Fonction elliptique de Weierstrass

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d'existence</span> </button> <ul id="toc-Le_théorème_d'existence-sublist" class="vector-toc-list"> <li id="toc-Remarque" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Remarque"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Remarque</span> </div> </a> <ul id="toc-Remarque-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Une_relation_algébrique_fondamentale" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Une_relation_algébrique_fondamentale"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Une relation algébrique fondamentale</span> </div> </a> <ul id="toc-Une_relation_algébrique_fondamentale-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-La_cubique_de_Weierstrass_vue_comme_courbe_elliptique" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#La_cubique_de_Weierstrass_vue_comme_courbe_elliptique"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>La cubique de Weierstrass vue comme courbe elliptique</span> </div> </a> <ul id="toc-La_cubique_de_Weierstrass_vue_comme_courbe_elliptique-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">6</span> <span>Références</span> </div> </a> <ul id="toc-Références-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">7</span> <span>Voir aussi</span> </div> </a> <button aria-controls="toc-Voir_aussi-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Afficher / masquer la sous-section Voir aussi</span> </button> <ul id="toc-Voir_aussi-sublist" class="vector-toc-list"> <li id="toc-Bibliographie" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bibliographie"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Bibliographie</span> </div> </a> <ul id="toc-Bibliographie-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Articles_connexes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Articles_connexes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Articles connexes</span> </div> </a> <ul id="toc-Articles_connexes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lien_externe" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lien_externe"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Lien externe</span> </div> </a> <ul id="toc-Lien_externe-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="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">Fonction elliptique de Weierstrass</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 15 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-15" 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">15 langues</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Funci%C3%B3_el%C2%B7l%C3%ADptica_de_Weierstrass" title="Funció el·líptica de Weierstrass – catalan" lang="ca" hreflang="ca" data-title="Funció el·líptica de Weierstrass" data-language-autonym="Català" data-language-local-name="catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Weierstra%C3%9Fsche_%E2%84%98-Funktion" title="Weierstraßsche ℘-Funktion – allemand" lang="de" hreflang="de" data-title="Weierstraßsche ℘-Funktion" data-language-autonym="Deutsch" data-language-local-name="allemand" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CE%BB%CE%BB%CE%B5%CE%B9%CF%80%CF%84%CE%B9%CE%BA%CE%AE_%CF%83%CF%85%CE%BD%CE%AC%CF%81%CF%84%CE%B7%CF%83%CE%B7_%CE%92%CE%AC%CE%B9%CE%B5%CF%81%CF%83%CF%84%CF%81%CE%B1%CF%82" title="Ελλειπτική συνάρτηση Βάιερστρας – grec" lang="el" hreflang="el" data-title="Ελλειπτική συνάρτηση Βάιερστρας" data-language-autonym="Ελληνικά" data-language-local-name="grec" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Weierstrass_elliptic_function" title="Weierstrass elliptic function – anglais" lang="en" hreflang="en" data-title="Weierstrass elliptic function" 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/Funciones_el%C3%ADpticas_de_Weierstra%C3%9F" title="Funciones elípticas de Weierstraß – espagnol" lang="es" hreflang="es" data-title="Funciones elípticas de Weierstraß" data-language-autonym="Español" data-language-local-name="espagnol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Weierstrassin_elliptinen_funktio" title="Weierstrassin elliptinen funktio – finnois" lang="fi" hreflang="fi" data-title="Weierstrassin elliptinen funktio" data-language-autonym="Suomi" data-language-local-name="finnois" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Funzioni_ellittiche_di_Weierstrass" title="Funzioni ellittiche di Weierstrass – italien" lang="it" hreflang="it" data-title="Funzioni ellittiche di Weierstrass" 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%83%B4%E3%82%A1%E3%82%A4%E3%82%A8%E3%83%AB%E3%82%B7%E3%83%A5%E3%83%88%E3%83%A9%E3%82%B9%E3%81%AE%E6%A5%95%E5%86%86%E5%87%BD%E6%95%B0" title="ヴァイエルシュトラスの楕円函数 – japonais" lang="ja" hreflang="ja" data-title="ヴァイエルシュトラスの楕円函数" 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-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B0%94%EC%9D%B4%EC%96%B4%EC%8A%88%ED%8A%B8%EB%9D%BC%EC%8A%A4_%ED%83%80%EC%9B%90%ED%95%A8%EC%88%98" title="바이어슈트라스 타원함수 – coréen" lang="ko" hreflang="ko" data-title="바이어슈트라스 타원함수" data-language-autonym="한국어" data-language-local-name="coréen" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Funkcje_eliptyczne_Weierstrassa" title="Funkcje eliptyczne Weierstrassa – polonais" lang="pl" hreflang="pl" data-title="Funkcje eliptyczne Weierstrassa" 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/Fun%C3%A7%C3%B5es_el%C3%ADpticas_de_Weierstrass" title="Funções elípticas de Weierstrass – portugais" lang="pt" hreflang="pt" data-title="Funções elípticas de Weierstrass" 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%AD%D0%BB%D0%BB%D0%B8%D0%BF%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B5_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%B8_%D0%92%D0%B5%D0%B9%D0%B5%D1%80%D1%88%D1%82%D1%80%D0%B0%D1%81%D1%81%D0%B0" title="Эллиптические функции Вейерштрасса – russe" lang="ru" hreflang="ru" data-title="Эллиптические функции Вейерштрасса" data-language-autonym="Русский" data-language-local-name="russe" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Weierstrass_elliptiska_funktion" title="Weierstrass elliptiska funktion – suédois" lang="sv" hreflang="sv" data-title="Weierstrass elliptiska funktion" data-language-autonym="Svenska" data-language-local-name="suédois" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%95%D0%BB%D1%96%D0%BF%D1%82%D0%B8%D1%87%D0%BD%D1%96_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%97_%D0%92%D0%B5%D1%94%D1%80%D1%88%D1%82%D1%80%D0%B0%D1%81%D1%81%D0%B0" title="Еліптичні функції Веєрштрасса – ukrainien" lang="uk" hreflang="uk" data-title="Еліптичні функції Веєрштрасса" data-language-autonym="Українська" data-language-local-name="ukrainien" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%AD%8F%E7%88%BE%E6%96%AF%E7%89%B9%E6%8B%89%E6%96%AF%E6%A9%A2%E5%9C%93%E5%87%BD%E6%95%B8" title="魏爾斯特拉斯橢圓函數 – chinois" lang="zh" hreflang="zh" data-title="魏爾斯特拉斯橢圓函數" data-language-autonym="中文" data-language-local-name="chinois" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet 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class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">masquer</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Un article de Wikipédia, l'encyclopédie libre.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirigé depuis <a href="/w/index.php?title=Discriminant_modulaire&redirect=no" class="mw-redirect" title="Discriminant modulaire">Discriminant modulaire</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="fr" dir="ltr"><p>En <a href="/wiki/Analyse_complexe" title="Analyse complexe">analyse complexe</a>, les <b>fonctions elliptiques de <a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Weierstrass</a></b> forment une classe importante de <a href="/wiki/Fonction_elliptique" title="Fonction elliptique">fonctions elliptiques</a> c'est-à-dire de <a href="/wiki/Fonction_m%C3%A9romorphe" title="Fonction méromorphe">fonctions méromorphes</a> doublement <a href="/wiki/Fonction_p%C3%A9riodique" title="Fonction périodique">périodiques</a>. Toute fonction elliptique peut être exprimée à l'aide de celles-ci. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fonction_elliptique_de_Weierstrass&veaction=edit&section=1" title="Modifier la section : Introduction" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Fonction_elliptique_de_Weierstrass&action=edit&section=1" title="Modifier le code source de la section : Introduction"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Fabrication_de_fonctions_périodiques"><span id="Fabrication_de_fonctions_p.C3.A9riodiques"></span>Fabrication de fonctions périodiques</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fonction_elliptique_de_Weierstrass&veaction=edit&section=2" title="Modifier la section : Fabrication de fonctions périodiques" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Fonction_elliptique_de_Weierstrass&action=edit&section=2" title="Modifier le code source de la section : Fabrication de fonctions périodiques"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Supposons que l'on souhaite fabriquer une telle fonction de période 1. On peut prendre une fonction quelconque, définie sur [0, 1] et telle que <i>f</i>(0) = <i>f</i>(1) et la prolonger convenablement. Un tel procédé a des limites. Par exemple, on obtiendra rarement des <a href="/wiki/Fonction_analytique" title="Fonction analytique">fonctions analytiques</a> de cette façon. </p><p>Une idée plus sophistiquée est de prendre une fonction <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> définie sur <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> et d'introduire la fonction <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> définie par <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=\sum _{n\in \mathbb {Z} }f(x+n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)=\sum _{n\in \mathbb {Z} }f(x+n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c717428a211faefc03df9a546172f5949758ac11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.748ex; height:5.676ex;" alt="{\displaystyle g(x)=\sum _{n\in \mathbb {Z} }f(x+n)}"></span>. </p><p>Un exemple simple est donné par </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\in \mathbb {Z} }(x+n)^{-k},k\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>,</mo> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n\in \mathbb {Z} }(x+n)^{-k},k\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7c091b35de6576fcd91b9443b8f74459c6ba513" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.86ex; height:5.676ex;" alt="{\displaystyle \sum _{n\in \mathbb {Z} }(x+n)^{-k},k\in \mathbb {N} }"></span>.</dd></dl> <p>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 k>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cda43bd4034dc2d04cd562005d0af81d3d2dbc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k>1}"></span>, on obtient une fonction infiniment dérivable, définie sur ℝ\ℤ et de période 1. 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 k=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c035ffa69b5bca8bf2d16c3da3aaad79a8bcbfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=1}"></span>, la série ne converge pas, mais on peut introduire à la place </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 {1}{x}}+\lim _{n\rightarrow +\infty }\sum _{0<\vert m\vert <n}{\frac {1}{x+m}}-{\frac {1}{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>+</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo><</mo> <mo fence="false" stretchy="false">|</mo> <mi>m</mi> <mo fence="false" stretchy="false">|</mo> <mo><</mo> <mi>n</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>+</mo> <mi>m</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>m</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{x}}+\lim _{n\rightarrow +\infty }\sum _{0<\vert m\vert <n}{\frac {1}{x+m}}-{\frac {1}{m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e069bd5f7b769ed79b12fea26c0587a325e116b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:30.818ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{x}}+\lim _{n\rightarrow +\infty }\sum _{0<\vert m\vert <n}{\frac {1}{x+m}}-{\frac {1}{m}}}"></span>,</dd></dl> <p>qui s'écrit aussi </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 {1}{x}}+\sum _{n\in \mathbb {N} ^{*}}{\frac {1}{x+n}}+{\frac {1}{x-n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>+</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>−</mo> <mi>n</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{x}}+\sum _{n\in \mathbb {N} ^{*}}{\frac {1}{x+n}}+{\frac {1}{x-n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db5281c5a5c5ffb0b2e144e36151e53f38f487ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:25.136ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{x}}+\sum _{n\in \mathbb {N} ^{*}}{\frac {1}{x+n}}+{\frac {1}{x-n}}}"></span>,</dd></dl> <p>ou encore </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 {1}{x}}+\sum _{n=1}^{\infty }{\frac {2x}{x^{2}-n^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>x</mi> </mrow> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{x}}+\sum _{n=1}^{\infty }{\frac {2x}{x^{2}-n^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/022d7dfa0f5a3c39feb73aeb4b1309f24d06c64f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.258ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{x}}+\sum _{n=1}^{\infty }{\frac {2x}{x^{2}-n^{2}}}}"></span>.</dd></dl> <p>C'est a priori la plus intéressante du lot, puisque les autres en sont (à des facteurs constants près) les dérivées successives. Dans le cadre de la théorie des <a href="/wiki/Fonction_holomorphe" title="Fonction holomorphe">fonctions holomorphes</a>, on démontre que cette série converge uniformément sur tout compact vers la fonction <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 \pi \cot(\pi z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \cot(\pi z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ce0cbc8a71b67d5f92c9e19ed518da4734b5f4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.048ex; height:2.843ex;" alt="{\displaystyle \pi \cot(\pi z)}"></span>. </p><p>Dans son petit livre sur les fonctions elliptiques dont cette introduction est inspirée, <a href="/wiki/Andr%C3%A9_Weil" title="André Weil">André Weil</a><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> reprend les travaux de <a href="/wiki/Gotthold_Eisenstein" title="Gotthold Eisenstein">Gotthold Eisenstein</a>, fait semblant d'ignorer les fonctions trigonométriques, et les retrouve avec des méthodes élémentaires ingénieuses à partir des séries ci-dessus. </p> <div class="mw-heading mw-heading3"><h3 id="Fonctions_périodiques,_fonctions_doublement_périodiques"><span id="Fonctions_p.C3.A9riodiques.2C_fonctions_doublement_p.C3.A9riodiques"></span>Fonctions périodiques, fonctions doublement périodiques</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fonction_elliptique_de_Weierstrass&veaction=edit&section=3" title="Modifier la section : Fonctions périodiques, fonctions doublement périodiques" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Fonction_elliptique_de_Weierstrass&action=edit&section=3" title="Modifier le code source de la section : Fonctions périodiques, fonctions doublement périodiques"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Une période d'une fonction continue <i>f </i>est un nombre non nul <i>T </i>tel que, pour tout réel <i>x</i>, on ait <i>f</i>(<i>x + T</i>) = <i>f</i>(<i>x</i>). La différence de deux périodes est une période, de sorte que les périodes forment un <a href="/wiki/Sous-groupe" title="Sous-groupe">sous-groupe</a> du <a href="/wiki/Groupe_(math%C3%A9matiques)" title="Groupe (mathématiques)">groupe</a> (ℝ, +), <a href="/wiki/Ferm%C3%A9_(topologie)" title="Fermé (topologie)">fermé</a> en raison de la continuité de <i>f</i>. Un tel sous-groupe, s'il n'est pas réduit à zéro, est soit égal à ℝ tout entier (la fonction <i>f </i>est alors constante, cas trivial) soit de la forme <i>a</i>ℤ pour un réel <i>a </i>> 0, que les physiciens appellent la plus petite période de <i>f</i>. </p><p>Une fonction doublement périodique est une fonction dont le groupe des périodes est <a href="/wiki/Isomorphisme" title="Isomorphisme">isomorphe</a> à ℤ<sup>2</sup>. D'après ce qui précède, de telles fonctions continues d'une variable réelle <i>n'existent pas</i>. Il faut prendre des fonctions de deux <a href="/wiki/Variable_(math%C3%A9matiques)" title="Variable (mathématiques)">variables</a>, ou, plus intéressant, des fonctions d'une <a href="/wiki/Variable_complexe" class="mw-redirect" title="Variable complexe">variable