Resíduo (análise complexa) – Wikipédia, a enciclopédia livre

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="pt" dir="ltr"> <head> <meta charset="UTF-8"> <title>Resíduo (análise complexa) – Wikipédia, a enciclopédia livre</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )ptwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":[",\t."," \t,"],"wgDigitTransformTable":["",""], "wgDefaultDateFormat":"dmy","wgMonthNames":["","janeiro","fevereiro","março","abril","maio","junho","julho","agosto","setembro","outubro","novembro","dezembro"],"wgRequestId":"0eb653d2-28ec-4405-b6f6-dcbc2ce71c0f","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Resíduo_(análise_complexa)","wgTitle":"Resíduo (análise complexa)","wgCurRevisionId":67550386,"wgRevisionId":67550386,"wgArticleId":1421347,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["!Artigos que carecem de notas de rodapé desde janeiro de 2016","!Artigos que carecem de notas de rodapé sem indicação de tema","Análise complexa","Números complexos"],"wgPageViewLanguage":"pt","wgPageContentLanguage":"pt","wgPageContentModel":"wikitext","wgRelevantPageName":"Resíduo_(análise_complexa)","wgRelevantArticleId":1421347,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit": [],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"pt","pageLanguageDir":"ltr","pageVariantFallbacks":"pt"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":true,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":4000,"wgRelatedArticlesCompat":[],"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q753375","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":true,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false,"wgSiteNoticeId": "2.30"};RLSTATE={"ext.gadget.FeedbackHighlight-base":"ready","ext.gadget.keepPDU":"ready","ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready","ext.dismissableSiteNotice.styles":"ready"};RLPAGEMODULES=["mediawiki.page.media","site","mediawiki.page.ready","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.Topicon","ext.gadget.Metacaixa","ext.gadget.TitleRewrite","ext.gadget.ElementosOcultaveis","ext.gadget.FeedbackHighlight","ext.gadget.ReferenceTooltips","ext.gadget.NewVillagePump","ext.gadget.wikibugs","ext.gadget.charinsert", "ext.gadget.requestForAdminship","ext.gadget.WikiMiniAtlas","ext.gadget.PagesForDeletion","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.quicksurveys.init","ext.growthExperiments.SuggestedEditSession","oojs-ui.styles.icons-media","oojs-ui-core.icons","ext.dismissableSiteNotice"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=pt&modules=ext.dismissableSiteNotice.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=pt&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=pt&modules=ext.gadget.FeedbackHighlight-base%2CkeepPDU&only=styles&skin=vector-2022"> <link rel="stylesheet" href="/w/load.php?lang=pt&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.6"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/3/33/Gamma_abs_3D.png/1200px-Gamma_abs_3D.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="938"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/3/33/Gamma_abs_3D.png/800px-Gamma_abs_3D.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="625"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/3/33/Gamma_abs_3D.png/640px-Gamma_abs_3D.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="500"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Resíduo (análise complexa) – Wikipédia, a enciclopédia livre"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//pt.m.wikipedia.org/wiki/Res%C3%ADduo_(an%C3%A1lise_complexa)"> <link rel="alternate" type="application/x-wiki" title="Editar" href="/w/index.php?title=Res%C3%ADduo_(an%C3%A1lise_complexa)&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipédia (pt)"> <link rel="EditURI" type="application/rsd+xml" href="//pt.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://pt.wikipedia.org/wiki/Res%C3%ADduo_(an%C3%A1lise_complexa)"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.pt"> <link rel="alternate" type="application/atom+xml" title="''Feed'' Atom Wikipédia" href="/w/index.php?title=Especial:Mudan%C3%A7as_recentes&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Resíduo_análise_complexa rootpage-Resíduo_análise_complexa skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Saltar para o conteúdo</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="''Site''"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Menu principal" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-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-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Menu principal</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Menu principal</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">mover para a barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">ocultar</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navegação </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Wikip%C3%A9dia:P%C3%A1gina_principal" title="Visitar a página principal [z]" accesskey="z"><span>Página principal</span></a></li><li id="n-featuredcontent" class="mw-list-item"><a href="/wiki/Portal:Conte%C3%BAdo_destacado"><span>Conteúdo destacado</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Eventos_atuais" title="Informação temática sobre eventos atuais"><span>Eventos atuais</span></a></li><li id="n-villagepump" class="mw-list-item"><a href="/wiki/Wikip%C3%A9dia:Esplanada"><span>Esplanada</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Especial:Aleat%C3%B3ria" title="Carregar página aleatória [x]" accesskey="x"><span>Página aleatória</span></a></li><li id="n-portals" class="mw-list-item"><a href="/wiki/Portal:%C3%8Dndice"><span>Portais</span></a></li><li id="n-bug_in_article" class="mw-list-item"><a href="/wiki/Wikip%C3%A9dia:Informe_um_erro"><span>Informar um erro</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Colaboração </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-welcome" class="mw-list-item"><a href="/wiki/Wikip%C3%A9dia:Boas-vindas"><span>Boas-vindas</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/Ajuda:P%C3%A1gina_principal" title="Um local reservado para auxílio."