Elliptic integral - Wikipedia

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Elliptic integral - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"6d71f516-2796-449c-b36c-3bd9bc72bbaa","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Elliptic_integral","wgTitle":"Elliptic integral","wgCurRevisionId":1251384191,"wgRevisionId":1251384191,"wgArticleId":9960,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["CS1 German-language sources (de)","Articles with short description","Short description is different from Wikidata","Use American English from January 2019","All Wikipedia articles written in American English","All articles with unsourced statements","Articles with unsourced statements from February 2024","Articles with unsourced statements from January 2017","Commons category link is on Wikidata","Elliptic functions","Special hypergeometric functions"],"wgPageViewLanguage":"en", "wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Elliptic_integral","wgRelevantArticleId":9960,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":40000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q1126603", "wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.math.styles":"ready","ext.cite.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","ext.scribunto.logs","site","mediawiki.page.ready","jquery.makeCollapsible", "mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.quicksurveys.init","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Elliptic integral - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Elliptic_integral"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Elliptic_integral&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Elliptic_integral"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Elliptic_integral rootpage-Elliptic_integral skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Elliptic+integral" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Elliptic+integral" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Elliptic+integral" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Elliptic+integral" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Argument_notation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Argument_notation"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Argument notation</span> </div> </a> <ul id="toc-Argument_notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Incomplete_elliptic_integral_of_the_first_kind" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Incomplete_elliptic_integral_of_the_first_kind"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Incomplete elliptic integral of the first kind</span> </div> </a> <ul id="toc-Incomplete_elliptic_integral_of_the_first_kind-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Incomplete_elliptic_integral_of_the_second_kind" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Incomplete_elliptic_integral_of_the_second_kind"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Incomplete elliptic integral of the second kind</span> </div> </a> <ul id="toc-Incomplete_elliptic_integral_of_the_second_kind-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Incomplete_elliptic_integral_of_the_third_kind" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Incomplete_elliptic_integral_of_the_third_kind"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Incomplete elliptic integral of the third kind</span> </div> </a> <ul id="toc-Incomplete_elliptic_integral_of_the_third_kind-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complete_elliptic_integral_of_the_first_kind" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Complete_elliptic_integral_of_the_first_kind"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Complete elliptic integral of the first kind</span> </div> </a> <button aria-controls="toc-Complete_elliptic_integral_of_the_first_kind-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>Toggle Complete elliptic integral of the first kind subsection</span> </button> <ul id="toc-Complete_elliptic_integral_of_the_first_kind-sublist" class="vector-toc-list"> <li id="toc-Relation_to_the_gamma_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_to_the_gamma_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Relation to the gamma function</span> </div> </a> <ul id="toc-Relation_to_the_gamma_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Asymptotic_expressions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Asymptotic_expressions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Asymptotic expressions</span> </div> </a> <ul id="toc-Asymptotic_expressions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Differential_equation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Differential_equation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Differential equation</span> </div> </a> <ul id="toc-Differential_equation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Continued_fraction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Continued_fraction"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Continued fraction</span> </div> </a> <ul id="toc-Continued_fraction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inverting_the_period_ratio" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inverting_the_period_ratio"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Inverting the period ratio</span> </div> </a> <ul id="toc-Inverting_the_period_ratio-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Complete_elliptic_integral_of_the_second_kind" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Complete_elliptic_integral_of_the_second_kind"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Complete elliptic integral of the second kind</span> </div> </a> <button aria-controls="toc-Complete_elliptic_integral_of_the_second_kind-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>Toggle Complete elliptic integral of the second kind subsection</span> </button> <ul id="toc-Complete_elliptic_integral_of_the_second_kind-sublist" class="vector-toc-list"> <li id="toc-Computation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computation"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Computation</span> </div> </a> <ul id="toc-Computation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivative_and_differential_equation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivative_and_differential_equation"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Derivative and differential equation</span> </div> </a> <ul id="toc-Derivative_and_differential_equation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Complete_elliptic_integral_of_the_third_kind" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Complete_elliptic_integral_of_the_third_kind"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Complete elliptic integral of the third kind</span> </div> </a> <button aria-controls="toc-Complete_elliptic_integral_of_the_third_kind-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>Toggle Complete elliptic integral of the third kind subsection</span> </button> <ul id="toc-Complete_elliptic_integral_of_the_third_kind-sublist" class="vector-toc-list"> <li id="toc-Partial_derivatives" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Partial_derivatives"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Partial derivatives</span> </div> </a> <ul id="toc-Partial_derivatives-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Jacobi_zeta_function" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Jacobi_zeta_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Jacobi zeta function</span> </div> </a> <ul id="toc-Jacobi_zeta_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Legendre's_relation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Legendre's_relation"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Legendre's relation</span> </div> </a> <button aria-controls="toc-Legendre's_relation-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>Toggle Legendre's relation subsection</span> </button> <ul id="toc-Legendre's_relation-sublist" class="vector-toc-list"> <li id="toc-Special_identity_for_the_lemniscatic_case" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Special_identity_for_the_lemniscatic_case"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Special identity for the lemniscatic case</span> </div> </a> <ul id="toc-Special_identity_for_the_lemniscatic_case-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalization_for_the_overall_case" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generalization_for_the_overall_case"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.2</span> <span>Generalization for the overall case</span> </div> </a> <ul id="toc-Generalization_for_the_overall_case-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>References</span> </div> </a> <button aria-controls="toc-References-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>Toggle References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.1</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#References_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.2</span> <span>References</span> </div> </a> <ul id="toc-References_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.3</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" 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="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</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">Elliptic integral</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="Go to an article in another language. Available in 26 languages" > <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 languages</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 – Arabic" lang="ar" hreflang="ar" data-title="تكامل ناقصي" data-language-autonym="العربية" data-language-local-name="Arabic" 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 – Azerbaijani" lang="az" hreflang="az" data-title="Elliptik inteqral" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" 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 – Catalan" lang="ca" hreflang="ca" data-title="Integral el·líptica" data-language-autonym="Català" data-language-local-name="Catalan" 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 – Chuvash" lang="cv" hreflang="cv" data-title="Эллипсла интеграл" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" 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 – German" lang="de" hreflang="de" data-title="Elliptische Integrale" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Integral_el%C3%ADptica" title="Integral elíptica – Spanish" lang="es" hreflang="es" data-title="Integral elíptica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</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 – Persian" lang="fa" hreflang="fa" data-title="انتگرال بیضوی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</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 – French" lang="fr" hreflang="fr" data-title="Intégrale elliptique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</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 – Korean" lang="ko" hreflang="ko" data-title="타원 적분" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Integrale_ellittico" title="Integrale ellittico – Italian" lang="it" hreflang="it" data-title="Integrale ellittico" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</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 – Hebrew" lang="he" hreflang="he" data-title="אינטגרל אליפטי" data-language-autonym="עברית" data-language-local-name="Hebrew" 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 – Hungarian" lang="hu" hreflang="hu" data-title="Elliptikus integrál" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Elliptische_integraal" title="Elliptische integraal – Dutch" lang="nl" hreflang="nl" data-title="Elliptische integraal" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</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 – Japanese" lang="ja" hreflang="ja" data-title="楕円積分" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Elliptisk_integral" title="Elliptisk integral – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Elliptisk integral" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Elliptisk_integral" title="Elliptisk integral – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Elliptisk integral" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</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 – Polish" lang="pl" hreflang="pl" data-title="Całki eliptyczne" data-language-autonym="Polski" data-language-local-name="Polish" 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 – Portuguese" lang="pt" hreflang="pt" data-title="Integral elíptica" data-language-autonym="Português" data-language-local-name="Portuguese" 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 – Romanian" lang="ro" hreflang="ro" data-title="Integrală eliptică" data-language-autonym="Română" data-language-local-name="Romanian" 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 – Russian" lang="ru" hreflang="ru" data-title="Эллиптический интеграл" data-language-autonym="Русский" data-language-local-name="Russian" 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 – Serbian" lang="sr" hreflang="sr" data-title="Елиптички интеграли" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Elliptinen_integraali" title="Elliptinen integraali – Finnish" lang="fi" hreflang="fi" data-title="Elliptinen integraali" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Elliptisk_integral" title="Elliptisk integral – Swedish" lang="sv" hreflang="sv" data-title="Elliptisk integral" data-language-autonym="Svenska" data-language-local-name="Swedish" 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 – Ukrainian" lang="uk" hreflang="uk" data-title="Еліптичні інтеграли" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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 – Chinese" lang="zh" hreflang="zh" data-title="椭圆积分" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1126603#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Elliptic_integral" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Elliptic_integral" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Elliptic_integral"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Elliptic_integral&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Elliptic_integral&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Elliptic_integral"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Elliptic_integral&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Elliptic_integral&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Elliptic_integral" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Elliptic_integral" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Elliptic_integral&oldid=1251384191" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Elliptic_integral&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Elliptic_integral&id=1251384191&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FElliptic_integral"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FElliptic_integral"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Elliptic_integral&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Elliptic_integral&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Elliptic_integral" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1126603" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <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">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</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">From Wikipedia, the free encyclopedia</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="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Special function defined by an integral</div> <p> In <a href="/wiki/Integral_calculus" class="mw-redirect" title="Integral calculus">integral calculus</a>, an <b>elliptic integral</b> is one of a number of related functions defined as the value of certain integrals, which were first studied by <a href="/wiki/Giulio_Carlo_de%27_Toschi_di_Fagnano" title="Giulio Carlo de' Toschi di Fagnano">Giulio Fagnano</a> and <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> (<abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 1750</span>). Their name originates from their originally arising in connection with the problem of finding the <a href="/wiki/Arc_length" title="Arc length">arc length</a> of an <a href="/wiki/Ellipse" title="Ellipse">ellipse</a>. </p><p>Modern mathematics defines an "elliptic integral" as any <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> <span class="texhtml"><i>f</i></span> which can be expressed in the form </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\int _{c}^{x}R{\left({\textstyle 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 class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <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> </mstyle> </mrow> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\int _{c}^{x}R{\left({\textstyle t,{\sqrt {P(t)}}}\right)}\,dt,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc8899bb757ea36132340346c8ce30eeecf471de" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.115ex; height:5.843ex;" alt="{\displaystyle f(x)=\int _{c}^{x}R{\left({\textstyle t,{\sqrt {P(t)}}}\right)}\,dt,}"></span> </p><p>where <span class="texhtml"><i>R</i></span> is a <a href="/wiki/Rational_function" title="Rational function">rational function</a> of its two arguments, <span class="texhtml"><i>P</i></span> is a <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> of degree 3 or 4 with no repeated roots, and <span class="texhtml"><i>c</i></span> is a constant. </p><p>In general, integrals in this form cannot be expressed in terms of <a href="/wiki/Elementary_function" title="Elementary function">elementary functions</a>. Exceptions to this general rule are when <span class="texhtml"><i>P</i></span> has repeated roots, or when <span class="texhtml"><i>R</i>(<i>x</i>, <i>y</i>)</span> contains no odd powers of <span class="texhtml"><i>y</i></span> or if the integral is pseudo-elliptic. However, with the appropriate <a href="/wiki/Integration_by_reduction_formulae" title="Integration by reduction formulae">reduction formula</a>, every elliptic integral can be brought into a form that involves integrals over rational functions and the three <a href="/wiki/Legendre_form" title="Legendre form">Legendre canonical forms</a>, also known as the elliptic integrals of the first, second and third kind. </p><p>Besides the Legendre form given below, the elliptic integrals may also be expressed in <a href="/wiki/Carlson_symmetric_form" title="Carlson symmetric form">Carlson symmetric form</a>. Additional insight into the theory of the elliptic integral may be gained through the study of the <a href="/wiki/Schwarz%E2%80%93Christoffel_mapping" title="Schwarz–Christoffel mapping">Schwarz–Christoffel mapping</a>. Historically, <a href="/wiki/Elliptic_functions" class="mw-redirect" title="Elliptic functions">elliptic functions</a> were discovered as inverse functions of elliptic integrals. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Argument_notation">Argument notation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elliptic_integral&action=edit&section=1" title="Edit section: Argument notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Incomplete elliptic integrals</i> are functions of two arguments; <i>complete elliptic integrals</i> are functions of a single argument. These arguments are expressed in a variety of different but equivalent ways as they give the same elliptic integral. Most texts adhere to a canonical naming scheme, using the following naming conventions. </p><p>For expressing one argument: </p> <ul><li><span class="texhtml"><i>α</i></span>, the <i><a href="/wiki/Modular_angle" class="mw-redirect" title="Modular angle">modular angle</a></i></li> <li><span class="texhtml"><i>k</i> = sin <i>α</i></span>, the <i>elliptic modulus</i> or <i><a href="/wiki/Eccentricity_(mathematics)" title="Eccentricity (mathematics)">eccentricity</a></i></li> <li><span class="texhtml"><i>m</i> = <i>k</i><sup>2</sup> = sin<sup>2</sup> <i>α</i></span>, the <i>parameter</i></li></ul> <p>Each of the above three quantities is completely determined by any of the others (given that they are non-negative). Thus, they can be used interchangeably. </p><p>The other argument can likewise be expressed as <span class="texhtml"><i>φ</i></span>, the <i>amplitude</i>, or as <span class="texhtml"><i>x</i></span> or <span class="texhtml"><i>u</i></span>, where <span class="texhtml"><i>x</i> = sin <i>φ</i> = sn <i>u</i></span> and <span class="texhtml">sn</span> is one of the <a href="/wiki/Jacobian_elliptic_functions" class="mw-redirect" title="Jacobian elliptic functions">Jacobian elliptic functions</a>. </p><p>Specifying the value of any one of these quantities determines the others. Note that <span class="texhtml"><i>u</i></span> also depends on <span class="texhtml"><i>m</i></span>. Some additional relationships involving <span class="texhtml"><i>u</i></span> include <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \varphi =\operatorname {cn} u,\quad {\textrm {and}}\quad {\sqrt {1-m\sin ^{2}\varphi }}=\operatorname {dn} u.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mi>cn</mi> <mo>⁡</mo> <mi>u</mi> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mi>m</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>φ</mi> </msqrt> </mrow> <mo>=</mo> <mi>dn</mi> <mo>⁡</mo> <mi>u</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \varphi =\operatorname {cn} u,\quad {\textrm {and}}\quad {\sqrt {1-m\sin ^{2}\varphi }}=\operatorname {dn} u.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fce984b4ecde9420b4bb2b28fe48ada78b50d74a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:44.203ex; height:4.843ex;" alt="{\displaystyle \cos \varphi =\operatorname {cn} u,\quad {\textrm {and}}\quad {\sqrt {1-m\sin ^{2}\varphi }}=\operatorname {dn} u.}"></span> </p><p>The latter is sometimes called the <i>delta amplitude</i> and written as <span class="texhtml">Δ(<i>φ</i>) = dn <i>u</i></span>. Sometimes the literature also refers to the <i>complementary parameter</i>, the <i>complementary modulus,</i> or the <i>complementary modular angle</i>. These are further defined in the article on <a href="/wiki/Quarter_period" title="Quarter period">quarter periods</a>. </p><p>In this notation, the use of a vertical bar as delimiter indicates that the argument following it is the "parameter" (as defined above), while the backslash indicates that it is the modular angle. The use of a semicolon implies that the argument preceding it is the sine of the amplitude: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(\varphi ,\sin \alpha )=F\left(\varphi \mid \sin ^{2}\alpha \right)=F(\varphi \setminus \alpha )=F(\sin \varphi ;\sin \alpha ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>,</mo> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> <mrow> <mo>(</mo> <mrow> <mi>φ</mi> <mo>∣</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>α</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo class="MJX-variant">∖</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>;</mo> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(\varphi ,\sin \alpha )=F\left(\varphi \mid \sin ^{2}\alpha \right)=F(\varphi \setminus \alpha )=F(\sin \varphi ;\sin \alpha ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ddcfdeb9a0882621c568c70e82f0addde887722" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:57.106ex; height:3.343ex;" alt="{\displaystyle F(\varphi ,\sin \alpha )=F\left(\varphi \mid \sin ^{2}\alpha \right)=F(\varphi \setminus \alpha )=F(\sin \varphi ;\sin \alpha ).