Integral elíptica - Wikipedia, la enciclopedia libre

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especie</span> </div> </a> <button aria-controls="toc-Integral_elíptica_de_primera_especie-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar subsección Integral elíptica de primera especie</span> </button> <ul id="toc-Integral_elíptica_de_primera_especie-sublist" class="vector-toc-list"> <li id="toc-Integral_elíptica_completa_de_primera_especie" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integral_elíptica_completa_de_primera_especie"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Integral elíptica completa de primera especie</span> </div> </a> <ul id="toc-Integral_elíptica_completa_de_primera_especie-sublist" class="vector-toc-list"> <li id="toc-Ecuación_diferencial" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Ecuación_diferencial"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>Ecuación diferencial</span> </div> </a> <ul id="toc-Ecuación_diferencial-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Integral_elíptica_incompleta_de_primera_especie" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integral_elíptica_incompleta_de_primera_especie"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Integral elíptica incompleta de primera especie</span> </div> </a> <ul id="toc-Integral_elíptica_incompleta_de_primera_especie-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transformación_de_Landen" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transformación_de_Landen"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Transformación de Landen</span> </div> </a> <ul id="toc-Transformación_de_Landen-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Integral_elíptica_de_segunda_especie" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Integral_elíptica_de_segunda_especie"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Integral elíptica de segunda especie</span> </div> </a> <button aria-controls="toc-Integral_elíptica_de_segunda_especie-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar subsección Integral elíptica de segunda especie</span> </button> <ul id="toc-Integral_elíptica_de_segunda_especie-sublist" class="vector-toc-list"> <li id="toc-Integral_elíptica_completa_de_segunda_especie" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integral_elíptica_completa_de_segunda_especie"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Integral elíptica completa de segunda especie</span> </div> </a> <ul id="toc-Integral_elíptica_completa_de_segunda_especie-sublist" class="vector-toc-list"> <li id="toc-Derivada_y_ecuación_diferencial" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Derivada_y_ecuación_diferencial"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1</span> <span>Derivada y ecuación diferencial</span> </div> </a> <ul id="toc-Derivada_y_ecuación_diferencial-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Integral_elíptica_incompleta_de_segunda_especie" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integral_elíptica_incompleta_de_segunda_especie"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Integral elíptica incompleta de segunda especie</span> </div> </a> <ul id="toc-Integral_elíptica_incompleta_de_segunda_especie-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Integral_elíptica_de_tercera_especie" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Integral_elíptica_de_tercera_especie"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Integral elíptica de tercera especie</span> </div> </a> <button aria-controls="toc-Integral_elíptica_de_tercera_especie-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar subsección Integral elíptica de tercera especie</span> </button> <ul id="toc-Integral_elíptica_de_tercera_especie-sublist" class="vector-toc-list"> <li id="toc-Aplicaciones" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Aplicaciones"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Aplicaciones</span> </div> </a> <ul id="toc-Aplicaciones-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Véase_también" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Véase_también"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Véase también</span> </div> </a> <ul id="toc-Véase_también-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">6</span> <span>Referencias</span> </div> </a> <button aria-controls="toc-Referencias-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar subsección Referencias</span> </button> <ul id="toc-Referencias-sublist" class="vector-toc-list"> <li id="toc-Bibliografía" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bibliografía"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Bibliografía</span> </div> </a> <ul id="toc-Bibliografía-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contenidos" 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="Cambiar a la tabla de contenidos" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button 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Disponible en 26 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-26" 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">26 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%AA%D9%83%D8%A7%D9%85%D9%84_%D9%86%D8%A7%D9%82%D8%B5%D9%8A" title="تكامل ناقصي (árabe)" lang="ar" hreflang="ar" data-title="تكامل ناقصي" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Elliptik_inteqral" title="Elliptik