><span>Ajuda</span></a></li><li id="n-Páginas-de-testes-públicas" class="mw-list-item"><a href="/wiki/Ajuda:P%C3%A1gina_de_testes"><span>Páginas de testes públicas</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikip%C3%A9dia:Portal_comunit%C3%A1rio" title="Sobre o projeto"><span>Portal comunitário</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Especial:Mudan%C3%A7as_recentes" title="Uma lista de mudanças recentes nesta wiki [r]" accesskey="r"><span>Mudanças recentes</span></a></li><li id="n-maintenance" class="mw-list-item"><a href="/wiki/Wikip%C3%A9dia:Manuten%C3%A7%C3%A3o"><span>Manutenção</span></a></li><li id="n-createpage" class="mw-list-item"><a href="/wiki/Ajuda:Guia_de_edi%C3%A7%C3%A3o/Como_come%C3%A7ar_uma_p%C3%A1gina"><span>Criar página</span></a></li><li id="n-newpages-description" class="mw-list-item"><a href="/wiki/Especial:P%C3%A1ginas_novas"><span>Páginas novas</span></a></li><li id="n-contact-description" class="mw-list-item"><a href="/wiki/Wikip%C3%A9dia:Contato"><span>Contato</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Wikip%C3%A9dia:P%C3%A1gina_principal" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipédia" src="/static/images/mobile/copyright/wikipedia-wordmark-fr.svg" style="width: 7.4375em; height: 1.125em;"> <img class="mw-logo-tagline" alt="" src="/static/images/mobile/copyright/wikipedia-tagline-pt.svg" width="120" height="13" style="width: 7.5em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Especial:Pesquisar" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Pesquisar na Wikipédia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Busca</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Pesquisar na Wikipédia" aria-label="Pesquisar na Wikipédia" autocapitalize="sentences" title="Pesquisar na Wikipédia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Especial:Pesquisar"> </div> <button class="cdx-button cdx-search-input__end-button">Pesquisar</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Ferramentas pessoais"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Aspeto"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Aspeto" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Aspeto</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=pt.wikipedia.org&uselang=pt" class=""><span>Donativos</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Especial:Criar_conta&returnto=Res%C3%ADduo+%28an%C3%A1lise+complexa%29" title="É encorajado a criar uma conta e iniciar sessão; no entanto, não é obrigatório" class=""><span>Criar uma conta</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Especial:Entrar&returnto=Res%C3%ADduo+%28an%C3%A1lise+complexa%29" title="Aconselhamos-lhe a criar uma conta na Wikipédia, embora tal não seja obrigatório. [o]" accesskey="o" class=""><span>Entrar</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Mais opções" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Ferramentas pessoais" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Ferramentas pessoais</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Menu do utilizador" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=pt.wikipedia.org&uselang=pt"><span>Donativos</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Especial:Criar_conta&returnto=Res%C3%ADduo+%28an%C3%A1lise+complexa%29" title="É encorajado a criar uma conta e iniciar sessão; no entanto, não é obrigatório"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Criar uma conta</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Especial:Entrar&returnto=Res%C3%ADduo+%28an%C3%A1lise+complexa%29" title="Aconselhamos-lhe a criar uma conta na Wikipédia, embora tal não seja obrigatório. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Entrar</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Páginas para editores sem sessão iniciada <a href="/wiki/Ajuda:Introduction" aria-label="Saiba mais sobre edição"><span>saber mais</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Especial:Minhas_contribui%C3%A7%C3%B5es" title="Uma lista de edições feitas a partir deste endereço IP [y]" accesskey="y"><span>Contribuições</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Especial:Minha_discuss%C3%A3o" title="Discussão sobre edições feitas a partir deste endereço IP [n]" accesskey="n"><span>Discussão</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"><div id="mw-dismissablenotice-anonplace"></div><script>(function(){var node=document.getElementById("mw-dismissablenotice-anonplace");if(node){node.outerHTML="\u003Cdiv class=\"mw-dismissable-notice\"\u003E\u003Cdiv class=\"mw-dismissable-notice-close\"\u003E[\u003Ca tabindex=\"0\" role=\"button\"\u003Eocultar\u003C/a\u003E]\u003C/div\u003E\u003Cdiv class=\"mw-dismissable-notice-body\"\u003E\u003C!-- CentralNotice --\u003E\u003Cdiv id=\"localNotice\" data-nosnippet=\"\"\u003E\u003Cdiv class=\"anonnotice\" lang=\"pt\" dir=\"ltr\"\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E";}}());</script></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="''Site''"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Conteúdo" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Conteúdo</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mover para a barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ocultar</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Início</div> </a> </li> <li id="toc-Motivação" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Motivação"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Motivação</span> </div> </a> <ul id="toc-Motivação-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definição" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definição"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Definição</span> </div> </a> <ul id="toc-Definição-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ver_também" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ver_também"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Ver também</span> </div> </a> <ul id="toc-Ver_também-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ligações_externas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ligações_externas"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Ligações externas</span> </div> </a> <ul id="toc-Ligações_externas-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="Conteúdo" 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="Alternar o índice" > <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">Alternar o índice</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">Resíduo (análise complexa)</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Ir para um artigo noutra língua. Disponível em 24 línguas" > <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-24" 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">24 línguas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B1%D8%A7%D8%B3%D8%A8_(%D8%AA%D8%AD%D9%84%D9%8A%D9%84_%D9%85%D8%B1%D9%83%D8%A8)" title="راسب (تحليل مركب) — árabe" lang="ar" hreflang="ar" data-title="راسب (تحليل