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Función analítica - Wikipedia, la enciclopedia libre

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class="vector-toc-numb">1.2</span> <span>Funciones holomorfas</span> </div> </a> <ul id="toc-Funciones_holomorfas-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Funciones_suaves_no_analíticas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Funciones_suaves_no_analíticas"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Funciones suaves no analíticas</span> </div> </a> <ul id="toc-Funciones_suaves_no_analíticas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referencias" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referencias"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Referencias</span> </div> </a> <ul id="toc-Referencias-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Enlaces_externos" class="vector-toc-list-item vector-toc-level-1 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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">Cambiar a la tabla de contenidos</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">Función analítica</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 a un artículo en otro idioma. Disponible en 38 idiomas" > <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-38" 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">38 idiomas</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%AF%D8%A7%D9%84%D8%A9_%D8%AA%D8%AD%D9%84%D9%8A%D9%84%D9%8A%D8%A9" title="دالة تحليلية – árabe" lang="ar" hreflang="ar" data-title="دالة تحليلية" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D1%96%D1%82%D1%8B%D1%87%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%8B%D1%8F" title="Аналітычная функцыя – bielorruso" lang="be" hreflang="be" data-title="Аналітычная функцыя" data-language-autonym="Беларуская" data-language-local-name="bielorruso" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D0%B8%D1%82%D0%B8%D1%87%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Аналитична функция – búlgaro" lang="bg" hreflang="bg" data-title="Аналитична функция" data-language-autonym="Български" data-language-local-name="búlgaro" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Funci%C3%B3_anal%C3%ADtica" title="Funció analítica – catalán" lang="ca" hreflang="ca" data-title="Funció analítica" data-language-autonym="Català" data-language-local-name="catalán" 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/Analytick%C3%A1_funkce" title="Analytická funkce – checo" lang="cs" hreflang="cs" data-title="Analytická funkce" 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/Analytische_Funktion" title="Analytische Funktion – alemán" lang="de" hreflang="de" data-title="Analytische Funktion" data-language-autonym="Deutsch" data-language-local-name="alemán" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%BD%CE%B1%CE%BB%CF%85%CF%84%CE%B9%CE%BA%CE%AE_%CF%83%CF%85%CE%BD%CE%AC%CF%81%CF%84%CE%B7%CF%83%CE%B7" title="Αναλυτική συνάρτηση – griego" lang="el" hreflang="el" data-title="Αναλυτική συνάρτηση" data-language-autonym="Ελληνικά" data-language-local-name="griego" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Analytic_function" title="Analytic function – inglés" lang="en" hreflang="en" data-title="Analytic function" data-language-autonym="English" data-language-local-name="inglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Anal%C3%BC%C3%BCtiline_funktsioon" title="Analüütiline funktsioon – estonio" lang="et" hreflang="et" data-title="Analüütiline funktsioon" data-language-autonym="Eesti" data-language-local-name="estonio" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%A7%D8%A8%D8%B9_%D8%AA%D8%AD%D9%84%DB%8C%D9%84%DB%8C" title="تابع تحلیلی – persa" lang="fa" hreflang="fa" data-title="تابع تحلیلی" 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/Analyyttinen_funktio" title="Analyyttinen funktio – finés" lang="fi" hreflang="fi" data-title="Analyyttinen funktio" data-language-autonym="Suomi" data-language-local-name="finés" 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/Fonction_analytique" title="Fonction analytique – francés" lang="fr" hreflang="fr" data-title="Fonction analytique" data-language-autonym="Français" data-language-local-name="francés" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Funci%C3%B3n_anal%C3%ADtica" title="Función analítica – gallego" lang="gl" hreflang="gl" data-title="Función analítica" data-language-autonym="Galego" data-language-local-name="gallego" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%90%D7%A0%D7%9C%D7%99%D7%98%D7%99%D7%AA" title="פונקציה