {d} ^{2}}{\mathrm {d} z^{2}}}\ln \Gamma (z)=\sum _{k=0}^{\infty }{\frac {1}{(z+k)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</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> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>k</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{1}(z)={\frac {\mathrm {d} ^{2}}{\mathrm {d} z^{2}}}\ln \Gamma (z)=\sum _{k=0}^{\infty }{\frac {1}{(z+k)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a754e5f5aedbd0eb9a7978b9640fd307af964f4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.58ex; height:7.009ex;" alt="{\displaystyle \psi _{1}(z)={\frac {\mathrm {d} ^{2}}{\mathrm {d} z^{2}}}\ln \Gamma (z)=\sum _{k=0}^{\infty }{\frac {1}{(z+k)^{2}}}}"></span>. </p><p>La formule d'Euler Maclaurin amène à un <a href="/wiki/D%C3%A9veloppement_asymptotique" title="Développement asymptotique">développement asymptotique</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 \psi _{1}(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{1}(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68b482f10339e8d27125edcc1ab412367db38865" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.465ex; height:2.843ex;" alt="{\displaystyle \psi _{1}(z)}"></span>, lequel permet une estimation précise de l'erreur de la <a href="/wiki/Formule_de_Stirling" title="Formule de Stirling">formule de Stirling</a> pour la fonction Gamma : </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 \ln \Gamma (z)=z\ln z-z-{\frac {1}{2}}\ln {z}+{\frac {1}{2}}\ln(2\pi )+\sum _{n=1}^{N}{\frac {B_{2n}}{2n(2n-1)z^{2n-1}}}+R_{N}(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>z</mi> <mi>ln</mi> <mo>⁡</mo> <mi>z</mi> <mo>−</mo> <mi>z</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mrow> <mn>2</mn> <mi>n</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln \Gamma (z)=z\ln z-z-{\frac {1}{2}}\ln {z}+{\frac {1}{2}}\ln(2\pi )+\sum _{n=1}^{N}{\frac {B_{2n}}{2n(2n-1)z^{2n-1}}}+R_{N}(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8eb7ba6c56a232301d07df24a88ad8e2b77ffb2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:71.473ex; height:7.343ex;" alt="{\displaystyle \ln \Gamma (z)=z\ln z-z-{\frac {1}{2}}\ln {z}+{\frac {1}{2}}\ln(2\pi )+\sum _{n=1}^{N}{\frac {B_{2n}}{2n(2n-1)z^{2n-1}}}+R_{N}(z)}"></span>.</center> <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 \Gamma (z)={\sqrt {\frac {2\pi }{z}}}\,{\left({\frac {z}{e}}\right)}^{z}\left(1+O\left({\frac {1}{z}}\right)\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>z</mi> </mfrac> </msqrt> </mrow> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mi>e</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>O</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (z)={\sqrt {\frac {2\pi }{z}}}\,{\left({\frac {z}{e}}\right)}^{z}\left(1+O\left({\frac {1}{z}}\right)\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed4a7a21192f91f42e90e0f2dbc5cbc851a089a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.617ex; height:6.343ex;" alt="{\displaystyle \Gamma (z)={\sqrt {\frac {2\pi }{z}}}\,{\left({\frac {z}{e}}\right)}^{z}\left(1+O\left({\frac {1}{z}}\right)\right).}"></span></center> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&veaction=edit&section=21" title="Modifier la section : Notes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&action=edit&section=21" title="Modifier le code source de la section : Notes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="references-small decimal" style=""><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Cohen22-2"><span class="mw-cite-backlink noprint">↑ <sup><a href="#cite_ref-Cohen22_2-0">a</a> et <a href="#cite_ref-Cohen22_2-1">b</a></sup> </span><span class="reference-text"><a href="#Cohen2007">Cohen 2007</a>, <abbr class="abbr" title="page(s)">p.