Talk:Fourier series/Archive 2

<!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>Talk:Fourier series/Archive 2 - 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":"46e68b75-30a2-478b-9461-e76e2c34a3af","wgCanonicalNamespace":"Talk","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":1,"wgPageName":"Talk:Fourier_series/Archive_2","wgTitle":"Fourier series/Archive 2","wgCurRevisionId":1140628542,"wgRevisionId":1140628542,"wgArticleId":27824697,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":[],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Talk:Fourier_series/Archive_2","wgRelevantArticleId":27824697,"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":100000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgDiscussionToolsFeaturesEnabled":{"replytool":false,"newtopictool":true,"sourcemodetoolbar":true,"topicsubscription":false,"autotopicsub":false,"visualenhancements":false,"visualenhancements_reply":false,"visualenhancements_pageframe":false},"wgDiscussionToolsFallbackEditMode":"source","wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":true,"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","mediawiki.action.styles":"ready","mediawiki.interface.helpers.styles":"ready","mediawiki.codex.messagebox.styles":"ready","ext.math.styles":"ready","ext.cite.styles":"ready","ext.discussionTools.init.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","site","mediawiki.page.ready","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.discussionTools.init","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession"];</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.discussionTools.init.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cmediawiki.action.styles%7Cmediawiki.codex.messagebox.styles%7Cmediawiki.interface.helpers.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles&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.5"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="noindex,nofollow,max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Talk:Fourier series/Archive 2 - 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/Talk:Fourier_series/Archive_2"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Talk:Fourier_series/Archive_2&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/Talk:Fourier_series/Archive_2"> <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="ext-discussiontools-newtopictool-enabled ext-discussiontools-sourcemodetoolbar-enabled skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-1 ns-talk mw-editable page-Talk_Fourier_series_Archive_2 rootpage-Talk_Fourier_series 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=Talk%3AFourier+series%2FArchive+2&returntoquery=oldid%3D1140628542" 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=Talk%3AFourier+series%2FArchive+2&returntoquery=oldid%3D1140628542" 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=Talk%3AFourier+series%2FArchive+2&returntoquery=oldid%3D1140628542" 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=Talk%3AFourier+series%2FArchive+2&returntoquery=oldid%3D1140628542" 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-Possible_Error" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Possible_Error"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Possible Error</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">1 comment</span> <ul id="toc-Possible_Error-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Format_of_equations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Format_of_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Format of equations</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">6 comments</span> <ul id="toc-Format_of_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-relationship_between_real_and_complex_forms" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#relationship_between_real_and_complex_forms"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>relationship between real and complex forms</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">1 comment</span> <ul id="toc-relationship_between_real_and_complex_forms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Still_very_convoluted" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Still_very_convoluted"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Still very convoluted</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">4 comments</span> <ul id="toc-Still_very_convoluted-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Error" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Error"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Error</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">2 comments</span> <ul id="toc-Error-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Trigonometric_series_redirect?" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Trigonometric_series_redirect?"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Trigonometric series redirect?</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">3 comments</span> <ul id="toc-Trigonometric_series_redirect?-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-"real_Fourier_coefficients"" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#"real_Fourier_coefficients""> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>"real Fourier coefficients"</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">1 comment</span> <ul id="toc-"real_Fourier_coefficients"-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Making_a_mountain_out_of_a_molehill_is_not_a_good_example." class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Making_a_mountain_out_of_a_molehill_is_not_a_good_example."> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Making a mountain out of a molehill is not a good example.</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">2 comments</span> <ul id="toc-Making_a_mountain_out_of_a_molehill_is_not_a_good_example.-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-inconsistent_use_of_"L"" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#inconsistent_use_of_"L""> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>inconsistent use of "L"</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">1 comment</span> <button aria-controls="toc-inconsistent_use_of_"L"-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 inconsistent use of "L" subsection</span> </button> <ul id="toc-inconsistent_use_of_"L"-sublist" class="vector-toc-list"> <li id="toc-The_wave_equation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_wave_equation"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>The wave equation</span> </div> </a> <ul id="toc-The_wave_equation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Interpretation?" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Interpretation?"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Interpretation?</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">1 comment</span> <ul id="toc-Interpretation?