inteqral (azerbaiyano)" lang="az" hreflang="az" data-title="Elliptik inteqral" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaiyano" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Integral_el%C2%B7l%C3%ADptica" title="Integral el·líptica (catalán)" lang="ca" hreflang="ca" data-title="Integral el·líptica" data-language-autonym="Català" data-language-local-name="catalán" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%AD%D0%BB%D0%BB%D0%B8%D0%BF%D1%81%D0%BB%D0%B0_%D0%B8%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB" title="Эллипсла интеграл (chuvasio)" lang="cv" hreflang="cv" data-title="Эллипсла интеграл" data-language-autonym="Чӑвашла" data-language-local-name="chuvasio" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Elliptische_Integrale" title="Elliptische Integrale (alemán)" lang="de" hreflang="de" data-title="Elliptische Integrale" data-language-autonym="Deutsch" data-language-local-name="alemán" 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/Elliptic_integral" title="Elliptic integral (inglés)" lang="en" hreflang="en" data-title="Elliptic integral" data-language-autonym="English" data-language-local-name="inglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Elipsa_integralo" title="Elipsa integralo (esperanto)" lang="eo" hreflang="eo" data-title="Elipsa integralo" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D9%86%D8%AA%DA%AF%D8%B1%D8%A7%D9%84_%D8%A8%DB%8C%D8%B6%D9%88%DB%8C" title="انتگرال بیضوی (persa)" lang="fa" hreflang="fa" data-title="انتگرال بیضوی" 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/Elliptinen_integraali" title="Elliptinen integraali (finés)" lang="fi" hreflang="fi" data-title="Elliptinen integraali" 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/Int%C3%A9grale_elliptique" title="Intégrale elliptique (francés)" lang="fr" hreflang="fr" data-title="Intégrale elliptique" data-language-autonym="Français" data-language-local-name="francés" 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%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C_%D7%90%D7%9C%D7%99%D7%A4%D7%98%D7%99" title="אינטגרל אליפטי (hebreo)" lang="he" hreflang="he" data-title="אינטגרל אליפטי" data-language-autonym="עברית" data-language-local-name="hebreo" 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/Elliptikus_integr%C3%A1l" title="Elliptikus integrál (húngaro)" lang="hu" hreflang="hu" data-title="Elliptikus integrál" data-language-autonym="Magyar" data-language-local-name="húngaro" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Integrale_ellittico" title="Integrale ellittico (italiano)" lang="it" hreflang="it" data-title="Integrale ellittico" 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/%E6%A5%95%E5%86%86%E7%A9%8D%E5%88%86" title="楕円積分 (japonés)" lang="ja" hreflang="ja" data-title="楕円積分" 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-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%83%80%EC%9B%90_%EC%A0%81%EB%B6%84" title="타원 적분 (coreano)" lang="ko" hreflang="ko" data-title="타원 적분" 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/Elliptische_integraal" title="Elliptische integraal (neerlandés)" lang="nl" hreflang="nl" data-title="Elliptische integraal" 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/Elliptisk_integral" title="Elliptisk integral (noruego nynorsk)" lang="nn" hreflang="nn" data-title="Elliptisk integral" 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/Elliptisk_integral" title="Elliptisk integral (noruego bokmal)" lang="nb" hreflang="nb" data-title="Elliptisk integral" 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/Ca%C5%82ki_eliptyczne" title="Całki eliptyczne (polaco)" lang="pl" hreflang="pl" data-title="Całki eliptyczne" 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/Integral_el%C3%ADptica" title="Integral elíptica (portugués)" lang="pt" hreflang="pt" data-title="Integral elíptica" 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/Integral%C4%83_eliptic%C4%83" title="Integrală eliptică (rumano)" lang="ro" hreflang="ro" data-title="Integrală eliptică" 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%AD%D0%BB%D0%BB%D0%B8%D0%BF%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B9_%D0%B8%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB" title="Эллиптический интеграл (ruso)" lang="ru" hreflang="ru" data-title="Эллиптический интеграл" data-language-autonym="Русский" data-language-local-name="ruso" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%95%D0%BB%D0%B8%D0%BF%D1%82%D0%B8%D1%87%D0%BA%D0%B8_%D0%B8%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB%D0%B8" title="Елиптички интеграли (serbio)" lang="sr" hreflang="sr" data-title="Елиптички интеграли" data-language-autonym="Српски / srpski" data-language-local-name="serbio" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Elliptisk_integral" title="Elliptisk integral (sueco)" lang="sv" hreflang="sv" data-title="Elliptisk integral" data-language-autonym="Svenska" data-language-local-name="sueco" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%95%D0%BB%D1%96%D0%BF%D1%82%D0%B8%D1%87%D0%BD%D1%96_%D1%96%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB%D0%B8" title="Еліптичні інтеграли (ucraniano)" lang="uk" hreflang="uk" data-title="Еліптичні інтеграли" data-language-autonym="Українська" data-language-local-name="ucraniano" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%A4%AD%E5%9C%86%E7%A7%AF%E5%88%86" title="椭圆积分 (chino)" lang="zh" hreflang="zh" data-title="椭圆积分" data-language-autonym="中文" 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especie</a>»)</span></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/C%C3%A1lculo_infinitesimal" title="Cálculo infinitesimal">cálculo</a>, una <b>integral elíptica</b> es una función <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> de la forma </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(x)=\int _{c}^{x}R\left(t,{\sqrt {P(t)}}\right)dt}"> <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> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mi>R</mi> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>P</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\int _{c}^{x}R\left(t,{\sqrt {P(t)}}\right)dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d377a35bbcf22a57b4caa9ab1b185ec440780c69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.501ex; height:5.843ex;" alt="{\displaystyle f(x)=\int _{c}^{x}R\left(t,{\sqrt {P(t)}}\right)dt}"></span></dd></dl> <p>donde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> es una <a href="/wiki/Funci%C3%B3n_racional" title="Función racional">función racional</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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> es un <a href="/wiki/Polinomio" title="Polinomio">polinomio</a> sin raíces repetidas y <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 c\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d47ef490c028656282fd8b18c44c4939bbfff750" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.526ex; height:2.176ex;" alt="{\displaystyle c\in \mathbb {R} }"></span>. </p><p>La denominación <i>integral elíptica</i> parte de los primeros problemas donde tuvieron lugar estas integrales, relacionados con el cálculo de la longitud de segmentos de <a href="/wiki/Elipse" title="Elipse">elipse</a>. </p><p>Las integrales elípticas pueden verse como generalizaciones de las <a href="/wiki/Funci%C3%B3n_trigonom%C3%A9trica_inversa" title="Función trigonométrica inversa">funciones trigonométricas inversas</a>. Las integrales elípticas proporcionan soluciones a una clase de problemas algo más amplia que las funciones trigonométricas inversas elementales, por ejemplo el cálculo de la <a href="/wiki/Longitud_de_arco" title="Longitud de arco">longitud de arco</a> de una <a href="/wiki/Circunferencia" title="Circunferencia">circunferencia</a> solo requiere de las funciones trigonométricas inversas, pero el cálculo de la longitud de arco de una elipse requiere de integrales elípticas. Otro buen ejemplo es el <a href="/wiki/P%C3%A9ndulo" title="Péndulo">péndulo</a>, cuyo movimiento para pequeñas oscilaciones puede representarse por funciones trigonométricas, pero para oscilaciones más grandes requiere el uso de <a href="/wiki/Funci%C3%B3n_el%C3%ADptica" title="Función elíptica">funciones elípticas</a> basadas en las integrales elípticas. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Cálculo"><span id="C.C3.A1lculo"></span>Cálculo</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral_el%C3%ADptica&action=edit&section=1" title="Editar sección: Cálculo"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Todas las integrales elípticas del tipo anterior pueden ser reescritas en términos en forma de suma de <a href="/wiki/Funci%C3%B3n_elemental" title="Función elemental">funciones elementales</a> y tres tipos "básicos" de integrales elípticas (llamados de primera especie, de segunda especie y de tercera especie). Para ver esto escribamos la integral elíptica en la forma: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int {\frac {P(w,x)}{Q(w,x)}}\ dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mtext> </mtext> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int {\frac {P(w,x)}{Q(w,x)}}\ dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99b5d5a6c016bd47e3b8b7dc57b23186aa263d40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.219ex; height:6.509ex;" alt="{\displaystyle \int {\frac {P(w,x)}{Q(w,x)}}\ dx}"></span></dd></dl> <p>Donde <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 w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}"></span> es una función 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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, tal que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8c089e90e5002149853ef25553813b8ba07e610" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.718ex; height:2.676ex;" alt="{\displaystyle w^{2}}"></span> es un polinomio de tercer o cuarto grado, que contiene al menos una potencia impar 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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>.