مركب)" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Residu_(an%C3%A0lisi_complexa)" title="Residu (anàlisi complexa) — catalão" lang="ca" hreflang="ca" data-title="Residu (anàlisi complexa)" data-language-autonym="Català" data-language-local-name="catalão" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Reziduum_(matematika)" title="Reziduum (matematika) — checo" lang="cs" hreflang="cs" data-title="Reziduum (matematika)" data-language-autonym="Čeština" data-language-local-name="checo" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Residuum_(Funktionentheorie)" title="Residuum (Funktionentheorie) — alemão" lang="de" hreflang="de" data-title="Residuum (Funktionentheorie)" data-language-autonym="Deutsch" data-language-local-name="alemão" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Residue_(complex_analysis)" title="Residue (complex analysis) — inglês" lang="en" hreflang="en" data-title="Residue (complex analysis)" data-language-autonym="English" data-language-local-name="inglês" 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/Residuo_(an%C3%A1lisis_complejo)" title="Residuo (análisis complejo) — espanhol" lang="es" hreflang="es" data-title="Residuo (análisis complejo)" data-language-autonym="Español" data-language-local-name="espanhol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%A7%D9%86%D8%AF%D9%87_(%D8%A2%D9%86%D8%A7%D9%84%DB%8C%D8%B2_%D9%85%D8%AE%D8%AA%D9%84%D8%B7)" title="مانده (آنالیز مختلط) — persa" lang="fa" hreflang="fa" data-title="مانده (آنالیز مختلط)" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Residy" title="Residy — finlandês" lang="fi" hreflang="fi" data-title="Residy" data-language-autonym="Suomi" data-language-local-name="finlandês" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/R%C3%A9sidu_(analyse_complexe)" title="Résidu (analyse complexe) — francês" lang="fr" hreflang="fr" data-title="Résidu (analyse complexe)" data-language-autonym="Français" data-language-local-name="francês" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A9%D7%90%D7%A8%D7%99%D7%AA_(%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%9E%D7%A8%D7%95%D7%9B%D7%91%D7%AA)" title="שארית (אנליזה מרוכבת) — hebraico" lang="he" hreflang="he" data-title="שארית (אנליזה מרוכבת)" data-language-autonym="עברית" data-language-local-name="hebraico" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Reziduum" title="Reziduum — húngaro" lang="hu" hreflang="hu" data-title="Reziduum" data-language-autonym="Magyar" data-language-local-name="húngaro" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%B6%D5%A1%D6%81%D6%84_(%D5%AF%D5%B8%D5%B4%D5%BA%D5%AC%D5%A5%D6%84%D5%BD_%D5%A1%D5%B6%D5%A1%D5%AC%D5%AB%D5%A6)" title="Մնացք (կոմպլեքս անալիզ) — arménio" lang="hy" hreflang="hy" data-title="Մնացք (կոմպլեքս անալիզ)" data-language-autonym="Հայերեն" data-language-local-name="arménio" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Residuo_(analisi_complessa)" title="Residuo (analisi complessa) — italiano" lang="it" hreflang="it" data-title="Residuo (analisi complessa)" data-language-autonym="Italiano" data-language-local-name="italiano" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%95%99%E6%95%B0" title="留数 — japonês" lang="ja" hreflang="ja" data-title="留数" data-language-autonym="日本語" data-language-local-name="japonês" 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/%EC%9C%A0%EC%88%98_(%EB%B3%B5%EC%86%8C%ED%95%B4%EC%84%9D%ED%95%99)" title="유수 (복소해석학) — coreano" lang="ko" hreflang="ko" data-title="유수 (복소해석학)" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Residu_(functietheorie)" title="Residu (functietheorie) — neerlandês" lang="nl" hreflang="nl" data-title="Residu (functietheorie)" data-language-autonym="Nederlands" data-language-local-name="neerlandês" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Residuum_funkcji_holomorficznej" title="Residuum funkcji holomorficznej — polaco" lang="pl" hreflang="pl" data-title="Residuum funkcji holomorficznej" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%92%D1%8B%D1%87%D0%B5%D1%82_(%D0%BA%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D1%8B%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7)" title="Вычет (комплексный анализ) — russo" lang="ru" hreflang="ru" data-title="Вычет (комплексный анализ)" data-language-autonym="Русский" data-language-local-name="russo" 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/Residy" title="Residy — sueco" lang="sv" hreflang="sv" data-title="Residy" data-language-autonym="Svenska" data-language-local-name="sueco" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kal%C4%B1nt%C4%B1_(karma%C5%9F%C4%B1k_analiz)" title="Kalıntı (karmaşık analiz) — turco" lang="tr" hreflang="tr" data-title="Kalıntı (karmaşık analiz)" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%A7%D0%B8%D0%B3%D0%B5%D1%80%D0%B5%D1%88_(%D0%BA%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7)" title="Чигереш (комплекс анализ) — tatar" lang="tt" hreflang="tt" data-title="Чигереш (комплекс анализ)" data-language-autonym="Татарча / tatarça" data-language-local-name="tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9B%D0%B8%D1%88%D0%BE%D0%BA" title="Лишок — ucraniano" lang="uk" hreflang="uk" data-title="Лишок" data-language-autonym="Українська" data-language-local-name="ucraniano" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Th%E1%BA%B7ng_d%C6%B0_(gi%E1%BA%A3i_t%C3%ADch_ph%E1%BB%A9c)" title="Thặng dư (giải tích phức) — vietnamita" lang="vi" hreflang="vi" data-title="Thặng dư (giải tích phức)" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%95%99%E6%95%B0" title="留数 — chinês" lang="zh" hreflang="zh" data-title="留数" data-language-autonym="中文" data-language-local-name="chinês" 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/Q753375#sitelinks-wikipedia" title="Editar hiperligações interlínguas" class="wbc-editpage">Editar hiperligações</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="Espaços nominais"> <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/Res%C3%ADduo_(an%C3%A1lise_complexa)" title="Ver a página de conteúdo [c]" accesskey="c"><span>Artigo</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Discuss%C3%A3o:Res%C3%ADduo_(an%C3%A1lise_complexa)&action=edit&redlink=1" rel="discussion" class="new" title="Discussão sobre o conteúdo da página (página não existe) [t]" accesskey="t"><span>Discussão</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Mudar a variante da língua" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">português</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Vistas"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Res%C3%ADduo_(an%C3%A1lise_complexa)"><span>Ler</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Res%C3%ADduo_(an%C3%A1lise_complexa)&veaction=edit" title="Editar