}"></span> This potentially confusing use of different argument delimiters is traditional in elliptic integrals and much of the notation is compatible with that used in the reference book by <a href="/wiki/Abramowitz_and_Stegun" title="Abramowitz and Stegun">Abramowitz and Stegun</a> and that used in the integral tables by <a href="/wiki/Gradshteyn_and_Ryzhik" title="Gradshteyn and Ryzhik">Gradshteyn and Ryzhik</a>. </p><p>There are still other conventions for the notation of elliptic integrals employed in the literature. The notation with interchanged arguments, <span class="texhtml"><i>F</i>(<i>k</i>, <i>φ</i>)</span>, is often encountered; and similarly <span class="texhtml"><i>E</i>(<i>k</i>, <i>φ</i>)</span> for the integral of the second kind. <a href="/wiki/Abramowitz_and_Stegun" title="Abramowitz and Stegun">Abramowitz and Stegun</a> substitute the integral of the first kind, <span class="texhtml"><i>F</i>(<i>φ</i>, <i>k</i>)</span>, for the argument <span class="texhtml mvar" style="font-style:italic;">φ</span> in their definition of the integrals of the second and third kinds, unless this argument is followed by a vertical bar: i.e. <span class="texhtml"><i>E</i>(<i>F</i>(<i>φ</i>, <i>k</i>) | <i>k</i><sup>2</sup>)</span> for <span class="texhtml"><i>E</i>(<i>φ</i> | <i>k</i><sup>2</sup>)</span>. Moreover, their complete integrals employ the <i>parameter</i> <span class="texhtml"><i>k</i><sup>2</sup></span> as argument in place of the modulus <span class="texhtml"><i>k</i></span>, i.e. <span class="texhtml"><i>K</i>(<i>k</i><sup>2</sup>)</span> rather than <span class="texhtml"><i>K</i>(<i>k</i>)</span>. And the integral of the third kind defined by <a href="/wiki/Gradshteyn_and_Ryzhik" title="Gradshteyn and Ryzhik">Gradshteyn and Ryzhik</a>, <span class="texhtml">Π(<i>φ</i>, <i>n</i>, <i>k</i>)</span>, puts the amplitude <span class="texhtml mvar" style="font-style:italic;">φ</span> first and not the "characteristic" <span class="texhtml mvar" style="font-style:italic;">n</span>. </p><p>Thus one must be careful with the notation when using these functions, because various reputable references and software packages use different conventions in the definitions of the elliptic functions. For example, <a href="/wiki/Wolfram_Research" title="Wolfram Research">Wolfram</a>'s <a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a> software and <a href="/wiki/Wolfram_Alpha" class="mw-redirect" title="Wolfram Alpha">Wolfram Alpha</a> define the complete elliptic integral of the first kind in terms of the parameter <span class="texhtml"><i>m</i></span>, instead of the elliptic modulus <span class="texhtml"><i>k</i></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Incomplete_elliptic_integral_of_the_first_kind">Incomplete elliptic integral of the first kind</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elliptic_integral&action=edit&section=2" title="Edit section: Incomplete elliptic integral of the first kind"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>incomplete elliptic integral of the first kind</b> <span class="texhtml mvar" style="font-style:italic;">F</span> is defined as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(\varphi ,k)=F\left(\varphi \mid k^{2}\right)=F(\sin \varphi ;k)=\int _{0}^{\varphi }{\frac {d\theta }{\sqrt {1-k^{2}\sin ^{2}\theta }}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> <mrow> <mo>(</mo> <mrow> <mi>φ</mi> <mo>∣</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</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"> <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>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(\varphi ,k)=F\left(\varphi \mid k^{2}\right)=F(\sin \varphi ;k)=\int _{0}^{\varphi }{\frac {d\theta }{\sqrt {1-k^{2}\sin ^{2}\theta }}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0efeeeb0c4f8a40198d3b907a622c719eaa1751f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:57.157ex; height:6.676ex;" alt="{\displaystyle F(\varphi ,k)=F\left(\varphi \mid k^{2}\right)=F(\sin \varphi ;k)=\int _{0}^{\varphi }{\frac {d\theta }{\sqrt {1-k^{2}\sin ^{2}\theta }}}.}"></span> </p><p>This is Legendre's trigonometric form of the elliptic integral; substituting <span class="texhtml"><i>t</i> = sin <i>θ</i></span> and <span class="texhtml"><i>x</i> = sin <i>φ</i></span>, one obtains Jacobi's algebraic form: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x;k)=\int _{0}^{x}{\frac {dt}{\sqrt {\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</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"> <mi>x</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>t</mi> </mrow> <msqrt> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>k</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> </mrow> <mo>)</mo> </mrow> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x;k)=\int _{0}^{x}{\frac {dt}{\sqrt {\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c17190c06f2e586ea105953ee3dbbdb3ce22ade9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:36.118ex; height:6.676ex;" alt="{\displaystyle F(x;k)=\int _{0}^{x}{\frac {dt}{\sqrt {\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}}.}"></span> </p><p>Equivalently, in terms of the amplitude and modular angle one has: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(\varphi \setminus \alpha )=F(\varphi ,\sin \alpha )=\int _{0}^{\varphi }{\frac {d\theta }{\sqrt {1-\left(\sin \theta \sin \alpha \right)^{2}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo class="MJX-variant">∖</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>,</mo> <mi>sin</mi> <mo>⁡</mo> <mi>α</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> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(\varphi \setminus \alpha )=F(\varphi ,\sin \alpha )=\int _{0}^{\varphi }{\frac {d\theta }{\sqrt {1-\left(\sin \theta \sin \alpha \right)^{2}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1649e22c3dbf0c847973ec949d261584c38c5ced" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:50.066ex; height:8.176ex;" alt="{\displaystyle F(\varphi \setminus \alpha )=F(\varphi ,\sin \alpha )=\int _{0}^{\varphi }{\frac {d\theta }{\sqrt {1-\left(\sin \theta \sin \alpha \right)^{2}}}}.}"></span> </p><p>With <span class="texhtml"><i>x</i> = sn(<i>u</i>, <i>k</i>)</span> one has: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x;k)=u;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>u</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x;k)=u;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d5a1ae81d9e00edaa38aa2367bf6f6583981ac6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.2ex; height:2.843ex;" alt="{\displaystyle F(x;k)=u;}"></span> demonstrating that this <a href="/wiki/Jacobian_elliptic_function" class="mw-redirect" title="Jacobian elliptic function">Jacobian elliptic function</a> is a simple inverse of the incomplete elliptic integral of the first kind. </p><p>The incomplete elliptic integral of the first kind has following addition theorem<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (February 2024)">citation needed</span></a></i>]</sup>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F{\bigl [}\arctan(x),k{\bigr ]}+F{\bigl [}\arctan(y),k{\bigr ]}=F\left[\arctan \left({\frac {x{\sqrt {k'^{2}y^{2}+1}}}{\sqrt {y^{2}+1}}}\right)+\arctan \left({\frac {y{\sqrt {k'^{2}x^{2}+1}}}{\sqrt {x^{2}+1}}}\right),k\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> <mo>+</mo> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> <mo>=</mo> <mi>F</mi> <mrow> <mo>[</mo> <mrow> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>k</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <msqrt> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>k</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mi>k</mi> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F{\bigl [}\arctan(x),k{\bigr ]}+F{\bigl [}\arctan(y),k{\bigr ]}=F\left[\arctan \left({\frac {x{\sqrt {k'^{2}y^{2}+1}}}{\sqrt {y^{2}+1}}}\right)+\arctan \left({\frac {y{\sqrt {k'^{2}x^{2}+1}}}{\sqrt {x^{2}+1}}}\right),k\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57910fd4c508794e391acf88c5a0773167da8c3a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:94.663ex; height:7.509ex;" alt="{\displaystyle F{\bigl [}\arctan(x),k{\bigr ]}+F{\bigl [}\arctan(y),k{\bigr ]}=F\left[\arctan \left({\frac {x{\sqrt {k'^{2}y^{2}+1}}}{\sqrt {y^{2}+1}}}\right)+\arctan \left({\frac {y{\sqrt {k'^{2}x^{2}+1}}}{\sqrt {x^{2}+1}}}\right),k\right]}"></span> </p><p>The elliptic modulus can be transformed that way: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F{\bigl [}\arcsin(x),k{\bigr ]}={\frac {2}{1+{\sqrt {1-k^{2}}}}}F\left[\arcsin \left({\frac {\left(1+{\sqrt {1-k^{2}}}\right)x}{1+{\sqrt {1-k^{2}x^{2}}}}}\right),{\frac {1-{\sqrt {1-k^{2}}}}{1+{\sqrt {1-k^{2}}}}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mn>1</mn> <mo>+</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> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mi>F</mi> <mrow> <mo>[</mo> <mrow> <mi>arcsin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</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> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mi>x</mi> </mrow> <mrow> <mn>1</mn> <mo>+</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>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</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> </msqrt> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</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> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F{\bigl [}\arcsin(x),k{\bigr ]}={\frac {2}{1+{\sqrt {1-k^{2}}}}}F\left[\arcsin \left({\frac {\left(1+{\sqrt {1-k^{2}}}\right)x}{1+{\sqrt {1-k^{2}x^{2}}}}}\right),{\frac {1-{\sqrt {1-k^{2}}}}{1+{\sqrt {1-k^{2}}}}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76eea8fbafa5746e507a2b9aa62ba30683aa47ad" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:79.589ex; height:10.176ex;" alt="{\displaystyle F{\bigl [}\arcsin(x),k{\bigr ]}={\frac {2}{1+{\sqrt {1-k^{2}}}}}F\left[\arcsin \left({\frac {\left(1+{\sqrt {1-k^{2}}}\right)x}{1+{\sqrt {1-k^{2}x^{2}}}}}\right),{\frac {1-{\sqrt {1-k^{2}}}}{1+{\sqrt {1-k^{2}}}}}\right]}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Incomplete_elliptic_integral_of_the_second_kind">Incomplete elliptic integral of the second kind</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elliptic_integral&action=edit&section=3" title="Edit section: Incomplete elliptic integral of the second kind"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>incomplete elliptic integral of the second kind</b> <span class="texhtml"><i>E</i></span> in Legendre's trigonometric form is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(\varphi ,k)=E\left(\varphi \,|\,k^{2}\right)=E(\sin \varphi ;k)=\int _{0}^{\varphi }{\sqrt {1-k^{2}\sin ^{2}\theta }}\,d\theta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>E</mi> <mrow> <mo>(</mo> <mrow> <mi>φ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</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"> <mi>φ</mi> </mrow> </msubsup> <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>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>θ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(\varphi ,k)=E\left(\varphi \,|\,k^{2}\right)=E(\sin \varphi ;k)=\int _{0}^{\varphi }{\sqrt {1-k^{2}\sin ^{2}\theta }}\,d\theta .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1bcc3ab6912f07c1fefa24902e3c7b793b886fb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:58.603ex; height:5.843ex;" alt="{\displaystyle E(\varphi ,k)=E\left(\varphi \,|\,k^{2}\right)=E(\sin \varphi ;k)=\int _{0}^{\varphi }{\sqrt {1-k^{2}\sin ^{2}\theta }}\,d\theta .}"></span> </p><p>Substituting <span class="texhtml"><i>t</i> = sin <i>θ</i></span> and <span class="texhtml"><i>x</i> = sin <i>φ</i></span>, one obtains Jacobi's algebraic form: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(x;k)=\int _{0}^{x}{\frac {\sqrt {1-k^{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>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"> <mi>x</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>k</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="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(x;k)=\int _{0}^{x}{\frac {\sqrt {1-k^{2}t^{2}}}{\sqrt {1-t^{2}}}}\,dt.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/394e744a658222e47a0593d2761025ced90de7de" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.693ex; height:7.509ex;" alt="{\displaystyle E(x;k)=\int _{0}^{x}{\frac {\sqrt {1-k^{2}t^{2}}}{\sqrt {1-t^{2}}}}\,dt.}"></span> </p><p>Equivalently, in terms of the amplitude and modular angle: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(\varphi \setminus \alpha )=E(\varphi ,\sin \alpha )=\int _{0}^{\varphi }{\sqrt {1-\left(\sin \theta \sin \alpha \right)^{2}}}\,d\theta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo class="MJX-variant">∖</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>,</mo> <mi>sin</mi> <mo>⁡</mo> <mi>α</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> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>θ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(\varphi \setminus \alpha )=E(\varphi ,\sin \alpha )=\int _{0}^{\varphi }{\sqrt {1-\left(\sin \theta \sin \alpha \right)^{2}}}\,d\theta .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4660f0d98788cb54e2070786328b7410c381a9f7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:51.993ex; height:5.843ex;" alt="{\displaystyle E(\varphi \setminus \alpha )=E(\varphi ,\sin \alpha )=\int _{0}^{\varphi }{\sqrt {1-\left(\sin \theta \sin \alpha \right)^{2}}}\,d\theta .}"></span> </p><p>Relations with the <a href="/wiki/Jacobi_elliptic_functions" title="Jacobi elliptic functions">Jacobi elliptic functions</a> include <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}E{\left(\operatorname {sn} (u;k);k\right)}=\int _{0}^{u}\operatorname {dn} ^{2}(w;k)\,dw&=u-k^{2}\int _{0}^{u}\operatorname {sn} ^{2}(w;k)\,dw\\[1ex]&=\left(1-k^{2}\right)u+k^{2}\int _{0}^{u}\operatorname {cn} ^{2}(w;k)\,dw.\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="0.73em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mi>sn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>;</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>;</mo> <mi>k</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msubsup> <msup> <mi>dn</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>w</mi> <mo>;</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>w</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>u</mi> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msubsup> <msup> <mi>sn</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>w</mi> <mo>;</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>w</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>u</mi> <mo>+</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msubsup> <msup> <mi>cn</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>w</mi> <mo>;</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>w</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}E{\left(\operatorname {sn} (u;k);k\right)}=\int _{0}^{u}\operatorname {dn} ^{2}(w;k)\,dw&=u-k^{2}\int _{0}^{u}\operatorname {sn} ^{2}(w;k)\,dw\\[1ex]&=\left(1-k^{2}\right)u+k^{2}\int _{0}^{u}\operatorname {cn} ^{2}(w;k)\,dw.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3daa5ea5ba73b833fb145588e70b71f71415c222" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:69.663ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}E{\left(\operatorname {sn} (u;k);k\right)}=\int _{0}^{u}\operatorname {dn} ^{2}(w;k)\,dw&=u-k^{2}\int _{0}^{u}\operatorname {sn} ^{2}(w;k)\,dw\\[1ex]&=\left(1-k^{2}\right)u+k^{2}\int _{0}^{u}\operatorname {cn} ^{2}(w;k)\,dw.\end{aligned}}}"></span> </p><p>The <a href="/wiki/Meridian_arc" title="Meridian arc">meridian arc</a> length from the <a href="/wiki/Equator" title="Equator">equator</a> to <a href="/wiki/Latitude" title="Latitude">latitude</a> <span class="texhtml"><i>φ</i></span> is written in terms of <span class="texhtml"><i>E</i></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m(\varphi )=a\left(E(\varphi ,e)+{\frac {d^{2}}{d\varphi ^{2}}}E(\varphi ,e)\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>,</mo> <mi>e</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>,</mo> <mi>e</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m(\varphi )=a\left(E(\varphi ,e)+{\frac {d^{2}}{d\varphi ^{2}}}E(\varphi ,e)\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d00ccce41ae57e6ff1f2c2b7a2117dc07edf775" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.452ex; height:6.343ex;" alt="{\displaystyle m(\varphi )=a\left(E(\varphi ,e)+{\frac {d^{2}}{d\varphi ^{2}}}E(\varphi ,e)\right),}"></span> where <span class="texhtml"><i>a</i></span> is the <a href="/wiki/Semi-major_axis" class="mw-redirect" title="Semi-major axis">semi-major axis</a>, and <span class="texhtml"><i>e</i></span> is the <a href="/wiki/Eccentricity_(mathematics)" title="Eccentricity (mathematics)">eccentricity</a>. </p><p>The incomplete elliptic integral of the second kind has following addition theorem<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (February 2024)">citation needed</span></a></i>]</sup>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E{\left[\arctan(x),k\right]}+E{\left[\arctan(y),k\right]}=E{\left[\arctan \left({\frac {x{\sqrt {k'^{2}y^{2}+1}}}{\sqrt {y^{2}+1}}}\right)+\arctan \left({\frac {y{\sqrt {k'^{2}x^{2}+1}}}{\sqrt {x^{2}+1}}}\right),k\right]}+{\frac {k^{2}xy}{k'^{2}x^{2}y^{2}+x^{2}+y^{2}+1}}\left({\frac {x{\sqrt {k'^{2}y^{2}+1}}}{\sqrt {y^{2}+1}}}+{\frac {y{\sqrt {k'^{2}x^{2}+1}}}{\sqrt {x^{2}+1}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mrow> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>k</mi> </mrow> <mo>]</mo> </mrow> </mrow> <mo>+</mo> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mrow> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>k</mi> </mrow> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mrow> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>k</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <msqrt> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>k</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mi>k</mi> </mrow> <mo>]</mo> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> <mi>y</mi> </mrow> <mrow> <msup> <mi>k</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>k</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <msqrt> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </msqrt> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>k</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </msqrt> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E{\left[\arctan(x),k\right]}+E{\left[\arctan(y),k\right]}=E{\left[\arctan \left({\frac {x{\sqrt {k'^{2}y^{2}+1}}}{\sqrt {y^{2}+1}}}\right)+\arctan \left({\frac {y{\sqrt {k'^{2}x^{2}+1}}}{\sqrt {x^{2}+1}}}\right),k\right]}+{\frac {k^{2}xy}{k'^{2}x^{2}y^{2}+x^{2}+y^{2}+1}}\left({\frac {x{\sqrt {k'^{2}y^{2}+1}}}{\sqrt {y^{2}+1}}}+{\frac {y{\sqrt {k'^{2}x^{2}+1}}}{\sqrt {x^{2}+1}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7296585a0f08d2d545ea65df758a897730f2db9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:151.345ex; height:7.509ex;" alt="{\displaystyle E{\left[\arctan(x),k\right]}+E{\left[\arctan(y),k\right]}=E{\left[\arctan \left({\frac {x{\sqrt {k'^{2}y^{2}+1}}}{\sqrt {y^{2}+1}}}\right)+\arctan \left({\frac {y{\sqrt {k'^{2}x^{2}+1}}}{\sqrt {x^{2}+1}}}\right),k\right]}+{\frac {k^{2}xy}{k'^{2}x^{2}y^{2}+x^{2}+y^{2}+1}}\left({\frac {x{\sqrt {k'^{2}y^{2}+1}}}{\sqrt {y^{2}+1}}}+{\frac {y{\sqrt {k'^{2}x^{2}+1}}}{\sqrt {x^{2}+1}}}\right)}"></span> </p><p>The elliptic modulus can be transformed that way: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E{\left[\arcsin(x),k\right]}=\left(1+{\sqrt {1-k^{2}}}\right)E{\left[\arcsin \left({\frac {\left(1+{\sqrt {1-k^{2}}}\right)x}{1+{\sqrt {1-k^{2}x^{2}}}}}\right),{\frac {1-{\sqrt {1-k^{2}}}}{1+{\sqrt {1-k^{2}}}}}\right]}-{\sqrt {1-k^{2}}}F{\left[\arcsin(x),k\right]}+{\frac {k^{2}x{\sqrt {1-x^{2}}}}{1+{\sqrt {1-k^{2}x^{2}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mrow> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>k</mi> </mrow> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</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> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mrow> <mi>arcsin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</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> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mi>x</mi> </mrow> <mrow> <mn>1</mn> <mo>+</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>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</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> </msqrt> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</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> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> </mrow> <mo>−</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> </msqrt> </mrow> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mrow> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>k</mi> </mrow> <mo>]</mo> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> <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> </msqrt> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</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>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E{\left[\arcsin(x),k\right]}=\left(1+{\sqrt {1-k^{2}}}\right)E{\left[\arcsin \left({\frac {\left(1+{\sqrt {1-k^{2}}}\right)x}{1+{\sqrt {1-k^{2}x^{2}}}}}\right),{\frac {1-{\sqrt {1-k^{2}}}}{1+{\sqrt {1-k^{2}}}}}\right]}-{\sqrt {1-k^{2}}}F{\left[\arcsin(x),k\right]}+{\frac {k^{2}x{\sqrt {1-x^{2}}}}{1+{\sqrt {1-k^{2}x^{2}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80ba9d8ee4d78dd93dfb0ad86ebdef17a98c7a1a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:125.034ex; height:10.176ex;" alt="{\displaystyle E{\left[\arcsin(x),k\right]}=\left(1+{\sqrt {1-k^{2}}}\right)E{\left[\arcsin \left({\frac {\left(1+{\sqrt {1-k^{2}}}\right)x}{1+{\sqrt {1-k^{2}x^{2}}}}}\right),{\frac {1-{\sqrt {1-k^{2}}}}{1+{\sqrt {1-k^{2}}}}}\right]}-{\sqrt {1-k^{2}}}F{\left[\arcsin(x),k\right]}+{\frac {k^{2}x{\sqrt {1-x^{2}}}}{1+{\sqrt {1-k^{2}x^{2}}}}}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Incomplete_elliptic_integral_of_the_third_kind">Incomplete elliptic integral of the third kind</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elliptic_integral&action=edit&section=4" title="Edit section: Incomplete elliptic integral of the third kind"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>incomplete elliptic integral of the third kind</b> <span class="texhtml">Π</span> is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi (n;\varphi \setminus \alpha )=\int _{0}^{\varphi }{\frac {1}{1-n\sin ^{2}\theta }}{\frac {d\theta }{\sqrt {1-\left(\sin \theta \sin \alpha \right)^{2}}}}}"> <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>φ</mi> <mo class="MJX-variant">∖</mo> <mi>α</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> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>n</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>θ</mi> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi (n;\varphi \setminus \alpha )=\int _{0}^{\varphi }{\frac {1}{1-n\sin ^{2}\theta }}{\frac {d\theta }{\sqrt {1-\left(\sin \theta \sin \alpha \right)^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7139789e847bae1ee03c9067841b54514c8f494" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:49.925ex; height:8.176ex;" alt="{\displaystyle \Pi (n;\varphi \setminus \alpha )=\int _{0}^{\varphi }{\frac {1}{1-n\sin ^{2}\theta }}{\frac {d\theta }{\sqrt {1-\left(\sin \theta \sin \alpha \right)^{2}}}}}"></span> </p><p>or </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi (n;\varphi \,|\,m)=\int _{0}^{\sin \varphi }{\frac {1}{1-nt^{2}}}{\frac {dt}{\sqrt {\left(1-mt^{2}\right)\left(1-t^{2}\right)}}}.