Type exponentiel — Wikipédia

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class="vector-toc-link" href="#Définition_formelle"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Définition formelle</span> </div> </a> <button aria-controls="toc-Définition_formelle-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 Définition formelle</span> </button> <ul id="toc-Définition_formelle-sublist" class="vector-toc-list"> <li id="toc-Exemples" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exemples"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Exemples</span> </div> </a> <ul id="toc-Exemples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Propriétés" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Propriétés"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Propriétés</span> </div> </a> <ul id="toc-Propriétés-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Type_exponentiel_sur_un_ensemble_convexe_symétrique" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Type_exponentiel_sur_un_ensemble_convexe_symétrique"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Type exponentiel sur un ensemble convexe symétrique</span> </div> </a> <ul id="toc-Type_exponentiel_sur_un_ensemble_convexe_symétrique-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Espace_de_Fréchet" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Espace_de_Fréchet"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Espace de Fréchet</span> </div> </a> <ul id="toc-Espace_de_Fréchet-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voir_aussi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voir_aussi"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Voir aussi</span> </div> </a> <ul id="toc-Voir_aussi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Références" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Références"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Références</span> </div> </a> <ul id="toc-Références-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Sommaire" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <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">Type exponentiel</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input 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Disponible en 3 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-3" 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">3 langues</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Exponential_type" title="Exponential type – anglais" lang="en" hreflang="en" data-title="Exponential type" data-language-autonym="English" data-language-local-name="anglais" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83" title="Функція експоненційного типу – ukrainien" lang="uk" hreflang="uk" data-title="Функція експоненційного типу" 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/%E6%8C%87%E6%95%B0%E5%9E%8B" title="指数型 – chinois" lang="zh" hreflang="zh" data-title="指数型" 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 after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q5421531#sitelinks-wikipedia" title="Modifier les liens interlangues" class="wbc-editpage">Modifier les liens</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Espaces de noms"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Type_exponentiel" title="Voir le contenu de la page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Discussion:Type_exponentiel&action=edit&redlink=1" rel="discussion" class="new" title="Discussion au sujet de cette page de contenu (page inexistante) [t]" accesskey="t"><span>Discussion</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" 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<div class="vector-pinnable-header-label">Outils</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">déplacer vers la barre latérale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">masquer</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Plus d’options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Type_exponentiel"><span>Lire</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Type_exponentiel&veaction=edit" title="Modifier cette page [v]" accesskey="v"><span>Modifier</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Type_exponentiel&action=edit" title="Modifier le wikicode de cette page [e]" accesskey="e"><span>Modifier le code</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Type_exponentiel&action=history"><span>Voir l’historique</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Général </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Sp%C3%A9cial:Pages_li%C3%A9es/Type_exponentiel" title="Liste des pages liées qui pointent sur celle-ci [j]" accesskey="j"><span>Pages liées</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Sp%C3%A9cial:Suivi_des_liens/Type_exponentiel" rel="nofollow" title="Liste