-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General_case" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#General_case"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>General case</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">5 comments</span> <ul id="toc-General_case-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Article_is_of_very_low_quality." class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Article_is_of_very_low_quality."> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Article is of very low quality.</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">6 comments</span> <ul id="toc-Article_is_of_very_low_quality.-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Looking_for_an_image_of_a_vibrating_drum" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Looking_for_an_image_of_a_vibrating_drum"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Looking for an image of a vibrating drum</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">4 comments</span> <ul id="toc-Looking_for_an_image_of_a_vibrating_drum-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Contradiction" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Contradiction"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>Contradiction</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">1 comment</span> <ul id="toc-Contradiction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_new_figure_is_not_as_good_as_the_old_one" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#The_new_figure_is_not_as_good_as_the_old_one"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</span> <span>The new figure is not as good as the old one</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">2 comments</span> <ul id="toc-The_new_figure_is_not_as_good_as_the_old_one-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-incorrect_use_of_"Fourier_transform"" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#incorrect_use_of_"Fourier_transform""> <div class="vector-toc-text"> <span class="vector-toc-numb">16</span> <span>incorrect use of "Fourier transform"</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">4 comments</span> <ul id="toc-incorrect_use_of_"Fourier_transform"-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Simplified_example" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Simplified_example"> <div class="vector-toc-text"> <span class="vector-toc-numb">17</span> <span>Simplified example</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">1 comment</span> <ul id="toc-Simplified_example-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Now_sections_are_in_the_wrong_order" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Now_sections_are_in_the_wrong_order"> <div class="vector-toc-text"> <span class="vector-toc-numb">18</span> <span>Now sections are in the wrong order</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">5 comments</span> <ul id="toc-Now_sections_are_in_the_wrong_order-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-This_article_lacks_basic_explanation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#This_article_lacks_basic_explanation"> <div class="vector-toc-text"> <span class="vector-toc-numb">19</span> <span>This article lacks basic explanation</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">4 comments</span> <ul id="toc-This_article_lacks_basic_explanation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dead_link" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Dead_link"> <div class="vector-toc-text"> <span class="vector-toc-numb">20</span> <span>Dead link</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">1 comment</span> <ul id="toc-Dead_link-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Introductory_Sentence." class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Introductory_Sentence."> <div class="vector-toc-text"> <span class="vector-toc-numb">21</span> <span>Introductory Sentence.</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">1 comment</span> <ul id="toc-Introductory_Sentence.-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-periodic_functions_as_tempered_distriutions." class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#periodic_functions_as_tempered_distriutions."> <div class="vector-toc-text"> <span class="vector-toc-numb">22</span> <span>periodic functions as tempered distriutions.</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">5 comments</span> <ul id="toc-periodic_functions_as_tempered_distriutions.-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vibrating_Drum" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Vibrating_Drum"> <div class="vector-toc-text"> <span class="vector-toc-numb">23</span> <span>Vibrating Drum</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">3 comments</span> <ul id="toc-Vibrating_Drum-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ping-pong_sections" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Ping-pong_sections"> <div class="vector-toc-text"> <span class="vector-toc-numb">24</span> <span>Ping-pong sections</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">8 comments</span> <ul id="toc-Ping-pong_sections-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example:_a_simple_Fourier_series" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Example:_a_simple_Fourier_series"> <div class="vector-toc-text"> <span class="vector-toc-numb">25</span> <span>Example: a simple Fourier series</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">1 comment</span> <ul id="toc-Example:_a_simple_Fourier_series-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sines_and_cosines_more_"accessible"_than_sinusoids?" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Sines_and_cosines_more_"accessible"_than_sinusoids?"> <div class="vector-toc-text"> <span class="vector-toc-numb">26</span> <span>Sines and cosines more "accessible" than sinusoids?</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">2 comments</span> <ul id="toc-Sines_and_cosines_more_"accessible"_than_sinusoids?