complexe</a>. Le groupe des périodes d'une telle fonction est un <a href="/wiki/R%C3%A9seau" title="Réseau">réseau</a>, c’est-à-dire un sous-groupe de (ℂ, +) engendré par deux éléments indépendants sur ℝ. </p><p>Toute <a href="/wiki/Fonction_holomorphe" title="Fonction holomorphe">fonction holomorphe</a> doublement périodique est constante, puisqu'une telle fonction est nécessairement bornée sur ℂ (<a href="/wiki/Th%C3%A9or%C3%A8me_de_Liouville_(variable_complexe)" title="Théorème de Liouville (variable complexe)">théorème de Liouville</a>). C'est à l'occasion de recherches sur les fonctions elliptiques que <a href="/wiki/Joseph_Liouville" title="Joseph Liouville">Joseph Liouville</a> a été amené à formuler et démontrer ce théorème. Il faut donc travailler avec des <a href="/wiki/Fonction_m%C3%A9romorphe" title="Fonction méromorphe">fonctions méromorphes</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Définition"><span id="D.C3.A9finition"></span>Définition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fonction_elliptique_de_Weierstrass&veaction=edit&section=4" title="Modifier la section : Définition" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Fonction_elliptique_de_Weierstrass&action=edit&section=4" title="Modifier le code source de la section : Définition"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Soit <i>L </i>un <a href="/wiki/R%C3%A9seau_(g%C3%A9om%C3%A9trie)" title="Réseau (géométrie)">réseau</a> du plan complexe, de <a href="/wiki/Module_libre" title="Module libre">base</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 \omega _{1},\omega _{2}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{1},\omega _{2}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb24613f6930fb5bc66362083009b15147026481" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.421ex; height:2.009ex;" alt="{\displaystyle \omega _{1},\omega _{2}\,}"></span>. Par analogie avec l'introduction, on est amené à considérer les fonctions </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 _{\lambda \in L}(z-\lambda )^{-k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo>∈</mo> <mi>L</mi> </mrow> </munder> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mi>λ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{\lambda \in L}(z-\lambda )^{-k},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91d7a643ab8be29e269b78a76d5aedc2dce341bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:13.462ex; height:5.676ex;" alt="{\displaystyle \sum _{\lambda \in L}(z-\lambda )^{-k},}"></span></dd></dl> <p>qui s'écrivent </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 _{(m_{1},m_{2})\in \mathbb {Z} \times \mathbb {Z} }(z-m_{1}\omega _{1}-m_{2}\omega _{2})^{-k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{(m_{1},m_{2})\in \mathbb {Z} \times \mathbb {Z} }(z-m_{1}\omega _{1}-m_{2}\omega _{2})^{-k}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07c35748a9bfe14985f8133f2bf4b4beff69448d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:33.634ex; height:6.009ex;" alt="{\displaystyle \sum _{(m_{1},m_{2})\in \mathbb {Z} \times \mathbb {Z} }(z-m_{1}\omega _{1}-m_{2}\omega _{2})^{-k}.}"></span></dd></dl> <p>Si l'entier <i>k </i>vaut au moins 3, elles convergent. Il s'agit d'une <a href="/wiki/Convergence_uniforme" title="Convergence uniforme">convergence uniforme sur tout compact</a> ne rencontrant pas le réseau. Cela résulte d'une série de remarques. </p> <ul><li><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 \vert x_{1}\omega _{1}+x_{2}\omega _{2}\vert }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">|</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">|</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vert x_{1}\omega _{1}+x_{2}\omega _{2}\vert }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39b3df9fc431308a591675431ff0952e60d597b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.902ex; height:2.843ex;" alt="{\displaystyle \vert x_{1}\omega _{1}+x_{2}\omega _{2}\vert }"></span> est une <a href="/wiki/Norme_(math%C3%A9matiques)" title="Norme (mathématiques)">norme</a> sur ℝ<sup>2</sup>.<br />Elle est <a href="/wiki/Norme_%C3%A9quivalente" title="Norme équivalente">équivalente</a> 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 {\sqrt {x_{1}^{2}+x_{2}^{2}}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {x_{1}^{2}+x_{2}^{2}}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5cbb0afb3946c6d284dc4d9a884f102585f4edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.319ex; height:4.843ex;" alt="{\displaystyle {\sqrt {x_{1}^{2}+x_{2}^{2}}}\,}"></span>. Il existe donc un <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 \alpha >0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>></mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha >0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e62dc722bdf4930eb2ceebbd1b78bf2a54314d01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.136ex; height:2.176ex;" alt="{\displaystyle \alpha >0\,}"></span> tel que, quels que soient <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 m_{1}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d613e7088bed138657375a2241f31cb6ba8c4bb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.482ex; height:2.009ex;" alt="{\displaystyle m_{1}\,}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{2}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{2}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10492057ef0b1e66719434a7b1442a9ad1f1f960" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.482ex; height:2.009ex;" alt="{\displaystyle m_{2}\,}"></span>, on ait <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 \vert m_{1}\omega _{1}+m_{2}\omega _{2}\vert \geq \alpha {\sqrt {m_{1}^{2}+m_{2}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">|</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">|</mo> <mo>≥</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vert m_{1}\omega _{1}+m_{2}\omega _{2}\vert \geq \alpha {\sqrt {m_{1}^{2}+m_{2}^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa3970e1db1b983f500234585fc4466299704f26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.263ex; height:4.843ex;" alt="{\displaystyle \vert m_{1}\omega _{1}+m_{2}\omega _{2}\vert \geq \alpha {\sqrt {m_{1}^{2}+m_{2}^{2}}}}"></span>.<br />Donc tout disque fermé ne contient qu'un nombre fini d'éléments de <i>L</i>.</li> <li>On prend <i>z </i>dans le disque fermé <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 \vert z\vert \leq R\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">|</mo> <mi>z</mi> <mo fence="false" stretchy="false">|</mo> <mo>≤</mo> <mi>R</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vert z\vert \leq R\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1543dd75adb9eb08cbf64b9a8b567d33f57a6df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.631ex; height:2.843ex;" alt="{\displaystyle \vert z\vert \leq R\,}"></span>.</li></ul> <p>En ce qui concerne la convergence, il suffit de considérer les <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 \lambda \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/988b7b8a22b11081bc97378c30391f573535c21c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.742ex; height:2.176ex;" alt="{\displaystyle \lambda \,}"></span> tels 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 \vert \lambda \vert \geq 2R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">|</mo> <mi>λ</mi> <mo fence="false" stretchy="false">|</mo> <mo>≥</mo> <mn>2</mn> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vert \lambda \vert \geq 2R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb274ac771b37f37f386e42fefe00d60eb0bd653" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.674ex; height:2.843ex;" alt="{\displaystyle \vert \lambda \vert \geq 2R}"></span> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \geq R\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≥</mo> <mi>R</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \geq R\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9cb16f0216a2613483f4c5432a4311a01760333" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.604ex; height:2.343ex;" alt="{\displaystyle \geq R\,}"></span> suffirait apparemment, il s'agit ici d'une astuce technique). </p> <ul><li>Dans ces