des modifications récentes des pages appelées par celle-ci [k]" accesskey="k"><span>Suivi des pages liées</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Aide:Importer_un_fichier" title="Téléverser des fichiers [u]" accesskey="u"><span>Téléverser un fichier</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Sp%C3%A9cial:Pages_sp%C3%A9ciales" title="Liste de toutes les pages spéciales [q]" accesskey="q"><span>Pages spéciales</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Type_exponentiel&oldid=191576321" title="Adresse permanente de cette version de cette page"><span>Lien permanent</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Type_exponentiel&action=info" title="Davantage d’informations sur cette page"><span>Informations sur la page</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Sp%C3%A9cial:Citer&page=Type_exponentiel&id=191576321&wpFormIdentifier=titleform" title="Informations sur la manière de citer cette page"><span>Citer cette page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Sp%C3%A9cial:UrlShortener&url=https%3A%2F%2Ffr.wikipedia.org%2Fwiki%2FType_exponentiel"><span>Obtenir l'URL raccourcie</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Sp%C3%A9cial:QrCode&url=https%3A%2F%2Ffr.wikipedia.org%2Fwiki%2FType_exponentiel"><span>Télécharger le code QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Imprimer / exporter </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Sp%C3%A9cial:Livre&bookcmd=book_creator&referer=Type+exponentiel"><span>Créer un livre</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Sp%C3%A9cial:DownloadAsPdf&page=Type_exponentiel&action=show-download-screen"><span>Télécharger comme PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Type_exponentiel&printable=yes" title="Version imprimable de cette page [p]" accesskey="p"><span>Version imprimable</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> Dans d’autres projets </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q5421531" title="Lien vers l’élément dans le dépôt de données connecté [g]" accesskey="g"><span>Élément Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Outils de la page"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Apparence"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Apparence</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">déplacer vers la barre latérale</button> <button 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"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="fr" dir="ltr"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier:ExtremeGaussian.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/ExtremeGaussian.png/220px-ExtremeGaussian.png" decoding="async" width="220" height="136" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/ExtremeGaussian.png/330px-ExtremeGaussian.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b2/ExtremeGaussian.png/440px-ExtremeGaussian.png 2x" data-file-width="1017" data-file-height="629" /></a><figcaption>Représentation graphique de 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 {\rm {e}}^{-\pi z^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {e}}^{-\pi z^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09164c89c2ac2c03875be3dea6f8e962398bc29e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.088ex; height:3.009ex;" alt="{\displaystyle {\rm {e}}^{-\pi z^{2}}}"></span>, une fonction gaussienne sur l'axe réel. Cette fonction n'a pas de type exponentiel, mais peut être approchée de chaque côté par deux fonction (en rouge et bleu) de type exponentiel <span class="texhtml">2π</span>.</figcaption></figure> <p>En <a href="/wiki/Analyse_complexe" title="Analyse complexe">analyse complexe</a>, une <a href="/wiki/Fonction_holomorphe" title="Fonction holomorphe">fonction holomorphe</a> est dite de <b>type exponentiel C</b> si sa <a href="/w/index.php?title=Fonction_%C3%A0_croissance_born%C3%A9e&action=edit&redlink=1" class="new" title="Fonction à croissance bornée (page inexistante)">croissance est bornée</a> par la <a href="/wiki/Fonction_exponentielle" title="Fonction exponentielle">fonction exponentielle</a> <span class="texhtml">e<sup><i>C</i>|<i>z</i>|</sup></span> avec une constante réelle <span class="texhtml mvar" style="font-style:italic;">C</span>, pour |<i>z</i>| → ∞. Quand une fonction est bornée de la sorte, il est alors possible de l'exprimer comme une somme convergente de série d'autres fonctions complexes, de même qu'il est possible d'appliquer des techniques comme la <a href="/wiki/Sommation_de_Borel" title="Sommation de Borel">sommation de Borel</a>, ou, par exemple, d'appliquer la <a href="/wiki/Transformation_de_Mellin" title="Transformation de Mellin">transformation de Mellin</a>, ou d'obtenir des approximations comme la <a href="/wiki/Formule_d%27Euler-Maclaurin" title="Formule d'Euler-Maclaurin">formule d'Euler-Maclaurin</a>. Le cas général est décrit par le <a href="/w/index.php?title=Th%C3%A9or%C3%A8me_de_Nachbin&action=edit&redlink=1" class="new" title="Théorème de Nachbin (page inexistante)">théorème de Nachbin</a>, qui utilise la notion analogue de <b>type <span class="texhtml">Ψ</span></b> pour une fonction générale <span class="texhtml">Ψ(<i>z</i>)</span> à la place d'une fonction exponentielle. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Principe">Principe</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Type_exponentiel&veaction=edit&section=1" title="Modifier la section : Principe" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Type_exponentiel&action=edit&section=1" title="Modifier le code source de la section : Principe"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Une fonction <span class="texhtml"><i>f</i>(<i>z</i>)</span> définie sur le <a href="/wiki/Plan_complexe" title="Plan complexe">plan complexe</a> est dite de type exponentiel s'il existe des constantes réelles <span class="texhtml mvar" style="font-style:italic;">M</span> et <span class="texhtml">τ</span> telles que </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 \left|f\left(r{\rm {e}}^{{\rm {i}}\theta }\right)\right|\leq M{\rm {e}}^{\tau r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>r</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>θ</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <mi>M</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> <mi>r</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|f\left(r{\rm {e}}^{{\rm {i}}\theta }\right)\right|\leq M{\rm {e}}^{\tau r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc6f1c3ef23c9549834cbdfccdcd8b8110acaa3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:17.028ex; height:3.509ex;" alt="{\displaystyle \left|f\left(r{\rm {e}}^{{\rm {i}}\theta }\right)\right|\leq M{\rm {e}}^{\tau r}}"></span></dd></dl> <p>dans le cas où <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcd3a85ea2e3d6b4027434e502cace4177d7a3e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.986ex; height:1.843ex;" alt="{\displaystyle r\to \infty }"></span>. Ici, la <a href="/wiki/Variable_complexe" class="mw-redirect" title="Variable complexe">variable complexe</a> <span class="texhtml mvar" style="font-style:italic;">z</span> est écrite sous la forme <span class="texhtml"><i>z</i> = <i>r</i>e<sup>i<i>θ</i></sup></span> pour indiquer que la limite est indépendante de la direction <span class="texhtml mvar" style="font-style:italic;">θ</span>. En notant <span class="texhtml mvar" style="font-style:italic;">τ</span> (de façon abusive) l'<a href="/wiki/Infimum" class="mw-redirect" title="Infimum">infimum</a> de tous les <span class="texhtml mvar" style="font-style:italic;">τ</span> qu conviennent, on dit que la fonction <span class="texhtml mvar" style="font-style:italic;">f</span> est de <i>type exponentiel <span class="texhtml mvar" style="font-style:italic;">τ</span></i>. </p><p>Considérons par exemple <span class="texhtml"><i>f</i>(<i>z</i>) = sin(π <i>z</i>)</span>. Alors on dit que <span class="texhtml mvar" style="font-style:italic;">f</span> est de type exponentiel <span class="texhtml">π</span>, car <span class="texhtml">π</span> est le plus petit nombre qui borne la croissance de <span class="texhtml">sin(π <i>z</i>)</span> sur l'axe imaginaire. Ainsi, dans ce cas, le <a href="/wiki/Th%C3%A9or%C3%A8me_de_Carlson" title="Théorème de Carlson">théorème de Carlson</a> ne s'applique pas, car il n'est vrai que pour des fonctions de type exponentiel inférieur à <span class="texhtml">π</span>. De même, la <a href="/wiki/Formule_d%27Euler-Maclaurin" title="Formule d'Euler-Maclaurin">formule d'Euler-Maclaurin</a> ne s'applique pas non plus, dans la mesure où elle est liée à un théorème lié à la théorie des <a href="/wiki/Diff%C3%A9rence_finie" title="Différence finie">différences finies</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Définition_formelle"><span id="D.C3.A9finition_formelle"></span>Définition formelle</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Type_exponentiel&veaction=edit&section=2" title="Modifier la section : Définition formelle" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Type_exponentiel&action=edit&section=2" title="Modifier le code source de la section : Définition formelle"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Une <a href="/wiki/Fonction_holomorphe" title="Fonction holomorphe">fonction holomorphe</a> <span class="texhtml"><i>F</i>(<i>z</i>)</span> est dite de <b>type exponentiel</b> <span class="texhtml">σ > 0</span> si </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 \forall \varepsilon >0,\ \exists A\in \mathbb {R} ,\ |F(z)|\leq A{\rm {e}}^{(\sigma +\varepsilon )|z|}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>ε</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mi mathvariant="normal">∃</mi> <mi>A</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>A</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>σ</mi> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \varepsilon >0,\ \exists A\in \mathbb {R} ,\ |F(z)|\leq A{\rm {e}}^{(\sigma +\varepsilon )|z|}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84737199b14b59c0562e391ee39fe3d3157d822d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.407ex; height:3.343ex;" alt="{\displaystyle \forall \varepsilon >0,\ \exists A\in \mathbb {R} ,\ |F(z)|\leq A{\rm {e}}^{(\sigma +\varepsilon )|z|}}"></span></dd></dl> <p>quand <span class="mwe-math-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|\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/878409f5f9775844a8d31df9dd60e41eadd1f889" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.32ex; height:2.843ex;" alt="{\displaystyle |z|\to \infty }"></span> avec <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\in \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/169fae60c23a2027ece2aa7fd4b5047492887e91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:2.176ex;" alt="{\displaystyle z\in \mathbb {C} }"></span>. On dira que <span class="texhtml"><i>F</i>(<i>z</i>)</span> est de type exponentiel si <span class="texhtml"><i>F</i>(<i>z</i>)</span> est de type exponentiel <span class="texhtml">σ</span> pour un certain <span class="texhtml">σ > 0</span>. </p><p>Le nombre </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 \tau (F)=\sigma =\displaystyle \limsup _{|z|\rightarrow \infty }{\frac {\log |F(z)|}{|z|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>σ</mi> <mo>=</mo> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau (F)=\sigma =\displaystyle \limsup _{|z|\rightarrow \infty }{\frac {\log |F(z)|}{|z|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe8c8b1c96dad2cb0ce89e069f996f48c24f41c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.911ex; height:6.843ex;" alt="{\displaystyle \tau (F)=\sigma =\displaystyle \limsup _{|z|\rightarrow \infty }{\frac {\log |F(z)|}{|z|}}}"></span></dd></dl> <p>est le type exponentiel de <span class="texhtml"><i>F</i>(<i>z</i>)</span>. La <a href="/wiki/Limite_sup%C3%A9rieure" class="mw-redirect" title="Limite supérieure">limite supérieure</a> désigne ici la limite du <a href="/wiki/Supremum" class="mw-redirect" title="Supremum">supremum</a> du rapport au-delà d'un rayon donné alors que le rayon tend vers l'infini. Cette limite supérieure peut exister même si le maximum au rayon <span class="texhtml mvar" style="font-style:italic;">r</span> n'a pas de limite quand <span class="texhtml mvar" style="font-style:italic;">r</span> tend vers l'infini. Par exemple, pour la fonction </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(z)=\sum _{n=1}^{\infty }{\frac {z^{10^{n!}}}{(10^{n!})!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>!</mo> </mrow> </msup> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>!</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(z)=\sum _{n=1}^{\infty }{\frac {z^{10^{n!}}}{(10^{n!})!}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a3442c9a990ed006ae52b3e6d1d6c655276f71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.772ex; height:7.176ex;" alt="{\displaystyle F(z)=\sum _{n=1}^{\infty }{\frac {z^{10^{n!}}}{(10^{n!})!}}}"></span></dd></dl> <p>La valeur de </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 {\max _{|z|=r}\log |F(z)|}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>r</mi> </mrow> </munder> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\max _{|z|=r}\log |F(z)|}{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bf9379d3574e58e8187578292946f43363a35c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.776ex; height:5.843ex;" alt="{\displaystyle {\frac {\max _{|z|=r}\log |F(z)|}{r}}}"></span></dd></dl> <p>pour <span class="texhtml"><i>r</i> = 10<sup><i>n</i>!-1</sup></span> est majoré par le <span class="texhtml"><i>n</i>- 1</span><sup>e</sup> terme donc on a les expressions asymptotiques : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\left(\max _{|z|=10^{n!-1}}\log |F(z)|\right)/10^{n!-1}&\sim \left(\log {\frac {(10^{n!-1})^{10^{(n-1)!}}}{(10^{(n-1)!})!}}\right)/10^{n!-1}\\&\sim (\log 10)\left[(n!-1)10^{(n-1)!}-10^{(n-1)!}(n-1)!\right]/10^{n!-1}\\&\sim (\log 10)(n!-1-(n-1)!)/10^{n!-1-(n-1)!}\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow> <mo>(</mo> <mrow> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>!</mo> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </munder> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>!</mo> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>∼</mo> <mrow> <mo>(</mo> <mrow> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>!