</abbr> 23. La formule avec son reste est donnée dans le cas, plus général, où les extrémités de l'intervalle sont des nombres réels <i>a</i> et <i>a</i>+<i>N</i> qui diffèrent d'un nombre entier naturel <i>N</i>. Cette formule est un cas particulier d'une formule donnée page 22 où les nombres de Bernoulli <span class="texhtml mvar" style="font-style:italic;">b<sub>j</sub></span> sont remplacés par les valeurs des polynômes de Bernoulli <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{r}(x-\lfloor x\rfloor )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mo fence="false" stretchy="false">⌊</mo> <mi>x</mi> <mo fence="false" stretchy="false">⌋</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{r}(x-\lfloor x\rfloor )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a1ac5c885cf342b919dbc4ab8738d687f3a9529" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.112ex; height:2.843ex;" alt="{\displaystyle B_{r}(x-\lfloor x\rfloor )}"></span> aux extrémités <i>a</i> et <i>b</i>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink noprint"><a href="#cite_ref-4">↑</a> </span><span class="reference-text"><a href="#Dieudonné1980">Dieudonné 1980</a>, <abbr class="abbr" title="page(s)">p.</abbr> 303. Selon Dieudonné, il suffit que <span class="texhtml mvar" style="font-style:italic;">f</span> admette une dérivée (2<i>k</i> + 1)-ème continue par morceaux.</span> </li> <li id="cite_note-D301-7"><span class="mw-cite-backlink noprint"><a href="#cite_ref-D301_7-0">↑</a> </span><span class="reference-text"><a href="#Dieudonné1980">Dieudonné 1980</a>, <abbr class="abbr" title="page(s)">p.</abbr> 301. Dieudonné note <span class="texhtml mvar" style="font-style:italic;">B<sub>k</sub></span> les coefficients <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |b_{2k}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |b_{2k}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9a4b4e5de2caa015c859451d3bdf5d8ba50d689" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.202ex; height:2.843ex;" alt="{\displaystyle |b_{2k}|}"></span>.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink noprint"><a href="#cite_ref-8">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Lehmer1940"><span class="ouvrage" id="Derrick_Lehmer1940"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <a href="/wiki/Derrick_Lehmer" title="Derrick Lehmer">Derrick <span class="nom_auteur">Lehmer</span></a>, « <cite style="font-style:normal" lang="en">On the maxima and minima of Bernoulli polynomials</cite> », <i><span class="lang-en" lang="en"><a href="/wiki/The_American_Mathematical_Monthly" title="The American Mathematical Monthly">The American Mathematical Monthly</a></span></i>, <abbr class="abbr" title="volume">vol.</abbr> 47, <abbr class="abbr" title="numéro">n<sup>o</sup></abbr> 8,‎ <time>1940</time>, <abbr class="abbr" title="pages">p.</abbr> <span class="nowrap">533–538</span> <small style="line-height:1em;">(<a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">DOI</a> <span class="plainlinks noarchive nowrap"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.2307/2303833">10.2307/2303833</a></span>, <a href="/wiki/JSTOR" title="JSTOR">JSTOR</a> <span class="plainlinks noarchive nowrap"><a rel="nofollow" class="external text" href="https://jstor.org/stable/2303833">2303833</a></span>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=On+the+maxima+and+minima+of+Bernoulli+polynomials&rft.jtitle=The+American+Mathematical+Monthly&rft.issue=8&rft.aulast=Lehmer&rft.aufirst=Derrick&rft.date=1940&rft.volume=47&rft.pages=533%E2%80%93538&rft_id=info%3Adoi%2F10.2307%2F2303833&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AFormule+d%27Euler-Maclaurin"></span></span></span>.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink noprint"><a href="#cite_ref-9">↑</a> </span><span class="reference-text">Selon <a href="#Bourbaki,_FVR">Bourbaki, FVR</a>, <abbr class="abbr" title="page(s)">p.</abbr> VI.26, il suffit que <span class="texhtml mvar" style="font-style:italic;">f</span> soit 2<i>k</i> + 1 fois dérivable et que la dérivée d'ordre 2<i>k</i> + 1 soit monotone.