conditions, <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 \vert z+\lambda \vert \geq \vert \lambda \vert -\vert z\vert \geq {\frac {\vert \lambda \vert }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">|</mo> <mi>z</mi> <mo>+</mo> <mi>λ</mi> <mo fence="false" stretchy="false">|</mo> <mo>≥</mo> <mo fence="false" stretchy="false">|</mo> <mi>λ</mi> <mo fence="false" stretchy="false">|</mo> <mo>−</mo> <mo fence="false" stretchy="false">|</mo> <mi>z</mi> <mo fence="false" stretchy="false">|</mo> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false">|</mo> <mi>λ</mi> <mo fence="false" stretchy="false">|</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vert z+\lambda \vert \geq \vert \lambda \vert -\vert z\vert \geq {\frac {\vert \lambda \vert }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72e279c8e5e92c7fc2308749d9d270e74e2aba12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.131ex; height:5.676ex;" alt="{\displaystyle \vert z+\lambda \vert \geq \vert \lambda \vert -\vert z\vert \geq {\frac {\vert \lambda \vert }{2}}}"></span>. On est donc ramené à la convergence de la série <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 _{\lambda \in L\setminus \{0\}}\vert \lambda \vert ^{-k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo>∈</mo> <mi>L</mi> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mrow> </munder> <mo fence="false" stretchy="false">|</mo> <mi>λ</mi> <msup> <mo fence="false" stretchy="false">|</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{\lambda \in L\setminus \{0\}}\vert \lambda \vert ^{-k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5616e0584fbfe4672082014cb330133afdc27d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:11.865ex; height:6.009ex;" alt="{\displaystyle \sum _{\lambda \in L\setminus \{0\}}\vert \lambda \vert ^{-k}}"></span>,</li></ul> <p>qui se ramène elle-même, d'après notre première remarque, à la convergence de la <a href="/wiki/S%C3%A9rie_de_Riemann" title="Série de Riemann">série de Riemann</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 \sum _{(m_{1},m_{2})\in \mathbb {Z} ^{2}\setminus \{0\}}^{+\infty }(m_{1}^{2}+m_{2}^{2})^{-k/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{(m_{1},m_{2})\in \mathbb {Z} ^{2}\setminus \{0\}}^{+\infty }(m_{1}^{2}+m_{2}^{2})^{-k/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1cf83589f510896d32b1bf4e8503fe12097803f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:27.448ex; height:8.009ex;" alt="{\displaystyle \sum _{(m_{1},m_{2})\in \mathbb {Z} ^{2}\setminus \{0\}}^{+\infty }(m_{1}^{2}+m_{2}^{2})^{-k/2}}"></span>. </p><p>Cette dernière <a href="/wiki/Famille_sommable#Exemples" title="Famille sommable">converge</a> si <i>k </i>> 2. </p><p>Pour <i>k </i>= 2, en revanche, cet argument est en défaut. Toujours par analogie avec l'introduction, on introduit la série modifiée </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 {1}{z^{2}}}+\sum _{\lambda \in L\setminus \{0\}}{\frac {1}{(z+\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\ (1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo>∈</mo> <mi>L</mi> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>λ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mtext> </mtext> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{z^{2}}}+\sum _{\lambda \in L\setminus \{0\}}{\frac {1}{(z+\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\ (1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fc516bde213327c83d0c76bf15b1876f17bb88f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:31.292ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{z^{2}}}+\sum _{\lambda \in L\setminus \{0\}}{\frac {1}{(z+\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\ (1)}"></span>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Le_théorème_d'existence"><span id="Le_th.C3.A9or.C3.A8me_d.27existence"></span>Le théorème d'existence</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fonction_elliptique_de_Weierstrass&veaction=edit&section=5" title="Modifier la section : Le théorème d'existence" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Fonction_elliptique_de_Weierstrass&action=edit&section=5" title="Modifier le code source de la section : Le théorème d'existence"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La série (1) converge uniformément sur tout compact ne rencontrant pas <i>L</i>. Sa somme est une fonction méromorphe <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 \wp \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">℘</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wp \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecfa35189a1f54c1e734fea345b0a3bf33c19991" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.865ex; height:2.176ex;" alt="{\displaystyle \wp \,}"></span>, qui admet des pôles doubles aux points de <i>L</i>. Elle est paire et <i>L</i>-périodique. Elle a été découverte par <a href="/wiki/Weierstrass" class="mw-redirect" title="Weierstrass">Weierstrass</a>, d'où son nom de "fonction <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 \wp \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">℘</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wp \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecfa35189a1f54c1e734fea345b0a3bf33c19991" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.865ex; height:2.176ex;" alt="{\displaystyle \wp \,}"></span> de Weierstrass" ou fonction elliptique de Weierstrass. </p><p>On se place dans un disque <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 \vert z\vert \leq R\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">|</mo> <mi>z</mi> <mo fence="false" stretchy="false">|</mo> <mo>≤</mo> <mi>R</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vert z\vert \leq R\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1543dd75adb9eb08cbf64b9a8b567d33f57a6df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.631ex; height:2.843ex;" alt="{\displaystyle \vert z\vert \leq R\,}"></span>, qui ne contient qu'un nombre fini d'éléments de <i>L</i>, qu'on peut enlever à la série sans dommage pour l'étude de sa convergence. </p><p>Pour les autres, on a </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 (z+\lambda )^{-2}=\lambda ^{-2}(1+{\frac {z}{\lambda }})^{-2}={\frac {1}{\lambda ^{2}}}-{\frac {2z}{\lambda ^{3}}}+O\left({\frac {1}{\lambda ^{4}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>λ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mi>λ</mi> </mfrac> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>z</mi> </mrow> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mi>O</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (z+\lambda )^{-2}=\lambda ^{-2}(1+{\frac {z}{\lambda }})^{-2}={\frac {1}{\lambda ^{2}}}-{\frac {2z}{\lambda ^{3}}}+O\left({\frac {1}{\lambda ^{4}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09a801ae70b21a9fb763ba67f26a5f135b7b2d0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:50.646ex; height:6.176ex;" alt="{\displaystyle (z+\lambda )^{-2}=\lambda ^{-2}(1+{\frac {z}{\lambda }})^{-2}={\frac {1}{\lambda ^{2}}}-{\frac {2z}{\lambda ^{3}}}+O\left({\frac {1}{\lambda ^{4}}}\right)}"></span></dd></dl> <p><i>uniformément par rapport 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 z\in D(0,R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>∈</mo> <mi>D</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\in D(0,R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/289424f54cb318b1c8e7d3a663701c4738539731" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.623ex; height:2.843ex;" alt="{\displaystyle z\in D(0,R)}"></span>.