</mo> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </msup> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>!</mo> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>∼</mo> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mn>10</mn> <mo stretchy="false">)</mo> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>!</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </msup> <mo>−</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>!</mo> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>∼</mo> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mn>10</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>!</mo> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>!</mo> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left(\max _{|z|=10^{n!-1}}\log |F(z)|\right)/10^{n!-1}&\sim \left(\log {\frac {(10^{n!-1})^{10^{(n-1)!}}}{(10^{(n-1)!})!}}\right)/10^{n!-1}\\&\sim (\log 10)\left[(n!-1)10^{(n-1)!}-10^{(n-1)!}(n-1)!\right]/10^{n!-1}\\&\sim (\log 10)(n!-1-(n-1)!)/10^{n!-1-(n-1)!}\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb345699cbded1628936f050b1e96b956b388622" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:83.543ex; height:15.843ex;" alt="{\displaystyle {\begin{aligned}\left(\max _{|z|=10^{n!-1}}\log |F(z)|\right)/10^{n!-1}&\sim \left(\log {\frac {(10^{n!-1})^{10^{(n-1)!}}}{(10^{(n-1)!})!}}\right)/10^{n!-1}\\&\sim (\log 10)\left[(n!-1)10^{(n-1)!}-10^{(n-1)!}(n-1)!\right]/10^{n!-1}\\&\sim (\log 10)(n!-1-(n-1)!)/10^{n!-1-(n-1)!}\\\end{aligned}}}"></span></dd></dl> <p>et tend vers 0 pour <span class="texhtml mvar" style="font-style:italic;">n</span> tend vers l'infini<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>, mais <span class="texhtml"><i>F</i>(<i>z</i>)</span> est tout de même de type exponentiel 1, comme on peut le voir aux points <span class="texhtml"><i>z</i> = 10<sup><i>n</i>!</sup></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Exemples">Exemples</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Type_exponentiel&veaction=edit&section=3" title="Modifier la section : Exemples" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Type_exponentiel&action=edit&section=3" title="Modifier le code source de la section : Exemples"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Les fonctions constantes sont de type exponentiel. </p> <div class="mw-heading mw-heading3"><h3 id="Propriétés"><span id="Propri.C3.A9t.C3.A9s"></span>Propriétés</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Type_exponentiel&veaction=edit&section=4" title="Modifier la section : Propriétés" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Type_exponentiel&action=edit&section=4" title="Modifier le code source de la section : Propriétés"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Le produit de deux fonctions de type exponentiel est également de type exponentiel<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup>. </p> <div class="mw-heading mw-heading2"><h2 id="Type_exponentiel_sur_un_ensemble_convexe_symétrique"><span id="Type_exponentiel_sur_un_ensemble_convexe_sym.C3.A9trique"></span>Type exponentiel sur un ensemble convexe symétrique</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Type_exponentiel&veaction=edit&section=5" title="Modifier la section : Type exponentiel sur un ensemble convexe symétrique" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Type_exponentiel&action=edit&section=5" title="Modifier le code source de la section : Type exponentiel sur un ensemble convexe symétrique"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>(<a href="#Stein1957">Stein 1957</a>) a donné une généralisation de type exponentiel pour les <a href="/wiki/Fonction_enti%C3%A8re" title="Fonction entière">fonctions entières</a> de plusieurs variables complexes. Soit <span class="texhtml mvar" style="font-style:italic;">K</span> un sous-<a href="/wiki/Ensemble_convexe" title="Ensemble convexe">ensemble convexe</a>, <a href="/wiki/Espace_compact" class="mw-redirect" title="Espace compact">compact</a> et <a href="/wiki/Espace_sym%C3%A9trique" title="Espace symétrique">symétrique</a> de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>. On sait que pour tout <span class="texhtml mvar" style="font-style:italic;">K</span>, il existe une <a href="/wiki/Norme_(math%C3%A9matiques)" title="Norme (mathématiques)">norme</a> associé <span class="mwe-math-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 \|\cdot \|_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{K}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d415781ab15cfce5b0d7e59fb959f19f49050f28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.697ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|_{K}}"></span> pour laquelle <span class="texhtml mvar" style="font-style:italic;">K</span> est la boule unité 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 \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> : </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 K=\{x\in \mathbb {R} ^{n}:\|x\|_{K}\leq 1\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>:</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo>≤</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=\{x\in \mathbb {R} ^{n}:\|x\|_{K}\leq 1\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07e913a74eec08e03264be330571a6a089f5901f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.749ex; height:2.843ex;" alt="{\displaystyle K=\{x\in \mathbb {R} ^{n}:\|x\|_{K}\leq 1\}.