esta página [v]" accesskey="v"><span>Editar</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Res%C3%ADduo_(an%C3%A1lise_complexa)&action=edit" title="Editar o código-fonte desta página [e]" accesskey="e"><span>Editar código-fonte</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Res%C3%ADduo_(an%C3%A1lise_complexa)&action=history" title="Edições anteriores desta página. [h]" accesskey="h"><span>Ver histórico</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Ferramentas de página"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Ferramentas" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Ferramentas</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Ferramentas</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">mover para a barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">ocultar</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Mais opções" > <div class="vector-menu-heading"> Operações </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/Res%C3%ADduo_(an%C3%A1lise_complexa)"><span>Ler</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Res%C3%ADduo_(an%C3%A1lise_complexa)&veaction=edit" title="Editar esta página [v]" accesskey="v"><span>Editar</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Res%C3%ADduo_(an%C3%A1lise_complexa)&action=edit" title="Editar o código-fonte desta página [e]" accesskey="e"><span>Editar código-fonte</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Res%C3%ADduo_(an%C3%A1lise_complexa)&action=history"><span>Ver histórico</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Geral </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Especial:P%C3%A1ginas_afluentes/Res%C3%ADduo_(an%C3%A1lise_complexa)" title="Lista de todas as páginas que contêm hiperligações para esta [j]" accesskey="j"><span>Páginas afluentes</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Especial:Altera%C3%A7%C3%B5es_relacionadas/Res%C3%ADduo_(an%C3%A1lise_complexa)" rel="nofollow" title="Mudanças recentes nas páginas para as quais esta contém hiperligações [k]" accesskey="k"><span>Alterações relacionadas</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:Carregar_ficheiro" title="Carregar ficheiros [u]" accesskey="u"><span>Carregar ficheiro</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Especial:P%C3%A1ginas_especiais" title="Lista de páginas especiais [q]" accesskey="q"><span>Páginas especiais</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Res%C3%ADduo_(an%C3%A1lise_complexa)&oldid=67550386" title="Hiperligação permanente para esta revisão desta página"><span>Hiperligação permanente</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Res%C3%ADduo_(an%C3%A1lise_complexa)&action=info" title="Mais informações sobre esta página"><span>Informações da página</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Especial:Citar&page=Res%C3%ADduo_%28an%C3%A1lise_complexa%29&id=67550386&wpFormIdentifier=titleform" title="Informação sobre como citar esta página"><span>Citar esta página</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Especial:UrlQ%C4%B1sald%C4%B1c%C4%B1s%C4%B1&url=https%3A%2F%2Fpt.wikipedia.org%2Fwiki%2FRes%25C3%25ADduo_%28an%25C3%25A1lise_complexa%29"><span>Obter URL encurtado</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Especial:QrKodu&url=https%3A%2F%2Fpt.wikipedia.org%2Fwiki%2FRes%25C3%25ADduo_%28an%25C3%25A1lise_complexa%29"><span>Descarregar código 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"> Imprimir/exportar </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=Especial:Livro&bookcmd=book_creator&referer=Res%C3%ADduo+%28an%C3%A1lise+complexa%29"><span>Criar um livro</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Especial:DownloadAsPdf&page=Res%C3%ADduo_%28an%C3%A1lise_complexa%29&action=show-download-screen"><span>Descarregar como PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Res%C3%ADduo_(an%C3%A1lise_complexa)&printable=yes" title="Versão para impressão desta página [p]" accesskey="p"><span>Versão para impressão</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"> Noutros projetos </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/Q753375" title="Hiperligação para o elemento do repositório de dados [g]" accesskey="g"><span>Elemento 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="Ferramentas de página"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Aspeto"> <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">Aspeto</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mover para a barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ocultar</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">Origem: Wikipédia, a enciclopédia livre.</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="pt" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r68971778">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}.mw-parser-output .cmbox{margin:3px 0;border-collapse:collapse;border:1px solid #a2a9b1;background-color:#dfe8ff;box-sizing:border-box;color:var(--color-base)}.mw-parser-output .cmbox-speedy{border:4px solid #b32424;background-color:#ffdbdb}.mw-parser-output .cmbox-delete{background-color:#ffdbdb}.mw-parser-output .cmbox-content{background-color:#ffe7ce}.mw-parser-output .cmbox-style{background-color:#fff9db}.mw-parser-output .cmbox-move{background-color:#e4d8ff}.mw-parser-output .cmbox-protection{background-color:#efefe1}.mw-parser-output .cmbox .mbox-text{border:none;padding:0.25em 0.9em;width:100%}.mw-parser-output .cmbox .mbox-image{border:none;padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .cmbox .mbox-imageright{border:none;padding:2px 0.9em 2px 0;text-align:center}.mw-parser-output .cmbox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .cmbox .mbox-invalid-type{text-align:center}@media(min-width:720px){.mw-parser-output .cmbox{margin:3px 10%}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .cmbox{background-color:#0d1a27}html.skin-theme-clientpref-night .mw-parser-output .cmbox-speedy,html.skin-theme-clientpref-night .mw-parser-output .cmbox-delete{background-color:#300}html.skin-theme-clientpref-night .mw-parser-output .cmbox-content{background-color:#331a00}html.skin-theme-clientpref-night .mw-parser-output .cmbox-style{background-color:#332b00}html.skin-theme-clientpref-night .mw-parser-output .cmbox-move{background-color:#08001a}html.skin-theme-clientpref-night .mw-parser-output .cmbox-protection{background-color:#212112}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cmbox{background-color:#0d1a27}html.skin-theme-clientpref-os .mw-parser-output .cmbox-speedy,html.skin-theme-clientpref-os .mw-parser-output .cmbox-delete{background-color:#300}html.skin-theme-clientpref-os .mw-parser-output .cmbox-content{background-color:#331a00}html.skin-theme-clientpref-os .mw-parser-output .cmbox-style{background-color:#332b00}html.skin-theme-clientpref-os .mw-parser-output .cmbox-move{background-color:#08001a}html.skin-theme-clientpref-os .mw-parser-output .cmbox-protection{background-color:#212112}}.mw-parser-output .fmbox{clear:both;margin:0.2em 0;width:100%;border:1px solid #a2a9b1;background-color:var(--background-color-interactive-subtle,#f8f9fa);box-sizing:border-box;color:var(--color-base,#202122)}.mw-parser-output .fmbox-warning{border:1px solid #bb7070;background-color:#ffdbdb}.mw-parser-output .fmbox-editnotice{background-color:transparent}.mw-parser-output .fmbox .mbox-text{border:none;padding:0.25em 0.9em;width:100%}.mw-parser-output .fmbox .mbox-image{border:none;padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .fmbox .mbox-imageright{border:none;padding:2px 0.9em 2px 0;text-align:center}.mw-parser-output .fmbox .mbox-invalid-type{text-align:center}@media screen{html.skin-theme-clientpref-night .mw-parser-output .fmbox-warning{background-color:#683131}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .fmbox-warning{background-color:#683131}}.mw-parser-output .imbox{margin:4px 0;border-collapse:collapse;border:3px solid #36c;background-color:var(--background-color-interactive-subtle,#f8f9fa);box-sizing:border-box}.mw-parser-output .imbox .mbox-text .imbox{margin:0 -0.5em;display:block}.mw-parser-output .imbox-speedy{border:3px solid #b32424;background-color:#fee7e6}.mw-parser-output .imbox-delete{border:3px solid #b32424}.mw-parser-output .imbox-content{border:3px solid #f28500}.mw-parser-output .imbox-style{border:3px solid #fc3}.mw-parser-output .imbox-move{border:3px solid #9932cc}.mw-parser-output .imbox-protection{border:3px solid #a2a9b1}.mw-parser-output .imbox-license{border:3px solid #88a}.mw-parser-output .imbox-featured{border:3px solid #cba135}.mw-parser-output .imbox .mbox-text{border:none;padding:0.25em 0.9em;width:100%}.mw-parser-output .imbox .mbox-image{border:none;padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .imbox .mbox-imageright{border:none;padding:2px 0.9em 2px 0;text-align:center}.mw-parser-output .imbox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .imbox .mbox-invalid-type{text-align:center}@media(min-width:720px){.mw-parser-output .imbox{margin:4px 10%}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .imbox-speedy{background-color:#310402}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .imbox-speedy{background-color:#310402}}.mw-parser-output .ombox{margin:4px 0;border-collapse:collapse;background-color:var(--background-color-neutral-subtle,#f8f9fa);box-sizing:border-box;border:1px solid #a2a9b1;color:var(--color-base,#202122)}.mw-parser-output .ombox.mbox-small{font-size:88%;line-height:1.25em}.mw-parser-output .ombox-speedy{border:2px solid #b32424;background-color:#fee7e6}.mw-parser-output .ombox-delete{border:2px solid #b32424}.mw-parser-output .ombox-content{border:1px solid #f28500}.mw-parser-output .ombox-style{border:1px solid #fc3}.mw-parser-output .ombox-move{border:1px solid #9932cc}.mw-parser-output .ombox-protection{border:2px solid #a2a9b1}.mw-parser-output .ombox .mbox-text{border:none;padding:0.25em 0.9em;width:100%}.mw-parser-output .ombox .mbox-image{border:none;padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .ombox .mbox-imageright{border:none;padding:2px 0.9em 2px 0;text-align:center}.mw-parser-output .ombox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ombox .mbox-invalid-type{text-align:center}@media(min-width:720px){.mw-parser-output .ombox{margin:4px 10%}.mw-parser-output .ombox.mbox-small{clear:right;float:right;margin:4px 0 4px 1em;width:238px}}body.skin--responsive .mw-parser-output table.ombox img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .ombox-speedy{background-color:#310402}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .ombox-speedy{background-color:#310402}}.mw-parser-output .tmbox{margin:4px 0;border-collapse:collapse;border:1px solid #c0c090;background-color:#f8eaba;box-sizing:border-box}.mw-parser-output .tmbox.mbox-small{font-size:88%;line-height:1.25em}.mw-parser-output .tmbox-speedy{border:2px solid #b32424;background-color:#fee7e6}.mw-parser-output .tmbox-delete{border:2px solid #b32424}.mw-parser-output .tmbox-content{border:1px solid #c0c090}.mw-parser-output .tmbox-style{border:2px solid #fc3}.mw-parser-output .tmbox-move{border:2px solid #9932cc}.mw-parser-output .tmbox .mbox-text{border:none;padding:0.25em 0.9em;width:100%}.mw-parser-output .tmbox .mbox-image{border:none;padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .tmbox .mbox-imageright{border:none;padding:2px 0.9em 2px 0;text-align:center}.mw-parser-output .tmbox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .tmbox .mbox-invalid-type{text-align:center}@media(min-width:720px){.mw-parser-output .tmbox{margin:4px 10%}.mw-parser-output .tmbox.mbox-small{clear:right;float:right;margin:4px 0 4px 1em;width:238px}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmbox{background-color:#2e2505}html.skin-theme-clientpref-night .mw-parser-output .tmbox-speedy{background-color:#310402}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmbox{background-color:#2e2505}html.skin-theme-clientpref-os .mw-parser-output .tmbox-speedy{background-color:#310402}}body.skin--responsive .mw-parser-output table.tmbox img{max-width:none!important}</style><table class="box-Sem_notas plainlinks metadata ambox ambox-content ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span typeof="mw:File"><a href="/wiki/Wikip%C3%A9dia:Livro_de_estilo/Cite_as_fontes" title="Wikipédia:Livro de estilo/Cite as fontes"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">Este artigo ou secção contém <a href="/wiki/Wikip%C3%A9dia:Livro_de_estilo/Cite_as_fontes#Referências_no_fim" title="Wikipédia:Livro de estilo/Cite as fontes">uma lista de referências no fim do texto</a>, mas as suas fontes não são claras porque <b>não são <a href="/wiki/Wikip%C3%A9dia:Livro_de_estilo/Cite_as_fontes#Referências_no_texto" title="Wikipédia:Livro de estilo/Cite as fontes">citadas no corpo do artigo</a></b>, o que <b>compromete a <a href="/wiki/Wikip%C3%A9dia:V" class="mw-redirect" title="Wikipédia:V">confiabilidade</a></b> das informações.