אנליטית – hebreo" lang="he" hreflang="he" data-title="פונקציה אנליטית" data-language-autonym="עברית" data-language-local-name="hebreo" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A5%88%E0%A4%B6%E0%A5%8D%E2%80%8D%E0%A4%B2%E0%A5%87%E0%A4%B7%E0%A4%BF%E0%A4%95_%E0%A4%AB%E0%A4%B2%E0%A4%A8" title="वैश्‍लेषिक फलन – hindi" lang="hi" hreflang="hi" data-title="वैश्‍लेषिक फलन" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Val%C3%B3s_analitikus_f%C3%BCggv%C3%A9ny" title="Valós analitikus függvény – húngaro" lang="hu" hreflang="hu" data-title="Valós analitikus függvény" 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/%D4%B1%D5%B6%D5%A1%D5%AC%D5%AB%D5%BF%D5%AB%D5%AF_%D6%86%D5%B8%D6%82%D5%B6%D5%AF%D6%81%D5%AB%D5%A1" title="Անալիտիկ ֆունկցիա – armenio" lang="hy" hreflang="hy" data-title="Անալիտիկ ֆունկցիա" data-language-autonym="Հայերեն" data-language-local-name="armenio" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/F%C3%A1ga%C3%B0_fall" title="Fágað fall – islandés" lang="is" hreflang="is" data-title="Fágað fall" data-language-autonym="Íslenska" data-language-local-name="islandés" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Funzione_analitica" title="Funzione analitica – italiano" lang="it" hreflang="it" data-title="Funzione analitica" 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/%E8%A7%A3%E6%9E%90%E9%96%A2%E6%95%B0" title="解析関数 – japonés" lang="ja" hreflang="ja" data-title="解析関数" data-language-autonym="日本語" data-language-local-name="japonés" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D0%B8%D1%82%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D2%9B_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Аналитикалық функция – kazajo" lang="kk" hreflang="kk" data-title="Аналитикалық функция" data-language-autonym="Қазақша" data-language-local-name="kazajo" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%95%B4%EC%84%9D_%ED%95%A8%EC%88%98" title="해석 함수 – coreano" lang="ko" hreflang="ko" data-title="해석 함수" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D0%B8%D1%82%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D0%BA_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Аналитикалык функция – kirguís" lang="ky" hreflang="ky" data-title="Аналитикалык функция" data-language-autonym="Кыргызча" data-language-local-name="kirguís" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Analizin%C4%97_funkcija" title="Analizinė funkcija – lituano" lang="lt" hreflang="lt" data-title="Analizinė funkcija" data-language-autonym="Lietuvių" data-language-local-name="lituano" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B5%E0%B4%BF%E0%B4%B6%E0%B5%8D%E0%B4%B2%E0%B5%87%E0%B4%B7%E0%B4%95%E0%B4%AB%E0%B4%B2%E0%B4%A8%E0%B4%82" title="വിശ്ലേഷകഫലനം – malayálam" lang="ml" hreflang="ml" data-title="വിശ്ലേഷകഫലനം" data-language-autonym="മലയാളം" data-language-local-name="malayálam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Analytische_functie" title="Analytische functie – neerlandés" lang="nl" hreflang="nl" data-title="Analytische functie" data-language-autonym="Nederlands" data-language-local-name="neerlandés" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Analytisk_funksjon" title="Analytisk funksjon – noruego nynorsk" lang="nn" hreflang="nn" data-title="Analytisk funksjon" data-language-autonym="Norsk nynorsk" data-language-local-name="noruego nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Analytisk_funksjon" title="Analytisk funksjon – noruego bokmal" lang="nb" hreflang="nb" data-title="Analytisk funksjon" data-language-autonym="Norsk bokmål" data-language-local-name="noruego bokmal" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Funkcja_analityczna" title="Funkcja analityczna – polaco" lang="pl" hreflang="pl" data-title="Funkcja analityczna" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Fun%C3%A7%C3%A3o_anal%C3%ADtica" title="Função analítica – portugués" lang="pt" hreflang="pt" data-title="Função analítica" data-language-autonym="Português" data-language-local-name="portugués" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Func%C8%9Bie_analitic%C4%83" title="Funcție analitică – rumano" lang="ro" hreflang="ro" data-title="Funcție analitică" data-language-autonym="Română" data-language-local-name="rumano" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D0%B8%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Аналитическая функция – ruso" lang="ru" hreflang="ru" data-title="Аналитическая