}"> <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>φ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mi>m</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>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>n</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>t</mi> </mrow> <msqrt> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>m</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi (n;\varphi \,|\,m)=\int _{0}^{\sin \varphi }{\frac {1}{1-nt^{2}}}{\frac {dt}{\sqrt {\left(1-mt^{2}\right)\left(1-t^{2}\right)}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b72db16511d2152447edbf5f8e4029d51ba8bcdc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:50.4ex; height:7.009ex;" alt="{\displaystyle \Pi (n;\varphi \,|\,m)=\int _{0}^{\sin \varphi }{\frac {1}{1-nt^{2}}}{\frac {dt}{\sqrt {\left(1-mt^{2}\right)\left(1-t^{2}\right)}}}.}"></span> </p><p>The number <span class="texhtml"><i>n</i></span> is called the <b>characteristic</b> and can take on any value, independently of the other arguments. Note though that the value <span class="texhtml">Π(1; <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">π</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> | <i>m</i>)</span> is infinite, for any <span class="texhtml"><i>m</i></span>. </p><p>A relation with the Jacobian elliptic functions is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi {\bigl (}n;\,\operatorname {am} (u;k);\,k{\bigr )}=\int _{0}^{u}{\frac {dw}{1-n\,\operatorname {sn} ^{2}(w;k)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>n</mi> <mo>;</mo> <mspace width="thinmathspace" /> <mi>am</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>;</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>;</mo> <mspace width="thinmathspace" /> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>w</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>n</mi> <mspace width="thinmathspace" /> <msup> <mi>sn</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>w</mi> <mo>;</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi {\bigl (}n;\,\operatorname {am} (u;k);\,k{\bigr )}=\int _{0}^{u}{\frac {dw}{1-n\,\operatorname {sn} ^{2}(w;k)}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9eda78236ad85fd2e881a682787a61275215bd8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:41.562ex; height:6.176ex;" alt="{\displaystyle \Pi {\bigl (}n;\,\operatorname {am} (u;k);\,k{\bigr )}=\int _{0}^{u}{\frac {dw}{1-n\,\operatorname {sn} ^{2}(w;k)}}.}"></span> </p><p>The meridian arc length from the equator to latitude <span class="texhtml"><i>φ</i></span> is also related to a special case of <span class="texhtml">Π</span>: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m(\varphi )=a\left(1-e^{2}\right)\Pi \left(e^{2};\varphi \,|\,e^{2}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi mathvariant="normal">Π</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>;</mo> <mi>φ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m(\varphi )=a\left(1-e^{2}\right)\Pi \left(e^{2};\varphi \,|\,e^{2}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c8a1250feaaf5427a9e2e66a30239922e8d7733" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.287ex; height:3.343ex;" alt="{\displaystyle m(\varphi )=a\left(1-e^{2}\right)\Pi \left(e^{2};\varphi \,|\,e^{2}\right).}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Complete_elliptic_integral_of_the_first_kind">Complete elliptic integral of the first kind</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elliptic_integral&action=edit&section=5" title="Edit section: Complete elliptic integral of the first kind"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Mplwp_complete_ellipticKk.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Mplwp_complete_ellipticKk.svg/300px-Mplwp_complete_ellipticKk.svg.png" decoding="async" width="300" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Mplwp_complete_ellipticKk.svg/450px-Mplwp_complete_ellipticKk.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Mplwp_complete_ellipticKk.svg/600px-Mplwp_complete_ellipticKk.svg.png 2x" data-file-width="600" data-file-height="400" /></a><figcaption>Plot of the complete elliptic integral of the first kind <span class="texhtml"><i>K</i>(<i>k</i>)</span></figcaption></figure> <p>Elliptic Integrals are said to be 'complete' when the amplitude <span class="texhtml"><i>φ</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">π</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span> and therefore <span class="texhtml"><i>x</i> = 1</span>. The <b>complete elliptic integral of the first kind</b> <span class="texhtml"><i>K</i></span> may thus be defined as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(k)=\int _{0}^{\tfrac {\pi }{2}}{\frac {d\theta }{\sqrt {1-k^{2}\sin ^{2}\theta }}}=\int _{0}^{1}{\frac {dt}{\sqrt {\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</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"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </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>sin</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>t</mi> </mrow> <msqrt> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>k</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> </mrow> <mo>)</mo> </mrow> </msqrt> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(k)=\int _{0}^{\tfrac {\pi }{2}}{\frac {d\theta }{\sqrt {1-k^{2}\sin ^{2}\theta }}}=\int _{0}^{1}{\frac {dt}{\sqrt {\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/101c6a731f2c1462e215de00137a56501bd6354f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:57.124ex; height:7.676ex;" alt="{\displaystyle K(k)=\int _{0}^{\tfrac {\pi }{2}}{\frac {d\theta }{\sqrt {1-k^{2}\sin ^{2}\theta }}}=\int _{0}^{1}{\frac {dt}{\sqrt {\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}},}"></span> or more compactly in terms of the incomplete integral of the first kind as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(k)=F\left({\tfrac {\pi }{2}},k\right)=F\left({\tfrac {\pi }{2}}\,|\,k^{2}\right)=F(1;k).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mi>k</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>F</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>;</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(k)=F\left({\tfrac {\pi }{2}},k\right)=F\left({\tfrac {\pi }{2}}\,|\,k^{2}\right)=F(1;k).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3225ae0ce011c25e6c99030c5ff9a830d302f199" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:39.989ex; height:3.509ex;" alt="{\displaystyle K(k)=F\left({\tfrac {\pi }{2}},k\right)=F\left({\tfrac {\pi }{2}}\,|\,k^{2}\right)=F(1;k).}"></span> </p><p>It can be expressed as a <a href="/wiki/Power_series" title="Power series">power series</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(k)={\frac {\pi }{2}}\sum _{n=0}^{\infty }\left({\frac {(2n)!}{2^{2n}(n!)^{2}}}\right)^{2}k^{2n}={\frac {\pi }{2}}\sum _{n=0}^{\infty }{\bigl (}P_{2n}(0){\bigr )}^{2}k^{2n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>k</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>k</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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(k)={\frac {\pi }{2}}\sum _{n=0}^{\infty }\left({\frac {(2n)!}{2^{2n}(n!)^{2}}}\right)^{2}k^{2n}={\frac {\pi }{2}}\sum _{n=0}^{\infty }{\bigl (}P_{2n}(0){\bigr )}^{2}k^{2n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e2a18cd2790d207ad6f17f9d1f219a9dcf6b174" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:54.396ex; height:8.009ex;" alt="{\displaystyle K(k)={\frac {\pi }{2}}\sum _{n=0}^{\infty }\left({\frac {(2n)!}{2^{2n}(n!)^{2}}}\right)^{2}k^{2n}={\frac {\pi }{2}}\sum _{n=0}^{\infty }{\bigl (}P_{2n}(0){\bigr )}^{2}k^{2n},}"></span> </p><p>where <span class="texhtml"><i>P</i><sub><i>n</i></sub></span> is the <a href="/wiki/Legendre_polynomials" title="Legendre polynomials">Legendre polynomials</a>, which is equivalent to </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(k)={\frac {\pi }{2}}\left(1+\left({\frac {1}{2}}\right)^{2}k^{2}+\left({\frac {1\cdot 3}{2\cdot 4}}\right)^{2}k^{4}+\cdots +\left({\frac {\left(2n-1\right)!!}{\left(2n\right)!!}}\right)^{2}k^{2n}+\cdots \right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>k</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>k</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>k</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> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>!</mo> <mo>!</mo> </mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mo>)</mo> </mrow> <mo>!</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(k)={\frac {\pi }{2}}\left(1+\left({\frac {1}{2}}\right)^{2}k^{2}+\left({\frac {1\cdot 3}{2\cdot 4}}\right)^{2}k^{4}+\cdots +\left({\frac {\left(2n-1\right)!!}{\left(2n\right)!!}}\right)^{2}k^{2n}+\cdots \right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf665f8188ed4086516fbee600793645cb9b4281" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:74.813ex; height:7.509ex;" alt="{\displaystyle K(k)={\frac {\pi }{2}}\left(1+\left({\frac {1}{2}}\right)^{2}k^{2}+\left({\frac {1\cdot 3}{2\cdot 4}}\right)^{2}k^{4}+\cdots +\left({\frac {\left(2n-1\right)!!}{\left(2n\right)!!}}\right)^{2}k^{2n}+\cdots \right),}"></span> </p><p>where <span class="texhtml"><i>n</i>!!</span> denotes the <a href="/wiki/Double_factorial" title="Double factorial">double factorial</a>. In terms of the Gauss <a href="/wiki/Hypergeometric_function" title="Hypergeometric function">hypergeometric function</a>, the complete elliptic integral of the first kind can be expressed as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(k)={\tfrac {\pi }{2}}\,{}_{2}F_{1}\left({\tfrac {1}{2}},{\tfrac {1}{2}};1;k^{2}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>;</mo> <mn>1</mn> <mo>;</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(k)={\tfrac {\pi }{2}}\,{}_{2}F_{1}\left({\tfrac {1}{2}},{\tfrac {1}{2}};1;k^{2}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adfc94d8f9998f8434bc18f33641c29302bf854d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:27.35ex; height:3.509ex;" alt="{\displaystyle K(k)={\tfrac {\pi }{2}}\,{}_{2}F_{1}\left({\tfrac {1}{2}},{\tfrac {1}{2}};1;k^{2}\right).}"></span> </p><p>The complete elliptic integral of the first kind is sometimes called the <a href="/wiki/Quarter_period" title="Quarter period">quarter period</a>. It can be computed very efficiently in terms of the <a href="/wiki/Arithmetic%E2%80%93geometric_mean" title="Arithmetic–geometric mean">arithmetic–geometric mean</a>:<sup id="cite_ref-FOOTNOTECarlson201019.8_1-0" class="reference"><a href="#cite_note-FOOTNOTECarlson201019.8-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(k)={\frac {\pi }{2\operatorname {agm} \left(1,{\sqrt {1-k^{2}}}\right)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mrow> <mn>2</mn> <mi>agm</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>,</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> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(k)={\frac {\pi }{2\operatorname {agm} \left(1,{\sqrt {1-k^{2}}}\right)}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b79a7475320fe15291003ddfcb7f33b80a9cdcf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:29.042ex; height:7.343ex;" alt="{\displaystyle K(k)={\frac {\pi }{2\operatorname {agm} \left(1,{\sqrt {1-k^{2}}}\right)}}.}"></span> </p><p>Therefore, the modulus can be transformed as: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}K(k)&={\frac {\pi }{2\operatorname {agm} \left(1,{\sqrt {1-k^{2}}}\right)}}\\[4pt]&={\frac {\pi }{2\operatorname {agm} \left({\frac {1}{2}}+{\frac {\sqrt {1-k^{2}}}{2}},{\sqrt[{4}]{1-k^{2}}}\right)}}\\[4pt]&={\frac {\pi }{\left(1+{\sqrt {1-k^{2}}}\right)\operatorname {agm} \left(1,{\frac {2{\sqrt[{4}]{1-k^{2}}}}{\left(1+{\sqrt {1-k^{2}}}\right)}}\right)}}\\[4pt]&={\frac {2}{1+{\sqrt {1-k^{2}}}}}K\left({\frac {1-{\sqrt {1-k^{2}}}}{1+{\sqrt {1-k^{2}}}}}\right)\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="0.7em 0.7em 0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>K</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mrow> <mn>2</mn> <mi>agm</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>,</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> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mrow> <mn>2</mn> <mi>agm</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</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> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mi>agm</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</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> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mn>1</mn> <mo>+</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> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mi>K</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</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> </msqrt> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</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> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}K(k)&={\frac {\pi }{2\operatorname {agm} \left(1,{\sqrt {1-k^{2}}}\right)}}\\[4pt]&={\frac {\pi }{2\operatorname {agm} \left({\frac {1}{2}}+{\frac {\sqrt {1-k^{2}}}{2}},{\sqrt[{4}]{1-k^{2}}}\right)}}\\[4pt]&={\frac {\pi }{\left(1+{\sqrt {1-k^{2}}}\right)\operatorname {agm} \left(1,{\frac {2{\sqrt[{4}]{1-k^{2}}}}{\left(1+{\sqrt {1-k^{2}}}\right)}}\right)}}\\[4pt]&={\frac {2}{1+{\sqrt {1-k^{2}}}}}K\left({\frac {1-{\sqrt {1-k^{2}}}}{1+{\sqrt {1-k^{2}}}}}\right)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6746059c2ef65a4af7d2856a18b5c5ebe7d179fb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -19.505ex; width:46.842ex; height:40.176ex;" alt="{\displaystyle {\begin{aligned}K(k)&={\frac {\pi }{2\operatorname {agm} \left(1,{\sqrt {1-k^{2}}}\right)}}\\[4pt]&={\frac {\pi }{2\operatorname {agm} \left({\frac {1}{2}}+{\frac {\sqrt {1-k^{2}}}{2}},{\sqrt[{4}]{1-k^{2}}}\right)}}\\[4pt]&={\frac {\pi }{\left(1+{\sqrt {1-k^{2}}}\right)\operatorname {agm} \left(1,{\frac {2{\sqrt[{4}]{1-k^{2}}}}{\left(1+{\sqrt {1-k^{2}}}\right)}}\right)}}\\[4pt]&={\frac {2}{1+{\sqrt {1-k^{2}}}}}K\left({\frac {1-{\sqrt {1-k^{2}}}}{1+{\sqrt {1-k^{2}}}}}\right)\end{aligned}}}"></span> </p><p>This expression is valid for all <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\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }"></span> and <span class="texhtml">0 ≤ <i>k</i> ≤ 1</span>: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(k)=n\left[\sum _{a=1}^{n}\operatorname {dn} \left({\frac {2a}{n}}K(k);k\right)\right]^{-1}K\left[k^{n}\prod _{a=1}^{n}\operatorname {sn} \left({\frac {2a-1}{n}}K(k);k\right)^{2}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <msup> <mrow> <mo>[</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>dn</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>a</mi> </mrow> <mi>n</mi> </mfrac> </mrow> <mi>K</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>;</mo> <mi>k</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>K</mi> <mrow> <mo>[</mo> <mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>sn</mi> <mo>⁡</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>a</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>n</mi> </mfrac> </mrow> <mi>K</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>;</mo> <mi>k</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(k)=n\left[\sum _{a=1}^{n}\operatorname {dn} \left({\frac {2a}{n}}K(k);k\right)\right]^{-1}K\left[k^{n}\prod _{a=1}^{n}\operatorname {sn} \left({\frac {2a-1}{n}}K(k);k\right)^{2}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69e4dd3df58277155b3c0a3a472621affaaedad5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:67.903ex; height:8.009ex;" alt="{\displaystyle K(k)=n\left[\sum _{a=1}^{n}\operatorname {dn} \left({\frac {2a}{n}}K(k);k\right)\right]^{-1}K\left[k^{n}\prod _{a=1}^{n}\operatorname {sn} \left({\frac {2a-1}{n}}K(k);k\right)^{2}\right]}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Relation_to_the_gamma_function">Relation to the gamma function</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elliptic_integral&action=edit&section=6" title="Edit section: Relation to the gamma function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <span class="texhtml"><i>k</i><sup>2</sup> = <i>λ</i>(<i>i</i><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>r</i></span></span>)</span> and <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\in \mathbb {Q} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\in \mathbb {Q} ^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f609e38525092d8b949c7f68e901a339ee527b9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.208ex; height:2.843ex;" alt="{\displaystyle r\in \mathbb {Q} ^{+}}"></span> (where <span class="texhtml mvar" style="font-style:italic;">λ</span> is the <a href="/wiki/Modular_lambda_function" title="Modular lambda function">modular lambda function</a>), then <span class="texhtml"><i>K</i>(<i>k</i>)</span> is expressible in closed form in terms of the <a href="/wiki/Gamma_function" title="Gamma function">gamma function</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> For example, <span class="texhtml"><i>r</i> = 2</span>, <span class="texhtml"><i>r</i> = 3</span> and <span class="texhtml"><i>r</i> = 7</span> give, respectively,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\left({\sqrt {2}}-1\right)={\frac {\Gamma \left({\frac {1}{8}}\right)\Gamma \left({\frac {3}{8}}\right){\sqrt {{\sqrt {2}}+1}}}{8{\sqrt[{4}]{2}}{\sqrt {\pi }}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <mrow> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\left({\sqrt {2}}-1\right)={\frac {\Gamma \left({\frac {1}{8}}\right)\Gamma \left({\frac {3}{8}}\right){\sqrt {{\sqrt {2}}+1}}}{8{\sqrt[{4}]{2}}{\sqrt {\pi }}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7af68e2bb757bf83cb31b8c621daba7d08d25233" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.012ex; height:9.176ex;" alt="{\displaystyle K\left({\sqrt {2}}-1\right)={\frac {\Gamma \left({\frac {1}{8}}\right)\Gamma \left({\frac {3}{8}}\right){\sqrt {{\sqrt {2}}+1}}}{8{\sqrt[{4}]{2}}{\sqrt {\pi }}}},}"></span> </p><p>and </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\left({\frac {{\sqrt {3}}-1}{2{\sqrt {2}}}}\right)={\frac {1}{8\pi }}{\sqrt[{4}]{3}}\,{\sqrt[{3}]{4}}\,\Gamma {\biggl (}{\frac {1}{3}}{\biggr )}^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>8</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <mspace width="thinmathspace" /> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\left({\frac {{\sqrt {3}}-1}{2{\sqrt {2}}}}\right)={\frac {1}{8\pi }}{\sqrt[{4}]{3}}\,{\sqrt[{3}]{4}}\,\Gamma {\biggl (}{\frac {1}{3}}{\biggr )}^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91fa028e9c294b1850c7a3ed45260a6ecfcf0cee" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:35.138ex; height:6.843ex;" alt="{\displaystyle K\left({\frac {{\sqrt {3}}-1}{2{\sqrt {2}}}}\right)={\frac {1}{8\pi }}{\sqrt[{4}]{3}}\,{\sqrt[{3}]{4}}\,\Gamma {\biggl (}{\frac {1}{3}}{\biggr )}^{3}}"></span> </p><p>and </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\left({\frac {3-{\sqrt {7}}}{4{\sqrt {2}}}}\right)={\frac {\Gamma \left({\frac {1}{7}}\right)\Gamma \left({\frac {2}{7}}\right)\Gamma \left({\frac {4}{7}}\right)}{4{\sqrt[{4}]{7}}\pi }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> </mrow> <mrow> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>7</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>7</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\left({\frac {3-{\sqrt {7}}}{4{\sqrt {2}}}}\right)={\frac {\Gamma \left({\frac {1}{7}}\right)\Gamma \left({\frac {2}{7}}\right)\Gamma \left({\frac {4}{7}}\right)}{4{\sqrt[{4}]{7}}\pi }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71967f390fee21bad3fb4f6a4422fb095cd6c06d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:37.988ex; height:8.509ex;" alt="{\displaystyle K\left({\frac {3-{\sqrt {7}}}{4{\sqrt {2}}}}\right)={\frac {\Gamma \left({\frac {1}{7}}\right)\Gamma \left({\frac {2}{7}}\right)\Gamma \left({\frac {4}{7}}\right)}{4{\sqrt[{4}]{7}}\pi }}.}"></span> </p><p>More generally, the condition that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {iK'}{K}}={\frac {iK\left({\sqrt {1-k^{2}}}\right)}{K(k)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <msup> <mi>K</mi> <mo>′</mo> </msup> </mrow> <mi>K</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi>K</mi> <mrow> <mo>(</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> </msqrt> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>K</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {iK'}{K}}={\frac {iK\left({\sqrt {1-k^{2}}}\right)}{K(k)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0b17d821e183620849aefcc36396e2267ef5b17" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.975ex; height:8.343ex;" alt="{\displaystyle {\frac {iK'}{K}}={\frac {iK\left({\sqrt {1-k^{2}}}\right)}{K(k)}}}"></span> be in an <a href="/wiki/Quadratic_field" title="Quadratic field">imaginary quadratic field</a><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> is sufficient.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> For instance, if <span class="texhtml"><i>k</i> = <i>e</i><sup>5<i>πi</i>/6</sup></span>, then <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>iK</i><span class="nowrap" style="padding-left:0.15em;">′</span></span><span class="sr-only">/</span><span class="den"><i>K</i></span></span>⁠</span> = <i>e</i><sup>2<i>πi</i>/3</sup></span> and<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\left(e^{5\pi i/6}\right)={\frac {e^{-\pi i/12}\Gamma ^{3}\left({\frac {1}{3}}\right){\sqrt[{4}]{3}}}{4{\sqrt[{3}]{2}}\pi }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mi>π</mi> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>6</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>12</mn> </mrow> </msup> <msup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> </mrow> <mrow> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\left(e^{5\pi i/6}\right)={\frac {e^{-\pi i/12}\Gamma ^{3}\left({\frac {1}{3}}\right){\sqrt[{4}]{3}}}{4{\sqrt[{3}]{2}}\pi }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af9aeeb4847d98060f6d067a35f0d07b04e7f308" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:32.484ex; height:8.509ex;" alt="{\displaystyle K\left(e^{5\pi i/6}\right)={\frac {e^{-\pi i/12}\Gamma ^{3}\left({\frac {1}{3}}\right){\sqrt[{4}]{3}}}{4{\sqrt[{3}]{2}}\pi }}.