</i> Dans ces conditions, <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 \left|{\frac {1}{(z+\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right|\leq C(R)\vert \lambda \vert ^{-3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>λ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">|</mo> <mi>λ</mi> <msup> <mo fence="false" stretchy="false">|</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|{\frac {1}{(z+\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right|\leq C(R)\vert \lambda \vert ^{-3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0376109285bb2639b986403a3287c6945a15dbe7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.783ex; height:6.509ex;" alt="{\displaystyle \left|{\frac {1}{(z+\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right|\leq C(R)\vert \lambda \vert ^{-3}}"></span>, et l'on est ramené à une question déjà résolue. </p><p>D'après ce qui précède et des résultats classiques de <a href="/wiki/S%C3%A9rie_enti%C3%A8re#Dérivation" title="Série entière">dérivation terme à terme</a>, </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 \wp ^{\prime }(z)=-2\sum _{\lambda \in L}{\frac {1}{(z+\lambda )^{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">℘</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>2</mn> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo>∈</mo> <mi>L</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>λ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wp ^{\prime }(z)=-2\sum _{\lambda \in L}{\frac {1}{(z+\lambda )^{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ae32d0e6a66ecf6fec798985105196a26bef73f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.242ex; height:6.509ex;" alt="{\displaystyle \wp ^{\prime }(z)=-2\sum _{\lambda \in L}{\frac {1}{(z+\lambda )^{3}}}}"></span>,</dd></dl> <p>ce qui montre que la fonction <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 \wp ^{\prime }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">℘</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wp ^{\prime }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb717ffdea8a7b394edc7a2403ee3f8e8458b852" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.55ex; height:3.009ex;" alt="{\displaystyle \wp ^{\prime }\,}"></span> est impaire et <i>L</i>-périodique. Donc <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 \wp }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">℘</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wp }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4050ebf63686af152bf1ef5caabcdf2a2d812cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.478ex; height:2.176ex;" alt="{\displaystyle \wp }"></span> est paire. En raison de la périodicité 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 \wp ^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">℘</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wp ^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8132be6e13c42e1330b0f2182a9fa849e67bcf5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.163ex; height:3.009ex;" alt="{\displaystyle \wp ^{\prime }}"></span>, la fonction <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 \wp (z+\omega _{i})-\wp (z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">℘</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi mathvariant="normal">℘</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wp (z+\omega _{i})-\wp (z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cc804d7039541eca2286eca60eb2d1660aa28f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.678ex; height:2.843ex;" alt="{\displaystyle \wp (z+\omega _{i})-\wp (z)}"></span> (pour <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 i=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fb857369fbc28dab4b5d76bb835e6f196b1d1cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.063ex; height:2.176ex;" alt="{\displaystyle i=1}"></span> ou <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 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span>) est constante, et cette constante vaut <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 \wp (\omega _{i}/2)-\wp (-\omega _{i}/2)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">℘</mi> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi mathvariant="normal">℘</mi> <mo stretchy="false">(</mo> <mo>−</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wp (\omega _{i}/2)-\wp (-\omega _{i}/2)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d8a479771550b5aaa9f27ea0c9c67d58f39cd1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.626ex; height:2.843ex;" alt="{\displaystyle \wp (\omega _{i}/2)-\wp (-\omega _{i}/2)=0}"></span> . </p> <div class="mw-heading mw-heading3"><h3 id="Remarque">Remarque</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fonction_elliptique_de_Weierstrass&veaction=edit&section=6" title="Modifier la section : Remarque" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Fonction_elliptique_de_Weierstrass&action=edit&section=6" title="Modifier le code source de la section : Remarque"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>On peut montrer que toute fonction méromorphe <i>L</i>-périodique qui admet des pôles doubles aux points de <i>L </i>est de la forme <i>a</i>℘ + <i>b</i> et, plus généralement, que toute fonction méromorphe <i>L</i>-périodique est une <a href="/wiki/Fraction_rationnelle#Fraction_rationnelle_à_plusieurs_variables" title="Fraction rationnelle">fraction rationnelle en ℘ et ℘'</a> (à coefficients complexes). </p> <div class="mw-heading mw-heading2"><h2 id="Une_relation_algébrique_fondamentale"><span id="Une_relation_alg.C3.A9brique_fondamentale"></span>Une relation algébrique fondamentale</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fonction_elliptique_de_Weierstrass&veaction=edit&section=7" title="Modifier la section : Une relation algébrique fondamentale" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Fonction_elliptique_de_Weierstrass&action=edit&section=7" title="Modifier le code source de la section : Une relation algébrique fondamentale"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La fonction <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 \wp }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">℘</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wp }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4050ebf63686af152bf1ef5caabcdf2a2d812cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.478ex; height:2.176ex;" alt="{\displaystyle \wp }"></span> vérifie l'équation différentielle </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 \wp ^{\prime }(z)^{2}=4\wp (z)^{3}-g_{2}\wp (z)-g_{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">℘</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>4</mn> <mi mathvariant="normal">℘</mi> <mo stretchy="false">(</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi mathvariant="normal">℘</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wp ^{\prime }(z)^{2}=4\wp (z)^{3}-g_{2}\wp (z)-g_{3}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/111604b2d9fc0ceca57276a9a2e5fc5a6b7a91f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.836ex; height:3.176ex;" alt="{\displaystyle \wp ^{\prime }(z)^{2}=4\wp (z)^{3}-g_{2}\wp (z)-g_{3}.}"></span></center> <p>Ici, on a posé (les notations relèvent d'une tradition vénérable) </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 g_{2}=60\sum _{\lambda \in L\setminus \{0\}}{\frac {1}{\lambda ^{4}}}\quad {\textrm {et}}\quad g_{3}=140\sum _{\lambda \in L\setminus \{0\}}{\frac {1}{\lambda ^{6}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>60</mn> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo>∈</mo> <mi>L</mi> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>et</mtext> </mrow> </mrow> <mspace width="1em" /> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>140</mn> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo>∈</mo> <mi>L</mi> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{2}=60\sum _{\lambda \in L\setminus \{0\}}{\frac {1}{\lambda ^{4}}}\quad {\textrm {et}}\quad g_{3}=140\sum _{\lambda \in L\setminus \{0\}}{\frac {1}{\lambda ^{6}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f296d928d8bf1d06bd9bc7f9676644f1768dfe1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:44.528ex; height:6.843ex;" alt="{\displaystyle g_{2}=60\sum _{\lambda \in L\setminus \{0\}}{\frac {1}{\lambda ^{4}}}\quad {\textrm {et}}\quad g_{3}=140\sum _{\lambda \in L\setminus \{0\}}{\frac {1}{\lambda ^{6}}}.