функция" data-language-autonym="Русский" data-language-local-name="ruso" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AE%95%E0%AF%81%E0%AE%AE%E0%AF%81%E0%AE%B1%E0%AF%88%E0%AE%9A%E0%AF%8D_%E0%AE%9A%E0%AE%BE%E0%AE%B0%E0%AF%8D%E0%AE%AA%E0%AF%81" title="பகுமுறைச் சார்பு – tamil" lang="ta" hreflang="ta" data-title="பகுமுறைச் சார்பு" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Analitik_fonksiyon" title="Analitik fonksiyon – turco" lang="tr" hreflang="tr" data-title="Analitik fonksiyon" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D1%96%D1%82%D0%B8%D1%87%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F" title="Аналітична функція – ucraniano" lang="uk" hreflang="uk" data-title="Аналітична функція" data-language-autonym="Українська" data-language-local-name="ucraniano" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Analitik_funksiya" title="Analitik funksiya – uzbeko" lang="uz" hreflang="uz" data-title="Analitik funksiya" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbeko" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%C3%A0m_gi%E1%BA%A3i_t%C3%ADch" title="Hàm giải tích – vietnamita" lang="vi" hreflang="vi" data-title="Hàm giải tích" 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/%E8%A7%A3%E6%9E%90%E5%87%BD%E6%95%B0" title="解析函数 – chino" lang="zh" hreflang="zh" data-title="解析函数" data-language-autonym="中文" data-language-local-name="chino" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet 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class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Página de herramientas"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Apariencia"> <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">Apariencia</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mover a la 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">De Wikipedia, la enciclopedia libre</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="es" dir="ltr"><p>En <a href="/wiki/Matem%C3%A1ticas" title="Matemáticas">matemáticas</a> una <b>función analítica</b> es aquella que puede expresarse como una <a href="/wiki/Serie_de_potencias" title="Serie de potencias">serie de potencias</a> <a href="/wiki/Serie_convergente" title="Serie convergente">convergente</a>. Una función analítica es <a href="/wiki/Funci%C3%B3n_suave" class="mw-redirect" title="Función suave">suave</a> si tiene infinitas derivadas. La noción de función analítica puede definirse para funciones <a href="/wiki/Funci%C3%B3n_real" title="Función real">reales</a> o <a href="/wiki/Funci%C3%B3n_compleja" class="mw-redirect" title="Función compleja">complejas</a>, aunque ambos conjuntos tienen propiedades distintas. Las funciones <i>complejas</i> derivables en un <a href="/wiki/Conjunto_abierto" title="Conjunto abierto">conjunto abierto</a> siempre son localmente analíticas, y se denominan <a href="/wiki/Funci%C3%B3n_holomorfa" title="Función holomorfa">funciones holomorfas</a>. Esto se demuestra en el artículo <a href="/wiki/Analiticidad_de_las_funciones_holomorfas" title="Analiticidad de las funciones holomorfas">Analiticidad de las funciones holomorfas</a>. Sin embargo, una función real infinitamente derivable no es necesariamente analítica. Cabe dejar constancia que las clases más importantes de funciones que ocurren en el análisis clásico y en sus aplicaciones a los problemas de mecánica y física son analíticas, salvo en algunos puntos singulares de estas funciones. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definición"><span id="Definici.C3.B3n"></span>Definición</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3n_anal%C3%ADtica&amp;action=edit&amp;section=1" title="Editar sección: Definición"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La <a href="/wiki/Definici%C3%B3n_(matem%C3%A1tica)" title="Definición (matemática)">definición</a> de función analítica es idéntica para los casos real y complejo: </p> <table style="margin-right:4em; min-width:50%; max-width:77%"> <tbody><tr> <td><blockquote style="padding-right:2em; padding-left:1.5em; padding-bottom:0.5em; padding-top:0.5em; border:1px solid; font-family:Georgia,serif; border-color: #880000; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122);"> <p>Una función real (compleja) <i>f</i> es <b>analítica</b> en un punto <i>x</i><sub>0</sub> de su dominio si existe una <a href="/wiki/Serie_de_potencias" title="Serie de potencias">serie de potencias</a> centrada en <i>x</i><sub>0</sub>: </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }a_{n}(x-x_{0})^{n}=a_{0}+a_{1}(x-x_{0})+a_{2}(x-x_{0})^{2}+\ldots \,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">&#x221E;<!