}"></span></dd></dl> <p>Alors l'ensemble </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 K^{*}=\{y\in \mathbb {R} ^{n}:x\cdot y\leq 1{\text{ pour tout }}x\in {K}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>y</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>:</mo> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> <mo>≤</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> pour tout </mtext> </mrow> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K^{*}=\{y\in \mathbb {R} ^{n}:x\cdot y\leq 1{\text{ pour tout }}x\in {K}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a9724d261d838c159ea73dbc514b246d9e938" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.728ex; height:2.843ex;" alt="{\displaystyle K^{*}=\{y\in \mathbb {R} ^{n}:x\cdot y\leq 1{\text{ pour tout }}x\in {K}\}}"></span></dd></dl> <p>est appelé l'<a href="/wiki/Ensemble_polaire" title="Ensemble polaire">ensemble polaire</a> de <span class="texhtml mvar" style="font-style:italic;">K</span><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite_crochet">[</span>3<span class="cite_crochet">]</span></a></sup> C'est aussi un sous-ensemble convexe, compact et symétrique 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 \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>. De plus, on peut écrire </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|_{K}=\displaystyle \sup _{y\in K^{*}}|x\cdot y|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo>=</mo> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>∈</mo> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|_{K}=\displaystyle \sup _{y\in K^{*}}|x\cdot y|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f679139d2e95b83fd1fc53bd683645e96b16479" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.164ex; height:4.843ex;" alt="{\displaystyle \|x\|_{K}=\displaystyle \sup _{y\in K^{*}}|x\cdot y|.}"></span></dd></dl> <p>On étend <span class="mwe-math-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 \|\cdot \|_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{K}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d415781ab15cfce5b0d7e59fb959f19f49050f28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.697ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|_{K}}"></span> 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} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> vers <span class="mwe-math-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 {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}"></span> 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 \|z\|_{K}=\displaystyle \sup _{y\in K^{*}}|z\cdot y|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>z</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo>=</mo> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>∈</mo> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mo>⋅</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|z\|_{K}=\displaystyle \sup _{y\in K^{*}}|z\cdot y|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40f8572a14bc836db0eb0ba511eb7b07455178bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:18.68ex; height:4.843ex;" alt="{\displaystyle \|z\|_{K}=\displaystyle \sup _{y\in K^{*}}|z\cdot y|.}"></span></dd></dl> <p>Une fonction entière <span class="texhtml"><i>F</i>(<i>z</i>)</span> de <span class="texhtml mvar" style="font-style:italic;">n</span> variables complexes est dite de type exponentiel par rapport à <span class="texhtml mvar" style="font-style:italic;">K</span> si : </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 \forall \varepsilon >0,\ \exists A\in \mathbb {R} ,\ \forall z\in \mathbb {C} ^{n}\,|F(z)|<A{\rm {e}}^{2\pi (1+\varepsilon )\|z\|_{K}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>ε</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mi mathvariant="normal">∃</mi> <mi>A</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mtext> </mtext> <mi mathvariant="normal">∀</mi> <mi>z</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>A</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖</mo> <mi>z</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \varepsilon >0,\ \exists A\in \mathbb {R} ,\ \forall z\in \mathbb {C} ^{n}\,|F(z)|<A{\rm {e}}^{2\pi (1+\varepsilon )\|z\|_{K}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3c9c79f0709b921e2984bcc4d86257c3d400cbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.284ex; height:3.343ex;" alt="{\displaystyle \forall \varepsilon >0,\ \exists A\in \mathbb {R} ,\ \forall z\in \mathbb {C} ^{n}\,|F(z)|<A{\rm {e}}^{2\pi (1+\varepsilon )\|z\|_{K}}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Espace_de_Fréchet"><span id="Espace_de_Fr.C3.A9chet"></span>Espace de Fréchet</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Type_exponentiel&veaction=edit&section=6" title="Modifier la section : Espace de Fréchet" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Type_exponentiel&action=edit&section=6" title="Modifier le code source de la section : Espace de Fréchet"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Des collections de fonctions de type exponentiel <span class="texhtml mvar" style="font-style:italic;">τ</span> peuvent former un <a href="/wiki/Espace_complet" title="Espace complet">espace complet</a> <a href="/wiki/Espace_uniforme" title="Espace uniforme">uniforme</a>, qu'on appelle <a href="/wiki/Espace_de_Fr%C3%A9chet" title="Espace de Fréchet">espace de Fréchet</a>, par la <a href="/wiki/Espace_topologique" title="Espace topologique">topologie</a> induite par la famille dénombrable des <a href="/wiki/Norme_(math%C3%A9matiques)" title="Norme (mathématiques)">normes</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 \|f\|_{n}=\sup _{z\in \mathbb {C} }\exp \left[-\left(\tau +{\frac {1}{n}}\right)|z|\right]|f(z)|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mrow> </munder> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mi>τ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{n}=\sup _{z\in \mathbb {C} }\exp \left[-\left(\tau +{\frac {1}{n}}\right)|z|\right]|f(z)|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0afab228894be9dfa1e50ff655815efb5392c58a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.591ex; height:6.176ex;" alt="{\displaystyle \|f\|_{n}=\sup _{z\in \mathbb {C} }\exp \left[-\left(\tau +{\frac {1}{n}}\right)|z|\right]|f(z)|.}"></span></dd></dl> <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=Type_exponentiel&veaction=edit&section=7" title="Modifier la section : Voir aussi" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Type_exponentiel&action=edit&section=7" title="Modifier le code source de la section : Voir aussi"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Th%C3%A9or%C3%A8me_de_Paley-Wiener" title="Théorème de Paley-Wiener">Théorème de Paley-Wiener</a></li> <li><a href="/w/index.php?title=Espace_de_Paley-Wiener&action=edit&redlink=1" class="new" title="Espace de Paley-Wiener (page inexistante)">Espace de Paley-Wiener</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Références"><span id="R.C3.A9f.C3.A9rences"></span>Références</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Type_exponentiel&veaction=edit&section=8" 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=Type_exponentiel&action=edit&section=8" title="Modifier le code source de la section : Références"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Cet article est partiellement ou en totalité issu de l’article de Wikipédia en anglais intitulé <span class="plainlinks">« <a class="external text" href="https://en.wikipedia.org/wiki/Exponential_type?oldid=1069100629">Exponential type</a> » <small>(<a class="external text" href="https://en.wikipedia.org/wiki/Exponential_type?action=history">voir la liste des auteurs</a>)</small></span>.</li></ul> <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">En fait, même <span class="mwe-math-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 (\max _{|z|=r}\log \log |F(z)|)/(\log r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>r</mi> </mrow> </munder> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\max _{|z|=r}\log \log |F(z)|)/(\log r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29e2d7e930c3a76b8b133b8347296379feaee28f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.551ex; height:4.509ex;" alt="{\displaystyle (\max _{|z|=r}\log \log |F(z)|)/(\log r)}"></span> tend vers 0 en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=10^{n!-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>!</mo> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=10^{n!-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b650de5b25f20c9c77518e82cf13909f0ecf5da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.248ex; height:2.676ex;" alt="{\displaystyle r=10^{n!-1}}"></span> pour <i>n</i> tendant vers l'infini.