<sup id="cite_ref-1" class="reference separada"><a href="#cite_note-1"><span class="corchete-llamada">[</span>1<span class="corchete-llamada">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Integral_elíptica_de_primera_especie"><span id="Integral_el.C3.ADptica_de_primera_especie"></span>Integral elíptica de primera especie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral_el%C3%ADptica&action=edit&section=2" title="Editar sección: Integral elíptica de primera especie"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Una <b>integral elíptica de primera especie</b> es un caso particular de la <b>integral elíptica</b>. Existen integrales elípticas de primera especie, completas e incompletas. Las primeras dependen de una sola variable y las segundas dependen de dos variables. </p> <div class="mw-heading mw-heading3"><h3 id="Integral_elíptica_completa_de_primera_especie"><span id="Integral_el.C3.ADptica_completa_de_primera_especie"></span>Integral elíptica completa de primera especie</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral_el%C3%ADptica&action=edit&section=3" title="Editar sección: Integral elíptica completa de primera especie"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La integral elíptica completa de primera especie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> se define 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 K(x)=\int _{0}^{\pi /2}{\frac {d\theta }{\sqrt {1-x^{2}\operatorname {sen} ^{2}\theta }}}=\int _{0}^{1}{\frac {dv}{\sqrt {(1-v^{2})(1-x^{2}v^{2})}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>θ</mi> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sen</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mfrac> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>v</mi> </mrow> <msqrt> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x)=\int _{0}^{\pi /2}{\frac {d\theta }{\sqrt {1-x^{2}\operatorname {sen} ^{2}\theta }}}=\int _{0}^{1}{\frac {dv}{\sqrt {(1-v^{2})(1-x^{2}v^{2})}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7117988b04e39a7c2cb4b79374113cacf961e98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:58.215ex; height:7.176ex;" alt="{\displaystyle K(x)=\int _{0}^{\pi /2}{\frac {d\theta }{\sqrt {1-x^{2}\operatorname {sen} ^{2}\theta }}}=\int _{0}^{1}{\frac {dv}{\sqrt {(1-v^{2})(1-x^{2}v^{2})}}}}"></span></dd></dl> <p>y puede expresarse como una <a href="/wiki/Serie_de_potencias" title="Serie de potencias">serie de potencias</a> 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 K(x)={\frac {\pi }{2}}\sum _{n=0}^{\infty }\left({\frac {(2n)!}{2^{2n}(n!)^{2}}}\right)^{2}x^{2n}={\frac {\pi }{2}}\sum _{n=0}^{\infty }\left(P_{2n}(0)\right)^{2}x^{2n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo>!</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x)={\frac {\pi }{2}}\sum _{n=0}^{\infty }\left({\frac {(2n)!}{2^{2n}(n!)^{2}}}\right)^{2}x^{2n}={\frac {\pi }{2}}\sum _{n=0}^{\infty }\left(P_{2n}(0)\right)^{2}x^{2n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8048cd08ef4f22d2523a19afc6c476a4738a4d6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:53.784ex; height:8.009ex;" alt="{\displaystyle K(x)={\frac {\pi }{2}}\sum _{n=0}^{\infty }\left({\frac {(2n)!}{2^{2n}(n!)^{2}}}\right)^{2}x^{2n}={\frac {\pi }{2}}\sum _{n=0}^{\infty }\left(P_{2n}(0)\right)^{2}x^{2n}}"></span></dd></dl> <p>donde <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 P_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5949c8b1de44005a1af3a11188361f2a830842d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.711ex; height:2.509ex;" alt="{\displaystyle P_{n}}"></span> es el polinomio de Legendre, la expresión anterior es equivalente 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 K(x)={\frac {\pi }{2}}\left(1+\left({\frac {1}{2}}\right)^{2}x^{2}+\left({\frac {1\cdot 3}{2\cdot 4}}\right)^{2}x^{4}+\cdots +\left({\frac {(2n-1)!!}{(2n)!!}}\right)^{2}x^{2n}+\cdots \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>3</mn> </mrow> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>4</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x)={\frac {\pi }{2}}\left(1+\left({\frac {1}{2}}\right)^{2}x^{2}+\left({\frac {1\cdot 3}{2\cdot 4}}\right)^{2}x^{4}+\cdots +\left({\frac {(2n-1)!!}{(2n)!!}}\right)^{2}x^{2n}+\cdots \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b475d667654d957afcffa650351de31427f9122f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:74.253ex; height:7.509ex;" alt="{\displaystyle K(x)={\frac {\pi }{2}}\left(1+\left({\frac {1}{2}}\right)^{2}x^{2}+\left({\frac {1\cdot 3}{2\cdot 4}}\right)^{2}x^{4}+\cdots +\left({\frac {(2n-1)!!}{(2n)!!}}\right)^{2}x^{2n}+\cdots \right)}"></span></dd></dl> <p>donde <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 n!!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/511717d541dba5357928e8d8631f1b4d4f8d5b31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.688ex; height:2.176ex;" alt="{\displaystyle n!!}"></span> denota el doble factorial. </p> <div class="mw-heading mw-heading4"><h4 id="Ecuación_diferencial"><span id="Ecuaci.C3.B3n_diferencial"></span>Ecuación diferencial</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral_el%C3%ADptica&action=edit&section=4" title="Editar sección: Ecuación diferencial"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La <a href="/wiki/Ecuaci%C3%B3n_diferencial" title="Ecuación diferencial">ecuación diferencial</a> para la integral elíptica de primera especie es </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dx}}\left(x(1-x^{2}){\frac {dK(x)}{dx}}\right)=xK(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>x</mi> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}\left(x(1-x^{2}){\frac {dK(x)}{dx}}\right)=xK(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b6a8c8b60541abd7c56088efbfe58a15ba257eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.606ex; height:6.343ex;" alt="{\displaystyle {\frac {d}{dx}}\left(x(1-x^{2}){\frac {dK(x)}{dx}}\right)=xK(x)}"></span></dd></dl> <p>Una segunda solución para esta ecuación es <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\left(1-x^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\left(1-x^{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7c43d45fe3d128b755d6c09a4c4819321302b57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.97ex; height:3.343ex;" alt="{\displaystyle K\left(1-x^{2}\right)}"></span>, esta solución satisface la relación </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dx}}K(x)={\frac {E(x)}{x(1-x^{2})}}-{\frac {K(x)}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}K(x)={\frac {E(x)}{x(1-x^{2})}}-{\frac {K(x)}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/196ba218cff2da319797135c4d2d8c85d6efe394" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.928ex; height:6.509ex;" alt="{\displaystyle {\frac {d}{dx}}K(x)={\frac {E(x)}{x(1-x^{2})}}-{\frac {K(x)}{x}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Integral_elíptica_incompleta_de_primera_especie"><span id="Integral_el.C3.ADptica_incompleta_de_primera_especie"></span>Integral elíptica incompleta de primera especie</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral_el%C3%ADptica&action=edit&section=5" title="Editar sección: Integral elíptica incompleta de primera especie"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La integral elíptica incompleta de primera especie <i>F</i> se define 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 u=F(x,\varphi )=\int _{0}^{\varphi }{\frac {d\theta }{\sqrt {1-x^{2}\operatorname {sen} ^{2}\theta }}}=\int _{0}^{\operatorname {sen} \varphi }{\frac {dv}{\sqrt {(1-v^{2})(1-x^{2}v^{2})}}}=F_{x}(\varphi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>θ</mi> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sen</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mfrac> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>sen</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>v</mi> </mrow> <msqrt> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </msqrt> </mfrac> </mrow> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u=F(x,\varphi )=\int _{0}^{\varphi }{\frac {d\theta }{\sqrt {1-x^{2}\operatorname {sen} ^{2}\theta }}}=\int _{0}^{\operatorname {sen} \varphi }{\frac {dv}{\sqrt {(1-v^{2})(1-x^{2}v^{2})}}}=F_{x}(\varphi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4116b19d2b3b0e41c07a7d2b35d043be8ad984df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:75.388ex; height:6.676ex;" alt="{\displaystyle u=F(x,\varphi )=\int _{0}^{\varphi }{\frac {d\theta }{\sqrt {1-x^{2}\operatorname {sen} ^{2}\theta }}}=\int _{0}^{\operatorname {sen} \varphi }{\frac {dv}{\sqrt {(1-v^{2})(1-x^{2}v^{2})}}}=F_{x}(\varphi )}"></span></dd></dl> <p>En este caso el parámetro <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 \varphi =\operatorname {am} (u)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mi>am</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =\operatorname {am} (u)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/613f3a2fd39e6128e1f0129105fae4954b41fb86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.856ex; height:2.843ex;" alt="{\displaystyle \varphi =\operatorname {am} (u)}"></span> se llama "amplitud" y si se toma <i>x</i> como un parámetro. Esta "amplitud" viene dada por la inversa de la función anterior <i>F</i>. Las <a href="/wiki/Funci%C3%B3n_el%C3%ADptica_de_Jacobi" title="Función elíptica de Jacobi">funciones elípticas de Jacobi</a> se definen a partir de esta amplitud. </p> <div class="mw-heading mw-heading3"><h3 id="Transformación_de_Landen"><span id="Transformaci.C3.B3n_de_Landen"></span>Transformación de Landen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral_el%C3%ADptica&action=edit&section=6" title="Editar sección: Transformación de Landen"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La transformación de Landen permite expresar integrales elípticas incompletas de un parámetro en integrales elípticas de otro parámetro diferente. Puede probarse que si definimos una nueva amplitud φ<sub>1</sub> y un nuevo parámetro <i>k</i><sub>1</sub>, relacionadas con la antigua amplitud φ y el antiguo parámetro <i>k</i> mediante: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{1}={\frac {2{\sqrt {k}}}{1+k}}\qquad \tan \varphi ={\frac {\operatorname {sen} 2\varphi _{1}}{k+\cos 2\varphi _{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>k</mi> </msqrt> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>k</mi> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mi>tan</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sen</mi> <mo>⁡</mo> <mn>2</mn> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mn>2</mn> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{1}={\frac {2{\sqrt {k}}}{1+k}}\qquad \tan \varphi ={\frac {\operatorname {sen} 2\varphi _{1}}{k+\cos 2\varphi _{1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a424205a08a5d3ec56d7d96017f939d00e2414ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.935ex; height:6.509ex;" alt="{\displaystyle k_{1}={\frac {2{\sqrt {k}}}{1+k}}\qquad \tan \varphi ={\frac {\operatorname {sen} 2\varphi _{1}}{k+\cos 2\varphi _{1}}}}"></span></dd></dl> <p>Entonces existe una relación simple entre las integrales elípticas incompletas asociadas a los parámetros (<i>k</i><sub>1</sub>,φ<sub>1</sub>) y (<i>k</i>,φ) dada por: </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(k,\varphi )=\int _{0}^{\varphi }{\frac {d\theta }{\sqrt {1-k^{2}\operatorname {sen} ^{2}\theta }}}={\frac {1}{1+k}}\int _{0}^{\varphi _{1}}{\frac {d\theta _{1}}{\sqrt {1-k_{1}^{2}\operatorname {sen} ^{2}\theta _{1}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>θ</mi> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sen</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mi>k</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>sen</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(k,\varphi )=\int _{0}^{\varphi }{\frac {d\theta }{\sqrt {1-k^{2}\operatorname {sen} ^{2}\theta }}}={\frac {1}{1+k}}\int _{0}^{\varphi _{1}}{\frac {d\theta _{1}}{\sqrt {1-k_{1}^{2}\operatorname {sen} ^{2}\theta _{1}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10e15a288371a65e221ccd6cb9edeb826855a0a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:61.329ex; height:8.176ex;" alt="{\displaystyle F(k,\varphi )=\int _{0}^{\varphi }{\frac {d\theta }{\sqrt {1-k^{2}\operatorname {sen} ^{2}\theta }}}={\frac {1}{1+k}}\int _{0}^{\varphi _{1}}{\frac {d\theta _{1}}{\sqrt {1-k_{1}^{2}\operatorname {sen} ^{2}\theta _{1}}}}}"></span></dd></dl> <p>Este resultado puede aplicarse iterativamente para calcular las integrales elípticas incompletas en términos de funciones elementales y límites. Si definimos las sucesiones: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{i}={\frac {2{\sqrt {k_{i+1}}}}{1+k_{i}}}\qquad \tan \varphi _{i}={\frac {\operatorname {sen} 2\varphi _{i+1}}{k+\cos 2\varphi _{i+1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </msqrt> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sen</mi> <mo>⁡</mo> <mn>2</mn> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mn>2</mn> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{i}={\frac {2{\sqrt {k_{i+1}}}}{1+k_{i}}}\qquad \tan \varphi _{i}={\frac {\operatorname {sen} 2\varphi _{i+1}}{k+\cos 2\varphi _{i+1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c9ba5f8e8f94286845ca04c1e3a101903dae37e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.709ex; height:6.843ex;" alt="{\displaystyle k_{i}={\frac {2{\sqrt {k_{i+1}}}}{1+k_{i}}}\qquad \tan \varphi _{i}={\frac {\operatorname {sen} 2\varphi _{i+1}}{k+\cos 2\varphi _{i+1}}}}"></span></dd></dl> <p>Entonces tenemos 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 F(k_{0},\varphi _{0})={\sqrt {\frac {k_{1}k_{2}k_{3}\dots }{k_{0}}}}\int _{0}^{\Phi }{\frac {d\theta }{\sqrt {1-\operatorname {sen} ^{2}\theta }}}={\sqrt {\frac {k_{1}k_{2}k_{3}\dots }{k_{0}}}}\ln \tan \left({\frac {\pi }{4}}+{\frac {\Phi }{2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>…</mo> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </msqrt> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Φ</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>θ</mi> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>sen</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>…</mo> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </msqrt> </mrow> <mi>ln</mi> <mo>⁡</mo> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">Φ</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(k_{0},\varphi _{0})={\sqrt {\frac {k_{1}k_{2}k_{3}\dots }{k_{0}}}}\int _{0}^{\Phi }{\frac {d\theta }{\sqrt {1-\operatorname {sen} ^{2}\theta }}}={\sqrt {\frac {k_{1}k_{2}k_{3}\dots }{k_{0}}}}\ln \tan \left({\frac {\pi }{4}}+{\frac {\Phi }{2}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65bb023739277399b175082db19b88799076426b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:76.364ex; height:7.509ex;" alt="{\displaystyle F(k_{0},\varphi _{0})={\sqrt {\frac {k_{1}k_{2}k_{3}\dots }{k_{0}}}}\int _{0}^{\Phi }{\frac {d\theta }{\sqrt {1-\operatorname {sen} ^{2}\theta }}}={\sqrt {\frac {k_{1}k_{2}k_{3}\dots }{k_{0}}}}\ln \tan \left({\frac {\pi }{4}}+{\frac {\Phi }{2}}\right)}"></span></dd></dl> <p>Donde: </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 \Phi =\lim _{k\to \infty }\varphi _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi =\lim _{k\to \infty }\varphi _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6dfa640b2e109786d691e26fa9f200ca0fe69a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.915ex; height:3.843ex;" alt="{\displaystyle \Phi =\lim _{k\to \infty }\varphi _{k}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Integral_elíptica_de_segunda_especie"><span id="Integral_el.C3.ADptica_de_segunda_especie"></span>Integral elíptica de segunda especie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral_el%C3%ADptica&action=edit&section=7" title="Editar sección: Integral elíptica de segunda especie"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Una <b>Integral elíptica de segunda especie</b> es un caso particular de la integral elíptica. </p> <div class="mw-heading mw-heading3"><h3 id="Integral_elíptica_completa_de_segunda_especie"><span id="Integral_el.C3.ADptica_completa_de_segunda_especie"></span>Integral elíptica completa de segunda especie</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral_el%C3%ADptica&action=edit&section=8" title="Editar sección: Integral elíptica completa de segunda especie"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La integral elíptica completa de segunda especie <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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> se define 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 E(x)=\int _{0}^{\pi /2}{\sqrt {1-x^{2}\operatorname {sen} ^{2}\theta }}\ d\theta =\int _{0}^{1}{\frac {\sqrt {1-x^{2}t^{2}}}{\sqrt {1-t^{2}}}}\ dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sen</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mrow> <mtext> </mtext> <mi>d</mi> <mi>θ</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mtext> </mtext> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(x)=\int _{0}^{\pi /2}{\sqrt {1-x^{2}\operatorname {sen} ^{2}\theta }}\ d\theta =\int _{0}^{1}{\frac {\sqrt {1-x^{2}t^{2}}}{\sqrt {1-t^{2}}}}\ dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2c35e5f26c29609dab1b3051701f444f80ccccf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:52.521ex; height:7.509ex;" alt="{\displaystyle E(x)=\int _{0}^{\pi /2}{\sqrt {1-x^{2}\operatorname {sen} ^{2}\theta }}\ d\theta =\int _{0}^{1}{\frac {\sqrt {1-x^{2}t^{2}}}{\sqrt {1-t^{2}}}}\ dt}"></span></dd></dl> <p>La integral elíptica de segunda especie puede expresarse como la serie de potencias </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 E(x)={\frac {\pi }{2}}\sum _{n=0}^{\infty }\left({\frac {(2n)!}{2^{2n}(n!)^{2}}}\right)^{2}{\frac {x^{2n}}{1-2n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo>!</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mrow> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(x)={\frac {\pi }{2}}\sum _{n=0}^{\infty }\left({\frac {(2n)!}{2^{2n}(n!)^{2}}}\right)^{2}{\frac {x^{2n}}{1-2n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb7da895c4162b77784857f08f88486ccb7c6a86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.386ex; height:8.009ex;" alt="{\displaystyle E(x)={\frac {\pi }{2}}\sum _{n=0}^{\infty }\left({\frac {(2n)!}{2^{2n}(n!)^{2}}}\right)^{2}{\frac {x^{2n}}{1-2n}}}"></span></dd></dl> <p>que es equivalente 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 E(x)={\frac {\pi }{2}}\left(1-\left({\frac {1}{2}}\right)^{2}x^{2}-\left({\frac {1\cdot 3}{2\cdot 4}}\right)^{2}{\frac {x^{4}}{3}}-\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right)^{2}{\frac {x^{6}}{5}}-\dots \left({\frac {(2n-1)!!}{(2n)!!}}\right)^{2}{\frac {x^{2n}}{2n-1}}-\dots \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>3</mn> </mrow> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>4</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>3</mn> <mo>⋅</mo> <mn>5</mn> </mrow> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>4</mn> <mo>⋅</mo> <mn>6</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mn>5</mn> </mfrac> </mrow> <mo>−</mo> <mo>…</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mrow> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <mo>…</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(x)={\frac {\pi }{2}}\left(1-\left({\frac {1}{2}}\right)^{2}x^{2}-\left({\frac {1\cdot 3}{2\cdot 4}}\right)^{2}{\frac {x^{4}}{3}}-\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right)^{2}{\frac {x^{6}}{5}}-\dots \left({\frac {(2n-1)!!}{(2n)!!}}\right)^{2}{\frac {x^{2n}}{2n-1}}-\dots \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de0b3534a08543910c2cb45dbde461b5be97c8f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:94.589ex; height:7.509ex;" alt="{\displaystyle E(x)={\frac {\pi }{2}}\left(1-\left({\frac {1}{2}}\right)^{2}x^{2}-\left({\frac {1\cdot 3}{2\cdot 4}}\right)^{2}{\frac {x^{4}}{3}}-\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right)^{2}{\frac {x^{6}}{5}}-\dots \left({\frac {(2n-1)!!}{(2n)!!}}\right)^{2}{\frac {x^{2n}}{2n-1}}-\dots \right)}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Derivada_y_ecuación_diferencial"><span id="Derivada_y_ecuaci.C3.B3n_diferencial"></span>Derivada y ecuación diferencial</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral_el%C3%ADptica&action=edit&section=9" title="Editar sección: Derivada y ecuación diferencial"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\frac {dE(x)}{dx}}={\frac {E(x)-K(x)}{x}}\\&(x^{2}-1){\frac {d}{dx}}\left(x{\frac {dE(x)}{dx}}\right)=xE(x)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>E</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>x</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>E</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>x</mi> <mi>E</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\frac {dE(x)}{dx}}={\frac {E(x)-K(x)}{x}}\\&(x^{2}-1){\frac {d}{dx}}\left(x{\frac {dE(x)}{dx}}\right)=xE(x)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ad6b41ad363e71d88d01c6c8e804b90f0b8256" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:33.777ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}&{\frac {dE(x)}{dx}}={\frac {E(x)-K(x)}{x}}\\&(x^{2}-1){\frac {d}{dx}}\left(x{\frac {dE(x)}{dx}}\right)=xE(x)\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Integral_elíptica_incompleta_de_segunda_especie"><span id="Integral_el.C3.ADptica_incompleta_de_segunda_especie"></span>Integral elíptica incompleta de segunda especie</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral_el%C3%ADptica&action=edit&section=10" title="Editar sección: Integral elíptica incompleta de segunda especie"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La integral elíptica incompleta de segunda especie es una función de dos variables que generaliza a la integral completa: </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 E(x,\varphi )=\int _{0}^{\varphi }{\sqrt {1-x^{2}\operatorname {sen} ^{2}\theta }}\;d\theta =\int _{0}^{\operatorname {sen} \varphi }{\frac {\sqrt {1-x^{2}t^{2}}}{\sqrt {1-t^{2}}}}\;dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sen</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mrow> <mspace width="thickmathspace" /> <mi>d</mi> <mi>θ</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>sen</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mspace width="thickmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(x,\varphi )=\int _{0}^{\varphi }{\sqrt {1-x^{2}\operatorname {sen} ^{2}\theta }}\;d\theta =\int _{0}^{\operatorname {sen} \varphi }{\frac {\sqrt {1-x^{2}t^{2}}}{\sqrt {1-t^{2}}}}\;dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6ef43f2888b810582c10982337e4e8ac8510b26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:56.625ex; height:7.509ex;" alt="{\displaystyle E(x,\varphi )=\int _{0}^{\varphi }{\sqrt {1-x^{2}\operatorname {sen} ^{2}\theta }}\;d\theta =\int _{0}^{\operatorname {sen} \varphi }{\frac {\sqrt {1-x^{2}t^{2}}}{\sqrt {1-t^{2}}}}\;dt}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Integral_elíptica_de_tercera_especie"><span id="Integral_el.C3.ADptica_de_tercera_especie"></span>Integral elíptica de tercera especie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral_el%C3%ADptica&action=edit&section=11" title="Editar sección: Integral elíptica de tercera especie"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Una <b>integral elíptica de tercera especie</b> es un caso particular de la <b>integral elíptica</b>. Sea <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 0<k^{2}<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<k^{2}<1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85966f1763424e4e03f2986dfc89c8c0d2299d11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.787ex; height:2.676ex;" alt="{\displaystyle 0<k^{2}<1}"></span>, la integral elíptica completa de tercera especie se define 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 \Pi (n,k)=\int _{0}^{\frac {\pi }{2}}{d\theta \over (1-n\operatorname {sen} ^{2}\theta ){\sqrt {1-k^{2}\operatorname {sen} ^{2}\theta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Π</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>θ</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>n</mi> <msup> <mi>sen</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sen</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi (n,k)=\int _{0}^{\frac {\pi }{2}}{d\theta \over (1-n\operatorname {sen} ^{2}\theta ){\sqrt {1-k^{2}\operatorname {sen} ^{2}\theta }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4d4940ddb0141a3aeb304575bc2e4536d79b6da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:43.93ex; height:7.343ex;" alt="{\displaystyle \Pi (n,k)=\int _{0}^{\frac {\pi }{2}}{d\theta \over (1-n\operatorname {sen} ^{2}\theta ){\sqrt {1-k^{2}\operatorname {sen} ^{2}\theta }}}}"></span></dd></dl> <p>donde <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> es una constante. </p> <div class="mw-heading mw-heading3"><h3 id="Aplicaciones">Aplicaciones</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral_el%C3%ADptica&action=edit&section=12" title="Editar sección: Aplicaciones"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Las integrales elípticas de tercera especie aparecen de modo natural en la integración de las <a href="/wiki/Ecuaci%C3%B3n_de_movimiento" title="Ecuación de movimiento">ecuaciones de movimiento</a> de un <a href="/wiki/P%C3%A9ndulo#péndulo_esférico" title="Péndulo">péndulo esférico</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Véase_también"><span id="V.C3.A9ase_tambi.C3.A9n"></span>Véase también</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral_el%C3%ADptica&action=edit&section=13" title="Editar sección: Véase también"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Curva_el%C3%ADptica" title="Curva elíptica">Curva elíptica</a></li> <li><a href="/wiki/Funci%C3%B3n_el%C3%ADptica_de_Jacobi" title="Función elíptica de Jacobi">Función elíptica de Jacobi</a></li> <li><a href="/wiki/Funciones_el%C3%ADpticas_de_Weierstrass" class="mw-redirect" title="Funciones elípticas de Weierstrass">Funciones elípticas de Weierstrass</a></li></ul> <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=Integral_el%C3%ADptica&action=edit&section=14" title="Editar sección: Referencias"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="listaref" style="list-style-type: decimal;"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Abramowitz y Stegun, 1972, p. 589.</span> </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="Bibliografía"><span id="Bibliograf.C3.ADa"></span>Bibliografía</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integral_el%C3%ADptica&action=edit&section=15" title="Editar sección: Bibliografía"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span id="Reference-Mathworld-Integral_elíptica" 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/EllipticIntegral.html">«Integral elíptica»</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&rfr_id=info%3Asid%2Fes.wikipedia.org%3AIntegral+el%C3%ADptica&rft.atitle=Integral+el%C3%ADptica&rft.au=Weisstein%2C+Eric+W&rft.aulast=Weisstein%2C+Eric+W&rft.genre=article&rft.jtitle=MathWorld&rft.pub=Wolfram+Research&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FEllipticIntegral.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></li> <li>Abramowitz, M. & Stegun, I. A. (eds.): "Elliptic Integrals", Ch. 17. 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