-- ∞ --></mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>&#x2212;<!-- − --></mo> <msub> <mi>x</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> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>&#x2212;<!-- − --></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>&#x2212;<!-- − --></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>&#x2026;<!-- … --></mo> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }a_{n}(x-x_{0})^{n}=a_{0}+a_{1}(x-x_{0})+a_{2}(x-x_{0})^{2}+\ldots \,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0971990081dc28bea4e5412ae65f1613d933b535" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:56.169ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }a_{n}(x-x_{0})^{n}=a_{0}+a_{1}(x-x_{0})+a_{2}(x-x_{0})^{2}+\ldots \,,}"></span> </p> </blockquote> <p>que converge en un entorno <i>U</i> &#8838; <b>R</b> (<i>U</i> &#8838; <b>C</b>) de <i>x</i><sub>0</sub> y que coincide con la función en dicho entorno: </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\sum _{n=0}^{\infty }a_{n}(x-x_{0})^{n}{\text{ , para cada }}x\in U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">&#x221E;<!-- ∞ --></mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>&#x2212;<!-- − --></mo> <msub> <mi>x</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> <mrow class="MJX-TeXAtom-ORD"> <mtext>&#xA0;, para cada&#xA0;</mtext> </mrow> <mi>x</mi> <mo>&#x2208;<!-- ∈ --></mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\sum _{n=0}^{\infty }a_{n}(x-x_{0})^{n}{\text{ , para cada }}x\in U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b82cbeafb66e5361b8c7af227cae289626eb6897" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.389ex; height:6.843ex;" alt="{\displaystyle f(x)=\sum _{n=0}^{\infty }a_{n}(x-x_{0})^{n}{\text{ , para cada }}x\in U}"></span> </p> </blockquote> </blockquote> </td></tr></tbody></table> <p>De esta definición se puede demostrar la siguiente caracterización alternativa: </p> <table style="margin-right:4em; min-width:50%; max-width:77%"> <tbody><tr> <td><blockquote style="padding-right:2em; padding-left:1.5em; padding-bottom:0.5em; padding-top:0.5em; border:1px solid; font-family:Georgia,serif; border-color: #49768C; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122);"> <p>Una función analítica en <i>x</i><sub>0</sub> es <a href="/wiki/Funci%C3%B3n_suave" class="mw-redirect" title="Función suave">infinitamente derivable</a> en un cierto entorno <i>U</i> de dicho punto, en el que además su <a href="/wiki/Serie_de_Taylor" title="Serie de Taylor">serie de Taylor</a>: </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }{\frac {f^{(n)}(x_{0})}{n!}}(x-x_{0})^{n}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">&#x221E;<!-- ∞ --></mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>&#x2212;<!-- − --></mo> <msub> <mi>x</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> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\frac {f^{(n)}(x_{0})}{n!}}(x-x_{0})^{n}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25ceb14bfd015c83cd58196faab60a4304c4649e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.205ex; height:7.176ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\frac {f^{(n)}(x_{0})}{n!}}(x-x_{0})^{n}\,,}"></span> </p> </blockquote> <p>converge (y coincide con <i>f</i>). </p> </blockquote> </td></tr></tbody></table> <p>Una función se dice analítica en un conjunto <i>U</i> si es analítica en cada punto de <i>U</i>. El conjunto de todas las funciones analíticas en un cierto abierto <i>U</i> se denota por <i>C</i><sup>&#969;</sup>(<i>U</i>). </p> <div class="mw-heading mw-heading3"><h3 id="Varias_variables">Varias variables</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3n_anal%C3%ADtica&amp;action=edit&amp;section=2" title="Editar sección: Varias variables"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La definición de función analítica puede extenderse para funciones (reales o complejas) de varias variables (definidas en <b>R</b><sup>n</sup> o <b>C</b><sup>n</sup>), sin más que considerar series de potencias de varias variables: </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i_{1}\ldots i_{n}=0}^{\infty }a_{i_{1}\ldots i_{n}}\prod _{k=1}^{n}(x_{k}-c_{k})^{i_{k}}=a_{0\ldots 0}+a_{1\ldots 0}(x_{1}-c_{1})+\ldots +a_{0\ldots 1}(x_{n}-c_{n})+a_{2\ldots 0}(x_{1}-c_{1})^{2}+a_{11\ldots 0}(x_{1}-c_{1})(x_{2}-c_{2})+\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>&#x2026;<!-- … --></mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">&#x221E;<!