<span class="hide-when-compact"> Ajude a <span class="plainlinks"><a class="external text" href="https://pt.wikipedia.org/w/index.php?title=Res%C3%ADduo_(an%C3%A1lise_complexa)&action=edit">melhorar</a></span> este artigo <a href="/wiki/Wikip%C3%A9dia:Livro_de_estilo/Refer%C3%AAncias_e_notas_de_rodap%C3%A9" title="Wikipédia:Livro de estilo/Referências e notas de rodapé">inserindo citações no corpo do artigo</a>.</span> <small class="date-container"><i>(<span class="date">Janeiro de 2016</span>)</i></small></div></td></tr></tbody></table> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Gamma_abs_3D.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Gamma_abs_3D.png/220px-Gamma_abs_3D.png" decoding="async" width="220" height="172" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Gamma_abs_3D.png/330px-Gamma_abs_3D.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Gamma_abs_3D.png/440px-Gamma_abs_3D.png 2x" data-file-width="1280" data-file-height="1000" /></a><figcaption>Gráfico tridimensional do valor absoluto da função gama complexa</figcaption></figure> <p>Em <a href="/wiki/An%C3%A1lise_complexa" title="Análise complexa">análise complexa</a>, o <b>resíduo</b> de uma <a href="/wiki/Fun%C3%A7%C3%A3o_anal%C3%ADtica" title="Função analítica">função analítica</a> <i>f</i> numa <a href="/wiki/Singularidade_matem%C3%A1tica" title="Singularidade matemática">singularidade</a> <i>p</i> é um <a href="/wiki/N%C3%BAmero_complexo" title="Número complexo">número complexo</a> que permite calcular o valor de um <a href="/wiki/Integral_de_linha" title="Integral de linha">integral de linha</a> de <i>f</i> cuja imagem esteja na vizinhança de <i>p</i>. Há métodos simples de cálculo de resíduos e, por outro lado, o conhecimento dos resíduos de <i>f</i> permite calcular integrais de <i>f</i> ao longo de <a href="/wiki/Lacete" title="Lacete">lacetes</a> arbitrários, através do <a href="/wiki/Teorema_dos_res%C3%ADduos" title="Teorema dos resíduos">teorema dos resíduos</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Motivação"><span id="Motiva.C3.A7.C3.A3o"></span>Motivação</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Res%C3%ADduo_(an%C3%A1lise_complexa)&veaction=edit&section=1" title="Editar secção: Motivação" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Res%C3%ADduo_(an%C3%A1lise_complexa)&action=edit&section=1" title="Editar código-fonte da secção: Motivação"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Como exemplo, considere a integral de contorno </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 \oint _{C}{e^{z} \over z^{5}}\,dz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{C}{e^{z} \over z^{5}}\,dz}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7a77dad0024b956391a788fe86b39e0676d169f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.833ex; height:5.676ex;" alt="{\displaystyle \oint _{C}{e^{z} \over z^{5}}\,dz}"></span></dd></dl> <p>onde <i>C</i> é uma <a href="/wiki/Curva_de_Jordan" class="mw-redirect" title="Curva de Jordan">curva de Jordan</a> em torno de 0. </p><p>Agora calculamos essa integral utilizando os teoremas padrões de integral disponíveis. Assim, a <a href="/wiki/S%C3%A9rie_de_Taylor" title="Série de Taylor">série de Taylor</a> para <i>e</i><sup><i>z</i></sup> é conhecida, e podemos substituir esta série no integrando. A integral passa a ser </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 \oint _{C}{1 \over z^{5}}\left(1+z+{z^{2} \over 2!}+{z^{3} \over 3!}+{z^{4} \over 4!}+{z^{5} \over 5!}+{z^{6} \over 6!}+\cdots \right)\,dz.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>z</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mrow> <mn>5</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mrow> <mn>6</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{C}{1 \over z^{5}}\left(1+z+{z^{2} \over 2!}+{z^{3} \over 3!}+{z^{4} \over 4!}+{z^{5} \over 5!}+{z^{6} \over 6!}+\cdots \right)\,dz.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0000b5b221ac208372b7af248080a6ac0f28e9d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:53.435ex; height:6.343ex;" alt="{\displaystyle \oint _{C}{1 \over z^{5}}\left(1+z+{z^{2} \over 2!}+{z^{3} \over 3!}+{z^{4} \over 4!}+{z^{5} \over 5!}+{z^{6} \over 6!}+\cdots \right)\,dz.}"></span></dd></dl> <p>Trazendo o termo 1/<i>z</i><sup>5</sup> para dentro da série, obtemos </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 \oint _{C}\left({1 \over z^{5}}+{z \over z^{5}}+{z^{2} \over 2!\;z^{5}}+{z^{3} \over 3!\;z^{5}}+{z^{4} \over 4!\;z^{5}}+{z^{5} \over 5!\;z^{5}}+{z^{6} \over 6!\;z^{5}}+\cdots \right)\,dz=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo>!</mo> <mspace width="thickmathspace" /> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mn>3</mn> <mo>!</mo> <mspace width="thickmathspace" /> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow> <mn>4</mn> <mo>!</mo> <mspace width="thickmathspace" /> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mrow> <mn>5</mn> <mo>!</mo> <mspace width="thickmathspace" /> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mrow> <mn>6</mn> <mo>!</mo> <mspace width="thickmathspace" /> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{C}\left({1 \over z^{5}}+{z \over z^{5}}+{z^{2} \over 2!\;z^{5}}+{z^{3} \over 3!\;z^{5}}+{z^{4} \over 4!\;z^{5}}+{z^{5} \over 5!\;z^{5}}+{z^{6} \over 6!\;z^{5}}+\cdots \right)\,dz=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59c88dc96e41f597d11d475c83ff0bf4d1e30177" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:67.857ex; height:6.343ex;" alt="{\displaystyle \oint _{C}\left({1 \over z^{5}}+{z \over z^{5}}+{z^{2} \over 2!\;z^{5}}+{z^{3} \over 3!\;z^{5}}+{z^{4} \over 4!\;z^{5}}+{z^{5} \over 5!\;z^{5}}+{z^{6} \over 6!\;z^{5}}+\cdots \right)\,dz=}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{C}\left({1 \over \;z^{5}}+{1 \over \;z^{4}}+{1 \over 2!\;z^{3}}+{1 \over 3!\;z^{2}}+{1 \over 4!\;z}+{1 \over \;5!}+{z \over 6!}+\cdots \right)\,dz.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thickmathspace" /> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thickmathspace" /> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>!</mo> <mspace width="thickmathspace" /> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mo>!</mo> <mspace width="thickmathspace" /> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mo>!</mo> <mspace width="thickmathspace" /> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thickmathspace" /> <mn>5</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mrow> <mn>6</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{C}\left({1 \over \;z^{5}}+{1 \over \;z^{4}}+{1 \over 2!\;z^{3}}+{1 \over 3!\;z^{2}}+{1 \over 4!\;z}+{1 \over \;5!}+{z \over 6!}+\cdots \right)\,dz.