-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definition_of_Fourier_series" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Definition_of_Fourier_series"> <div class="vector-toc-text"> <span class="vector-toc-numb">27</span> <span>Definition of Fourier series</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">2 comments</span> <ul id="toc-Definition_of_Fourier_series-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formula_for_a0" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Formula_for_a0"> <div class="vector-toc-text"> <span class="vector-toc-numb">28</span> <span>Formula for <i>a</i><sub>0</sub></span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">2 comments</span> <ul id="toc-Formula_for_a0-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Merger_proposal" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Merger_proposal"> <div class="vector-toc-text"> <span class="vector-toc-numb">29</span> <span>Merger proposal</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">5 comments</span> <ul id="toc-Merger_proposal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reviewers_of_the_original_Fourier_article" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Reviewers_of_the_original_Fourier_article"> <div class="vector-toc-text"> <span class="vector-toc-numb">30</span> <span>Reviewers of the original Fourier article</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">2 comments</span> <ul id="toc-Reviewers_of_the_original_Fourier_article-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vibrating_Disc_doesn't_look_right" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Vibrating_Disc_doesn't_look_right"> <div class="vector-toc-text"> <span class="vector-toc-numb">31</span> <span>Vibrating Disc doesn't look right</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">2 comments</span> <ul id="toc-Vibrating_Disc_doesn't_look_right-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Too_technical" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Too_technical"> <div class="vector-toc-text"> <span class="vector-toc-numb">32</span> <span>Too technical</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">13 comments</span> <ul id="toc-Too_technical-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Very_first_example" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Very_first_example"> <div class="vector-toc-text"> <span class="vector-toc-numb">33</span> <span>Very first example</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">2 comments</span> <ul id="toc-Very_first_example-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quote_isn't_actually_Fourier's?" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Quote_isn't_actually_Fourier's?"> <div class="vector-toc-text"> <span class="vector-toc-numb">34</span> <span>Quote isn't actually Fourier's?</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">7 comments</span> <ul id="toc-Quote_isn't_actually_Fourier's?-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formula_incorrect" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Formula_incorrect"> <div class="vector-toc-text"> <span class="vector-toc-numb">35</span> <span>Formula incorrect</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">5 comments</span> <ul id="toc-Formula_incorrect-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formula_and_graph_axis_labelling" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Formula_and_graph_axis_labelling"> <div class="vector-toc-text"> <span class="vector-toc-numb">36</span> <span>Formula and graph axis labelling</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">6 comments</span> <ul id="toc-Formula_and_graph_axis_labelling-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Request_for_comment_by_readers_of_this_article" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Request_for_comment_by_readers_of_this_article"> <div class="vector-toc-text"> <span class="vector-toc-numb">37</span> <span>Request for comment by readers of this article</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">1 comment</span> <ul id="toc-Request_for_comment_by_readers_of_this_article-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Riemann_paper_and_1811_essay_competition" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Riemann_paper_and_1811_essay_competition"> <div class="vector-toc-text"> <span class="vector-toc-numb">38</span> <span>Riemann paper and 1811 essay competition</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">2 comments</span> <ul id="toc-Riemann_paper_and_1811_essay_competition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fourier's_theorem" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Fourier's_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">39</span> <span>Fourier's theorem</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">2 comments</span> <ul id="toc-Fourier's_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fourier_series_on_a_general_interval_[a,_a_+_τ]" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Fourier_series_on_a_general_interval_[a,_a_+_τ]"> <div class="vector-toc-text"> <span class="vector-toc-numb">40</span> <span>Fourier series on a general interval [a, a + τ]</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">2 comments</span> <ul id="toc-Fourier_series_on_a_general_interval_[a,_a_+_τ]-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fourier's_formulae_for_T-periodic_functions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Fourier's_formulae_for_T-periodic_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">41</span> <span>Fourier's formulae for T-periodic functions</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">8 comments</span> <ul id="toc-Fourier's_formulae_for_T-periodic_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-translation_of_paper's_title?" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#translation_of_paper's_title?"> <div class="vector-toc-text"> <span class="vector-toc-numb">42</span> <span>translation of paper's title?</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">1 comment</span> <ul id="toc-translation_of_paper's_title?-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_coefficient_in_"2.1_Example_1:_a_simple_Fourier_series"" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#The_coefficient_in_"2.1_Example_1:_a_simple_Fourier_series""> <div class="vector-toc-text"> <span class="vector-toc-numb">43</span> <span>The coefficient in "2.1 Example 1: a simple Fourier series"</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">3 comments</span> <ul id="toc-The_coefficient_in_"2.1_Example_1:_a_simple_Fourier_series"-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Why_are_we_using_a_finite_series_in_the_definition?" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Why_are_we_using_a_finite_series_in_the_definition?"> <div class="vector-toc-text"> <span class="vector-toc-numb">44</span> <span>Why are we using a finite series in the definition?</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">4 comments</span> <ul id="toc-Why_are_we_using_a_finite_series_in_the_definition?-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-layperson_readability" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#layperson_readability"> <div class="vector-toc-text"> <span class="vector-toc-numb">45</span> <span>layperson readability</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">3 comments</span> <ul id="toc-layperson_readability-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-1/2_[f(x0+)+f(x0-)]" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#1/2_[f(x0+)+f(x0-)]"> <div class="vector-toc-text"> <span class="vector-toc-numb">46</span> <span>1/2 [f(x0+)+f(x0-)]</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">2 comments</span> <ul id="toc-1/2_[f(x0+)+f(x0-)]-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Eye-catching_GIF" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Eye-catching_GIF"> <div class="vector-toc-text"> <span class="vector-toc-numb">47</span> <span>Eye-catching GIF</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">4 comments</span> <ul id="toc-Eye-catching_GIF-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-there_is_an_asterisk_(*)._What_does_it_mean?" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#there_is_an_asterisk_(*)._What_does_it_mean?"> <div class="vector-toc-text"> <span class="vector-toc-numb">48</span> <span>there is an asterisk (*). What does it mean?</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">3 comments</span> <ul id="toc-there_is_an_asterisk_(*)._What_does_it_mean?-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fourier_Series_definition_and_other_problems" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Fourier_Series_definition_and_other_problems"> <div class="vector-toc-text"> <span class="vector-toc-numb">49</span> <span>Fourier Series definition and other problems</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">3 comments</span> <ul id="toc-Fourier_Series_definition_and_other_problems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Label_for_the_cool_gif_is_unclear." class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Label_for_the_cool_gif_is_unclear."> <div class="vector-toc-text"> <span class="vector-toc-numb">50</span> <span>Label for the cool gif is unclear.</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">1 comment</span> <ul id="toc-Label_for_the_cool_gif_is_unclear.-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Coefficient_of_'"`UNIQ--postMath-00000070-QINU`"'" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Coefficient_of_'"`UNIQ--postMath-00000070-QINU`"'"> <div class="vector-toc-text"> <span class="vector-toc-numb">51</span> <span>Coefficient of '"`UNIQ--postMath-00000070-QINU`"'</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">2 comments</span> <ul id="toc-Coefficient_of_'"`UNIQ--postMath-00000070-QINU`"'-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-No_such_topic_yet" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#No_such_topic_yet"> <div class="vector-toc-text"> <span class="vector-toc-numb">52</span> <span>No such topic yet</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">1 comment</span> <ul id="toc-No_such_topic_yet-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_first_few_paragraphs_should_be_moved" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#The_first_few_paragraphs_should_be_moved"> <div class="vector-toc-text"> <span class="vector-toc-numb">53</span> <span>The first few paragraphs should be moved</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">2 comments</span> <ul id="toc-The_first_few_paragraphs_should_be_moved-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Should_this_article_definition_start_with_a_scary_correlation_function?" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Should_this_article_definition_start_with_a_scary_correlation_function?"> <div class="vector-toc-text"> <span class="vector-toc-numb">54</span> <span>Should this article definition start with a scary correlation function?</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">3 comments</span> <ul id="toc-Should_this_article_definition_start_with_a_scary_correlation_function?-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fourier's_Theorem_is_necessary_but_not_in_Wikipedia" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Fourier's_Theorem_is_necessary_but_not_in_Wikipedia"> <div class="vector-toc-text"> <span class="vector-toc-numb">55</span> <span>Fourier's Theorem is necessary but not in Wikipedia</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">2 comments</span> <ul id="toc-Fourier's_Theorem_is_necessary_but_not_in_Wikipedia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-I_brought_the_animations_to_be_fixed_in_between_paragraphs_of_the_intro_(using_align_=_none)." class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#I_brought_the_animations_to_be_fixed_in_between_paragraphs_of_the_intro_(using_align_=_none)."> <div class="vector-toc-text"> <span class="vector-toc-numb">56</span> <span>I brought the animations to be fixed in between paragraphs of the intro (using align = none).</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">1 comment</span> <ul id="toc-I_brought_the_animations_to_be_fixed_in_between_paragraphs_of_the_intro_(using_align_=_none).-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Maybe_use_'ξ'_instead_of_'f'_for_frequency_to_avoid_conflict_with_'f'_for_function" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Maybe_use_'ξ'_instead_of_'f'_for_frequency_to_avoid_conflict_with_'f'_for_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">57</span> <span>Maybe use 'ξ' instead of 'f' for frequency to avoid conflict with 'f' for function</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">6 comments</span> <ul id="toc-Maybe_use_'ξ'_instead_of_'f'_for_frequency_to_avoid_conflict_with_'f'_for_function-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-namespace">Talk</span><span class="mw-page-title-separator">:</span><span class="mw-page-title-main">Fourier series/Archive 2</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang mw-portlet-lang-icon-only" > <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-empty" aria-label="This article exist only in this language. Add the article for other 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--icon-only mw-portlet-lang-heading-empty" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language mw-ui-icon-wikimedia-language"></span> <span class="vector-dropdown-label-text">Add languages</span> </label> <div class="vector-dropdown-content"> <div class="mw-portlet-empty-language-selector-body"> Page contents not supported in other languages. </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="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Fourier_series/Archive_2&action=edit&redlink=1" class="new" title="View the content page (page does not exist) [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Fourier_series/Archive_2" 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/Talk:Fourier_series/Archive_2"><span>Read</span></a></li><li id="ca-edit" class="istalk vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Talk:Fourier_series/Archive_2&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=Talk:Fourier_series/Archive_2&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/Talk:Fourier_series/Archive_2"><span>Read</span></a></li><li id="ca-more-edit" class="istalk vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Talk:Fourier_series/Archive_2&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=Talk:Fourier_series/Archive_2&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/Talk:Fourier_series/Archive_2" 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/Talk:Fourier_series/Archive_2" 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=Talk:Fourier_series/Archive_2&oldid=1140628542" 