-- ∞ --></mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>&#x2026;<!-- … --></mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <munderover> <mo>&#x220F;<!-- ∏ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>&#x2212;<!-- − --></mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>&#x2026;<!-- … --></mo> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>&#x2026;<!-- … --></mo> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>&#x2212;<!-- − --></mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo>&#x2026;<!-- … --></mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>&#x2026;<!-- … --></mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>&#x2212;<!-- − --></mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>&#x2026;<!-- … --></mo> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>&#x2212;<!-- − --></mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> <mo>&#x2026;<!-- … --></mo> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>&#x2212;<!-- − --></mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>&#x2212;<!-- − --></mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo>&#x2026;<!-- … --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i_{1}\ldots i_{n}=0}^{\infty }a_{i_{1}\ldots i_{n}}\prod _{k=1}^{n}(x_{k}-c_{k})^{i_{k}}=a_{0\ldots 0}+a_{1\ldots 0}(x_{1}-c_{1})+\ldots +a_{0\ldots 1}(x_{n}-c_{n})+a_{2\ldots 0}(x_{1}-c_{1})^{2}+a_{11\ldots 0}(x_{1}-c_{1})(x_{2}-c_{2})+\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67827e823aa3a0d24958b03403a1fc898bcd01f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:126.267ex; height:7.176ex;" alt="{\displaystyle \sum _{i_{1}\ldots i_{n}=0}^{\infty }a_{i_{1}\ldots i_{n}}\prod _{k=1}^{n}(x_{k}-c_{k})^{i_{k}}=a_{0\ldots 0}+a_{1\ldots 0}(x_{1}-c_{1})+\ldots +a_{0\ldots 1}(x_{n}-c_{n})+a_{2\ldots 0}(x_{1}-c_{1})^{2}+a_{11\ldots 0}(x_{1}-c_{1})(x_{2}-c_{2})+\ldots }"></span> </p> </blockquote> <div class="mw-heading mw-heading3"><h3 id="Funciones_holomorfas">Funciones holomorfas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3n_anal%C3%ADtica&amp;action=edit&amp;section=3" title="Editar sección: Funciones holomorfas"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint AP rellink"><span style="font-size:88%">Artículo principal:</span>&#32;<i><a href="/wiki/Funci%C3%B3n_holomorfa" title="Función holomorfa"> Función holomorfa</a></i></div> <p>En el caso de las funciones complejas analíticas, existe un teorema que las caracteriza de manera mucho más sencilla, y que constituye uno de los rasgos fundamentales del <a href="/wiki/An%C3%A1lisis_complejo" title="Análisis complejo">análisis complejo</a>: </p> <table style="margin-right:4em; min-width:50%; max-width:77%"> <tbody><tr> <td><blockquote style="padding-right:2em; padding-left:1.5em; padding-bottom:0.5em; padding-top:0.5em; border:1px solid; font-family:Georgia,serif; border-color: #49768C; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122);"> <p>Una función compleja <i>f</i>&#160;: <i>D</i> &#8838; <b>C</b> &#8594; <b>C</b> derivable en un intervalo abierto <i>U</i>, es analítica en <i>U</i>. </p> </blockquote> </td></tr></tbody></table> <p>Un teorema similar se aplica en el caso de funciones complejas de varias variables que sean diferenciables: </p> <table style="margin-right:4em; min-width:50%; max-width:77%"> <tbody><tr> <td><blockquote style="padding-right:2em; padding-left:1.5em; padding-bottom:0.5em; padding-top:0.5em; border:1px solid; font-family:Georgia,serif; border-color: #49768C; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122);"> <p>Una función compleja <i>f</i>&#160;: <i>D</i> &#8838; <b>C</b><sup>n</sup> &#8594; <b>C</b> diferenciable en un intervalo abierto <i>U</i> es analítica en <i>U</i>. </p> </blockquote> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Funciones_suaves_no_analíticas"><span id="Funciones_suaves_no_anal.C3.ADticas"></span>Funciones suaves no analíticas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3n_anal%C3%ADtica&amp;action=edit&amp;section=4" title="Editar sección: Funciones suaves no analíticas"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En variable real pueden encontrarse funciones suaves que no son analíticas. Un ejemplo de ello es la función: </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\left\{{\begin{array}{l}e^{-1/x^{2}}{\text{ , si }}x\neq 0\\0{\text{ , si }}x=0\end{array}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>&#x2212;<!-- − --></mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>&#xA0;, si&#xA0;</mtext> </mrow> <mi>x</mi> <mo>&#x2260;<!