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1076c0e528fa62a3217f720006c194ffb778468b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:61.35ex; height:6.176ex;" alt="{\displaystyle \oint _{C}\left({1 \over \;z^{5}}+{1 \over \;z^{4}}+{1 \over 2!\;z^{3}}+{1 \over 3!\;z^{2}}+{1 \over 4!\;z}+{1 \over \;5!}+{z \over 6!}+\cdots \right)\,dz.}"></span></dd></dl> <p>A integral agora toma uma forma muito mais simples. Lembre-se 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 \oint _{C}{1 \over z^{a}}\,dz=0,\quad a\in \mathbb {Z} ,{\mbox{ para }}a\neq 1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mi>a</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> para </mtext> </mstyle> </mrow> <mi>a</mi> <mo>≠</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{C}{1 \over z^{a}}\,dz=0,\quad a\in \mathbb {Z} ,{\mbox{ para }}a\neq 1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2953e35a7281e0294bbc3aa9c9f43dc39320f4ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.98ex; height:5.676ex;" alt="{\displaystyle \oint _{C}{1 \over z^{a}}\,dz=0,\quad a\in \mathbb {Z} ,{\mbox{ para }}a\neq 1.}"></span></dd></dl> <p>Então, a integral em torno de <i>C</i> de todos os termos que não estão na forma <i>cz</i><sup>−1</sup> são iguais a zero e a integral é reduzida 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 \oint _{C}{1 \over 4!\;z}\,dz={1 \over 4!}\oint _{C}{1 \over z}\,dz={1 \over 4!}(2\pi i)={\pi i \over 12}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mo>!</mo> <mspace width="thickmathspace" /> <mi>z</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <mi>i</mi> </mrow> <mn>12</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{C}{1 \over 4!\;z}\,dz={1 \over 4!}\oint _{C}{1 \over z}\,dz={1 \over 4!}(2\pi i)={\pi i \over 12}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43c1bebaae80a9bc3ac0d03794abb6fba18a11c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:41.969ex; height:5.676ex;" alt="{\displaystyle \oint _{C}{1 \over 4!\;z}\,dz={1 \over 4!}\oint _{C}{1 \over z}\,dz={1 \over 4!}(2\pi i)={\pi i \over 12}.}"></span></dd></dl> <p>O valor 1/4! é conhecido como o <i>resíduo</i> de <i>e</i><sup><i>z</i></sup>/<i>z</i><sup>5</sup> em <i>z</i> = 0, e denotado como </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Res} (f,0),\mathrm {ou} \ \mathrm {Res} _{0}{e^{z} \over z^{5}}\mathrm {ou} \ \mathrm {Res} _{z=0}{e^{z} \over z^{5}},.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> </mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">u</mi> </mrow> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">u</mi> </mrow> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Res} (f,0),\mathrm {ou} \ \mathrm {Res} _{0}{e^{z} \over z^{5}}\mathrm {ou} \ \mathrm {Res} _{z=0}{e^{z} \over z^{5}},.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3efd6767b35bf87f4a721d2aaffef2f7ac639ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:35.166ex; height:5.509ex;" alt="{\displaystyle \mathrm {Res} (f,0),\mathrm {ou} \ \mathrm {Res} _{0}{e^{z} \over z^{5}}\mathrm {ou} \ \mathrm {Res} _{z=0}{e^{z} \over z^{5}},.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Definição"><span id="Defini.C3.A7.C3.A3o"></span>Definição</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Res%C3%ADduo_(an%C3%A1lise_complexa)&veaction=edit&section=2" title="Editar secção: Definição" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Res%C3%ADduo_(an%C3%A1lise_complexa)&action=edit&section=2" title="Editar código-fonte da secção: Definição"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Seja <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" 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 \Omega }"></span> um <a href="/wiki/Conjunto_aberto" title="Conjunto aberto">subconjunto aberto</a> do <a href="/wiki/Plano_complexo" title="Plano complexo">plano complexo</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 \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" 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 {C} }"></span>, e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e72d1d86e86355892b39b8eb32b964834e113bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{0}}"></span> um ponto 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 \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" 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 \Omega }"></span>. Seja </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:\Omega \setminus \{z_{0}\}\to \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi mathvariant="normal">Ω</mi> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\Omega \setminus \{z_{0}\}\to \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b51d3e1caa25ef9e544730744f5b4d2ad5802d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.841ex; height:2.843ex;" alt="{\displaystyle f:\Omega \setminus \{z_{0}\}\to \mathbb {C} }"></span></dd></dl> <p>uma <a href="/wiki/Fun%C3%A7%C3%A3o_holomorfa" title="Função holomorfa">função holomorfa</a>, que apresenta em <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_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e72d1d86e86355892b39b8eb32b964834e113bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{0}}"></span> uma <a href="/w/index.php?title=Singularidade_isolada&action=edit&redlink=1" class="new" title="Singularidade isolada (página não existe)">singularidade isolada</a> e possui uma única expansão local na <a href="/wiki/S%C3%A9rie_de_Laurent" title="Série de Laurent">série de Laurent</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(z)=\sum \limits _{n=-\infty }^{\infty }a_{n}(z-z_{0})^{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 movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z)=\sum \limits _{n=-\infty }^{\infty }a_{n}(z-z_{0})^{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1dc6deded6f8380c2b2530cfa8317d6fce959fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.035ex; height:6.843ex;" alt="{\displaystyle f(z)=\sum \limits _{n=-\infty }^{\infty }a_{n}(z-z_{0})^{n}.