</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink noprint"><a href="#cite_ref-16">↑</a> </span><span class="reference-text"><a href="#Deheuvels1980">Deheuvels 1980</a>, <abbr class="abbr" title="page(s)">p.</abbr> 189. Noter que dans la notation de Deheuvels, les nombres de Bernoulli sont considérés sans leur signe (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{2k}=|b_{2k}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{2k}=|b_{2k}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b65dcd1558d079d7a44b0a60b272b95bceaacb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.975ex; height:2.843ex;" alt="{\displaystyle B_{2k}=|b_{2k}|}"></span>).</span> </li> </ol></div> </div> <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-Maclaurin&veaction=edit&section=22" 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-Maclaurin&action=edit&section=22" title="Modifier le code source de la section : Voir aussi"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Formule_d%27Abel-Plana" title="Formule d'Abel-Plana">Formule d'Abel-Plana</a></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-Maclaurin&veaction=edit&section=23" 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-Maclaurin&action=edit&section=23" 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="references-small decimal" style="column-width:20em;"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink noprint"><a href="#cite_ref-1">↑</a> </span><span class="reference-text"><span class="ouvrage">« <a rel="nofollow" class="external text" href="http://www.bibmath.net/dico/index.php?action=affiche&quoi=./e/eulermaclaurin.html"><cite style="font-style:normal;">Formule d'Euler-MacLaurin [sic]</cite></a> », sur <span class="italique">bibmath.net/dico</span></span>.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink noprint"><a href="#cite_ref-3">↑</a> </span><span class="reference-text"><a href="#Tenenbaum2008">Tenenbaum 2008</a>, <abbr class="abbr" title="page(s)">p.</abbr> 23.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink noprint"><a href="#cite_ref-5">↑</a> </span><span class="reference-text"><a href="#Rombaldi2005">Rombaldi 2005</a>, <abbr class="abbr" title="page(s)">p.</abbr> 527 ; <a href="#Hardy1949">Hardy 1949</a>, <abbr class="abbr" title="page(s)">p.</abbr> 325.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink noprint"><a href="#cite_ref-6">↑</a> </span><span class="reference-text"><a href="#Deheuvels1980">Deheuvels 1980</a>, <abbr class="abbr" title="page(s)">p.</abbr> 185-187 et p. 195.</span> </li> <li id="cite_note-Cohen25-10"><span class="mw-cite-backlink noprint">↑ <sup><a href="#cite_ref-Cohen25_10-0">a</a> <a href="#cite_ref-Cohen25_10-1">b</a> et <a href="#cite_ref-Cohen25_10-2">c</a></sup> </span><span class="reference-text"><a href="#Cohen2007">Cohen 2007</a>, <abbr class="abbr" title="page(s)">p.</abbr> 25.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink noprint"><a href="#cite_ref-11">↑</a> </span><span class="reference-text"><a href="#Tenenbaum2008">Tenenbaum 2008</a>, <abbr class="abbr" title="page(s)">p.</abbr> 26.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink noprint"><a href="#cite_ref-12">↑</a> </span><span class="reference-text"><a href="#Rombaldi2005">Rombaldi 2005</a>, <abbr class="abbr" title="page(s)">p.</abbr> 330.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink noprint"><a href="#cite_ref-13">↑</a> </span><span class="reference-text"><a href="#Deheuvels1980">Deheuvels 1980</a>, <abbr class="abbr" title="page(s)">p.</abbr> 188.</span> </li> <li id="cite_note-Hardy-14"><span class="mw-cite-backlink noprint">↑ <sup><a href="#cite_ref-Hardy_14-0">a</a> et <a href="#cite_ref-Hardy_14-1">b</a></sup> </span><span class="reference-text"><a href="#Hardy1949">Hardy 1949</a>, <abbr class="abbr" title="page(s)">p.</abbr> 318.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink noprint"><a href="#cite_ref-15">↑</a> </span><span class="reference-text"><a href="#Hardy1949">Hardy 1949</a>, <abbr class="abbr" title="page(s)">p.