}"></span></center> <p>Le principe de la preuve est le suivant. La fonction <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 \wp ^{\prime }(z)^{2}-4\wp (z)^{3}+g_{2}\wp (z)+g_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">℘</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mi mathvariant="normal">℘</mi> <mo stretchy="false">(</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi mathvariant="normal">℘</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wp ^{\prime }(z)^{2}-4\wp (z)^{3}+g_{2}\wp (z)+g_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fd5ef437cbec0f4c84ddcd32cc5d97de1d5081b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.931ex; height:3.176ex;" alt="{\displaystyle \wp ^{\prime }(z)^{2}-4\wp (z)^{3}+g_{2}\wp (z)+g_{3}}"></span> est certainement méromorphe et <i>L</i>-périodique. Un argument de développement limité montre qu'elle est nulle à l'origine. Elle est donc holomorphe et bornée, donc constante (et ici identiquement nulle) d'après le théorème de Liouville. </p><p>L'application <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\mapsto {\big (}\wp (z),\wp ^{\prime }(z){\big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi mathvariant="normal">℘</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msup> <mi mathvariant="normal">℘</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\mapsto {\big (}\wp (z),\wp ^{\prime }(z){\big )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38ad0ec2646da286a267ed2f2492c8bea3bb9d08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.302ex; height:3.176ex;" alt="{\displaystyle z\mapsto {\big (}\wp (z),\wp ^{\prime }(z){\big )}}"></span> envoie ℂ dans la <a href="/wiki/Courbe_cubique" title="Courbe cubique">cubique</a> de ℂ×ℂ d'équation <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^{2}=4x^{3}-g_{2}x-g_{3}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>x</mi> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}=4x^{3}-g_{2}x-g_{3}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/984a8b6ee3d9c2ff3ad987b5ad2666958943f2fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.584ex; height:3.009ex;" alt="{\displaystyle y^{2}=4x^{3}-g_{2}x-g_{3}\,}"></span>. La courbe correspondante <span class="texhtml">E</span> du <a href="/wiki/Plan_projectif" title="Plan projectif">plan projectif complexe</a>, appelée <i>cubique de Weierstrass</i> est donnée en <a href="/wiki/Coordonn%C3%A9es_homog%C3%A8nes" title="Coordonnées homogènes">coordonnées homogènes</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 (x,y,t)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,t)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c45b25cdc52eb949db8edc22a3b4eb7138b94ea2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.589ex; height:2.843ex;" alt="{\displaystyle (x,y,t)\,}"></span> par l'équation <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^{2}t-4x^{3}+g_{2}xt^{2}+g_{3}t^{3}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>t</mi> <mo>−</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>x</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}t-4x^{3}+g_{2}xt^{2}+g_{3}t^{3}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55e053e4098c26636cb357064506f803c050bbb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.474ex; height:3.009ex;" alt="{\displaystyle y^{2}t-4x^{3}+g_{2}xt^{2}+g_{3}t^{3}=0.}"></span> On a alors une application continue (et même holomorphe à condition de savoir ce qu'est une <a href="/wiki/Vari%C3%A9t%C3%A9_complexe" title="Variété complexe">variété complexe</a>) de ℂ dans <span class="texhtml">E</span> donnée par <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\mapsto [{\big (}\wp (z),\wp ^{\prime }(z),1{\big )}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo stretchy="false">↦</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi mathvariant="normal">℘</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msup> <mi mathvariant="normal">℘</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\mapsto [{\big (}\wp (z),\wp ^{\prime }(z),1{\big )}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc039fafcfd11c1f333910a1bfe442daff48a5de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.792ex; height:3.176ex;" alt="{\displaystyle z\mapsto [{\big (}\wp (z),\wp ^{\prime }(z),1{\big )}]}"></span> qui se prolonge par continuité aux pôles 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 \wp \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">℘</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wp \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecfa35189a1f54c1e734fea345b0a3bf33c19991" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.865ex; height:2.176ex;" alt="{\displaystyle \wp \,}"></span>, qu'elle envoie sur le « point à l'infini » [(0, 1, 0)] de <span class="texhtml">E</span>. </p> <div class="mw-heading mw-heading2"><h2 id="La_cubique_de_Weierstrass_vue_comme_courbe_elliptique">La cubique de Weierstrass vue comme courbe elliptique</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fonction_elliptique_de_Weierstrass&veaction=edit&section=8" title="Modifier la section : La cubique de Weierstrass vue comme courbe elliptique" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Fonction_elliptique_de_Weierstrass&action=edit&section=8" title="Modifier le code source de la section : La cubique de Weierstrass vue comme courbe elliptique"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>De même que le <a href="/wiki/Groupe_quotient" title="Groupe quotient">quotient</a> ℝ/ℤ est <a href="/wiki/Hom%C3%A9omorphisme" title="Homéomorphisme">homéomorphe</a> au cercle, le quotient ℂ/<i>L</i> est homéomorphe à un <a href="/wiki/Tore" title="Tore">tore</a> de dimension 2 ; c'est aussi une <a href="/wiki/Surface_de_Riemann" title="Surface de Riemann">surface de Riemann</a> compacte. </p><p>On démontre que l'application <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\mapsto [{\big (}\wp (z),\wp ^{\prime }(z),1{\big )}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo stretchy="false">↦</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi mathvariant="normal">℘</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msup> <mi mathvariant="normal">℘</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\mapsto [{\big (}\wp (z),\wp ^{\prime }(z),1{\big )}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc039fafcfd11c1f333910a1bfe442daff48a5de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.792ex; height:3.176ex;" alt="{\displaystyle z\mapsto [{\big (}\wp (z),\wp ^{\prime }(z),1{\big )}]}"></span> vue plus haut définit par passage au quotient un <a href="/wiki/Hom%C3%A9omorphisme" title="Homéomorphisme">homéomorphisme</a> et une application biholomorphe de ℂ/<i>L</i> dans <span class="texhtml">E</span>. </p><p>Le <b>discriminant modulaire</b> Δ est défini comme le quotient par 16 du <a href="/wiki/Discriminant" title="Discriminant">discriminant</a> du polynôme <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 4x^{3}-g_{2}x-g_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>x</mi> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4x^{3}-g_{2}x-g_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64d633e6eff33f3bc2d30a191eeb115faae24bcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.883ex; height:3.009ex;" alt="{\displaystyle 4x^{3}-g_{2}x-g_{3}}"></span> qui apparaît dans l'équation différentielle ci-dessus : </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 \Delta =g_{2}^{3}-27g_{3}^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <msubsup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>−</mo> <mn>27</mn> <msubsup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =g_{2}^{3}-27g_{3}^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75aea810f524692943a624bb693bea38b79ab402" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.191ex; height:3.176ex;" alt="{\displaystyle \Delta =g_{2}^{3}-27g_{3}^{2}.