}"></span></dd></dl> <p>O <b>resíduo</b> 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 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> em <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_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e72d1d86e86355892b39b8eb32b964834e113bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{0}}"></span> é o coeficiente <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_{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2238e179fc0f9004707cbfdf6f42510b3c89ee3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.563ex; height:2.009ex;" alt="{\displaystyle a_{-1}}"></span> da série de Laurent. </p> <div class="mw-heading mw-heading2"><h2 id="Ver_também"><span id="Ver_tamb.C3.A9m"></span>Ver também</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Res%C3%ADduo_(an%C3%A1lise_complexa)&veaction=edit&section=3" title="Editar secção: Ver também" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Res%C3%ADduo_(an%C3%A1lise_complexa)&action=edit&section=3" title="Editar código-fonte da secção: Ver também"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Teorema_dos_res%C3%ADduos" title="Teorema dos resíduos">Teorema dos resíduos</a></li> <li><a href="/wiki/F%C3%B3rmula_integral_de_Cauchy" title="Fórmula integral de Cauchy">Fórmula integral de Cauchy</a></li> <li><a href="/w/index.php?title=Teorema_de_Cauchy-Goursat&action=edit&redlink=1" class="new" title="Teorema de Cauchy-Goursat (página não existe)">Teorema de Cauchy-Goursat</a></li> <li><a href="/wiki/Teorema_de_Morera" title="Teorema de Morera">Teorema de Morera</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Res%C3%ADduo_(an%C3%A1lise_complexa)&veaction=edit&section=4" title="Editar secção: Bibliografia" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Res%C3%ADduo_(an%C3%A1lise_complexa)&action=edit&section=4" title="Editar código-fonte da secção: Bibliografia"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>L. Ahlfors, Complex Analysis, McGraw Hill, 1979.</li> <li>Ruel V. Churchill, Complex variables and applications, McGrall Hill, 1960</li></ul> <div class="mw-heading mw-heading2"><h2 id="Ligações_externas"><span id="Liga.C3.A7.C3.B5es_externas"></span>Ligações externas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Res%C3%ADduo_(an%C3%A1lise_complexa)&veaction=edit&section=5" title="Editar secção: Ligações externas" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Res%C3%ADduo_(an%C3%A1lise_complexa)&action=edit&section=5" title="Editar código-fonte da secção: Ligações externas"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><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/ComplexResidue.html">Complex Residue</a> no <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="http://math.fullerton.edu/mathews/c2003/ResidueCalcMod.html">«Module for Residues por John H. Mathews»</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3ARes%C3duo+%28an%C3%A1lise+complexa%29&rft.btitle=Module+for+Residues+por+John+H.+Mathews&rft.genre=unknown&rft_id=http%3A%2F%2Fmath.fullerton.edu%2Fmathews%2Fc2003%2FResidueCalcMod.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Obtida de "<a dir="ltr" href="https://pt.wikipedia.org/w/index.php?title=Resíduo_(análise_complexa)&oldid=67550386">https://pt.wikipedia.org/w/index.php?title=Resíduo_(análise_complexa)&oldid=67550386</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Especial:Categorias" title="Especial:Categorias">Categorias</a>: <ul><li><a href="/wiki/Categoria:An%C3%A1lise_complexa" title="Categoria:Análise complexa">Análise complexa</a></li><li><a href="/wiki/Categoria:N%C3%BAmeros_complexos" title="Categoria:Números complexos">Números complexos</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categorias ocultas: <ul><li><a href="/wiki/Categoria:!Artigos_que_carecem_de_notas_de_rodap%C3%A9_desde_janeiro_de_2016" title="Categoria:!Artigos que carecem de notas de rodapé desde janeiro de 2016">!Artigos que carecem de notas de rodapé desde janeiro de 2016</a></li><li><a href="/wiki/Categoria:!Artigos_que_carecem_de_notas_de_rodap%C3%A9_sem_indica%C3%A7%C3%A3o_de_tema" title="Categoria:!Artigos que carecem de notas de rodapé sem indicação de tema">!Artigos que carecem de notas de rodapé sem indicação de tema</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"> Esta página foi editada pela última vez às 03h46min de 28 de fevereiro de 2024.</li> <li id="footer-info-copyright">Este texto é disponibilizado nos termos da licença <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.pt">Atribuição-CompartilhaIgual 4.0 Internacional (CC BY-SA 4.0) da Creative Commons</a>; pode estar sujeito a condições adicionais. Para mais detalhes, consulte as <a class="external text" href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use">condições de utilização</a>.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy/pt-br">Política de privacidade</a></li> <li id="footer-places-about"><a href="/wiki/Wikip%C3%A9dia:Sobre">Sobre a Wikipédia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikip%C3%A9dia:Aviso_geral">Avisos gerais</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Código de conduta</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Programadores</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/pt.wikipedia.org">Estatísticas</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Declaração sobre ''cookies''</a></li> <li id="footer-places-mobileview"><a href="//pt.m.wikipedia.org/w/index.php?title=Res%C3%ADduo_(an%C3%A1lise_complexa)&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Versão móvel</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-7bbfc6c4f5-ltjtf","wgBackendResponseTime":127,"wgPageParseReport":{"limitreport":{"cputime":"0.162","walltime":"0.508","ppvisitednodes":{"value":243,"limit":1000000},"postexpandincludesize":{"value":7801,"limit":2097152},"templateargumentsize":{"value":113,"limit":2097152},"expansiondepth":{"value":11,"limit":100},"expensivefunctioncount":{"value":2,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":9837,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 149.067 1 -total"," 76.63% 114.232 1 Predefinição:Sem_notas"," 63.25% 94.283 1 Predefinição:Ambox"," 22.93% 34.184 1 Predefinição:Link"," 21.92% 32.670 1 Predefinição:Citar_web"," 3.90% 5.819 1 Predefinição:Argvar"]},"scribunto":{"limitreport-timeusage":{"value":"0.101","limit":"10.000"},"limitreport-memusage":{"value":1704644,"limit":52428800}},"cachereport":{"origin":"mw-web.eqiad.main-d948c7fb8-jv7v2","timestamp":"20241208183055","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Res\u00edduo (an\u00e1lise complexa)","url":"https:\/\/pt.wikipedia.org\/wiki\/Res%C3%ADduo_(an%C3%A1lise_complexa)","sameAs":"http:\/\/www.wikidata.org\/entity\/Q753375","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q753375","author":{"@type":"Organization","name":"Contribuidores dos projetos da Wikimedia"},"publisher":{"@type":"Organization","name":"Funda\u00e7\u00e3o Wikimedia, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2007-11-02T21:56:44Z","dateModified":"2024-02-28T03:46:01Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/3\/33\/Gamma_abs_3D.png"}</script> </body> </html>

CINXE.COM

Resíduo (análise complexa) – Wikipédia, a enciclopédia livre