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink noprint"><a href="#cite_ref-2">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Martineau1963"><span class="ouvrage" id="André_Martineau1963">André Martineau, <cite class="italique">Sur les fonctionnelles analytiques et la transformation de Fourier-Borel</cite>, <time class="nowrap" datetime="1963-12" data-sort-value="1963-12">décembre 1963</time> <small style="line-height:1em;">(<a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">DOI</a> <span class="plainlinks noarchive nowrap"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.1007/bf02789982">10.1007/bf02789982</a></span>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sur+les+fonctionnelles+analytiques+et+la+transformation+de+Fourier-Borel&rft.aulast=Martineau&rft.aufirst=Andr%C3%A9&rft.date=1963-12&rft_id=info%3Adoi%2F10.1007%2Fbf02789982&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AType+exponentiel"></span></span></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink noprint"><a href="#cite_ref-3">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Lelong1966"><span class="ouvrage" id="Pierre_Lelong1966">Pierre Lelong, « <cite style="font-style:normal">Fonctions entières de type exponentiel dans C<sup><i>n</i></sup></cite> », <i>Annales de l’institut Fourier</i>, <abbr class="abbr" title="tome">t.</abbr> 16, <abbr class="abbr" title="numéro">n<sup>o</sup></abbr> 2,‎ <time>1966</time>, <abbr class="abbr" title="pages">p.</abbr> <span class="nowrap">269-318</span> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="http://www.numdam.org/item/10.5802/aif.244.pdf">lire en ligne</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Fonctions+enti%C3%A8res+de+type+exponentiel+dans+C%27%27n%27%27&rft.jtitle=Annales+de+l%E2%80%99institut+Fourier&rft.issue=2&rft.aulast=Lelong&rft.aufirst=Pierre&rft.date=1966&rft.pages=269-318&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AType+exponentiel"></span></span></span></span> </li> </ol></div> </div> <ul><li><span class="ouvrage" id="Stein1957"><span class="ouvrage" id="Elias_M._Stein1957"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Elias M. Stein, « <cite style="font-style:normal" lang="en">Functions of exponential type</cite> », <i><span class="lang-en" lang="en">Ann. of Math.</span></i>, <abbr class="abbr" title="deuxième">2<sup>e</sup></abbr> série, <abbr class="abbr" title="volume">vol.</abbr> 65,‎ <time>1957</time>, <abbr class="abbr" title="pages">p.</abbr> <span class="nowrap">582–592</span> <small style="line-height:1em;">(<a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">DOI</a> <span class="plainlinks noarchive nowrap"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.2307/1970066">10.2307/1970066</a></span>, <a href="/wiki/JSTOR" title="JSTOR">JSTOR</a> <span class="plainlinks noarchive nowrap"><a rel="nofollow" class="external text" href="https://jstor.org/stable/1970066">1970066</a></span>, <a href="/wiki/Mathematical_Reviews" title="Mathematical Reviews">MR</a> <span class="plainlinks noarchive nowrap"><a rel="nofollow" class="external text" href="https://www.ams.org/mathscinet-getitem?mr=0085342">0085342</a></span>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Functions+of+exponential+type&rft.jtitle=Ann.+of+Math.&rft.aulast=Stein&rft.aufirst=Elias+M.&rft.date=1957&rft.volume=65&rft.pages=582%E2%80%93592&rft_id=info%3Adoi%2F10.2307%2F1970066&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AType+exponentiel"></span></span></span></li></ul> <ul id="bandeau-portail" class="bandeau-portail"><li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer" typeof="mw:File"><a href="/wiki/Portail:Analyse" title="Portail de l'analyse"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Nuvola_apps_kmplot.svg/24px-Nuvola_apps_kmplot.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Nuvola_apps_kmplot.svg/36px-Nuvola_apps_kmplot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Nuvola_apps_kmplot.svg/48px-Nuvola_apps_kmplot.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span></span> <span class="bandeau-portail-texte"><a href="/wiki/Portail:Analyse" title="Portail:Analyse">Portail de l'analyse</a></span> </span></li> </ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Ce document provient de « <a dir="ltr" href="https://fr.wikipedia.org/w/index.php?title=Type_exponentiel&oldid=191576321">https://fr.wikipedia.org/w/index.php?title=Type_exponentiel&oldid=191576321</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">Catégorie</a> : <ul><li><a href="/wiki/Cat%C3%A9gorie:Analyse_complexe" title="Catégorie:Analyse complexe">Analyse complexe</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><li><a href="/wiki/Cat%C3%A9gorie:Page_qui_utilise_un_format_obsol%C3%A8te_des_balises_math%C3%A9matiques" title="Catégorie:Page qui utilise un format obsolète des balises mathématiques">Page qui utilise un format obsolète des balises mathématiques</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 3 mars 2022 à 16:04.</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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