</abbr> 319.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink noprint"><a href="#cite_ref-17">↑</a> </span><span class="reference-text"><a href="#Rombaldi2005">Rombaldi 2005</a>, <abbr class="abbr" title="page(s)">p.</abbr> 328.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink noprint"><a href="#cite_ref-18">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Pengelley2003"><span class="ouvrage" id="David_J._Pengelley2003"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> David J. <span class="nom_auteur">Pengelley</span>, <cite style="font-style:normal" lang="en">« Dances between continuous and discrete: Euler's summation formula »</cite>, dans Robert Bradley et Ed Sandifer, <cite class="italique" lang="en">Proceedings, Euler 2K+2 Conference (Rumford, Maine, 2002)</cite>, Euler Society, <time>2003</time> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="http://www.math.nmsu.edu/~davidp/euler2k2.pdf">lire en ligne</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=Proceedings%2C+Euler+2K%2B2+Conference+%28Rumford%2C+Maine%2C+2002%29&rft.atitle=Dances+between+continuous+and+discrete%3A+Euler%27s+summation+formula&rft.pub=Euler+Society&rft.aulast=Pengelley&rft.aufirst=David+J.&rft.date=2003&rft_id=http%3A%2F%2Fwww.math.nmsu.edu%2F~davidp%2Feuler2k2.pdf&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AFormule+d%27Euler-Maclaurin"></span></span></span>.</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink noprint"><a href="#cite_ref-19">↑</a> </span><span class="reference-text">Voir <a rel="nofollow" class="external text" href="http://www-fourier.ujf-grenoble.fr/~faure/enseignement/M1_math_pour_physique/Documents/Article_de_F_Pham_Borel.pdf">cette conférence de F. Pham</a></span> </li> </ol> </div> <div class="mw-heading mw-heading2"><h2 id="Bibliographie">Bibliographie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&veaction=edit&section=24" title="Modifier la section : Bibliographie" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Formule_d%27Euler-Maclaurin&action=edit&section=24" title="Modifier le code source de la section : Bibliographie"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span title="Document utilisé pour la rédaction de l’article"><span typeof="mw:File"><span><img alt="Document utilisé pour la rédaction de l’article" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Icon_flat_design_plume.svg/20px-Icon_flat_design_plume.svg.png" decoding="async" width="20" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Icon_flat_design_plume.svg/30px-Icon_flat_design_plume.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Icon_flat_design_plume.svg/40px-Icon_flat_design_plume.svg.png 2x" data-file-width="330" data-file-height="158" /></span></span></span> : document utilisé comme source pour la rédaction de cet article. </p> <ul><li><span class="ouvrage" id="Bourbaki,_FVR"><a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">N. Bourbaki</a>, <cite class="italique"><a href="/wiki/%C3%89l%C3%A9ments_de_math%C3%A9matique" title="Éléments de mathématique">Éléments de mathématique</a></cite>, <span class="italique">Fonctions d'une variable réelle</span>, <abbr class="abbr" title="chapitre(s)">chap.</abbr> VI<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%C3%89l%C3%A9ments+de+math%C3%A9matique&rft.au=N.+Bourbaki&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AFormule+d%27Euler-Maclaurin"></span></span><span title="Document utilisé pour la rédaction de l’article"><span typeof="mw:File"><span><img alt="Document utilisé pour la rédaction de l’article" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Icon_flat_design_plume.svg/20px-Icon_flat_design_plume.svg.png" decoding="async" width="20" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Icon_flat_design_plume.svg/30px-Icon_flat_design_plume.