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Asymptotic_expressions">Asymptotic expressions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elliptic_integral&action=edit&section=7" title="Edit section: Asymptotic expressions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\left(k\right)\approx {\frac {\pi }{2}}+{\frac {\pi }{8}}{\frac {k^{2}}{1-k^{2}}}-{\frac {\pi }{16}}{\frac {k^{4}}{1-k^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>8</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>16</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\left(k\right)\approx {\frac {\pi }{2}}+{\frac {\pi }{8}}{\frac {k^{2}}{1-k^{2}}}-{\frac {\pi }{16}}{\frac {k^{4}}{1-k^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c503695b96630d8add9cf88630761197465ef7e4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.959ex; height:6.176ex;" alt="{\displaystyle K\left(k\right)\approx {\frac {\pi }{2}}+{\frac {\pi }{8}}{\frac {k^{2}}{1-k^{2}}}-{\frac {\pi }{16}}{\frac {k^{4}}{1-k^{2}}}}"></span> This approximation has a relative precision better than <span class="nowrap"><span data-sort-value="6996300000000000000♠"></span>3<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−4</sup></span> for <span class="texhtml"><i>k</i> < <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>. Keeping only the first two terms is correct to 0.01 precision for <span class="texhtml"><i>k</i> < <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (January 2017)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Differential_equation">Differential equation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elliptic_integral&action=edit&section=8" title="Edit section: Differential equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The differential equation for the elliptic integral of the first kind is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dk}}\left(k\left(1-k^{2}\right){\frac {dK(k)}{dk}}\right)=k\,K(k)}"> <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>k</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>K</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>k</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>k</mi> <mspace width="thinmathspace" /> <mi>K</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dk}}\left(k\left(1-k^{2}\right){\frac {dK(k)}{dk}}\right)=k\,K(k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59dde9de73b43b0009d341ecd641fd0e8eab8ed8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.377ex; height:6.343ex;" alt="{\displaystyle {\frac {d}{dk}}\left(k\left(1-k^{2}\right){\frac {dK(k)}{dk}}\right)=k\,K(k)}"></span> </p><p>A second solution to this equation is <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({\sqrt {1-k^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mrow> <mo>(</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> </msqrt> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\left({\sqrt {1-k^{2}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d71ce299f87d8f4efeaa3e966837c903f208e69c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.821ex; height:4.843ex;" alt="{\displaystyle K\left({\sqrt {1-k^{2}}}\right)}"></span>. This solution satisfies the relation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dk}}K(k)={\frac {E(k)}{k\left(1-k^{2}\right)}}-{\frac {K(k)}{k}}.}"> <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>k</mi> </mrow> </mfrac> </mrow> <mi>K</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>k</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>K</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mi>k</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dk}}K(k)={\frac {E(k)}{k\left(1-k^{2}\right)}}-{\frac {K(k)}{k}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a197bdb12a20efa4350d147fb19c54446750141" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.37ex; height:6.509ex;" alt="{\displaystyle {\frac {d}{dk}}K(k)={\frac {E(k)}{k\left(1-k^{2}\right)}}-{\frac {K(k)}{k}}.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Continued_fraction">Continued fraction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elliptic_integral&action=edit&section=9" title="Edit section: Continued fraction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Continued_fraction" title="Continued fraction">continued fraction</a> expansion is:<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {K(k)}{2\pi }}=-{\frac {1}{4}}+\sum _{n=0}^{\infty }{\frac {q^{n}}{1+q^{2n}}}=-{\frac {1}{4}}+{\cfrac {1}{1-q+{\cfrac {\left(1-q\right)^{2}}{1-q^{3}+{\cfrac {q\left(1-q^{2}\right)^{2}}{1-q^{5}+{\cfrac {q^{2}\left(1-q^{3}\right)^{2}}{1-q^{7}+{\cfrac {q^{3}\left(1-q^{4}\right)^{2}}{1-q^{9}+\cdots }}}}}}}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>K</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>q</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {K(k)}{2\pi }}=-{\frac {1}{4}}+\sum _{n=0}^{\infty }{\frac {q^{n}}{1+q^{2n}}}=-{\frac {1}{4}}+{\cfrac {1}{1-q+{\cfrac {\left(1-q\right)^{2}}{1-q^{3}+{\cfrac {q\left(1-q^{2}\right)^{2}}{1-q^{5}+{\cfrac {q^{2}\left(1-q^{3}\right)^{2}}{1-q^{7}+{\cfrac {q^{3}\left(1-q^{4}\right)^{2}}{1-q^{9}+\cdots }}}}}}}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/847f9bf58cc2951d9fcadc254b27b9fd6b296503" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -18.338ex; width:88.486ex; height:22.509ex;" alt="{\displaystyle {\frac {K(k)}{2\pi }}=-{\frac {1}{4}}+\sum _{n=0}^{\infty }{\frac {q^{n}}{1+q^{2n}}}=-{\frac {1}{4}}+{\cfrac {1}{1-q+{\cfrac {\left(1-q\right)^{2}}{1-q^{3}+{\cfrac {q\left(1-q^{2}\right)^{2}}{1-q^{5}+{\cfrac {q^{2}\left(1-q^{3}\right)^{2}}{1-q^{7}+{\cfrac {q^{3}\left(1-q^{4}\right)^{2}}{1-q^{9}+\cdots }}}}}}}}}},}"></span> where the <a href="/wiki/Nome_(mathematics)" title="Nome (mathematics)">nome</a> is <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 q=q(k)=\exp[-\pi K'(k)/K(k)]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mi>π</mi> <msup> <mi>K</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>K</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=q(k)=\exp[-\pi K'(k)/K(k)]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb86726110efea04f613a63fbdb98f397fa30d15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.391ex; height:3.009ex;" alt="{\displaystyle q=q(k)=\exp[-\pi K'(k)/K(k)]}"></span> in its definition. </p> <div class="mw-heading mw-heading3"><h3 id="Inverting_the_period_ratio">Inverting the period ratio</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elliptic_integral&action=edit&section=10" title="Edit section: Inverting the period ratio"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Here, we use the complete elliptic integral of the first kind with the <i>parameter</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> instead, because the squaring function introduces problems when inverting in the complex plane. So let </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[m]=\int _{0}^{\pi /2}{\dfrac {d\theta }{\sqrt {1-m\sin ^{2}\theta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">[</mo> <mi>m</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"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>d</mi> <mi>θ</mi> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <mi>m</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[m]=\int _{0}^{\pi /2}{\dfrac {d\theta }{\sqrt {1-m\sin ^{2}\theta }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3007fef4e342b267f123e7b3e8a9c93f119d8a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.145ex; height:7.176ex;" alt="{\displaystyle K[m]=\int _{0}^{\pi /2}{\dfrac {d\theta }{\sqrt {1-m\sin ^{2}\theta }}}}"></span></dd></dl> <p>and let </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 \theta _{2}(\tau )=2e^{\pi i\tau /4}\sum _{n=0}^{\infty }q^{n(n+1)},\quad q=e^{\pi i\tau },\,\operatorname {Im} \tau >0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> <mi>i</mi> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> <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> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mi>q</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> <mi>i</mi> <mi>τ</mi> </mrow> </msup> <mo>,</mo> <mspace width="thinmathspace" /> <mi>Im</mi> <mo>⁡</mo> <mi>τ</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{2}(\tau )=2e^{\pi i\tau /4}\sum _{n=0}^{\infty }q^{n(n+1)},\quad q=e^{\pi i\tau },\,\operatorname {Im} \tau >0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d55e5622f89d70f12fa15c29b2432dfc83dd8d49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:47.422ex; height:6.843ex;" alt="{\displaystyle \theta _{2}(\tau )=2e^{\pi i\tau /4}\sum _{n=0}^{\infty }q^{n(n+1)},\quad q=e^{\pi i\tau },\,\operatorname {Im} \tau >0,}"></span></dd> <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 \theta _{3}(\tau )=1+2\sum _{n=1}^{\infty }q^{n^{2}},\quad q=e^{\pi i\tau },\,\operatorname {Im} \tau >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mi>q</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> <mi>i</mi> <mi>τ</mi> </mrow> </msup> <mo>,</mo> <mspace width="thinmathspace" /> <mi>Im</mi> <mo>⁡</mo> <mi>τ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{3}(\tau )=1+2\sum _{n=1}^{\infty }q^{n^{2}},\quad q=e^{\pi i\tau },\,\operatorname {Im} \tau >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e10c258fe693eab8badf6e2ffb509a42337f291" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.924ex; height:6.843ex;" alt="{\displaystyle \theta _{3}(\tau )=1+2\sum _{n=1}^{\infty }q^{n^{2}},\quad q=e^{\pi i\tau },\,\operatorname {Im} \tau >0}"></span></dd></dl> <p>be the <a href="/wiki/Theta_function" title="Theta function">theta functions</a>. </p><p>The equation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau =i{\frac {K[1-m]}{K[m]}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>K</mi> <mo stretchy="false">[</mo> <mn>1</mn> <mo>−</mo> <mi>m</mi> <mo stretchy="false">]</mo> </mrow> <mrow> <mi>K</mi> <mo stretchy="false">[</mo> <mi>m</mi> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau =i{\frac {K[1-m]}{K[m]}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e80a5226937cb73c0699926d4cfa3e10d4aca09d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.342ex; height:6.509ex;" alt="{\displaystyle \tau =i{\frac {K[1-m]}{K[m]}}}"></span></dd></dl> <p>can then be solved (provided that a solution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> exists) by </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 m={\frac {\theta _{2}(\tau )^{4}}{\theta _{3}(\tau )^{4}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>τ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mrow> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>τ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m={\frac {\theta _{2}(\tau )^{4}}{\theta _{3}(\tau )^{4}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21e53edb71ac05d2fcf46c693175a5b987d9e4e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:12.185ex; height:6.676ex;" alt="{\displaystyle m={\frac {\theta _{2}(\tau )^{4}}{\theta _{3}(\tau )^{4}}}}"></span></dd></dl> <p>which is in fact the <a href="/wiki/Modular_lambda_function" title="Modular lambda function">modular lambda function</a>. </p><p>For the purposes of computation, the error analysis is given by<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|{e}^{-\pi i\tau /4}\theta _{2}\!\left(\tau \right)-2\sum _{n=0}^{N-1}{q}^{n\left(n+1\right)}\right|\leq {\begin{cases}{\frac {2{\left|q\right|}^{N\left(N+1\right)}}{1-\left|q\right|^{2N+1}}},&\left|q\right|^{2N+1}<1\\\infty ,&{\text{otherwise}}\\\end{cases}}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> <mi>i</mi> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> <mo>−</mo> <mn>2</mn> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mi>q</mi> <mo>|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mrow> <mo>(</mo> <mrow> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>|</mo> <mi>q</mi> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> <mtd> <msup> <mrow> <mo>|</mo> <mi>q</mi> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo><</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∞</mi> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|{e}^{-\pi i\tau /4}\theta _{2}\!\left(\tau \right)-2\sum _{n=0}^{N-1}{q}^{n\left(n+1\right)}\right|\leq {\begin{cases}{\frac {2{\left|q\right|}^{N\left(N+1\right)}}{1-\left|q\right|^{2N+1}}},&\left|q\right|^{2N+1}<1\\\infty ,&{\text{otherwise}}\\\end{cases}}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/372acf19e5199c1ec1633c0898bd80da80a86713" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:57.423ex; height:8.509ex;" alt="{\displaystyle \left|{e}^{-\pi i\tau /4}\theta _{2}\!\left(\tau \right)-2\sum _{n=0}^{N-1}{q}^{n\left(n+1\right)}\right|\leq {\begin{cases}{\frac {2{\left|q\right|}^{N\left(N+1\right)}}{1-\left|q\right|^{2N+1}}},&\left|q\right|^{2N+1}<1\\\infty ,&{\text{otherwise}}\\\end{cases}}\;}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\theta _{3}\!\left(\tau \right)-\left(1+2\sum _{n=1}^{N-1}{q}^{n^{2}}\right)\right|\leq {\begin{cases}{\frac {2{\left|q\right|}^{N^{2}}}{1-\left|q\right|^{2N+1}}},&\left|q\right|^{2N+1}<1\\\infty ,&{\text{otherwise}}\\\end{cases}}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mn>2</mn> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mi>q</mi> <mo>|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>|</mo> <mi>q</mi> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> <mtd> <msup> <mrow> <mo>|</mo> <mi>q</mi> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo><</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∞</mi> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\theta _{3}\!\left(\tau \right)-\left(1+2\sum _{n=1}^{N-1}{q}^{n^{2}}\right)\right|\leq {\begin{cases}{\frac {2{\left|q\right|}^{N^{2}}}{1-\left|q\right|^{2N+1}}},&\left|q\right|^{2N+1}<1\\\infty ,&{\text{otherwise}}\\\end{cases}}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c7a73d0718476dfb3923edcba5fe53d1b9e1202" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:54.698ex; height:8.843ex;" alt="{\displaystyle \left|\theta _{3}\!\left(\tau \right)-\left(1+2\sum _{n=1}^{N-1}{q}^{n^{2}}\right)\right|\leq {\begin{cases}{\frac {2{\left|q\right|}^{N^{2}}}{1-\left|q\right|^{2N+1}}},&\left|q\right|^{2N+1}<1\\\infty ,&{\text{otherwise}}\\\end{cases}}\;}"></span></dd></dl> <p>where <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\in \mathbb {Z} _{\geq 1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>≥</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N\in \mathbb {Z} _{\geq 1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd9cd048ed44490b64be0599e6dcac65921c2e51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.787ex; height:2.676ex;" alt="{\displaystyle N\in \mathbb {Z} _{\geq 1}}"></span> and <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 \operatorname {Im} \tau >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Im</mi> <mo>⁡</mo> <mi>τ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Im} \tau >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3cbb8e8cc06392e462125f6bbe9c9cfe4906bae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.625ex; height:2.176ex;" alt="{\displaystyle \operatorname {Im} \tau >0}"></span>. </p><p>Also </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[m]={\frac {\pi }{2}}\theta _{3}(\tau )^{2},\quad \tau =i{\frac {K[1-m]}{K[m]}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">[</mo> <mi>m</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>τ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mi>τ</mi> <mo>=</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>K</mi> <mo stretchy="false">[</mo> <mn>1</mn> <mo>−</mo> <mi>m</mi> <mo stretchy="false">]</mo> </mrow> <mrow> <mi>K</mi> <mo stretchy="false">[</mo> <mi>m</mi> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[m]={\frac {\pi }{2}}\theta _{3}(\tau )^{2},\quad \tau =i{\frac {K[1-m]}{K[m]}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee3b2fe5e7899ee67a5c9f8d6364a9c7cb92b9d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:35.575ex; height:6.509ex;" alt="{\displaystyle K[m]={\frac {\pi }{2}}\theta _{3}(\tau )^{2},\quad \tau =i{\frac {K[1-m]}{K[m]}}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\in \mathbb {C} \setminus \{0,1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\in \mathbb {C} \setminus \{0,1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cd68f0d3b82acba4c739c659894c86a08b64caa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.438ex; height:2.843ex;" alt="{\displaystyle m\in \mathbb {C} \setminus \{0,1\}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Complete_elliptic_integral_of_the_second_kind">Complete elliptic integral of the second kind</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elliptic_integral&action=edit&section=11" title="Edit section: Complete elliptic integral of the second kind"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Mplwp_complete_ellipticEk.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Mplwp_complete_ellipticEk.svg/300px-Mplwp_complete_ellipticEk.svg.png" decoding="async" width="300" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Mplwp_complete_ellipticEk.svg/450px-Mplwp_complete_ellipticEk.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/Mplwp_complete_ellipticEk.svg/600px-Mplwp_complete_ellipticEk.svg.png 2x" data-file-width="600" data-file-height="400" /></a><figcaption>Plot of the complete elliptic integral of the second kind <span class="texhtml"><i>E</i>(<i>k</i>)</span></figcaption></figure> <p>The <b>complete elliptic integral of the second kind</b> <span class="texhtml"><i>E</i></span> is defined as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(k)=\int _{0}^{\tfrac {\pi }{2}}{\sqrt {1-k^{2}\sin ^{2}\theta }}\,d\theta =\int _{0}^{1}{\frac {\sqrt {1-k^{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>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"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </msubsup> <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>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mrow> <mspace width="thinmathspace" /> <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>k</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="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(k)=\int _{0}^{\tfrac {\pi }{2}}{\sqrt {1-k^{2}\sin ^{2}\theta }}\,d\theta =\int _{0}^{1}{\frac {\sqrt {1-k^{2}t^{2}}}{\sqrt {1-t^{2}}}}\,dt,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bde6d04200ddda7bffb5644e3e39c1db29d3131" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:51.232ex; height:7.676ex;" alt="{\displaystyle E(k)=\int _{0}^{\tfrac {\pi }{2}}{\sqrt {1-k^{2}\sin ^{2}\theta }}\,d\theta =\int _{0}^{1}{\frac {\sqrt {1-k^{2}t^{2}}}{\sqrt {1-t^{2}}}}\,dt,}"></span> </p><p>or more compactly in terms of the incomplete integral of the second kind <span class="texhtml"><i>E</i>(<i>φ</i>,<i>k</i>)</span> as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(k)=E\left({\tfrac {\pi }{2}},k\right)=E(1;k).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>E</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mi>k</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>;</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(k)=E\left({\tfrac {\pi }{2}},k\right)=E(1;k).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b339157c60ee79b69f614e396b2d576f3588c25" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:26.948ex; height:3.343ex;" alt="{\displaystyle E(k)=E\left({\tfrac {\pi }{2}},k\right)=E(1;k).}"></span> </p><p>For an ellipse with semi-major axis <span class="texhtml"><i>a</i></span> and semi-minor axis <span class="texhtml"><i>b</i></span> and eccentricity <span class="texhtml"><i>e</i> = <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">1 − <i>b</i><sup>2</sup>/<i>a</i><sup>2</sup></span></span></span>, the complete elliptic integral of the second kind <span class="texhtml"><i>E</i>(<i>e</i>)</span> is equal to one quarter of the <a href="/wiki/Ellipse#Circumference" title="Ellipse">circumference</a> <span class="texhtml"><i>C</i></span> of the ellipse measured in units of the semi-major axis <span class="texhtml"><i>a</i></span>. In other words: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=4aE(e).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mn>4</mn> <mi>a</mi> <mi>E</mi> <mo stretchy="false">(</mo> <mi>e</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=4aE(e).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aedf3b9b30c184bec464d8e591d0a76553bfbf7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.572ex; height:2.843ex;" alt="{\displaystyle C=4aE(e).}"></span> </p><p>The complete elliptic integral of the second kind can be expressed as a <a href="/wiki/Power_series" title="Power series">power series</a><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(k)={\frac {\pi }{2}}\sum _{n=0}^{\infty }\left({\frac {(2n)!}{2^{2n}\left(n!\right)^{2}}}\right)^{2}{\frac {k^{2n}}{1-2n}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>k</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> <msup> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>!</mo> </mrow> <mo>)</mo> </mrow> <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>k</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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(k)={\frac {\pi }{2}}\sum _{n=0}^{\infty }\left({\frac {(2n)!}{2^{2n}\left(n!\right)^{2}}}\right)^{2}{\frac {k^{2n}}{1-2n}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a7d4cdfa87ca4a434a7ade053b5ed9b0e261141" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.915ex; height:8.009ex;" alt="{\displaystyle E(k)={\frac {\pi }{2}}\sum _{n=0}^{\infty }\left({\frac {(2n)!}{2^{2n}\left(n!\right)^{2}}}\right)^{2}{\frac {k^{2n}}{1-2n}},}"></span> </p><p>which is equivalent to </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(k)={\frac {\pi }{2}}\left(1-\left({\frac {1}{2}}\right)^{2}{\frac {k^{2}}{1}}-\left({\frac {1\cdot 3}{2\cdot 4}}\right)^{2}{\frac {k^{4}}{3}}-\cdots -\left({\frac {(2n-1)!!}{(2n)!!}}\right)^{2}{\frac {k^{2n}}{2n-1}}-\cdots \right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>k</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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>1</mn> </mfrac> </mrow> <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>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>k</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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(k)={\frac {\pi }{2}}\left(1-\left({\frac {1}{2}}\right)^{2}{\frac {k^{2}}{1}}-\left({\frac {1\cdot 3}{2\cdot 4}}\right)^{2}{\frac {k^{4}}{3}}-\cdots -\left({\frac {(2n-1)!!}{(2n)!!}}\right)^{2}{\frac {k^{2n}}{2n-1}}-\cdots \right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5f0058c3be7c4e624142b2ea1b0c6bb4b164656" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:80.339ex; height:7.509ex;" alt="{\displaystyle E(k)={\frac {\pi }{2}}\left(1-\left({\frac {1}{2}}\right)^{2}{\frac {k^{2}}{1}}-\left({\frac {1\cdot 3}{2\cdot 4}}\right)^{2}{\frac {k^{4}}{3}}-\cdots -\left({\frac {(2n-1)!!}{(2n)!!}}\right)^{2}{\frac {k^{2n}}{2n-1}}-\cdots \right).}"></span> </p><p>In terms of the <a href="/wiki/Gauss_hypergeometric_function" class="mw-redirect" title="Gauss hypergeometric function">Gauss hypergeometric function</a>, the complete elliptic integral of the second kind can be expressed as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(k)={\tfrac {\pi }{2}}\,{}_{2}F_{1}\left({\tfrac {1}{2}},-{\tfrac {1}{2}};1;k^{2}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>;</mo> <mn>1</mn> <mo>;</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(k)={\tfrac {\pi }{2}}\,{}_{2}F_{1}\left({\tfrac {1}{2}},-{\tfrac {1}{2}};1;k^{2}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c977aff06d33963712d0dba1ddb2a8b168401b8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:28.868ex; height:3.509ex;" alt="{\displaystyle E(k)={\tfrac {\pi }{2}}\,{}_{2}F_{1}\left({\tfrac {1}{2}},-{\tfrac {1}{2}};1;k^{2}\right).}"></span> </p><p>The modulus can be transformed that way: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(k)=\left(1+{\sqrt {1-k^{2}}}\right)\,E\left({\frac {1-{\sqrt {1-k^{2}}}}{1+{\sqrt {1-k^{2}}}}}\right)-{\sqrt {1-k^{2}}}\,K(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</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> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>E</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</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> </msqrt> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</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> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</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> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>K</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(k)=\left(1+{\sqrt {1-k^{2}}}\right)\,E\left({\frac {1-{\sqrt {1-k^{2}}}}{1+{\sqrt {1-k^{2}}}}}\right)-{\sqrt {1-k^{2}}}\,K(k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cd5ad657b56cf33446048ecc42e2298ae211717" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:60.22ex; height:7.509ex;" alt="{\displaystyle E(k)=\left(1+{\sqrt {1-k^{2}}}\right)\,E\left({\frac {1-{\sqrt {1-k^{2}}}}{1+{\sqrt {1-k^{2}}}}}\right)-{\sqrt {1-k^{2}}}\,K(k)}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Computation">Computation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elliptic_integral&action=edit&section=12" title="Edit section: Computation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Like the integral of the first kind, the complete elliptic integral of the second kind can be computed very efficiently using the <a href="/wiki/Arithmetic-geometric_mean" class="mw-redirect" title="Arithmetic-geometric mean">arithmetic–geometric mean</a>.