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=Talk:Fourier_series/Archive_2&action=info" title="More information about this page"><span>Page information</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%2Fw%2Findex.php%3Ftitle%3DTalk%3AFourier_series%2FArchive_2%26oldid%3D1140628542"><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%2Fw%2Findex.php%3Ftitle%3DTalk%3AFourier_series%2FArchive_2%26oldid%3D1140628542"><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=Talk%3AFourier_series%2FArchive_2&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=Talk:Fourier_series/Archive_2&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 emptyPortlet" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </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 class="subpages">< <bdi dir="ltr"><a href="/wiki/Talk:Fourier_series" title="Talk:Fourier series">Talk:Fourier series</a></bdi></div><div class="cdx-message cdx-message--block cdx-message--warning mw-revision"><span class="cdx-message__icon"></span><div class="cdx-message__content"><div id="mw-revision-info-current"><p><b>This is the <a href="/wiki/Help:Page_history" title="Help:Page history">current revision</a> of this page, as edited by <span id="mw-revision-name"><a href="/wiki/User:Lowercase_sigmabot_III" class="mw-userlink" title="User:Lowercase sigmabot III" data-mw-revid="1140628542"><bdi>Lowercase sigmabot III</bdi></a> <span class="mw-usertoollinks">(<a href="/wiki/User_talk:Lowercase_sigmabot_III" class="mw-redirect mw-usertoollinks-talk" title="User talk:Lowercase sigmabot III">talk</a> | <a href="/wiki/Special:Contributions/Lowercase_sigmabot_III" class="mw-usertoollinks-contribs" title="Special:Contributions/Lowercase sigmabot III">contribs</a>)</span></span> at <span id="mw-revision-date">00:26, 21 February 2023</span><span id="mw-revision-summary"> <span class="comment">(Archiving 2 discussion(s) from <a href="/wiki/Talk:Fourier_series" title="Talk:Fourier series">Talk:Fourier series</a>) (bot)</span></span>. The present address (URL) is a <a href="/wiki/Help:Permanent_link" title="Help:Permanent link">permanent link</a> to this version.</b></p><div id="revision-info-current-plain" style="display: none;">Revision as of 00:26, 21 February 2023 by <a href="/wiki/User:Lowercase_sigmabot_III" class="mw-userlink" title="User:Lowercase sigmabot III" data-mw-revid="1140628542"><bdi>Lowercase sigmabot III</bdi></a> <span class="mw-usertoollinks">(<a href="/wiki/User_talk:Lowercase_sigmabot_III" class="mw-redirect mw-usertoollinks-talk" title="User talk:Lowercase sigmabot III">talk</a> | <a href="/wiki/Special:Contributions/Lowercase_sigmabot_III" class="mw-usertoollinks-contribs" title="Special:Contributions/Lowercase sigmabot III">contribs</a>)</span> <span class="comment">(Archiving 2 discussion(s) from <a href="/wiki/Talk:Fourier_series" title="Talk:Fourier series">Talk:Fourier series</a>) (bot)</span></div></div><div id="mw-revision-nav">(<a href="/w/index.php?title=Talk:Fourier_series/Archive_2&diff=prev&oldid=1140628542" title="Talk:Fourier series/Archive 2">diff</a>) <a href="/w/index.php?title=Talk:Fourier_series/Archive_2&direction=prev&oldid=1140628542" title="Talk:Fourier series/Archive 2">← Previous revision</a> | Latest revision (diff) | Newer revision → (diff)</div></div></div></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r1237879475">.mw-parser-output .tmbox{margin:4px 0;border-collapse:collapse;border:1px solid #c0c090;background-color:#f8eaba;box-sizing:border-box}.mw-parser-output .tmbox.mbox-small{font-size:88%;line-height:1.25em}.mw-parser-output .tmbox-speedy{border:2px solid #b32424;background-color:#fee7e6}.mw-parser-output .tmbox-delete{border:2px solid #b32424}.mw-parser-output .tmbox-content{border:2px solid #f28500}.mw-parser-output .tmbox-style{border:2px solid #fc3}.mw-parser-output .tmbox-move{border:2px solid #9932cc}.mw-parser-output .tmbox .mbox-text{border:none;padding:0.25em 0.9em;width:100%}.mw-parser-output .tmbox .mbox-image{border:none;padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .tmbox .mbox-imageright{border:none;padding:2px 0.9em 2px 0;text-align:center}.mw-parser-output .tmbox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .tmbox .mbox-invalid-type{text-align:center}@media(min-width:720px){.mw-parser-output .tmbox{margin:4px 10%}.mw-parser-output .tmbox.mbox-small{clear:right;float:right;margin:4px 0 4px 1em;width:238px}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmbox{background-color:#2e2505}html.skin-theme-clientpref-night .mw-parser-output .tmbox-speedy{background-color:#310402}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmbox{background-color:#2e2505}html.skin-theme-clientpref-os .mw-parser-output .tmbox-speedy{background-color:#310402}}body.skin--responsive .mw-parser-output table.tmbox img{max-width:none!important}</style><table class="plainlinks tmbox tmbox-notice" role="presentation"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/2/2a/Replacement_filing_cabinet.svg/40px-Replacement_filing_cabinet.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/2/2a/Replacement_filing_cabinet.svg/60px-Replacement_filing_cabinet.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/2/2a/Replacement_filing_cabinet.svg/80px-Replacement_filing_cabinet.svg.png 2x" data-file-width="200" data-file-height="200"/></span></span></td><td class="mbox-text" style="text-align:center">This is an <b><a href="/wiki/Help:Archiving_a_talk_page" title="Help:Archiving a talk page">archive</a></b> of past discussions about <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a>. <b>Do not edit the contents of this page.</b> If you wish to start a new discussion or revive an old one, please do so on the <a href="/wiki/Talk:Fourier_series" title="Talk:Fourier series">current talk page</a>.</td></tr></tbody></table> <table style="text-align:center;margin:0 auto 0.5em;max-width:30em;width:100%;width:-moz-available;width:-webkit-fill-available;width:stretch"><tbody><tr><td><a href="/wiki/Talk:Fourier_series/Archive_1" title="Talk:Fourier series/Archive 1">Archive 1</a></td><td><span style="font-size:115%;"><a class="mw-selflink selflink">Archive 2</a></span></td><td><a href="/wiki/Talk:Fourier_series/Archive_3" title="Talk:Fourier series/Archive 3">Archive 3</a></td></tr></tbody></table> <meta property="mw:PageProp/toc"/> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Possible_Error" data-mw-thread-id="h-Possible_Error-2006-11-28T15:15:00.000Z"><span data-mw-comment-start="" id="h-Possible_Error-2006-11-28T15:15:00.000Z"></span>Possible Error<span data-mw-comment-end="h-Possible_Error-2006-11-28T15:15:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-87.69.5.52-2006-11-28T15:15:00.000Z-Possible_Error"></span>In section 5.3, the one on convergence, the following setence appears: </p><p>"This is convergence in the norm of the space L2, which means that the series converges almost everywhere to f." </p><p>This doesn't seem correct. </p><p><small