}"></span></dd></dl> <p>La courbe <span class="texhtml">E</span> est <a href="/wiki/Vari%C3%A9t%C3%A9_alg%C3%A9brique_non_singuli%C3%A8re" title="Variété algébrique non singulière">lisse</a>, c’est-à-dire sans singularités. Il suffit pour le voir de montrer Δ est non nul. Ce qui est le cas, car ses trois racines sont distinctes. D'après la relation algébrique fondamentale, 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 \wp ^{\prime }(u)=0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">℘</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wp ^{\prime }(u)=0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e300a976aeddcd058128522ae6ca5afe0ae18917" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.95ex; height:3.009ex;" alt="{\displaystyle \wp ^{\prime }(u)=0\,}"></span>, alors <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=\wp (u)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi mathvariant="normal">℘</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\wp (u)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f19f71d6595530b0b486aa56dec1a7ff667b7da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.433ex; height:2.843ex;" alt="{\displaystyle x=\wp (u)\,}"></span> est une racine de ce <a href="/wiki/Polyn%C3%B4me" title="Polynôme">polynôme</a>. Mais la fonction <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 \wp ^{\prime }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">℘</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wp ^{\prime }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb717ffdea8a7b394edc7a2403ee3f8e8458b852" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.55ex; height:3.009ex;" alt="{\displaystyle \wp ^{\prime }\,}"></span>, impaire et <i>L</i>-périodique, s'annule 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 2u\in L\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>u</mi> <mo>∈</mo> <mi>L</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2u\in L\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f2aa647365f9ff217c605ae4abd626b98cf444" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.303ex; height:2.176ex;" alt="{\displaystyle 2u\in L\,}"></span>, donc pour <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 {\omega _{1}}{2}},{\frac {\omega _{2}}{2}},{\frac {\omega _{1}+\omega _{2}}{2}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\omega _{1}}{2}},{\frac {\omega _{2}}{2}},{\frac {\omega _{1}+\omega _{2}}{2}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f0bfe2c0e5fbd92111be09be04b248a8a1d16a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.804ex; height:5.009ex;" alt="{\displaystyle {\frac {\omega _{1}}{2}},{\frac {\omega _{2}}{2}},{\frac {\omega _{1}+\omega _{2}}{2}}\,}"></span>. En raison de la <a href="/wiki/Bijection" title="Bijection">bijection</a> entre ℂ/<i>L</i> et <span class="texhtml">E</span>, 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 \wp ({\frac {\omega _{1}}{2}})\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">℘</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wp ({\frac {\omega _{1}}{2}})\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe7f9bc674555cc8ec798722db3446c264fd9fee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.011ex; height:4.676ex;" alt="{\displaystyle \wp ({\frac {\omega _{1}}{2}})\,}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wp ({\frac {\omega _{2}}{2}})\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">℘</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wp ({\frac {\omega _{2}}{2}})\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b28f187617dafc8a81839e00cf83d14045bfa366" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.011ex; height:4.676ex;" alt="{\displaystyle \wp ({\frac {\omega _{2}}{2}})\,}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wp ({\frac {\omega _{1}+\omega _{2}}{2}})\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">℘</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wp ({\frac {\omega _{1}+\omega _{2}}{2}})\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f531e27f2fd869f72bd079cc59797458c91d8cfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.351ex; height:5.009ex;" alt="{\displaystyle \wp ({\frac {\omega _{1}+\omega _{2}}{2}})\,}"></span> sont distincts. </p><p>Une structure de groupe additif sur l'ensemble des points d'une telle cubique est décrite dans l'article « <a href="/wiki/Courbe_elliptique" title="Courbe elliptique">Courbe elliptique</a> ». L'élément neutre est le point à l'infini [(0, 1, 0)], et trois points <i>P</i>, <i>Q</i>, <i>R </i>sont alignés si et seulement si <i>P + Q + R </i>= 0. L'application <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\mapsto [{\big (}\wp (z),\wp ^{\prime }(z),1{\big )}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo stretchy="false">↦</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi mathvariant="normal">℘</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msup> <mi mathvariant="normal">℘</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\mapsto [{\big (}\wp (z),\wp ^{\prime }(z),1{\big )}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc039fafcfd11c1f333910a1bfe442daff48a5de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.792ex; height:3.176ex;" alt="{\displaystyle z\mapsto [{\big (}\wp (z),\wp ^{\prime }(z),1{\big )}]}"></span> est un isomorphisme de groupes entre ℂ/<i>L</i> et <span class="texhtml">E</span>. </p><p>Les éléments neutres se correspondent. Les points de <span class="texhtml">E</span> correspondant 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 u,v\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u,v\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/289f8e0f3897018de2a3902a7d6da47aaf08955f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.878ex; height:2.009ex;" alt="{\displaystyle u,v\,}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -(u+v)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>+</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -(u+v)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af58f016bfab8075abc19c07d1808e6fe10c6bfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.302ex; height:2.843ex;" alt="{\displaystyle -(u+v)\,}"></span> étant alignés, il s'agit d'une conséquence de la formule d'addition, citée sans preuve : </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 \wp (u)+\wp (v)+\wp (u+v)={\frac {1}{4}}\left({\frac {\wp ^{\prime }(u)-\wp ^{\prime }(v)}{\wp (u)-\wp (v)}}\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">℘</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi mathvariant="normal">℘</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi mathvariant="normal">℘</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>+</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">℘</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi mathvariant="normal">℘</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">℘</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi mathvariant="normal">℘</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wp (u)+\wp (v)+\wp (u+v)={\frac {1}{4}}\left({\frac {\wp ^{\prime }(u)-\wp ^{\prime }(v)}{\wp (u)-\wp (v)}}\right)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc65e5b4717236bfdcdf67c228d1bb3ad91ab191" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:46.95ex; height:7.009ex;" alt="{\displaystyle \wp (u)+\wp (v)+\wp (u+v)={\frac {1}{4}}\left({\frac {\wp ^{\prime }(u)-\wp ^{\prime }(v)}{\wp (u)-\wp (v)}}\right)^{2}}"></span></dd></dl> <p>« Structure de groupe sur <span class="texhtml">E</span> » et « structure de groupe sur ℂ/<i>L</i> » sont équivalents : un résultat profond, qui fait appel à la théorie des <a href="/wiki/Forme_modulaire" title="Forme modulaire"> formes modulaires</a> assure que pour toute cubique lisse <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^{2}=4x^{3}-ax-b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mi>a</mi> <mi>x</mi> <mo>−</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}=4x^{3}-ax-b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16299ea4fcd696c15c268f6b1cb9c8a06f1b7907" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.097ex; height:3.009ex;" alt="{\displaystyle y^{2}=4x^{3}-ax-b}"></span>, il existe un unique réseau <i>L</i> tel 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 a=g_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=g_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/845df1e00a57e3c081c07e8439bfd3aecaf913a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.492ex; height:2.009ex;" alt="{\displaystyle a=g_{2}}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=g_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=g_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31d850878a62d6e43e68b7c824e254e46bdb2d60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.259ex; height:2.509ex;" alt="{\displaystyle b=g_{3}}"></span> (voir le &6 du chapitre XII du livre de Godement cité dans la bibliographie). </p> <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=Fonction_elliptique_de_Weierstrass&veaction=edit&section=9" title="Modifier la section : Références" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Fonction_elliptique_de_Weierstrass&action=edit&section=9" title="Modifier le code source de la section : Références"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="references-small decimal" style=""><div class="mw-references-wrap"><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"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> A. Weil, <i><span class="lang-en" lang="en">Elliptic Functions According to Eisenstein and Kronecker</span></i>, Springer, chap. 1.