<sup id="cite_ref-FOOTNOTECarlson201019.8_1-1" class="reference"><a href="#cite_note-FOOTNOTECarlson201019.8-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Define sequences <span class="texhtml mvar" style="font-style:italic;">a<sub>n</sub></span> and <span class="texhtml mvar" style="font-style:italic;">g<sub>n</sub></span>, where <span class="texhtml"><i>a</i><sub>0</sub> = 1</span>, <span class="texhtml"><i>g</i><sub>0</sub> = <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">1 − <i>k</i><sup>2</sup></span></span> = <i>k</i><span class="nowrap" style="padding-left:0.15em;">′</span></span> and the recurrence relations <span class="texhtml"><i>a</i><sub><i>n</i> + 1</sub> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>a<sub>n</sub></i> + <i>g<sub>n</sub></i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>, <span class="texhtml"><i>g</i><sub><i>n</i> + 1</sub> = <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>a<sub>n</sub> g<sub>n</sub></i></span></span></span> hold. Furthermore, define <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{n}={\sqrt {\left|a_{n}^{2}-g_{n}^{2}\right|}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow> <mo>|</mo> <mrow> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>|</mo> </mrow> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{n}={\sqrt {\left|a_{n}^{2}-g_{n}^{2}\right|}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eeb5c497bf862695403f19201ce8649e58f23b79" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.204ex; height:4.843ex;" alt="{\displaystyle c_{n}={\sqrt {\left|a_{n}^{2}-g_{n}^{2}\right|}}.}"></span> </p><p>By definition, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{\infty }=\lim _{n\to \infty }a_{n}=\lim _{n\to \infty }g_{n}=\operatorname {agm} \left(1,{\sqrt {1-k^{2}}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>agm</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>,</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> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{\infty }=\lim _{n\to \infty }a_{n}=\lim _{n\to \infty }g_{n}=\operatorname {agm} \left(1,{\sqrt {1-k^{2}}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ce58f2b0469461e19072d9efb0c8d8d27f316e7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.967ex; height:4.843ex;" alt="{\displaystyle a_{\infty }=\lim _{n\to \infty }a_{n}=\lim _{n\to \infty }g_{n}=\operatorname {agm} \left(1,{\sqrt {1-k^{2}}}\right).}"></span> </p><p>Also </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }c_{n}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }c_{n}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef30148a2d6fa6b70faefaa180452a830f705808" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.793ex; height:3.676ex;" alt="{\displaystyle \lim _{n\to \infty }c_{n}=0.}"></span> </p><p>Then </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(k)={\frac {\pi }{2a_{\infty }}}\left(1-\sum _{n=0}^{\infty }2^{n-1}c_{n}^{2}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mrow> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <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> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(k)={\frac {\pi }{2a_{\infty }}}\left(1-\sum _{n=0}^{\infty }2^{n-1}c_{n}^{2}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2bf447b7f28cf012198eef177f627dcf6524372" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.552ex; height:7.509ex;" alt="{\displaystyle E(k)={\frac {\pi }{2a_{\infty }}}\left(1-\sum _{n=0}^{\infty }2^{n-1}c_{n}^{2}\right).}"></span> </p><p>In practice, the arithmetic-geometric mean would simply be computed up to some limit. This formula converges quadratically for all <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>k</i></span>| ≤ 1</span>. To speed up computation further, the relation <span class="texhtml"><i>c</i><sub><i>n</i> + 1</sub> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>c<sub>n</sub></i><sup>2</sup></span><span class="sr-only">/</span><span class="den">4<i>a</i><sub><i>n</i> + 1</sub></span></span>⁠</span></span> can be used. </p><p>Furthermore, if <span class="texhtml"><i>k</i><sup>2</sup> = <i>λ</i>(<i>i</i><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>r</i></span></span>)</span> and <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\in \mathbb {Q} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\in \mathbb {Q} ^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f609e38525092d8b949c7f68e901a339ee527b9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.208ex; height:2.843ex;" alt="{\displaystyle r\in \mathbb {Q} ^{+}}"></span> (where <span class="texhtml mvar" style="font-style:italic;">λ</span> is the <a href="/wiki/Modular_lambda_function" title="Modular lambda function">modular lambda function</a>), then <span class="texhtml"><i>E</i>(<i>k</i>)</span> is expressible in closed form in terms of <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(k)={\frac {\pi }{2\operatorname {agm} \left(1,{\sqrt {1-k^{2}}}\right)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mrow> <mn>2</mn> <mi>agm</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>,</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> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(k)={\frac {\pi }{2\operatorname {agm} \left(1,{\sqrt {1-k^{2}}}\right)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a8a8bebc00ac46c47f71d6b28f03375d0c25f9e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:28.395ex; height:7.343ex;" alt="{\displaystyle K(k)={\frac {\pi }{2\operatorname {agm} \left(1,{\sqrt {1-k^{2}}}\right)}}}"></span> and hence can be computed without the need for the infinite summation term. For example, <span class="texhtml"><i>r</i> = 1</span>, <span class="texhtml"><i>r</i> = 3</span> and <span class="texhtml"><i>r</i> = 7</span> give, respectively,<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\left({\frac {1}{\sqrt {2}}}\right)={\frac {1}{2}}K\left({\frac {1}{\sqrt {2}}}\right)+{\frac {\pi }{4K\left({\frac {1}{\sqrt {2}}}\right)}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>K</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mrow> <mn>4</mn> <mi>K</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\left({\frac {1}{\sqrt {2}}}\right)={\frac {1}{2}}K\left({\frac {1}{\sqrt {2}}}\right)+{\frac {\pi }{4K\left({\frac {1}{\sqrt {2}}}\right)}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e467d2fcb47db84c3c0adde67ef85c8173a67cef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:38.165ex; height:8.176ex;" alt="{\displaystyle E\left({\frac {1}{\sqrt {2}}}\right)={\frac {1}{2}}K\left({\frac {1}{\sqrt {2}}}\right)+{\frac {\pi }{4K\left({\frac {1}{\sqrt {2}}}\right)}},}"></span> </p><p>and </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\left({\frac {{\sqrt {3}}-1}{2{\sqrt {2}}}}\right)={\frac {3+{\sqrt {3}}}{6}}K\left({\frac {{\sqrt {3}}-1}{2{\sqrt {2}}}}\right)+{\frac {\pi {\sqrt {3}}}{12K\left({\frac {{\sqrt {3}}-1}{2{\sqrt {2}}}}\right)}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> <mn>6</mn> </mfrac> </mrow> <mi>K</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> <mrow> <mn>12</mn> <mi>K</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\left({\frac {{\sqrt {3}}-1}{2{\sqrt {2}}}}\right)={\frac {3+{\sqrt {3}}}{6}}K\left({\frac {{\sqrt {3}}-1}{2{\sqrt {2}}}}\right)+{\frac {\pi {\sqrt {3}}}{12K\left({\frac {{\sqrt {3}}-1}{2{\sqrt {2}}}}\right)}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/270cd2eae816bd9e52e865b4ea4656e2865ccdad" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:55.373ex; height:8.676ex;" alt="{\displaystyle E\left({\frac {{\sqrt {3}}-1}{2{\sqrt {2}}}}\right)={\frac {3+{\sqrt {3}}}{6}}K\left({\frac {{\sqrt {3}}-1}{2{\sqrt {2}}}}\right)+{\frac {\pi {\sqrt {3}}}{12K\left({\frac {{\sqrt {3}}-1}{2{\sqrt {2}}}}\right)}},}"></span> </p><p>and </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\left({\frac {3-{\sqrt {7}}}{4{\sqrt {2}}}}\right)={\frac {7+2{\sqrt {7}}}{14}}K\left({\frac {3-{\sqrt {7}}}{4{\sqrt {2}}}}\right)+{\frac {\pi {\sqrt {7}}}{28K\left({\frac {3-{\sqrt {7}}}{4{\sqrt {2}}}}\right)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> </mrow> <mrow> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>7</mn> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> </mrow> <mn>14</mn> </mfrac> </mrow> <mi>K</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> </mrow> <mrow> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> </mrow> <mrow> <mn>28</mn> <mi>K</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> </mrow> <mrow> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\left({\frac {3-{\sqrt {7}}}{4{\sqrt {2}}}}\right)={\frac {7+2{\sqrt {7}}}{14}}K\left({\frac {3-{\sqrt {7}}}{4{\sqrt {2}}}}\right)+{\frac {\pi {\sqrt {7}}}{28K\left({\frac {3-{\sqrt {7}}}{4{\sqrt {2}}}}\right)}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df820c0dcda43207f022f13919481794242b4e79" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:56.535ex; height:8.676ex;" alt="{\displaystyle E\left({\frac {3-{\sqrt {7}}}{4{\sqrt {2}}}}\right)={\frac {7+2{\sqrt {7}}}{14}}K\left({\frac {3-{\sqrt {7}}}{4{\sqrt {2}}}}\right)+{\frac {\pi {\sqrt {7}}}{28K\left({\frac {3-{\sqrt {7}}}{4{\sqrt {2}}}}\right)}}.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Derivative_and_differential_equation">Derivative and differential equation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elliptic_integral&action=edit&section=13" title="Edit section: Derivative and differential equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dE(k)}{dk}}={\frac {E(k)-K(k)}{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>E</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>k</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mi>k</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dE(k)}{dk}}={\frac {E(k)-K(k)}{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15238a06fca7d648f6508c288899911523015de6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.506ex; height:5.843ex;" alt="{\displaystyle {\frac {dE(k)}{dk}}={\frac {E(k)-K(k)}{k}}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(k^{2}-1\right){\frac {d}{dk}}\left(k\;{\frac {dE(k)}{dk}}\right)=kE(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>k</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>E</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>k</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>k</mi> <mi>E</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(k^{2}-1\right){\frac {d}{dk}}\left(k\;{\frac {dE(k)}{dk}}\right)=kE(k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6408f7f70b7d7032edf048c228c4d44f3a30c659" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.667ex; height:6.343ex;" alt="{\displaystyle \left(k^{2}-1\right){\frac {d}{dk}}\left(k\;{\frac {dE(k)}{dk}}\right)=kE(k)}"></span> </p><p>A second solution to this equation is <span class="texhtml"><i>E</i>(<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">1 − <i>k</i><sup>2</sup></span></span>) − <i>K</i>(<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">1 − <i>k</i><sup>2</sup></span></span>)</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Complete_elliptic_integral_of_the_third_kind">Complete elliptic integral of the third kind</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elliptic_integral&action=edit&section=14" title="Edit section: Complete elliptic integral of the third kind"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Mplwp_complete_ellipticPi_nfixed_k.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Mplwp_complete_ellipticPi_nfixed_k.svg/300px-Mplwp_complete_ellipticPi_nfixed_k.svg.png" decoding="async" width="300" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Mplwp_complete_ellipticPi_nfixed_k.svg/450px-Mplwp_complete_ellipticPi_nfixed_k.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Mplwp_complete_ellipticPi_nfixed_k.svg/600px-Mplwp_complete_ellipticPi_nfixed_k.svg.png 2x" data-file-width="600" data-file-height="400" /></a><figcaption>Plot of the complete elliptic integral of the third kind <span class="texhtml">Π(<i>n</i>,<i>k</i>)</span> with several fixed values of <span class="texhtml mvar" style="font-style:italic;">n</span></figcaption></figure> <p>The <b>complete elliptic integral of the third kind</b> <span class="texhtml">Π</span> can be defined as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi (n,k)=\int _{0}^{\frac {\pi }{2}}{\frac {d\theta }{\left(1-n\sin ^{2}\theta \right){\sqrt {1-k^{2}\sin ^{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> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>n</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <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>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi (n,k)=\int _{0}^{\frac {\pi }{2}}{\frac {d\theta }{\left(1-n\sin ^{2}\theta \right){\sqrt {1-k^{2}\sin ^{2}\theta }}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cc997c09bd47fbbb772a8afa21cff2aaa1e4e24" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:44.514ex; height:7.676ex;" alt="{\displaystyle \Pi (n,k)=\int _{0}^{\frac {\pi }{2}}{\frac {d\theta }{\left(1-n\sin ^{2}\theta \right){\sqrt {1-k^{2}\sin ^{2}\theta }}}}.}"></span> </p><p>Note that sometimes the elliptic integral of the third kind is defined with an inverse sign for the <i>characteristic</i> <span class="texhtml"><i>n</i></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi '(n,k)=\int _{0}^{\frac {\pi }{2}}{\frac {d\theta }{\left(1+n\sin ^{2}\theta \right){\sqrt {1-k^{2}\sin ^{2}\theta }}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Π</mi> <mo>′</mo> </msup> <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> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>n</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <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>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi '(n,k)=\int _{0}^{\frac {\pi }{2}}{\frac {d\theta }{\left(1+n\sin ^{2}\theta \right){\sqrt {1-k^{2}\sin ^{2}\theta }}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2781888a0c9638007c6099a46cb57de0ae0ad5e5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:45.198ex; height:7.676ex;" alt="{\displaystyle \Pi '(n,k)=\int _{0}^{\frac {\pi }{2}}{\frac {d\theta }{\left(1+n\sin ^{2}\theta \right){\sqrt {1-k^{2}\sin ^{2}\theta }}}}.}"></span> </p><p>Just like the complete elliptic integrals of the first and second kind, the complete elliptic integral of the third kind can be computed very efficiently using the arithmetic-geometric mean.<sup id="cite_ref-FOOTNOTECarlson201019.8_1-2" class="reference"><a href="#cite_note-FOOTNOTECarlson201019.8-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Partial_derivatives">Partial derivatives</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elliptic_integral&action=edit&section=15" title="Edit section: Partial derivatives"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {\partial \Pi (n,k)}{\partial n}}&={\frac {1}{2\left(k^{2}-n\right)(n-1)}}\left(E(k)+{\frac {1}{n}}\left(k^{2}-n\right)K(k)+{\frac {1}{n}}\left(n^{2}-k^{2}\right)\Pi (n,k)\right)\\[8pt]{\frac {\partial \Pi (n,k)}{\partial k}}&={\frac {k}{n-k^{2}}}\left({\frac {E(k)}{k^{2}-1}}+\Pi (n,k)\right)\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="1.1em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Π</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>n</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> <mi>K</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi mathvariant="normal">Π</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Π</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>k</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mrow> <mi>n</mi> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mi mathvariant="normal">Π</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\partial \Pi (n,k)}{\partial n}}&={\frac {1}{2\left(k^{2}-n\right)(n-1)}}\left(E(k)+{\frac {1}{n}}\left(k^{2}-n\right)K(k)+{\frac {1}{n}}\left(n^{2}-k^{2}\right)\Pi (n,k)\right)\\[8pt]{\frac {\partial \Pi (n,k)}{\partial k}}&={\frac {k}{n-k^{2}}}\left({\frac {E(k)}{k^{2}-1}}+\Pi (n,k)\right)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ebce786ea35739a3664eb489c14c514a1c4d602" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.531ex; margin-bottom: -0.307ex; width:82.375ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}{\frac {\partial \Pi (n,k)}{\partial n}}&={\frac {1}{2\left(k^{2}-n\right)(n-1)}}\left(E(k)+{\frac {1}{n}}\left(k^{2}-n\right)K(k)+{\frac {1}{n}}\left(n^{2}-k^{2}\right)\Pi (n,k)\right)\\[8pt]{\frac {\partial \Pi (n,k)}{\partial k}}&={\frac {k}{n-k^{2}}}\left({\frac {E(k)}{k^{2}-1}}+\Pi (n,k)\right)\end{aligned}}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Jacobi_zeta_function">Jacobi zeta function</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elliptic_integral&action=edit&section=16" title="Edit section: Jacobi zeta function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1829, Jacobi defined the <b>Jacobi zeta function</b>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(\varphi ,k)=E(\varphi ,k)-{\frac {E(k)}{K(k)}}F(\varphi ,k).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>K</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z(\varphi ,k)=E(\varphi ,k)-{\frac {E(k)}{K(k)}}F(\varphi ,k).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cfcb0f61e9b1c154ff5c75e3d9b63e2a760244c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:34.429ex; height:6.509ex;" alt="{\displaystyle Z(\varphi ,k)=E(\varphi ,k)-{\frac {E(k)}{K(k)}}F(\varphi ,k).}"></span> It is periodic in <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> with minimal period <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span>. It is related to the <a href="/wiki/Jacobi_elliptic_functions" title="Jacobi elliptic functions">Jacobi zn function</a> by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(\varphi ,k)=\operatorname {zn} (F(\varphi ,k),k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>zn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z(\varphi ,k)=\operatorname {zn} (F(\varphi ,k),k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efa0e22240b13e0155eef7974330a070fe2fec0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.048ex; height:2.843ex;" alt="{\displaystyle Z(\varphi ,k)=\operatorname {zn} (F(\varphi ,k),k)}"></span>. In the literature (e.g. Whittaker and Watson (1927)), sometimes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span> means Wikipedia's <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 \operatorname {zn} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>zn</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {zn} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90224ba7c4191cb4ce61e0a8226dfa8c30ed409b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:1.676ex;" alt="{\displaystyle \operatorname {zn} }"></span>. Some authors (e.g. King (1924)) use <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span> for both Wikipedia's <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span> and <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 \operatorname {zn} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>zn</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {zn} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90224ba7c4191cb4ce61e0a8226dfa8c30ed409b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:1.676ex;" alt="{\displaystyle \operatorname {zn} }"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Legendre's_relation"><span id="Legendre.27s_relation"></span>Legendre's relation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elliptic_integral&action=edit&section=17" title="Edit section: Legendre's relation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Legendre%27s_relation" title="Legendre's relation">Legendre's relation</a> or <i>Legendre Identity</i> shows the relation of the integrals K and E of an elliptic modulus and its anti-related counterpart<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> in an integral equation of second degree: </p><p>For two modules that are Pythagorean counterparts to each other, this relation is valid: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(\varepsilon )E\left({\sqrt {1-\varepsilon ^{2}}}\right)+E(\varepsilon )K\left({\sqrt {1-\varepsilon ^{2}}}\right)-K(\varepsilon )K\left({\sqrt {1-\varepsilon ^{2}}}\right)={\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mi>E</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mi>K</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mi>K</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(\varepsilon )E\left({\sqrt {1-\varepsilon ^{2}}}\right)+E(\varepsilon )K\left({\sqrt {1-\varepsilon ^{2}}}\right)-K(\varepsilon )K\left({\sqrt {1-\varepsilon ^{2}}}\right)={\frac {\pi }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/657cd1c9c656a1abe5e4fc352b8c89541869663b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:66.322ex; height:4.843ex;" alt="{\displaystyle K(\varepsilon )E\left({\sqrt {1-\varepsilon ^{2}}}\right)+E(\varepsilon )K\left({\sqrt {1-\varepsilon ^{2}}}\right)-K(\varepsilon )K\left({\sqrt {1-\varepsilon ^{2}}}\right)={\frac {\pi }{2}}}"></span> </p><p>For example: </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({\color {blueviolet}{\tfrac {3}{5}}})E({\color {blue}{\tfrac {4}{5}}})+E({\color {blueviolet}{\tfrac {3}{5}}})K({\color {blue}{\tfrac {4}{5}}})-K({\color {blueviolet}{\tfrac {3}{5}}})K({\color {blue}{\tfrac {4}{5}}})={\tfrac {1}{2}}\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="blueviolet"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <mo stretchy="false">)</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#0000ff"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="blueviolet"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <mo stretchy="false">)</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#0000ff"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="blueviolet"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <mo stretchy="false">)</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#0000ff"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K({\color {blueviolet}{\tfrac {3}{5}}})E({\color {blue}{\tfrac {4}{5}}})+E({\color {blueviolet}{\tfrac {3}{5}}})K({\color {blue}{\tfrac {4}{5}}})-K({\color {blueviolet}{\tfrac {3}{5}}})K({\color {blue}{\tfrac {4}{5}}})={\tfrac {1}{2}}\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c1ac127d144ecc773031c3967f0075a6954adde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:44.389ex; height:3.676ex;" alt="{\displaystyle K({\color {blueviolet}{\tfrac {3}{5}}})E({\color {blue}{\tfrac {4}{5}}})+E({\color {blueviolet}{\tfrac {3}{5}}})K({\color {blue}{\tfrac {4}{5}}})-K({\color {blueviolet}{\tfrac {3}{5}}})K({\color {blue}{\tfrac {4}{5}}})={\tfrac {1}{2}}\pi }"></span></dd></dl> <p>And for two modules that are tangential counterparts to each other, the following relationship is valid: </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 (1+\varepsilon )K(\varepsilon )E({\tfrac {1-\varepsilon }{1+\varepsilon }})+{\tfrac {2}{1+\varepsilon }}E(\varepsilon )K({\tfrac {1-\varepsilon }{1+\varepsilon }})-2K(\varepsilon )K({\tfrac {1-\varepsilon }{1+\varepsilon }})={\tfrac {1}{2}}\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>ε</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>ε</mi> </mrow> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mrow> <mn>1</mn> <mo>+</mo> <mi>ε</mi> </mrow> </mfrac> </mstyle> </mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>ε</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>ε</mi> </mrow> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mi>K</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>ε</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>ε</mi> </mrow> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1+\varepsilon )K(\varepsilon )E({\tfrac {1-\varepsilon }{1+\varepsilon }})+{\tfrac {2}{1+\varepsilon }}E(\varepsilon )K({\tfrac {1-\varepsilon }{1+\varepsilon }})-2K(\varepsilon )K({\tfrac {1-\varepsilon }{1+\varepsilon }})={\tfrac {1}{2}}\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27c0c074799b6beecbf6d01a8a9adb3423923b66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:60.56ex; height:3.843ex;" alt="{\displaystyle (1+\varepsilon )K(\varepsilon )E({\tfrac {1-\varepsilon }{1+\varepsilon }})+{\tfrac {2}{1+\varepsilon }}E(\varepsilon )K({\tfrac {1-\varepsilon }{1+\varepsilon }})-2K(\varepsilon )K({\tfrac {1-\varepsilon }{1+\varepsilon }})={\tfrac {1}{2}}\pi }"></span></dd></dl> <p>For example: </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 {\tfrac {4}{3}}K({\color {blue}{\tfrac {1}{3}}})E({\color {green}{\tfrac {1}{2}}})+{\tfrac {3}{2}}E({\color {blue}{\tfrac {1}{3}}})K({\color {green}{\tfrac {1}{2}}})-2K({\color {blue}{\tfrac {1}{3}}})K({\color {green}{\tfrac {1}{2}}})={\tfrac {1}{2}}\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mi>K</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#0000ff"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <mo stretchy="false">)</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#008000"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#0000ff"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <mo stretchy="false">)</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#008000"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mi>K</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#0000ff"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <mo stretchy="false">)</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#008000"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {4}{3}}K({\color {blue}{\tfrac {1}{3}}})E({\color {green}{\tfrac {1}{2}}})+{\tfrac {3}{2}}E({\color {blue}{\tfrac {1}{3}}})K({\color {green}{\tfrac {1}{2}}})-2K({\color {blue}{\tfrac {1}{3}}})K({\color {green}{\tfrac {1}{2}}})={\tfrac {1}{2}}\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30272aae336f11fae0affbf348dce9781b6a6723" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:48.868ex; height:3.676ex;" alt="{\displaystyle {\tfrac {4}{3}}K({\color {blue}{\tfrac {1}{3}}})E({\color {green}{\tfrac {1}{2}}})+{\tfrac {3}{2}}E({\color {blue}{\tfrac {1}{3}}})K({\color {green}{\tfrac {1}{2}}})-2K({\color {blue}{\tfrac {1}{3}}})K({\color {green}{\tfrac {1}{2}}})={\tfrac {1}{2}}\pi }"></span></dd></dl> <p>The Legendre's relation for tangential modular counterparts results directly from the Legendre's identity for Pythagorean modular counterparts by using the <a href="/wiki/Landen%27s_transformation" title="Landen's transformation">Landen modular transformation</a> on the Pythagorean counter modulus. </p> <div class="mw-heading mw-heading3"><h3 id="Special_identity_for_the_lemniscatic_case">Special identity for the lemniscatic case</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elliptic_integral&action=edit&section=18" title="Edit section: Special identity for the lemniscatic case"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For the lemniscatic case, the elliptic modulus or specific eccentricity ε is equal to half the square root of two. Legendre's identity for the lemniscatic case can be proved as follows: </p><p>According to the <a href="/wiki/Chain_rule" title="Chain rule">Chain rule</a> these derivatives hold: </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 {\mathrm {d} }{\mathrm {d} y}}\,K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-F{\biggl [}\arccos(xy);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}={\frac {{\sqrt {2}}\,x}{\sqrt {1-x^{4}y^{4}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>−</mo> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>x</mi> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} y}}\,K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-F{\biggl [}\arccos(xy);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}={\frac {{\sqrt {2}}\,x}{\sqrt {1-x^{4}y^{4}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dc1718190eb8439c5553637893ee9656401692e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:51.89ex; height:7.176ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} y}}\,K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-F{\biggl [}\arccos(xy);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}={\frac {{\sqrt {2}}\,x}{\sqrt {1-x^{4}y^{4}}}}}"></span></dd> <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 {\mathrm {d} }{\mathrm {d} y}}\,2E{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-2E{\biggl [}\arccos(xy);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}+F{\biggl [}\arccos(xy);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}={\frac {{\sqrt {2}}\,x^{3}y^{2}}{\sqrt {1-x^{4}y^{4}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mn>2</mn> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>−</mo> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>−</mo> <mn>2</mn> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mo>+</mo> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} y}}\,2E{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-2E{\biggl [}\arccos(xy);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}+F{\biggl [}\arccos(xy);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}={\frac {{\sqrt {2}}\,x^{3}y^{2}}{\sqrt {1-x^{4}y^{4}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51fd6c63925cf33e09dd212d2f2855ee54c35a9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:90.159ex; height:7.176ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} y}}\,2E{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-2E{\biggl [}\arccos(xy);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}+F{\biggl [}\arccos(xy);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}={\frac {{\sqrt {2}}\,x^{3}y^{2}}{\sqrt {1-x^{4}y^{4}}}}}"></span></dd></dl> <p>By using the <a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem of calculus</a> these formulas can be generated: </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{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-F{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}=\int _{0}^{1}{\frac {{\sqrt {2}}\,x}{\sqrt {1-x^{4}y^{4}}}}\,\mathrm {d} y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>−</mo> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </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> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>x</mi> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-F{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}=\int _{0}^{1}{\frac {{\sqrt {2}}\,x}{\sqrt {1-x^{4}y^{4}}}}\,\mathrm {d} y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58282761c03c55c3f20388ee7bf290e4dbc1df6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:53.804ex; height:7.176ex;" alt="{\displaystyle K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-F{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}=\int _{0}^{1}{\frac {{\sqrt {2}}\,x}{\sqrt {1-x^{4}y^{4}}}}\,\mathrm {d} y}"></span></dd> <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 2E{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-2E{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}+F{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}=\int _{0}^{1}{\frac {{\sqrt {2}}\,x^{3}y^{2}}{\sqrt {1-x^{4}y^{4}}}}\,\mathrm {d} y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>−</mo> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>−</mo> <mn>2</mn> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mo>+</mo> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </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> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2E{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-2E{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}+F{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}=\int _{0}^{1}{\frac {{\sqrt {2}}\,x^{3}y^{2}}{\sqrt {1-x^{4}y^{4}}}}\,\mathrm {d} y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/199cf8df44d0d8a8724c407a1bc765265fa06832" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:90.917ex; height:7.176ex;" alt="{\displaystyle 2E{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-2E{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}+F{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}=\int _{0}^{1}{\frac {{\sqrt {2}}\,x^{3}y^{2}}{\sqrt {1-x^{4}y^{4}}}}\,\mathrm {d} y}"></span></dd></dl> <p>The <a href="/wiki/Linear_combination" title="Linear combination">Linear combination</a> of the two now mentioned integrals leads to the following formula: </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 {\sqrt {2}}{\sqrt {1-x^{4}}}}{\biggl \{}2E{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-2E{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}+F{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}{\biggr \}}\,+}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">{</mo> </mrow> </mrow> <mn>2</mn> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>−</mo> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>−</mo> <mn>2</mn> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mo>+</mo> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">}</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sqrt {2}}{\sqrt {1-x^{4}}}}{\biggl \{}2E{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-2E{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}+F{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}{\biggr \}}\,+}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2728dee1248dd6d671e14e2a97c8ba1ea1f9dc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:84.545ex; height:7.176ex;" alt="{\displaystyle {\frac {\sqrt {2}}{\sqrt {1-x^{4}}}}{\biggl \{}2E{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-2E{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}+F{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}{\biggr \}}\,+}"></span></dd> <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 {{\sqrt {2}}\,x^{2}}{\sqrt {1-x^{4}}}}{\biggl \{}K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-F{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}{\biggr \}}=\int _{0}^{1}{\frac {2\,x^{3}(y^{2}+1)}{\sqrt {(1-x^{4})(1-x^{4}\,y^{4})}}}\,\mathrm {d} y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">{</mo> </mrow> </mrow> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>−</mo> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">}</mo> </mrow> </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> <mn>2</mn> <mspace width="thinmathspace" /> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <msqrt> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</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>4</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +\,{\frac {{\sqrt {2}}\,x^{2}}{\sqrt {1-x^{4}}}}{\biggl \{}K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-F{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}{\biggr \}}=\int _{0}^{1}{\frac {2\,x^{3}(y^{2}+1)}{\sqrt {(1-x^{4})(1-x^{4}\,y^{4})}}}\,\mathrm {d} y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b57d166320807a046a01e07b470de52bfd40a80e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:79.424ex; height:7.176ex;" alt="{\displaystyle +\,{\frac {{\sqrt {2}}\,x^{2}}{\sqrt {1-x^{4}}}}{\biggl \{}K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-F{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}{\biggr \}}=\int _{0}^{1}{\frac {2\,x^{3}(y^{2}+1)}{\sqrt {(1-x^{4})(1-x^{4}\,y^{4})}}}\,\mathrm {d} y}"></span></dd></dl> <p>By forming the original antiderivative related to x from the function now shown using the <a href="/wiki/Product_rule" title="Product rule">Product rule</a> this formula results: </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 {\biggl \{}K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-F{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}{\biggr \}}{\biggl \{}2E{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-2E{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}+F{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}{\biggr \}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">{</mo> </mrow> </mrow> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>−</mo> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">}</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">{</mo> </mrow> </mrow> <mn>2</mn> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>−</mo> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>−</mo> <mn>2</mn> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mo>+</mo> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">}</mo> </mrow> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\biggl \{}K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-F{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}{\biggr \}}{\biggl \{}2E{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-2E{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}+F{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}{\biggr \}}=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15e9f11c1f6b542a1f472fb7b974897d33203df1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:110.946ex; height:6.176ex;" alt="{\displaystyle {\biggl \{}K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-F{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}{\biggr \}}{\biggl \{}2E{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-2E{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}+F{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}{\biggr \}}=}"></span></dd> <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 _{0}^{1}{\frac {1}{y^{2}}}(y^{2}+1){\biggl [}{\text{artanh}}(y^{2})-{\text{artanh}}{\bigl (}{\frac {{\sqrt {1-x^{4}}}\,y^{2}}{\sqrt {1-x^{4}y^{4}}}}{\bigr )}{\biggr ]}\mathrm {d} y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <mn>1</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>y</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"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>artanh</mtext> </mrow> <mo stretchy="false">(</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>artanh</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace" /> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\int _{0}^{1}{\frac {1}{y^{2}}}(y^{2}+1){\biggl [}{\text{artanh}}(y^{2})-{\text{artanh}}{\bigl (}{\frac {{\sqrt {1-x^{4}}}\,y^{2}}{\sqrt {1-x^{4}y^{4}}}}{\bigr )}{\biggr ]}\mathrm {d} y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2503c8ea3948bd57fb32a6a33e02c60afcac539" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:56.935ex; height:7.509ex;" alt="{\displaystyle =\int _{0}^{1}{\frac {1}{y^{2}}}(y^{2}+1){\biggl [}{\text{artanh}}(y^{2})-{\text{artanh}}{\bigl (}{\frac {{\sqrt {1-x^{4}}}\,y^{2}}{\sqrt {1-x^{4}y^{4}}}}{\bigr )}{\biggr ]}\mathrm {d} y}"></span></dd></dl> <p>If the value <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=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee42176e76ae6b56d68c42ced807e08b962a2b54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=1}"></span> is inserted in this integral identity, then the following identity emerges: </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{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}{\biggl [}2\,E{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}{\biggr ]}=\int _{0}^{1}{\frac {1}{y^{2}}}(y^{2}+1)\,{\text{artanh}}(y^{2})\,\mathrm {d} y=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mn>2</mn> <mspace width="thinmathspace" /> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>−</mo> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </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> <mn>1</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>artanh</mtext> </mrow> <mo stretchy="false">(</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}{\biggl [}2\,E{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}{\biggr ]}=\int _{0}^{1}{\frac {1}{y^{2}}}(y^{2}+1)\,{\text{artanh}}(y^{2})\,\mathrm {d} y=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb6222e15184d9633c7d6c2406698e6e018464ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:68.94ex; height:6.343ex;" alt="{\displaystyle K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}{\biggl [}2\,E{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}{\biggr ]}=\int _{0}^{1}{\frac {1}{y^{2}}}(y^{2}+1)\,{\text{artanh}}(y^{2})\,\mathrm {d} y=}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\biggl [}2\arctan(y)-{\frac {1}{y}}(1-y^{2})\,{\text{artanh}}(y^{2}){\biggr ]}_{y=0}^{y=1}=2\arctan(1)={\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mn>2</mn> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>y</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>artanh</mtext> </mrow> <mo stretchy="false">(</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>=</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <mn>2</mn> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\biggl [}2\arctan(y)-{\frac {1}{y}}(1-y^{2})\,{\text{artanh}}(y^{2}){\biggr ]}_{y=0}^{y=1}=2\arctan(1)={\frac {\pi }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec66a7530dc2aacf368477e7fe3ee49ddc4c3273" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:62.394ex; height:6.843ex;" alt="{\displaystyle ={\biggl [}2\arctan(y)-{\frac {1}{y}}(1-y^{2})\,{\text{artanh}}(y^{2}){\biggr ]}_{y=0}^{y=1}=2\arctan(1)={\frac {\pi }{2}}}"></span></dd></dl> <p>This is how this lemniscatic excerpt from Legendre's identity appears: </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 2E{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}^{2}={\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>−</mo> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2E{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}^{2}={\frac {\pi }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58daa3b4afce71a13c58e553bc2b4b4a4a32a9e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.911ex; height:5.176ex;" alt="{\displaystyle 2E{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}^{2}={\frac {\pi }{2}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Generalization_for_the_overall_case">Generalization for the overall case</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elliptic_integral&action=edit&section=19" title="Edit section: Generalization for the overall case"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Now the modular general case<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> is worked out. For this purpose, the derivatives of the complete elliptic integrals are derived after the modulus <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 \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"></span> and then they are combined. And then the Legendre's identity balance is determined. </p><p>Because the derivative of the <i>circle function</i> is the negative product of the <i>identical mapping function</i> and the reciprocal of the circle function: </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 {\mathrm {d} }{\mathrm {d} \varepsilon }}{\sqrt {1-\varepsilon ^{2}}}=-\,{\frac {\varepsilon }{\sqrt {1-\varepsilon ^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ε</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mo>−</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ε</mi> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}{\sqrt {1-\varepsilon ^{2}}}=-\,{\frac {\varepsilon }{\sqrt {1-\varepsilon ^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13707f25d35c5dbf1a54abf04b02a2b1d37b1510" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.271ex; height:6.676ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}{\sqrt {1-\varepsilon ^{2}}}=-\,{\frac {\varepsilon }{\sqrt {1-\varepsilon ^{2}}}}}"></span></dd></dl> <p>These are the derivatives of K and E shown in this article in the sections above: </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 {\mathrm {d} }{\mathrm {d} \varepsilon }}K(\varepsilon )={\frac {1}{\varepsilon (1-\varepsilon ^{2})}}{\bigl [}E(\varepsilon )-(1-\varepsilon ^{2})K(\varepsilon ){\bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ε</mi> </mrow> </mfrac> </mrow> <mi>K</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>ε</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}K(\varepsilon )={\frac {1}{\varepsilon (1-\varepsilon ^{2})}}{\bigl [}E(\varepsilon )-(1-\varepsilon ^{2})K(\varepsilon ){\bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8257c714bf0c36a8da6b69b079fb6aa68ca8779" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:43.496ex; height:6.176ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}K(\varepsilon )={\frac {1}{\varepsilon (1-\varepsilon ^{2})}}{\bigl [}E(\varepsilon )-(1-\varepsilon ^{2})K(\varepsilon ){\bigr ]}}"></span></dd> <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 {\mathrm {d} }{\mathrm {d} \varepsilon }}E(\varepsilon )=-\,{\frac {1}{\varepsilon }}{\bigl [}K(\varepsilon )-E(\varepsilon ){\bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ε</mi> </mrow> </mfrac> </mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ε</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mi>K</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}E(\varepsilon )=-\,{\frac {1}{\varepsilon }}{\bigl [}K(\varepsilon )-E(\varepsilon ){\bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a9f0f2fa512b0426ff239625f6651b4cd519591" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:29.58ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}E(\varepsilon )=-\,{\frac {1}{\varepsilon }}{\bigl [}K(\varepsilon )-E(\varepsilon ){\bigr ]}}"></span></dd></dl> <p>In combination with the derivative of the circle function these derivatives are valid then: </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 {\mathrm {d} }{\mathrm {d} \varepsilon }}K({\sqrt {1-\varepsilon ^{2}}})={\frac {1}{\varepsilon (1-\varepsilon ^{2})}}{\bigl [}\varepsilon ^{2}K({\sqrt {1-\varepsilon ^{2}}})-E({\sqrt {1-\varepsilon ^{2}}}){\bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ε</mi> </mrow> </mfrac> </mrow> <mi>K</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>ε</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>K</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}K({\sqrt {1-\varepsilon ^{2}}})={\frac {1}{\varepsilon (1-\varepsilon ^{2})}}{\bigl [}\varepsilon ^{2}K({\sqrt {1-\varepsilon ^{2}}})-E({\sqrt {1-\varepsilon ^{2}}}){\bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb0a0cfe27981a29e18422cdd2fabf9f4fe241ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:59.826ex; height:6.176ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}K({\sqrt {1-\varepsilon ^{2}}})={\frac {1}{\varepsilon (1-\varepsilon ^{2})}}{\bigl [}\varepsilon ^{2}K({\sqrt {1-\varepsilon ^{2}}})-E({\sqrt {1-\varepsilon ^{2}}}){\bigr ]}}"></span></dd> <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 {\mathrm {d} }{\mathrm {d} \varepsilon }}E({\sqrt {1-\varepsilon ^{2}}})={\frac {\varepsilon }{1-\varepsilon ^{2}}}{\bigl [}K({\sqrt {1-\varepsilon ^{2}}})-E({\sqrt {1-\varepsilon ^{2}}}){\bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ε</mi> </mrow> </mfrac> </mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ε</mi> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mi>K</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}E({\sqrt {1-\varepsilon ^{2}}})={\frac {\varepsilon }{1-\varepsilon ^{2}}}{\bigl [}K({\sqrt {1-\varepsilon ^{2}}})-E({\sqrt {1-\varepsilon ^{2}}}){\bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/165f349634d534b3da323035a60dd50f249e26c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:54.505ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}E({\sqrt {1-\varepsilon ^{2}}})={\frac {\varepsilon }{1-\varepsilon ^{2}}}{\bigl [}K({\sqrt {1-\varepsilon ^{2}}})-E({\sqrt {1-\varepsilon ^{2}}}){\bigr ]}}"></span></dd></dl> <p>Legendre's identity includes products of any two complete elliptic integrals. For the derivation of the function side from the equation scale of Legendre's identity, the <a href="/wiki/Product_rule" title="Product rule">Product rule</a> is now applied in the following: </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 {\mathrm {d} }{\mathrm {d} \varepsilon }}K(\varepsilon )E({\sqrt {1-\varepsilon ^{2}}})={\frac {1}{\varepsilon (1-\varepsilon ^{2})}}{\bigl [}E(\varepsilon )E({\sqrt {1-\varepsilon ^{2}}})-K(\varepsilon )E({\sqrt {1-\varepsilon ^{2}}})+\varepsilon ^{2}K(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}}){\bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ε</mi> </mrow> </mfrac> </mrow> <mi>K</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>ε</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>K</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}K(\varepsilon )E({\sqrt {1-\varepsilon ^{2}}})={\frac {1}{\varepsilon (1-\varepsilon ^{2})}}{\bigl [}E(\varepsilon )E({\sqrt {1-\varepsilon ^{2}}})-K(\varepsilon )E({\sqrt {1-\varepsilon ^{2}}})+\varepsilon ^{2}K(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}}){\bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f346b9a656f98160dcbf47164685c470bf227ca1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:93.97ex; height:6.176ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}K(\varepsilon )E({\sqrt {1-\varepsilon ^{2}}})={\frac {1}{\varepsilon (1-\varepsilon ^{2})}}{\bigl [}E(\varepsilon )E({\sqrt {1-\varepsilon ^{2}}})-K(\varepsilon )E({\sqrt {1-\varepsilon ^{2}}})+\varepsilon ^{2}K(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}}){\bigr ]}}"></span></dd> <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 {\mathrm {d} }{\mathrm {d} \varepsilon }}E(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}})={\frac {1}{\varepsilon (1-\varepsilon ^{2})}}{\bigl [}-E(\varepsilon )E({\sqrt {1-\varepsilon ^{2}}})+E(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}})-(1-\varepsilon ^{2})K(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}}){\bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ε</mi> </mrow> </mfrac> </mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>ε</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mo>−</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}E(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}})={\frac {1}{\varepsilon (1-\varepsilon ^{2})}}{\bigl [}-E(\varepsilon )E({\sqrt {1-\varepsilon ^{2}}})+E(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}})-(1-\varepsilon ^{2})K(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}}){\bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/442ae2b56db1b68bee1c2b44535a248527fc2e32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:102.622ex; height:6.176ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}E(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}})={\frac {1}{\varepsilon (1-\varepsilon ^{2})}}{\bigl [}-E(\varepsilon )E({\sqrt {1-\varepsilon ^{2}}})+E(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}})-(1-\varepsilon ^{2})K(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}}){\bigr ]}}"></span></dd> <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 {\mathrm {d} }{\mathrm {d} \varepsilon }}K(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}})={\frac {1}{\varepsilon (1-\varepsilon ^{2})}}{\bigl [}E(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}})-K(\varepsilon )E({\sqrt {1-\varepsilon ^{2}}})-(1-2\varepsilon ^{2})K(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}}){\bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ε</mi> </mrow> </mfrac> </mrow> <mi>K</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>ε</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}K(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}})={\frac {1}{\varepsilon (1-\varepsilon ^{2})}}{\bigl [}E(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}})-K(\varepsilon )E({\sqrt {1-\varepsilon ^{2}}})-(1-2\varepsilon ^{2})K(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}}){\bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0981cb098ac4b82cb503bffa8964b0d92030ae8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:101.525ex; height:6.176ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}K(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}})={\frac {1}{\varepsilon (1-\varepsilon ^{2})}}{\bigl [}E(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}})-K(\varepsilon )E({\sqrt {1-\varepsilon ^{2}}})-(1-2\varepsilon ^{2})K(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}}){\bigr ]}}"></span></dd></dl> <p>Of these three equations, adding the top two equations and subtracting the bottom equation gives this result: </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 {\mathrm {d} }{\mathrm {d} \varepsilon }}{\bigl [}K(\varepsilon )E({\sqrt {1-\varepsilon ^{2}}})+E(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}})-K(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}}){\bigr ]}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ε</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mi>K</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}{\bigl [}K(\varepsilon )E({\sqrt {1-\varepsilon ^{2}}})+E(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}})-K(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}}){\bigr ]}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19a6abc97b33945523271bd18bd396c52280c64a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:66.407ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}{\bigl [}K(\varepsilon )E({\sqrt {1-\varepsilon ^{2}}})+E(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}})-K(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}}){\bigr ]}=0}"></span></dd></dl> <p>In relation to the <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 \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"></span> the equation balance constantly gives the value zero. </p><p>The previously determined result shall be combined with the Legendre equation to the modulus <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 \varepsilon =1/{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon =1/{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c64f4474aa3df449243c9ada07321d6e6856ca7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.605ex; height:3.176ex;" alt="{\displaystyle \varepsilon =1/{\sqrt {2}}}"></span> that is worked out in the section before: </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 2E{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}^{2}={\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>−</mo> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2E{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}^{2}={\frac {\pi }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58daa3b4afce71a13c58e553bc2b4b4a4a32a9e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.911ex; height:5.176ex;" alt="{\displaystyle 2E{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}^{2}={\frac {\pi }{2}}}"></span></dd></dl> <p>The combination of the last two formulas gives the following result: </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(\varepsilon )E({\sqrt {1-\varepsilon ^{2}}})+E(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}})-K(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}})={\tfrac {1}{2}}\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(\varepsilon )E({\sqrt {1-\varepsilon ^{2}}})+E(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}})-K(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}})={\tfrac {1}{2}}\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da254063949df698311fcac01140f329c6c020f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:63.084ex; height:3.843ex;" alt="{\displaystyle K(\varepsilon )E({\sqrt {1-\varepsilon ^{2}}})+E(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}})-K(\varepsilon )K({\sqrt {1-\varepsilon ^{2}}})={\tfrac {1}{2}}\pi }"></span></dd></dl> <p>Because if the derivative of a continuous function constantly takes the value zero, then the concerned function is a constant function. This means that this function results in the same function value for each abscissa value <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 \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"></span> and the associated function graph is therefore a horizontal straight line. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elliptic_integral&action=edit&section=20" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span><span class="portalbox-link"><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></span></li></ul> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li><a href="/wiki/Elliptic_curve" title="Elliptic curve">Elliptic curve</a></li> <li><a href="/wiki/Schwarz%E2%80%93Christoffel_mapping" title="Schwarz–Christoffel mapping">Schwarz–Christoffel mapping</a></li> <li><a href="/wiki/Carlson_symmetric_form" title="Carlson symmetric form">Carlson symmetric form</a></li> <li><a href="/wiki/Jacobi%27s_elliptic_functions" class="mw-redirect" title="Jacobi's elliptic functions">Jacobi's elliptic functions</a></li> <li><a href="/wiki/Weierstrass%27s_elliptic_functions" class="mw-redirect" title="Weierstrass's elliptic functions">Weierstrass's elliptic functions</a></li> <li><a href="/wiki/Theta_function" title="Theta function">Jacobi theta function</a></li> <li><a href="/wiki/Ramanujan_theta_function" title="Ramanujan theta function">Ramanujan theta function</a></li> <li><a href="/wiki/Arithmetic%E2%80%93geometric_mean" title="Arithmetic–geometric mean">Arithmetic–geometric mean</a></li> <li><a href="/wiki/Pendulum_(mathematics)#Arbitrary-amplitude_period" class="mw-redirect" title="Pendulum (mathematics)">Pendulum period</a></li> <li><a href="/wiki/Meridian_arc#Relation_to_elliptic_integrals" title="Meridian arc">Meridian arc</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elliptic_integral&action=edit&section=21" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elliptic_integral&action=edit&section=22" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><span class="texhtml mvar" style="font-style:italic;">K</span> can be <a href="/wiki/Analytic_continuation" title="Analytic continuation">analytically extended</a> to the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="References_2">References</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elliptic_integral&action=edit&section=23" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-FOOTNOTECarlson201019.8-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTECarlson201019.8_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTECarlson201019.8_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTECarlson201019.8_1-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFCarlson2010">Carlson 2010</a>, 19.8.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBorweinBorwein1987" class="citation book cs1">Borwein, Jonathan M.; Borwein, Peter B. (1987). <i>Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity</i> (First ed.). Wiley-Interscience. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-83138-7" title="Special:BookSources/0-471-83138-7"><bdi>0-471-83138-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Pi+and+the+AGM%3A+A+Study+in+Analytic+Number+Theory+and+Computational+Complexity&rft.edition=First&rft.pub=Wiley-Interscience&rft.date=1987&rft.isbn=0-471-83138-7&rft.aulast=Borwein&rft.aufirst=Jonathan+M.&rft.au=Borwein%2C+Peter+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+integral" class="Z3988"></span> p. 296</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBorweinBorwein1987" class="citation book cs1">Borwein, Jonathan M.; Borwein, Peter B. (1987). <i>Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity</i> (First ed.). Wiley-Interscience. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-83138-7" title="Special:BookSources/0-471-83138-7"><bdi>0-471-83138-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Pi+and+the+AGM%3A+A+Study+in+Analytic+Number+Theory+and+Computational+Complexity&rft.edition=First&rft.pub=Wiley-Interscience&rft.date=1987&rft.isbn=0-471-83138-7&rft.aulast=Borwein&rft.aufirst=Jonathan+M.&rft.au=Borwein%2C+Peter+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+integral" class="Z3988"></span> p. 298</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChowlaSelberg1949" class="citation journal cs1">Chowla, S.; Selberg, A. (1949). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1063041">"On Epstein's Zeta Function (I)"</a>. <i>Proceedings of the National Academy of Sciences</i>. <b>35</b> (7): 373. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1949PNAS...35..371C">1949PNAS...35..371C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1073%2FPNAS.35.7.371">10.1073/PNAS.35.7.371</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1063041">1063041</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16588908">16588908</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:45071481">45071481</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+National+Academy+of+Sciences&rft.atitle=On+Epstein%27s+Zeta+Function+%28I%29.&rft.volume=35&rft.issue=7&rft.pages=373&rft.date=1949&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1063041%23id-name%3DPMC&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A45071481%23id-name%3DS2CID&rft_id=info%3Abibcode%2F1949PNAS...35..371C&rft_id=info%3Apmid%2F16588908&rft_id=info%3Adoi%2F10.1073%2FPNAS.35.7.371&rft.aulast=Chowla&rft.aufirst=S.&rft.au=Selberg%2C+A.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1063041&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+integral" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChowlaSelberg1967" class="citation journal cs1">Chowla, S.; Selberg, A. (1967). <a rel="nofollow" class="external text" href="https://eudml.org/doc/150803">"On Epstein's Zeta-Function"</a>. <i>Journal für die Reine und Angewandte Mathematik</i>. <b>227</b>: 86–110.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+f%C3%BCr+die+Reine+und+Angewandte+Mathematik&rft.atitle=On+Epstein%27s+Zeta-Function&rft.volume=227&rft.pages=86-110&rft.date=1967&rft.aulast=Chowla&rft.aufirst=S.&rft.au=Selberg%2C+A.&rft_id=https%3A%2F%2Feudml.org%2Fdoc%2F150803&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+integral" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://fungrim.org/topic/Legendre_elliptic_integrals/">"Legendre elliptic integrals (Entry 175b7a)"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Legendre+elliptic+integrals+%28Entry+175b7a%29&rft_id=https%3A%2F%2Ffungrim.org%2Ftopic%2FLegendre_elliptic_integrals%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+integral" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">N.Bagis,L.Glasser.(2015)"Evaluations of a Continued fraction of Ramanujan". Rend.Sem.Mat.Univ.Padova, Vol.133 pp 1-10</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://fungrim.org/topic/Approximations_of_Jacobi_theta_functions/">"Approximations of Jacobi theta functions"</a>. <i>The Mathematical Functions Grimoire</i>. Fredrik Johansson<span class="reference-accessdate">. Retrieved <span class="nowrap">August 29,</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Mathematical+Functions+Grimoire&rft.atitle=Approximations+of+Jacobi+theta+functions&rft_id=https%3A%2F%2Ffungrim.org%2Ftopic%2FApproximations_of_Jacobi_theta_functions%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+integral" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://functions.wolfram.com/EllipticIntegrals/EllipticE/06/01/03/01/0003/">"Complete elliptic integral of the second kind: Series representations (Formula 08.01.06.0002)"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Complete+elliptic+integral+of+the+second+kind%3A+Series+representations+%28Formula+08.01.06.0002%29&rft_id=https%3A%2F%2Ffunctions.wolfram.com%2FEllipticIntegrals%2FEllipticE%2F06%2F01%2F03%2F01%2F0003%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+integral" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBorweinBorwein1987" class="citation book cs1">Borwein, Jonathan M.; Borwein, Peter B. (1987). <i>Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity</i> (First ed.). Wiley-Interscience. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-83138-7" title="Special:BookSources/0-471-83138-7"><bdi>0-471-83138-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Pi+and+the+AGM%3A+A+Study+in+Analytic+Number+Theory+and+Computational+Complexity&rft.edition=First&rft.pub=Wiley-Interscience&rft.date=1987&rft.isbn=0-471-83138-7&rft.aulast=Borwein&rft.aufirst=Jonathan+M.&rft.au=Borwein%2C+Peter+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+integral" class="Z3988"></span> p. 26, 161</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1 cs1-prop-foreign-lang-source"><a rel="nofollow" class="external text" href="https://www.spektrum.de/lexikon/mathematik/legendre-relation/5873">"Legendre-Relation"</a> (in German)<span class="reference-accessdate">. Retrieved <span class="nowrap">2022-11-29</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Legendre-Relation&rft_id=https%3A%2F%2Fwww.spektrum.de%2Flexikon%2Fmathematik%2Flegendre-relation%2F5873&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+integral" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://archive.lib.msu.edu/crcmath/math/math/l/l179.htm">"Legendre Relation"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-11-29</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Legendre+Relation&rft_id=https%3A%2F%2Farchive.lib.msu.edu%2Fcrcmath%2Fmath%2Fmath%2Fl%2Fl179.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+integral" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.stackexchange.com/questions/701515/proving-legendres-relation-for-elliptic-curves">"integration - Proving Legendres Relation for elliptic curves"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-02-10</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=integration+-+Proving+Legendres+Relation+for+elliptic+curves&rft_id=https%3A%2F%2Fmath.stackexchange.com%2Fquestions%2F701515%2Fproving-legendres-relation-for-elliptic-curves&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+integral" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFInternet_Archive1991" class="citation cs2">Internet Archive (1991), <a rel="nofollow" class="external text" href="https://archive.org/details/paulhalmoscelebr0000unse"><i>Paul Halmos celebrating 50 years of mathematics</i></a>, New York : Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-97509-8" title="Special:BookSources/0-387-97509-8"><bdi>0-387-97509-8</bdi></a><span class="reference-accessdate">, retrieved <span class="nowrap">2023-02-10</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Paul+Halmos+celebrating+50+years+of+mathematics&rft.pub=New+York+%3A+Springer-Verlag&rft.date=1991&rft.isbn=0-387-97509-8&rft.au=Internet+Archive&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fpaulhalmoscelebr0000unse&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+integral" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Sources">Sources</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elliptic_integral&action=edit&section=24" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAbramowitzStegun1983" class="citation book cs1"><a href="/wiki/Milton_Abramowitz" title="Milton Abramowitz">Abramowitz, Milton</a>; <a href="/wiki/Irene_Stegun" title="Irene Stegun">Stegun, Irene Ann</a>, eds. (1983) [June 1964]. <a rel="nofollow" class="external text" href="http://www.math.ubc.ca/~cbm/aands/page_587.htm">"Chapter 17"</a>. <a href="/wiki/Abramowitz_and_Stegun" title="Abramowitz and Stegun"><i>Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</i></a>. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 587. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-61272-0" title="Special:BookSources/978-0-486-61272-0"><bdi>978-0-486-61272-0</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/64-60036">64-60036</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0167642">0167642</a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://www.loc.gov/item/65012253">65-12253</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+17&rft.btitle=Handbook+of+Mathematical+Functions+with+Formulas%2C+Graphs%2C+and+Mathematical+Tables&rft.place=Washington+D.C.%3B+New+York&rft.series=Applied+Mathematics+Series&rft.pages=587&rft.edition=Ninth+reprint+with+additional+corrections+of+tenth+original+printing+with+corrections+%28December+1972%29%3B+first&rft.pub=United+States+Department+of+Commerce%2C+National+Bureau+of+Standards%3B+Dover+Publications&rft.date=1983&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0167642%23id-name%3DMR&rft_id=info%3Alccn%2F64-60036&rft.isbn=978-0-486-61272-0&rft_id=http%3A%2F%2Fwww.math.ubc.ca%2F~cbm%2Faands%2Fpage_587.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+integral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFByrdFriedman1971" class="citation book cs1">Byrd, P. F.; Friedman, M.D. (1971). <i>Handbook of Elliptic Integrals for Engineers and Scientists</i> (2nd ed.). New York: Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-05318-2" title="Special:BookSources/0-387-05318-2"><bdi>0-387-05318-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Elliptic+Integrals+for+Engineers+and+Scientists&rft.place=New+York&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1971&rft.isbn=0-387-05318-2&rft.aulast=Byrd&rft.aufirst=P.+F.&rft.au=Friedman%2C+M.D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+integral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarlson1995" class="citation journal cs1">Carlson, B. C. (1995). "Numerical Computation of Real or Complex Elliptic Integrals". <i>Numerical Algorithms</i>. <b>10</b> (1): 13–26. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/9409227">math/9409227</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1995NuAlg..10...13C">1995NuAlg..10...13C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02198293">10.1007/BF02198293</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:11580137">11580137</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Numerical+Algorithms&rft.atitle=Numerical+Computation+of+Real+or+Complex+Elliptic+Integrals&rft.volume=10&rft.issue=1&rft.pages=13-26&rft.date=1995&rft_id=info%3Aarxiv%2Fmath%2F9409227&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A11580137%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF02198293&rft_id=info%3Abibcode%2F1995NuAlg..10...13C&rft.aulast=Carlson&rft.aufirst=B.+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+integral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarlson2010" class="citation cs2">Carlson, B. C. (2010), <a rel="nofollow" class="external text" href="http://dlmf.nist.gov/19">"Elliptic integral"</a>, in <a href="/wiki/Frank_W._J._Olver" title="Frank W. J. Olver">Olver, Frank W. J.</a>; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), <i><a href="/wiki/Digital_Library_of_Mathematical_Functions" title="Digital Library of Mathematical Functions">NIST Handbook of Mathematical Functions</a></i>, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-19225-5" title="Special:BookSources/978-0-521-19225-5"><bdi>978-0-521-19225-5</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2723248">2723248</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Elliptic+integral&rft.btitle=NIST+Handbook+of+Mathematical+Functions&rft.pub=Cambridge+University+Press&rft.date=2010&rft.isbn=978-0-521-19225-5&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2723248%23id-name%3DMR&rft.aulast=Carlson&rft.aufirst=B.+C.