class="autosigned">— Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/Special:Contributions/87.69.5.52" title="Special:Contributions/87.69.5.52">87.69.5.52</a> (<a href="/w/index.php?title=User_talk:87.69.5.52&action=edit&redlink=1" class="new" title="User talk:87.69.5.52 (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-87.69.5.52-2006-11-28T15:15:00.000Z-Possible_Error" class="ext-discussiontools-init-timestamplink">15:15, 28 November 2006 (UTC)</a></small><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-87.69.5.52-2006-11-28T15:15:00.000Z-Possible_Error"></span><span data-mw-comment-end="c-87.69.5.52-2006-11-28T15:15:00.000Z-Possible_Error"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Format_of_equations" data-mw-thread-id="h-Format_of_equations-2006-05-24T17:41:00.000Z"><span data-mw-comment-start="" id="h-Format_of_equations-2006-05-24T17:41:00.000Z"></span>Format of equations<span data-mw-comment-end="h-Format_of_equations-2006-05-24T17:41:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Fresheneesz-2006-05-24T17:41:00.000Z-Format_of_equations"></span>I find the equations difficult to read in their current format. Having a sentence, then a TeX png, then another sentence, etc, makes it hard for me to see what parts are associated with which equation, and where those parts end. Not only that, it makes the section unnecessarily long. </p><p>The format that I find very useful is this: </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 equation\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>t</mi> <mi>i</mi> <mi>o</mi> <mi>n</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle equation\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1983e65dd3a78a53a6229203904ad3a55742b05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.458ex; height:2.509ex;" alt="{\displaystyle equation\ }"/></span> <dl><dd><dl><dd>where <dl><dd><ul><li><i>some variable</i> is this,</li> <li><i>some other variable</i> is that, and</li> <li><i>some variable</i> = <i>this other stuff</i> is this other thing.</li></ul></dd></dl></dd></dl></dd></dl></dd></dl> <p>Some people might find that ugly or whatever, but it makes it very easy to see what is associated with a single equation. Let me give my example in the format currently employed on this page: </p><p>The format that I <b>don't</b> find very useful is this: </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 equation\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>t</mi> <mi>i</mi> <mi>o</mi> <mi>n</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle equation\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1983e65dd3a78a53a6229203904ad3a55742b05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.458ex; height:2.509ex;" alt="{\displaystyle equation\ }"/></span></dd></dl> <p>where </p> <dl><dd><i>some variable</i></dd></dl> <p>is this, </p> <dl><dd><i>some other variable</i></dd></dl> <p>is that, and </p> <dl><dd><i>some variable</i> = <i>this other stuff</i></dd></dl> <p>is this other thing. </p><p>I find it especially confusing on this page, where more than one *separate* equation is written in one long string. I'm not saying we have to use my perferred format, but I do think that the format needs to be different to make it easier to read. Comments? <a href="/wiki/User:Fresheneesz" title="User:Fresheneesz">Fresheneesz</a> <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Fresheneesz-2006-05-24T17:41:00.000Z-Format_of_equations" class="ext-discussiontools-init-timestamplink">17:41, 24 May 2006 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Fresheneesz-2006-05-24T17:41:00.000Z-Format_of_equations"></span><span data-mw-comment-end="c-Fresheneesz-2006-05-24T17:41:00.000Z-Format_of_equations"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Oleg_Alexandrov-2006-05-25T01:04:00.000Z-Fresheneesz-2006-05-24T17:41:00.000Z"></span>I like more the formal format where everything is indented once. I don't think that the "staircasing" of formulas improves things that much, and is also nonstandard. I'd say we should be conservative and not invent new paradigms here. <a href="/wiki/User:Oleg_Alexandrov" title="User:Oleg Alexandrov">Oleg Alexandrov</a> (<a href="/wiki/User_talk:Oleg_Alexandrov" title="User talk:Oleg Alexandrov">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Oleg_Alexandrov-2006-05-25T01:04:00.000Z-Fresheneesz-2006-05-24T17:41:00.000Z" class="ext-discussiontools-init-timestamplink">01:04, 25 May 2006 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Oleg_Alexandrov-2006-05-25T01:04:00.000Z-Fresheneesz-2006-05-24T17:41:00.000Z"></span><span data-mw-comment-end="c-Oleg_Alexandrov-2006-05-25T01:04:00.000Z-Fresheneesz-2006-05-24T17:41:00.000Z"></span></dd></dl> <dl><dd><dl><dd><span data-mw-comment-start="" id="c-Fresheneesz-2006-05-25T02:51:00.000Z-Oleg_Alexandrov-2006-05-25T01:04:00.000Z"></span>The bulleting of variables in an equation is not a new paradigm, and I think it applies very nicely here. I started the staircasing of formulas, and so I won't push that if you don't like it. However, I disagree with keeping it the way it is now. If my format is "staircasing" then the current format is "laddering" - the variables, formulas, equations, and explanation, being strung out in a long, hard to read list.</dd></dl></dd></dl> <dl><dd><dl><dd>I wonder though, why you removed the bullets. I kept the same indenting, but I feel that the bullets help a reader more clearly see what goes with what. I can't accept the current format, but I would like to hear whatever ideas you have to make those equations more readible.</dd></dl></dd></dl> <dl><dd><dl><dd>Do you think its not useful for a reader to be able to understand the equations in a section without reading that whole section? <a href="/wiki/User:Fresheneesz" title="User:Fresheneesz">Fresheneesz</a> <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Fresheneesz-2006-05-25T02:51:00.000Z-Oleg_Alexandrov-2006-05-25T01:04:00.000Z" class="ext-discussiontools-init-timestamplink">02:51, 25 May 2006 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Fresheneesz-2006-05-25T02:51:00.000Z-Oleg_Alexandrov-2006-05-25T01:04:00.000Z"></span><span data-mw-comment-end="c-Fresheneesz-2006-05-25T02:51:00.000Z-Oleg_Alexandrov-2006-05-25T01:04:00.000Z"></span> <dl><dd><span data-mw-comment-start="" id="c-Oleg_Alexandrov-2006-05-25T15:12:00.000Z-Fresheneesz-2006-05-25T02:51:00.000Z"></span>I believe the equations are readable enough with just one indent, that's what used in math papers everywhere and people don't complain. :) However, if you do really positively want a star in front of a few equation for emphasis, well, it would not make sense for me to oppose that. Just not much staircasing. :) <a href="/wiki/User:Oleg_Alexandrov" title="User:Oleg Alexandrov">Oleg Alexandrov</a> (<a href="/wiki/User_talk:Oleg_Alexandrov" title="User talk:Oleg Alexandrov">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Oleg_Alexandrov-2006-05-25T15:12:00.000Z-Fresheneesz-2006-05-25T02:51:00.000Z" class="ext-discussiontools-init-timestamplink">15:12, 25 May 2006 