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Icon_flat_design_plume.svg/40px-Icon_flat_design_plume.svg.png 2x" data-file-width="330" data-file-height="158" /></span></span></span></li> <li><span class="ouvrage" id="Cohen2007"><span class="ouvrage" id="Henri_Cohen2007"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <a href="/wiki/Henri_Cohen_(math%C3%A9maticien)" title="Henri Cohen (mathématicien)">Henri Cohen</a>, <cite class="italique" lang="en">Number Theory</cite>, <abbr class="abbr" title="volume">vol.</abbr> II : <span class="lang-en italique" lang="en">Analytic and Modern Tools</span>, <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <time>2007</time><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number+Theory&rft.pub=Springer-Verlag&rft.aulast=Cohen&rft.aufirst=Henri&rft.date=2007&rft.volume=II&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AFormule+d%27Euler-Maclaurin"></span></span></span><span title="Document utilisé pour la rédaction de l’article"><span typeof="mw:File"><span><img alt="Document utilisé pour la rédaction de l’article" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Icon_flat_design_plume.svg/20px-Icon_flat_design_plume.svg.png" decoding="async" width="20" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Icon_flat_design_plume.svg/30px-Icon_flat_design_plume.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Icon_flat_design_plume.svg/40px-Icon_flat_design_plume.svg.png 2x" data-file-width="330" data-file-height="158" /></span></span></span></li> <li><span class="ouvrage" id="Deheuvels1980"><span class="ouvrage" id="Paul_Deheuvels1980"><a href="/wiki/Paul_Deheuvels" title="Paul Deheuvels">Paul Deheuvels</a>, <cite class="italique">L'Intégrale</cite>, <a href="/wiki/PUF" class="mw-redirect" title="PUF">PUF</a>, <abbr class="abbr" title="collection">coll.</abbr> « mathématiques », <time>1980</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%27Int%C3%A9grale&rft.pub=PUF&rft.aulast=Deheuvels&rft.aufirst=Paul&rft.date=1980&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AFormule+d%27Euler-Maclaurin"></span></span></span><span title="Document utilisé pour la rédaction de l’article"><span typeof="mw:File"><span><img alt="Document utilisé pour la rédaction de l’article" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Icon_flat_design_plume.svg/20px-Icon_flat_design_plume.svg.png" decoding="async" width="20" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Icon_flat_design_plume.svg/30px-Icon_flat_design_plume.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Icon_flat_design_plume.svg/40px-Icon_flat_design_plume.svg.png 2x" data-file-width="330" data-file-height="158" /></span></span></span></li> <li><a href="/wiki/Jean-Pierre_Demailly" title="Jean-Pierre Demailly">Jean-Pierre Demailly</a>, <i>Analyse numérique et équations différentielles</i>, <a href="/wiki/Presses_universitaires_de_Grenoble" title="Presses universitaires de Grenoble">Presses universitaires de Grenoble</a></li> <li><span class="ouvrage" id="Dieudonné1980"><span class="ouvrage" id="Jean_Dieudonné1980"><a href="/wiki/Jean_Dieudonn%C3%A9" title="Jean Dieudonné">Jean Dieudonné</a>, <cite class="italique">Calcul infinitésimal</cite>, Paris, <a href="/wiki/Hermann_(%C3%A9ditions)" class="mw-redirect" title="Hermann (éditions)">Hermann</a>, <time>1980</time><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calcul+infinit%C3%A9simal&rft.place=Paris&rft.pub=Hermann&rft.aulast=Dieudonn%C3%A9&rft.aufirst=Jean&rft.date=1980&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AFormule+d%27Euler-Maclaurin"></span></span></span><span title="Document utilisé pour la rédaction de l’article"><span typeof="mw:File"><span><img alt="Document utilisé pour la rédaction de l’article" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Icon_flat_design_plume.svg/20px-Icon_flat_design_plume.svg.png" decoding="async" width="20" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Icon_flat_design_plume.svg/30px-Icon_flat_design_plume.