&rft_id=http%3A%2F%2Fdlmf.nist.gov%2F19&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+integral" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFErdélyiMagnusOberhettingerTricomi1953" class="citation book cs1">Erdélyi, Arthur; <a href="/wiki/Wilhelm_Magnus" title="Wilhelm Magnus">Magnus, Wilhelm</a>; Oberhettinger, Fritz; Tricomi, Francesco G. (1953). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110714210423/http://apps.nrbook.com/bateman/Vol2.pdf"><i>Higher transcendental functions. Vol II</i></a> <span class="cs1-format">(PDF)</span>. McGraw-Hill Book Company, Inc., New York-Toronto-London. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0058756">0058756</a>. Archived from <a rel="nofollow" class="external text" href="http://apps.nrbook.com/bateman/Vol2.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2011-07-14<span class="reference-accessdate">. Retrieved <span class="nowrap">2016-07-24</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Higher+transcendental+functions.+Vol+II&rft.pub=McGraw-Hill+Book+Company%2C+Inc.%2C+New+York-Toronto-London&rft.date=1953&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0058756%23id-name%3DMR&rft.aulast=Erd%C3%A9lyi&rft.aufirst=Arthur&rft.au=Magnus%2C+Wilhelm&rft.au=Oberhettinger%2C+Fritz&rft.au=Tricomi%2C+Francesco+G.&rft_id=http%3A%2F%2Fapps.nrbook.com%2Fbateman%2FVol2.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+integral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGradshteynRyzhikGeronimusTseytlin2015" class="citation book cs1"><a href="/wiki/Izrail_Solomonovich_Gradshteyn" class="mw-redirect" title="Izrail Solomonovich Gradshteyn">Gradshteyn, Izrail Solomonovich</a>; <a href="/wiki/Iosif_Moiseevich_Ryzhik" class="mw-redirect" title="Iosif Moiseevich Ryzhik">Ryzhik, Iosif Moiseevich</a>; <a href="/wiki/Yuri_Veniaminovich_Geronimus" class="mw-redirect" title="Yuri Veniaminovich Geronimus">Geronimus, Yuri Veniaminovich</a>; <a href="/wiki/Michail_Yulyevich_Tseytlin" class="mw-redirect" title="Michail Yulyevich Tseytlin">Tseytlin, Michail Yulyevich</a>; Jeffrey, Alan (2015) [October 2014]. "8.1.". In Zwillinger, Daniel; <a href="/wiki/Victor_Hugo_Moll" class="mw-redirect" title="Victor Hugo Moll">Moll, Victor Hugo</a> (eds.). <a href="/wiki/Gradshteyn_and_Ryzhik" title="Gradshteyn and Ryzhik"><i>Table of Integrals, Series, and Products</i></a>. Translated by Scripta Technica, Inc. (8 ed.). <a href="/wiki/Academic_Press,_Inc." class="mw-redirect" title="Academic Press, Inc.">Academic Press, Inc.</a> <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-384933-5" title="Special:BookSources/978-0-12-384933-5"><bdi>978-0-12-384933-5</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/2014010276">2014010276</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=8.1.&rft.btitle=Table+of+Integrals%2C+Series%2C+and+Products&rft.edition=8&rft.pub=Academic+Press%2C+Inc.&rft.date=2015&rft_id=info%3Alccn%2F2014010276&rft.isbn=978-0-12-384933-5&rft.aulast=Gradshteyn&rft.aufirst=Izrail+Solomonovich&rft.au=Ryzhik%2C+Iosif+Moiseevich&rft.au=Geronimus%2C+Yuri+Veniaminovich&rft.au=Tseytlin%2C+Michail+Yulyevich&rft.au=Jeffrey%2C+Alan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+integral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGreenhill1892" class="citation book cs1"><a href="/wiki/Alfred_George_Greenhill" title="Alfred George Greenhill">Greenhill, Alfred George</a> (1892). <a rel="nofollow" class="external text" href="https://archive.org/details/applicationselli00greerich"><i>The applications of elliptic functions</i></a>. New York: Macmillan.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+applications+of+elliptic+functions&rft.place=New+York&rft.pub=Macmillan&rft.date=1892&rft.aulast=Greenhill&rft.aufirst=Alfred+George&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fapplicationselli00greerich&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+integral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHancock1910" class="citation book cs1"><a href="/wiki/Harris_Hancock" title="Harris Hancock">Hancock, Harris</a> (1910). <a rel="nofollow" class="external text" href="https://archive.org/details/lecturestheorell00hancrich"><i>Lectures on the Theory of Elliptic Functions</i></a>. New York: J. Wiley & sons.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+the+Theory+of+Elliptic+Functions&rft.place=New+York&rft.pub=J.+Wiley+%26+sons&rft.date=1910&rft.aulast=Hancock&rft.aufirst=Harris&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Flecturestheorell00hancrich&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+integral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKing1924" class="citation book cs1">King, Louis V. (1924). <a rel="nofollow" class="external text" href="https://archive.org/details/onthenumerical032686mbp"><i>On The Direct Numerical Calculation Of Elliptic Functions And Integrals</i></a>. Cambridge University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=On+The+Direct+Numerical+Calculation+Of+Elliptic+Functions+And+Integrals&rft.pub=Cambridge+University+Press&rft.date=1924&rft.aulast=King&rft.aufirst=Louis+V.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fonthenumerical032686mbp&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+integral" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPressTeukolskyVetterlingFlannery2007" class="citation cs2">Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. (2007), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110811154417/http://apps.nrbook.com/empanel/index.html#pg=309">"Section 6.12. Elliptic Integrals and Jacobian Elliptic Functions"</a>, <i>Numerical Recipes: The Art of Scientific Computing</i> (3rd ed.), New York: Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-88068-8" title="Special:BookSources/978-0-521-88068-8"><bdi>978-0-521-88068-8</bdi></a>, archived from <a rel="nofollow" class="external text" href="http://apps.nrbook.com/empanel/index.html#pg=309">the original</a> on 2011-08-11<span class="reference-accessdate">, retrieved <span class="nowrap">2011-08-09</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Section+6.12.+Elliptic+Integrals+and+Jacobian+Elliptic+Functions&rft.btitle=Numerical+Recipes%3A+The+Art+of+Scientific+Computing&rft.place=New+York&rft.edition=3rd&rft.pub=Cambridge+University+Press&rft.date=2007&rft.isbn=978-0-521-88068-8&rft.aulast=Press&rft.aufirst=W.+H.&rft.au=Teukolsky%2C+S.+A.&rft.au=Vetterling%2C+W.+T.&rft.au=Flannery%2C+B.+P.&rft_id=http%3A%2F%2Fapps.nrbook.com%2Fempanel%2Findex.html%23pg%3D309&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+integral" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elliptic_integral&action=edit&section=25" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Elliptic_integral" class="extiw" title="commons:Category:Elliptic integral">Elliptic integral</a></span>.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Elliptic_integral">"Elliptic integral"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Elliptic+integral&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DElliptic_integral&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElliptic+integral" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/EllipticIntegral.html">Eric W. Weisstein, "Elliptic Integral" (Mathworld)</a></li> <li><a rel="nofollow" class="external text" href="https://github.com/moiseevigor/elliptic">Matlab code for elliptic integrals evaluation</a> by elliptic project</li> <li><a rel="nofollow" class="external text" href="http://www.exstrom.com/math/elliptic/ellipint.html">Rational Approximations for Complete Elliptic Integrals</a> (Exstrom Laboratories)</li> <li><a rel="nofollow" class="external text" href="https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1148&context=rhumj">A Brief History of Elliptic Integral Addition Theorems</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Nonelementary_integrals" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Nonelementary_integrals" title="Template:Nonelementary integrals"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Nonelementary_integrals" title="Template talk:Nonelementary integrals"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Nonelementary_integrals" title="Special:EditPage/Template:Nonelementary integrals"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Nonelementary_integrals" style="font-size:114%;margin:0 4em"><a href="/wiki/Nonelementary_integral" title="Nonelementary integral">Nonelementary integrals</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Elliptic integral</a></li> <li><a href="/wiki/Error_function" title="Error function">Error function</a></li> <li><a href="/wiki/Exponential_integral" title="Exponential integral">Exponential integral</a></li> <li><a href="/wiki/Fresnel_integral" title="Fresnel integral">Fresnel integral</a></li> <li><a href="/wiki/Logarithmic_integral_function" title="Logarithmic integral function">Logarithmic integral function</a></li> <li><a href="/wiki/Trigonometric_integral" title="Trigonometric integral">Trigonometric integral</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Topics_in_algebraic_curves" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Algebraic_curves_navbox" title="Template:Algebraic curves navbox"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Algebraic_curves_navbox" title="Template talk:Algebraic curves navbox"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Algebraic_curves_navbox" title="Special:EditPage/Template:Algebraic curves navbox"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Topics_in_algebraic_curves" style="font-size:114%;margin:0 4em">Topics in <a href="/wiki/Algebraic_curve" title="Algebraic curve">algebraic curves</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Rational_curve" class="mw-redirect" title="Rational curve">Rational curves</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Five_points_determine_a_conic" title="Five points determine a conic">Five points determine a conic</a></li> <li><a href="/wiki/Projective_line" title="Projective line">Projective line</a></li> <li><a href="/wiki/Rational_normal_curve" title="Rational normal curve">Rational normal curve</a></li> <li><a href="/wiki/Riemann_sphere" title="Riemann sphere">Riemann sphere</a></li> <li><a href="/wiki/Twisted_cubic" title="Twisted cubic">Twisted cubic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Elliptic_curve" title="Elliptic curve">Elliptic curves</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Analytic theory</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Elliptic_function" title="Elliptic function">Elliptic function</a></li> <li><a class="mw-selflink selflink">Elliptic integral</a></li> <li><a href="/wiki/Fundamental_pair_of_periods" title="Fundamental pair of periods">Fundamental pair of periods</a></li> <li><a href="/wiki/Modular_form" title="Modular form">Modular form</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Arithmetic theory</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Counting_points_on_elliptic_curves" title="Counting points on elliptic curves">Counting points on elliptic curves</a></li> <li><a href="/wiki/Division_polynomials" title="Division polynomials">Division polynomials</a></li> <li><a href="/wiki/Hasse%27s_theorem_on_elliptic_curves" title="Hasse's theorem on elliptic curves">Hasse's theorem on elliptic curves</a></li> <li><a href="/wiki/Mazur%27s_torsion_theorem" class="mw-redirect" title="Mazur's torsion theorem">Mazur's torsion theorem</a></li> <li><a href="/wiki/Modular_elliptic_curve" title="Modular elliptic curve">Modular elliptic curve</a></li> <li><a href="/wiki/Modularity_theorem" title="Modularity theorem">Modularity theorem</a></li> <li><a href="/wiki/Mordell%E2%80%93Weil_theorem" title="Mordell–Weil theorem">Mordell–Weil theorem</a></li> <li><a href="/wiki/Nagell%E2%80%93Lutz_theorem" title="Nagell–Lutz theorem">Nagell–Lutz theorem</a></li> <li><a href="/wiki/Supersingular_elliptic_curve" title="Supersingular elliptic curve">Supersingular elliptic curve</a></li> <li><a href="/wiki/Schoof%27s_algorithm" title="Schoof's algorithm">Schoof's algorithm</a></li> <li><a href="/wiki/Schoof%E2%80%93Elkies%E2%80%93Atkin_algorithm" title="Schoof–Elkies–Atkin algorithm">Schoof–Elkies–Atkin algorithm</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Elliptic_curve_cryptography" class="mw-redirect" title="Elliptic curve cryptography">Elliptic curve cryptography</a></li> <li><a href="/wiki/Elliptic_curve_primality" title="Elliptic curve primality">Elliptic curve primality</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Higher genus</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/De_Franchis_theorem" title="De Franchis theorem">De Franchis theorem</a></li> <li><a href="/wiki/Faltings%27s_theorem" title="Faltings's theorem">Faltings's theorem</a></li> <li><a href="/wiki/Hurwitz%27s_automorphisms_theorem" title="Hurwitz's automorphisms theorem">Hurwitz's automorphisms theorem</a></li> <li><a href="/wiki/Hurwitz_surface" title="Hurwitz surface">Hurwitz surface</a></li> <li><a href="/wiki/Hyperelliptic_curve" title="Hyperelliptic curve">Hyperelliptic curve</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Plane_curve" title="Plane curve">Plane curves</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/AF%2BBG_theorem" title="AF+BG theorem">AF+BG theorem</a></li> <li><a href="/wiki/B%C3%A9zout%27s_theorem" title="Bézout's theorem">Bézout's theorem</a></li> <li><a href="/wiki/Bitangent" title="Bitangent">Bitangent</a></li> <li><a href="/wiki/Cayley%E2%80%93Bacharach_theorem" title="Cayley–Bacharach theorem">Cayley–Bacharach theorem</a></li> <li><a href="/wiki/Conic_section" title="Conic section">Conic section</a></li> <li><a href="/wiki/Cramer%27s_paradox" title="Cramer's paradox">Cramer's paradox</a></li> <li><a href="/wiki/Cubic_plane_curve" title="Cubic plane curve">Cubic plane curve</a></li> <li><a href="/wiki/Fermat_curve" title="Fermat curve">Fermat curve</a></li> <li><a href="/wiki/Genus%E2%80%93degree_formula" title="Genus–degree formula">Genus–degree formula</a></li> <li><a href="/wiki/Hilbert%27s_sixteenth_problem" title="Hilbert's sixteenth problem">Hilbert's sixteenth problem</a></li> <li><a href="/wiki/Nagata%27s_conjecture_on_curves" title="Nagata's conjecture on curves">Nagata's conjecture on curves</a></li> <li><a href="/wiki/Pl%C3%BCcker_formula" title="Plücker formula">Plücker formula</a></li> <li><a href="/wiki/Quartic_plane_curve" title="Quartic plane curve">Quartic plane curve</a></li> <li><a href="/wiki/Real_plane_curve" title="Real plane curve">Real plane curve</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surfaces</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Belyi%27s_theorem" title="Belyi's theorem">Belyi's theorem</a></li> <li><a href="/wiki/Bring%27s_curve" title="Bring's curve">Bring's curve</a></li> <li><a href="/wiki/Bolza_surface" title="Bolza surface">Bolza surface</a></li> <li><a href="/wiki/Compact_Riemann_surface" class="mw-redirect" title="Compact Riemann surface">Compact Riemann surface</a></li> <li><a href="/wiki/Dessin_d%27enfant" title="Dessin d'enfant">Dessin d'enfant</a></li> <li><a href="/wiki/Differential_of_the_first_kind" title="Differential of the first kind">Differential of the first kind</a></li> <li><a href="/wiki/Klein_quartic" title="Klein quartic">Klein quartic</a></li> <li><a href="/wiki/Riemann%27s_existence_theorem" class="mw-redirect" title="Riemann's existence theorem">Riemann's existence theorem</a></li> <li><a href="/wiki/Riemann%E2%80%93Roch_theorem" title="Riemann–Roch theorem">Riemann–Roch theorem</a></li> <li><a href="/wiki/Teichm%C3%BCller_space" title="Teichmüller space">Teichmüller space</a></li> <li><a href="/wiki/Torelli_theorem" title="Torelli theorem">Torelli theorem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constructions</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dual_curve" title="Dual curve">Dual curve</a></li> <li><a href="/wiki/Polar_curve" title="Polar curve">Polar curve</a></li> <li><a href="/wiki/Smooth_completion" title="Smooth completion">Smooth completion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Structure of curves</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Divisors on curves</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abel%E2%80%93Jacobi_map" title="Abel–Jacobi map">Abel–Jacobi map</a></li> <li><a href="/wiki/Brill%E2%80%93Noether_theory" title="Brill–Noether theory">Brill–Noether theory</a></li> <li><a href="/wiki/Clifford%27s_theorem_on_special_divisors" title="Clifford's theorem on special divisors">Clifford's theorem on special divisors</a></li> <li><a href="/wiki/Gonality_of_an_algebraic_curve" title="Gonality of an algebraic curve">Gonality of an algebraic curve</a></li> <li><a href="/wiki/Jacobian_variety" title="Jacobian variety">Jacobian variety</a></li> <li><a href="/wiki/Riemann%E2%80%93Roch_theorem" title="Riemann–Roch theorem">Riemann–Roch theorem</a></li> <li><a href="/wiki/Weierstrass_point" title="Weierstrass point">Weierstrass point</a></li> <li><a href="/wiki/Weil_reciprocity_law" title="Weil reciprocity law">Weil reciprocity law</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Moduli</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/ELSV_formula" title="ELSV formula">ELSV formula</a></li> <li><a href="/wiki/Gromov%E2%80%93Witten_invariant" title="Gromov–Witten invariant">Gromov–Witten invariant</a></li> <li><a href="/wiki/Hodge_bundle" title="Hodge bundle">Hodge bundle</a></li> <li><a href="/wiki/Moduli_of_algebraic_curves" title="Moduli of algebraic curves">Moduli of algebraic curves</a></li> <li><a href="/wiki/Stable_curve" title="Stable curve">Stable curve</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Morphisms</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hasse%E2%80%93Witt_matrix" title="Hasse–Witt matrix">Hasse–Witt matrix</a></li> <li><a href="/wiki/Riemann%E2%80%93Hurwitz_formula" title="Riemann–Hurwitz formula">Riemann–Hurwitz formula</a></li> <li><a href="/wiki/Prym_variety" title="Prym variety">Prym variety</a></li> <li><a href="/wiki/Weber%27s_theorem_(Algebraic_curves)" title="Weber's theorem (Algebraic curves)">Weber's theorem (Algebraic curves)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Singular_point_of_a_curve" title="Singular point of a curve">Singularities</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ak_singularity" title="Ak singularity"><i>A<sub>k</sub></i> singularity</a></li> <li><a href="/wiki/Acnode" title="Acnode">Acnode</a></li> <li><a href="/wiki/Crunode" title="Crunode">Crunode</a></li> <li><a href="/wiki/Cusp_(singularity)" title="Cusp (singularity)">Cusp</a></li> <li><a href="/wiki/Delta_invariant" class="mw-redirect" title="Delta invariant">Delta invariant</a></li> <li><a href="/wiki/Tacnode" title="Tacnode">Tacnode</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Vector_bundle" title="Vector bundle">Vector bundles</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Birkhoff%E2%80%93Grothendieck_theorem" title="Birkhoff–Grothendieck theorem">Birkhoff–Grothendieck theorem</a></li> <li><a href="/wiki/Stable_vector_bundle" title="Stable vector bundle">Stable vector bundle</a></li> <li><a href="/wiki/Vector_bundles_on_algebraic_curves" title="Vector bundles on algebraic curves">Vector bundles on algebraic curves</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q1126603#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4152029-4">Germany</a></span></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Elliptic_integral&oldid=1251384191">https://en.wikipedia.org/w/index.php?title=Elliptic_integral&oldid=1251384191</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Elliptic_functions" title="Category:Elliptic functions">Elliptic functions</a></li><li><a href="/wiki/Category:Special_hypergeometric_functions" title="Category:Special hypergeometric functions">Special hypergeometric functions</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_German-language_sources_(de)" title="Category:CS1 German-language sources (de)">CS1 German-language sources (de)</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Use_American_English_from_January_2019" title="Category:Use American English from January 2019">Use American English from January 2019</a></li><li><a href="/wiki/Category:All_Wikipedia_articles_written_in_American_English" title="Category:All Wikipedia articles written in American English">All Wikipedia articles written in American English</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_February_2024" title="Category:Articles with unsourced statements from February 2024">Articles with unsourced statements from February 2024</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_January_2017" title="Category:Articles with unsourced statements from January 2017">Articles with unsourced statements from January 2017</a></li><li><a href="/wiki/Category:Commons_category_link_is_on_Wikidata" title="Category:Commons category link is on Wikidata">Commons category link is on Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 15 October 2024, at 21:38<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Elliptic_integral&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-694cf4987f-zjhj9","wgBackendResponseTime":150,"wgPageParseReport":{"limitreport":{"cputime":"0.829","walltime":"1.170","ppvisitednodes":{"value":8010,"limit":1000000},"postexpandincludesize":{"value":115536,"limit":2097152},"templateargumentsize":{"value":12913,"limit":2097152},"expansiondepth":{"value":14,"limit":100},"expensivefunctioncount":{"value":5,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":102161,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 801.400 1 -total"," 18.53% 148.489 2 Template:Reflist"," 15.53% 124.440 10 Template:Cite_book"," 12.84% 102.887 87 Template:Math"," 12.63% 101.188 4 Template:Navbox"," 12.12% 97.151 1 Template:Short_description"," 11.52% 92.301 1 Template:Nonelementary_Integral"," 8.96% 71.806 2 Template:Pagetype"," 5.84% 46.835 3 Template:Citation_needed"," 5.69% 45.581 1 Template:Commons_category"]},"scribunto":{"limitreport-timeusage":{"value":"0.438","limit":"10.000"},"limitreport-memusage":{"value":10448313,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFBorweinBorwein1987\"] = 3,\n [\"CITEREFByrdFriedman1971\"] = 1,\n [\"CITEREFCarlson1995\"] = 1,\n [\"CITEREFChowlaSelberg1949\"] = 1,\n [\"CITEREFChowlaSelberg1967\"] = 1,\n [\"CITEREFErdélyiMagnusOberhettingerTricomi1953\"] = 1,\n [\"CITEREFGradshteynRyzhikGeronimusTseytlin2015\"] = 1,\n [\"CITEREFGreenhill1892\"] = 1,\n [\"CITEREFHancock1910\"] = 1,\n [\"CITEREFInternet_Archive1991\"] = 1,\n [\"CITEREFKing1924\"] = 1,\n [\"CITEREFPressTeukolskyVetterlingFlannery2007\"] = 1,\n}\ntemplate_list = table#1 {\n [\"!\"] = 3,\n [\"AS ref\"] = 1,\n [\"Abs\"] = 1,\n [\"Algebraic curves navbox\"] = 1,\n [\"Authority control\"] = 1,\n [\"Circa\"] = 1,\n [\"Citation\"] = 2,\n [\"Citation needed\"] = 3,\n [\"Cite book\"] = 9,\n [\"Cite journal\"] = 3,\n [\"Cite web\"] = 6,\n [\"Commons category\"] = 1,\n [\"Div col\"] = 1,\n [\"Div col end\"] = 1,\n [\"Dlmf\"] = 1,\n [\"Math\"] = 87,\n [\"Mvar\"] = 10,\n [\"Nonelementary Integral\"] = 1,\n [\"Portal\"] = 1,\n [\"Prime\"] = 2,\n [\"Refbegin\"] = 1,\n [\"Refend\"] = 1,\n [\"Reflist\"] = 2,\n [\"Sfn\"] = 3,\n [\"Sfrac\"] = 7,\n [\"Short description\"] = 1,\n [\"Springer\"] = 1,\n [\"Sqrt\"] = 7,\n [\"Use American English\"] = 1,\n [\"Val\"] = 1,\n}\narticle_whitelist = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-6b7f745dd4-r95xb","timestamp":"20241125135707","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Elliptic integral","url":"https:\/\/en.wikipedia.org\/wiki\/Elliptic_integral","sameAs":"http:\/\/www.wikidata.org\/entity\/Q1126603","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q1126603","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2002-02-25T15:51:15Z","dateModified":"2024-10-15T21:38:57Z","headline":"family of special functions defined by an integral"}</script> </body> </html>

CINXE.COM

Elliptic integral - Wikipedia