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Oleg_Alexandrov-2006-05-25T15:12:00.000Z-Fresheneesz-2006-05-25T02:51:00.000Z"></span><span data-mw-comment-end="c-Oleg_Alexandrov-2006-05-25T15:12:00.000Z-Fresheneesz-2006-05-25T02:51:00.000Z"></span> <dl><dd><span data-mw-comment-start="" id="c-Fresheneesz-2006-05-26T04:18:00.000Z-Oleg_Alexandrov-2006-05-25T15:12:00.000Z"></span>Alright thanks! (although.. where would people complain about the format of math papers..?) <a href="/wiki/User:Fresheneesz" title="User:Fresheneesz">Fresheneesz</a> <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Fresheneesz-2006-05-26T04:18:00.000Z-Oleg_Alexandrov-2006-05-25T15:12:00.000Z" class="ext-discussiontools-init-timestamplink">04:18, 26 May 2006 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Fresheneesz-2006-05-26T04:18:00.000Z-Oleg_Alexandrov-2006-05-25T15:12:00.000Z"></span><span data-mw-comment-end="c-Fresheneesz-2006-05-26T04:18:00.000Z-Oleg_Alexandrov-2006-05-25T15:12:00.000Z"></span></dd></dl></dd></dl></dd></dl></dd></dl> <dl><dd><dl><dd><span data-mw-comment-start="" id="c-85.220.116.138-2006-11-24T22:08:00.000Z-Oleg_Alexandrov-2006-05-25T01:04:00.000Z"></span>I think the last equation (in the "Modern derivations of Fourier series") is wrong; if f is a Riemann-integrable function then the Lebesgue integral (left side) equals the Riemann integral (right side), there's no need to multiply by a constant (2pi in this case).</dd> <dd><small class="autosigned">— Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/Special:Contributions/85.220.116.138" title="Special:Contributions/85.220.116.138">85.220.116.138</a> (<a href="/w/index.php?title=User_talk:85.220.116.138&action=edit&redlink=1" class="new" title="User talk:85.220.116.138 (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-85.220.116.138-2006-11-24T22:08:00.000Z-Oleg_Alexandrov-2006-05-25T01:04:00.000Z" class="ext-discussiontools-init-timestamplink">22:08, 24 November 2006 (UTC)</a><span data-mw-comment-end="c-85.220.116.138-2006-11-24T22:08:00.000Z-Oleg_Alexandrov-2006-05-25T01:04:00.000Z"></span></small><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-85.220.116.138-2006-11-24T22:08:00.000Z-Oleg_Alexandrov-2006-05-25T01:04:00.000Z"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="relationship_between_real_and_complex_forms" data-mw-thread-id="h-relationship_between_real_and_complex_forms-2006-05-26T21:20:00.000Z"><span data-mw-comment-start="" id="h-relationship_between_real_and_complex_forms-2006-05-26T21:20:00.000Z"></span>relationship between real and complex forms<span data-mw-comment-end="h-relationship_between_real_and_complex_forms-2006-05-26T21:20:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Fresheneesz-2006-05-26T21:20:00.000Z-relationship_between_real_and_complex_forms"></span>I was thinking about setting the real and complex forms equal, rather than writing them separately. I would guess people would think this way is "crowded" or "ugly". But I also have some related things I'd like to put up: </p><p>Some relationships between the variables in fourier series: </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 c_{n}={\frac {a_{n}-jb_{n}}{2}}}"> <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"> <mfrac> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>j</mi> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{n}={\frac {a_{n}-jb_{n}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f35efff6a465ea675f7b3373c5bcc2931d8461f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.623ex; height:5.343ex;" alt="{\displaystyle c_{n}={\frac {a_{n}-jb_{n}}{2}}}"/></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 c_{0}=a_{0}/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{0}=a_{0}/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aadd94046e456802fb0b152f0f6dfbd28dc57ed3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.769ex; height:2.843ex;" alt="{\displaystyle c_{0}=a_{0}/2}"/></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 a_{n}=c_{n}+c_{-n}=2Re[c_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>R</mi> <mi>e</mi> <mo stretchy="false">[</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}=c_{n}+c_{-n}=2Re[c_{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e77a659a3fe7306d695bf4e846e328381d6f47fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.744ex; height:2.843ex;" alt="{\displaystyle a_{n}=c_{n}+c_{-n}=2Re[c_{n}]}"/></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 b_{n}=c_{-n}-c_{n}=2Im[c_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>I</mi> <mi>m</mi> <mo stretchy="false">[</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}=c_{-n}-c_{n}=2Im[c_{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa2aad1a1446f97b0bf32b0141c648adc6fe8e5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.876ex; height:2.843ex;" alt="{\displaystyle b_{n}=c_{-n}-c_{n}=2Im[c_{n}]}"/></span></dd></dl> <p>and for real functions: </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 c_{-n}=c_{n}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{-n}=c_{n}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6e8e1bd54b6380481851f5b545ad92cdd5ce481" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.828ex; height:2.509ex;" alt="{\displaystyle c_{-n}=c_{n}^{*}}"/></span></dd></dl> <p><a href="/wiki/User:Fresheneesz" title="User:Fresheneesz">Fresheneesz</a> <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Fresheneesz-2006-05-26T21:20:00.000Z-relationship_between_real_and_complex_forms" class="ext-discussiontools-init-timestamplink">21:20, 26 May 2006 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Fresheneesz-2006-05-26T21:20:00.000Z-relationship_between_real_and_complex_forms"></span><span data-mw-comment-end="c-Fresheneesz-2006-05-26T21:20:00.000Z-relationship_between_real_and_complex_forms"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Still_very_convoluted" data-mw-thread-id="h-Still_very_convoluted-2006-07-02T19:08:00.000Z"><span data-mw-comment-start="" id="h-Still_very_convoluted-2006-07-02T19:08:00.000Z"></span>Still very convoluted<span data-mw-comment-end="h-Still_very_convoluted-2006-07-02T19:08:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Fresheneesz-2006-07-02T19:08:00.000Z-Still_very_convoluted"></span>I'm not new to maths, and I find this article difficult to understand. Possibly it needs to be re-written in a more coherent fashion? Starting with a global definition, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }[a_{n}cos({\frac {n\pi x}{L}})+b_{n}sin({\frac {n\pi x}{L}})]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>π</mi> <mi>x</mi> </mrow> <mi>L</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>π</mi> <mi>x</mi> </mrow> <mi>L</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }[a_{n}cos({\frac {n\pi x}{L}})+b_{n}sin({\frac {n\pi x}{L}})]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fad394ff168f8a4c48f0ef3b629d8f143f12003" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:45.546ex; height:6.843ex;" alt="{\displaystyle