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Icon_flat_design_plume.svg/40px-Icon_flat_design_plume.svg.png 2x" data-file-width="330" data-file-height="158" /></span></span></span></li> <li><span class="ouvrage" id="Hardy1949"><span class="ouvrage" id="G._H._Hardy1949"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <a href="/wiki/G._H._Hardy" class="mw-redirect" title="G. H. Hardy">G. H. <span class="nom_auteur">Hardy</span></a>, <cite class="italique" lang="en">Divergent Series</cite>, <a href="/wiki/Oxford_University_Press" title="Oxford University Press">OUP</a>, <time>1949</time><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Divergent+Series&rft.pub=OUP&rft.aulast=Hardy&rft.aufirst=G.+H.&rft.date=1949&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AFormule+d%27Euler-Maclaurin"></span></span></span><span title="Document utilisé pour la rédaction de l’article"><span typeof="mw:File"><span><img alt="Document utilisé pour la rédaction de l’article" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Icon_flat_design_plume.svg/20px-Icon_flat_design_plume.svg.png" decoding="async" width="20" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Icon_flat_design_plume.svg/30px-Icon_flat_design_plume.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Icon_flat_design_plume.svg/40px-Icon_flat_design_plume.svg.png 2x" data-file-width="330" data-file-height="158" /></span></span></span></li> <li><span class="ouvrage" id="Rombaldi2005"><span class="ouvrage" id="Jean-Étienne_Rombaldi2005">Jean-Étienne Rombaldi, <cite class="italique">Interpolation et approximation</cite>, <a href="/wiki/Vuibert" title="Vuibert">Vuibert</a>, <time>2005</time><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Interpolation+et+approximation&rft.pub=Vuibert&rft.aulast=Rombaldi&rft.aufirst=Jean-%C3%89tienne&rft.date=2005&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AFormule+d%27Euler-Maclaurin"></span></span></span><span title="Document utilisé pour la rédaction de l’article"><span typeof="mw:File"><span><img alt="Document utilisé pour la rédaction de l’article" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Icon_flat_design_plume.svg/20px-Icon_flat_design_plume.svg.png" decoding="async" width="20" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Icon_flat_design_plume.svg/30px-Icon_flat_design_plume.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Icon_flat_design_plume.svg/40px-Icon_flat_design_plume.svg.png 2x" data-file-width="330" data-file-height="158" /></span></span></span></li> <li><span class="ouvrage" id="Tenenbaum2008"><span class="ouvrage" id="Gérald_Tenenbaum2008"><a href="/wiki/G%C3%A9rald_Tenenbaum" title="Gérald Tenenbaum">Gérald Tenenbaum</a>, <cite class="italique">Introduction à la théorie analytique et probabiliste des nombres</cite>, <a href="/wiki/%C3%89ditions_Belin" class="mw-redirect" title="Éditions Belin">Belin</a>, <time>2008</time><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+%C3%A0+la+th%C3%A9orie+analytique+et+probabiliste+des+nombres&rft.pub=Belin&rft.aulast=Tenenbaum&rft.aufirst=G%C3%A9rald&rft.date=2008&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AFormule+d%27Euler-Maclaurin"></span></span></span><span title="Document utilisé pour la rédaction de l’article"><span typeof="mw:File"><span><img alt="Document utilisé pour la rédaction de l’article" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Icon_flat_design_plume.svg/20px-Icon_flat_design_plume.svg.png" decoding="async" width="20" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Icon_flat_design_plume.svg/30px-Icon_flat_design_plume.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Icon_flat_design_plume.svg/40px-Icon_flat_design_plume.svg.png 2x" data-file-width="330" data-file-height="158" /></span></span></span></li></ul> <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?type=1x1" alt="" width="1" height="1" style="border: none; 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