-- ≠ --></mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>&#xA0;, si&#xA0;</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\left\{{\begin{array}{l}e^{-1/x^{2}}{\text{ , si }}x\neq 0\\0{\text{ , si }}x=0\end{array}}\right.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed087607344a52a35cbccd44792b434f37b99242" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.563ex; height:6.509ex;" alt="{\displaystyle f(x)=\left\{{\begin{array}{l}e^{-1/x^{2}}{\text{ , si }}x\neq 0\\0{\text{ , si }}x=0\end{array}}\right.}"></span> </p> </blockquote> <p>Esta función es infinitamente derivable para cualquier <i>x</i> &#8712; <b>R</b>, y en particular todas sus derivadas en 0 son nulas: <i>f</i><sup>(n)</sup>(0) = 0. Por tanto, su serie de Taylor alrededor de 0 es idénticamente nula, y en ningún entorno de dicho punto coinciden la función y la serie de Taylor. </p> <div class="mw-heading mw-heading2"><h2 id="Referencias">Referencias</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3n_anal%C3%ADtica&amp;action=edit&amp;section=5" title="Editar sección: Referencias"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span id="CITAREFKrantzParks1992" class="citation libro">Krantz, Steven; Parks, Harold (1992). <i>A primer of real analytic functions</i> <span style="color:var(--color-subtle, #555 );">(en inglés)</span>. Birkhäuser Verlag. <small><a href="/wiki/ISBN" title="ISBN">ISBN</a>&#160;<a href="/wiki/Especial:FuentesDeLibros/3-7643-2768-5" title="Especial:FuentesDeLibros/3-7643-2768-5">3-7643-2768-5</a></small>.</span><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fes.wikipedia.org%3AFunci%C3%B3n+anal%C3%ADtica&amp;rft.au=Krantz%2C+Steven&amp;rft.au=Parks%2C+Harold&amp;rft.aufirst=Steven&amp;rft.aulast=Krantz&amp;rft.btitle=A+primer+of+real+analytic+functions&amp;rft.date=1992&amp;rft.genre=book&amp;rft.isbn=3-7643-2768-5&amp;rft.pub=Birkh%C3%A4user+Verlag&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span> Capítulo 1.</li> <li><span id="CITAREFScheidemann2005" class="citation libro">Scheidemann, Volker (2005). <i>Introduction to complex analysis in several variables</i> <span style="color:var(--color-subtle, #555 );">(en inglés)</span>. Birkhäuser Verlag. <small><a href="/wiki/ISBN" title="ISBN">ISBN</a>&#160;<a href="/wiki/Especial:FuentesDeLibros/3-7643-7490-X" title="Especial:FuentesDeLibros/3-7643-7490-X">3-7643-7490-X</a></small>.</span><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fes.wikipedia.org%3AFunci%C3%B3n+anal%C3%ADtica&amp;rft.au=Scheidemann%2C+Volker&amp;rft.aufirst=Volker&amp;rft.aulast=Scheidemann&amp;rft.btitle=Introduction+to+complex+analysis+in+several+variables&amp;rft.date=2005&amp;rft.genre=book&amp;rft.isbn=3-7643-7490-X&amp;rft.pub=Birkh%C3%A4user+Verlag&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span> Capítulo 1.</li> <li>Complex Analysis Ahlfors, Lars V. Mc Graw-Hill Company, Inc. Tokyo 1953</li></ul> <ul><li>Teoría de las funciones analíticas Tomo I, Markushevich, A. traducción de Emiliano Aparicio Bernardo (sic) A. Editorial Mir, Moscú, 1970</li></ul> <div class="mw-heading mw-heading2"><h2 id="Enlaces_externos">Enlaces externos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3n_anal%C3%ADtica&amp;action=edit&amp;section=6" title="Editar sección: Enlaces externos"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span id="Reference-Mathworld-Analytic_Function" class="citation web"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W</a>. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/AnalyticFunction.html">«Analytic Function»</a>. En Weisstein, Eric W, ed. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i> <span style="color:var(--color-subtle, #555 );">(en inglés)</span>. <a href="/wiki/Wolfram_Research" title="Wolfram Research">Wolfram Research</a>.</span><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fes.wikipedia.org%3AFunci%C3%B3n+anal%C3%ADtica&amp;rft.atitle=Analytic+Function&amp;rft.au=Weisstein%2C+Eric+W&amp;rft.aulast=Weisstein%2C+Eric+W&amp;rft.genre=article&amp;rft.jtitle=MathWorld&amp;rft.pub=Wolfram+Research&amp;rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FAnalyticFunction.html&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;">&#160;</span></span></li> <li><span id="Reference-Mathworld-Real_Analytic_Function" class="citation web"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W</a>. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/RealAnalyticFunction.html">«Real Analytic Function»</a>. 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