</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=Fonction_elliptique_de_Weierstrass&veaction=edit&section=10" 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=Fonction_elliptique_de_Weierstrass&action=edit&section=10" title="Modifier le code source de la section : Voir aussi"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Bibliographie">Bibliographie</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fonction_elliptique_de_Weierstrass&veaction=edit&section=11" 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=Fonction_elliptique_de_Weierstrass&action=edit&section=11" title="Modifier le code source de la section : Bibliographie"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Roger_Godement" title="Roger Godement">Roger Godement</a>, <i>Cours d'analyse</i>, Springer.</li> <li><a href="/wiki/Yves_Hellegouarch" title="Yves Hellegouarch">Yves Hellegouarch</a>, <i>Invitation aux mathématiques de Fermat-Wiles</i>, Masson 1997.</li> <li><span class="ouvrage" id="Finch2003"><span class="ouvrage" id="Steven_R._Finch2003"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Steven R. Finch, <cite class="italique" lang="en">Mathematical Constants</cite>, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <time>2003</time>, 602 <abbr class="abbr" title="pages">p.</abbr> <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/978-0-521-81805-6" title="Spécial:Ouvrages de référence/978-0-521-81805-6"><span class="nowrap">978-0-521-81805-6</span></a>, <a rel="nofollow" class="external text" href="//books.google.com/books?id=Pl5I2ZSI6uAC&pg=PA421">lire en ligne</a>)</small>, <abbr class="abbr" title="pages">p.</abbr> <span class="nowrap">421-422</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Constants&rft.pub=Cambridge+University+Press&rft.aulast=Finch&rft.aufirst=Steven+R.&rft.date=2003&rft.pages=421-422&rft.tpages=602&rft.isbn=978-0-521-81805-6&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AFonction+elliptique+de+Weierstrass"></span></span></span></li> <li><span class="ouvrage" id="E._PinchF._Swinnerton-Dyer1991"><span class="ouvrage" id="R._G._E._PinchH._P._F._Swinnerton-Dyer1991"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> R. G. E. Pinch et <a href="/wiki/Peter_Swinnerton-Dyer" title="Peter Swinnerton-Dyer">H. P. F. Swinnerton-Dyer</a>, <cite style="font-style:normal" lang="en">« Arithmetic of Diagonal Quartic Surfaces I »</cite>, dans <a href="/wiki/John_Coates_(math%C3%A9maticien)" title="John Coates (mathématicien)">J. Coates</a> et <a href="/wiki/Martin_J._Taylor" class="mw-redirect" title="Martin J. Taylor">Martin J. Taylor</a> (ed.), <cite class="italique" lang="en">L-Functions and Arithmetic</cite>, Cambridge University Press, <time>1991</time> <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/978-0-521-38619-7" title="Spécial:Ouvrages de référence/978-0-521-38619-7"><span class="nowrap">978-0-521-38619-7</span></a>, <a rel="nofollow" class="external text" href="//books.google.com/books?id=aKQhpm1h770C&pg=PA317">lire en ligne</a>)</small>, <abbr class="abbr" title="pages">p.</abbr> <span class="nowrap">317-338</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=L-Functions+and+Arithmetic&rft.atitle=Arithmetic+of+Diagonal+Quartic+Surfaces+I&rft.pub=Cambridge+University+Press&rft.aulast=E.+Pinch&rft.aufirst=R.+G.&rft.au=H.+P.+F.+Swinnerton-Dyer&rft.date=1991&rft.pages=317-338&rft.isbn=978-0-521-38619-7&rft_id=%2F%2Fbooks.google.com%2Fbooks%3Fid%3DaKQhpm1h770C%26pg%3DPA317&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AFonction+elliptique+de+Weierstrass"></span></span></span>, <abbr class="abbr" title="page">p.</abbr> 333</li></ul> <div class="mw-heading mw-heading3"><h3 id="Articles_connexes">Articles connexes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fonction_elliptique_de_Weierstrass&veaction=edit&section=12" title="Modifier la section : Articles connexes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Fonction_elliptique_de_Weierstrass&action=edit&section=12" title="Modifier le code source de la section : Articles connexes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Courbe_d%27Edwards" title="Courbe d'Edwards">Courbe d'Edwards</a></li> <li><a href="/wiki/Fonction_elliptique_de_Jacobi" title="Fonction elliptique de Jacobi">Fonction elliptique de Jacobi</a></li> <li><a href="/wiki/Fonction_z%C3%AAta_de_Weierstrass" title="Fonction zêta de Weierstrass">Fonction zêta de Weierstrass</a></li> <li><a href="/wiki/Forme_modulaire" title="Forme modulaire">Forme modulaire</a></li> <li><a href="/wiki/S%C3%A9rie_d%27Eisenstein" title="Série d'Eisenstein">Série d'Eisenstein</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Lien_externe">Lien externe</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fonction_elliptique_de_Weierstrass&veaction=edit&section=13" title="Modifier la section : Lien externe" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Fonction_elliptique_de_Weierstrass&action=edit&section=13" title="Modifier le code source de la section : Lien externe"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="ouvrage" id="Weisstein"><span class="ouvrage" id="Eric_W._Weisstein"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Eric W. Weisstein</a>, « <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/LemniscateCase.html"><cite style="font-style:normal;" lang="en"><span class="lang-en" lang="en">Lemniscate Case</span></cite></a> », sur <span class="italique"><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></span></span></span> </p> <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; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Ce document provient de « <a dir="ltr" href="https://fr.wikipedia.org/w/index.php?title=Fonction_elliptique_de_Weierstrass&oldid=218766779#La_cubique_de_Weierstrass_vue_comme_courbe_elliptique">https://fr.wikipedia.org/w/index.php?title=Fonction_elliptique_de_Weierstrass&oldid=218766779#La_cubique_de_Weierstrass_vue_comme_courbe_elliptique</a> ».</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Cat%C3%A9gorie:Accueil" title="Catégorie:Accueil">Catégories</a> : <ul><li><a href="/wiki/Cat%C3%A9gorie:Fonction_elliptique" title="Catégorie:Fonction elliptique">Fonction elliptique</a></li><li><a href="/wiki/Cat%C3%A9gorie:G%C3%A9om%C3%A9trie_alg%C3%A9brique" title="Catégorie:Géométrie algébrique">Géométrie algébrique</a></li><li><a href="/wiki/Cat%C3%A9gorie:Analyse_complexe" title="Catégorie:Analyse complexe">Analyse complexe</a></li><li><a href="/wiki/Cat%C3%A9gorie:Fonction_remarquable" title="Catégorie:Fonction remarquable">Fonction remarquable</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Catégories cachées : <ul><li><a href="/wiki/Cat%C3%A9gorie:Portail:Analyse/Articles_li%C3%A9s" title="Catégorie:Portail:Analyse/Articles liés">Portail:Analyse/Articles liés</a></li><li><a href="/wiki/Cat%C3%A9gorie:Portail:Math%C3%A9matiques/Articles_li%C3%A9s" title="Catégorie:Portail:Mathématiques/Articles liés">Portail:Mathématiques/Articles liés</a></li><li><a href="/wiki/Cat%C3%A9gorie:Portail:Sciences/Articles_li%C3%A9s" title="Catégorie:Portail:Sciences/Articles liés">Portail:Sciences/Articles liés</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> La dernière modification de cette page a été faite le 19 septembre 2024 à 21:37.</li> <li id="footer-info-copyright"><span style="white-space: normal"><a href="/wiki/Wikip%C3%A9dia:Citation_et_r%C3%A9utilisation_du_contenu_de_Wikip%C3%A9dia" title="Wikipédia:Citation et réutilisation du contenu de Wikipédia">Droit d'auteur</a> : les textes sont disponibles sous <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.fr">licence Creative Commons attribution, partage dans les mêmes conditions</a> ; d’autres conditions peuvent s’appliquer. 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