f(x)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }[a_{n}cos({\frac {n\pi x}{L}})+b_{n}sin({\frac {n\pi x}{L}})]}"/></span><br/> where<br/> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}={\frac {1}{L}}\int _{-L}^{L}f(x)cos({\frac {2n\pi x}{L}})dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>L</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> <mi>π</mi> <mi>x</mi> </mrow> <mi>L</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}={\frac {1}{L}}\int _{-L}^{L}f(x)cos({\frac {2n\pi x}{L}})dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7237c801682532c116e139ffaacb45347960746d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.715ex; height:6.343ex;" alt="{\displaystyle a_{n}={\frac {1}{L}}\int _{-L}^{L}f(x)cos({\frac {2n\pi x}{L}})dx}"/></span><br/> <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 b_{n}={\frac {1}{L}}\int _{-L}^{L}f(x)sin({\frac {2n\pi x}{L}})dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>L</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> <mi>π</mi> <mi>x</mi> </mrow> <mi>L</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}={\frac {1}{L}}\int _{-L}^{L}f(x)sin({\frac {2n\pi x}{L}})dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee2376737667cf3ed6775f32da648fbc055d35ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.545ex; height:6.343ex;" alt="{\displaystyle b_{n}={\frac {1}{L}}\int _{-L}^{L}f(x)sin({\frac {2n\pi x}{L}})dx}"/></span><br/> </p><p>Should the function be odd (link to odd function definition) the fourier cosine series may be used. This simplifies to<br/> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}={\frac {2}{L}}\int _{L_{1}}^{L_{2}}f(x)cos({\frac {2n\pi x}{L}})dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>L</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> <mi>π</mi> <mi>x</mi> </mrow> <mi>L</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}={\frac {2}{L}}\int _{L_{1}}^{L_{2}}f(x)cos({\frac {2n\pi x}{L}})dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1aec41ac25fda069b5cc5e2b4870a75f638c7d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.439ex; height:6.509ex;" alt="{\displaystyle a_{n}={\frac {2}{L}}\int _{L_{1}}^{L_{2}}f(x)cos({\frac {2n\pi x}{L}})dx}"/></span><br/> <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 b_{n}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecc141d40ca4a13ec7beafd6d264d5a435792a58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.477ex; height:2.509ex;" alt="{\displaystyle b_{n}=0}"/></span><br/> </p><p>Should the function be even (link to even function definition) the fourier sine series may be used. This simplifies to<br/> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a6dcbd7dfa3904fbbfef7745ab8b19904ccf009" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.709ex; height:2.509ex;" alt="{\displaystyle a_{n}=0}"/></span><br/> <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 b_{n}={\frac {2}{L}}\int _{L_{1}}^{L_{2}}f(x)sin({\frac {2n\pi x}{L}})dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>L</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> <mi>π</mi> <mi>x</mi> </mrow> <mi>L</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}={\frac {2}{L}}\int _{L_{1}}^{L_{2}}f(x)sin({\frac {2n\pi x}{L}})dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd239975478bf28b9bd0f9d88f60dd1c7663e72a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.27ex; height:6.509ex;" alt="{\displaystyle b_{n}={\frac {2}{L}}\int _{L_{1}}^{L_{2}}f(x)sin({\frac {2n\pi x}{L}})dx}"/></span><br/> </p><p>Then explain the wave (string) equation, complex, real, properties, then historical. </p><p>If I have made mathematical mistakes, please forgive me, because I have apparently conflicting sources. :) I am only beginning fourier analysis, so I may have missed formulae which are important to more advanced parts. </p><p>Edit: oops, everything *is* in there - I must have skimmed past it these last 10 times I viewed the page. However, I still find the omega, t and T notation confusing - wouldn't f(x) be easier? <a href="/wiki/User:Chrislewis.au" title="User:Chrislewis.au">Chrislewis.au</a> </p> <dl><dd><dl><dd>f(t) is used because in many (or most) cases, fourier series are implimented as functions of time. But you're right that f(x) might imply a more general use - however I think its a very minor thing, and might confuse people used to seeing f(t). Also, the use of t and T as period and time don't have an easily understood analog in other sets - like distance rather than time. So i really don't think that should be changed. I did make a note about what t1 and t2 are explicitely, rather than just compared to the period T. <a href="/wiki/User:Fresheneesz" title="User:Fresheneesz">Fresheneesz</a> <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Fresheneesz-2006-07-02T19:08:00.000Z-Still_very_convoluted" class="ext-discussiontools-init-timestamplink">19:08, 2 July 2006 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Fresheneesz-2006-07-02T19:08:00.000Z-Still_very_convoluted"></span><span data-mw-comment-end="c-Fresheneesz-2006-07-02T19:08:00.000Z-Still_very_convoluted"></span></dd></dl></dd></dl> <p><span data-mw-comment-start="" id="c-Michael_Hardy-2006-07-03T00:05:00.000Z-Still_very_convoluted"></span>Chrislewis, please notice the difference between these: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }[a_{n}cos({\frac {n\pi x}{L}})+b_{n}sin({\frac {n\pi x}{L}})]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>π</mi> <mi>x</mi> </mrow> <mi>L</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>π</mi> <mi>x</mi> </mrow> <mi>L</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }[a_{n}cos({\frac {n\pi x}{L}})+b_{n}sin({\frac {n\pi x}{L}})]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fad394ff168f8a4c48f0ef3b629d8f143f12003" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:45.546ex; height:6.843ex;" alt="{\displaystyle f(x)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }[a_{n}cos({\frac {n\pi x}{L}})+b_{n}sin({\frac {n\pi x}{L}})]}"/></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 f(x)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }\left[a_{n}\cos \left({\frac {n\pi x}{L}}\right)+b_{n}\sin \left({\frac {n\pi x}{L}}\right)\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow> <mo>[</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>π</mi> <mi>x</mi> </mrow> <mi>L</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>π</mi> <mi>x</mi> </mrow> <mi>L</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }\left[a_{n}\cos \left({\frac {n\pi x}{L}}\right)+b_{n}\sin \left({\frac {n\pi x}{L}}\right)\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da8a9927a50cae6edc3455830c8178be0ca564b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:48.995ex; height:6.843ex;" alt="{\displaystyle f(x)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }\left[a_{n}\cos \left({\frac {n\pi x}{L}}\right)+b_{n}\sin \left({\frac {n\pi x}{L}}\right)\right]}"/></span></dd></dl> <ul><li>The square brackets and round parentheses are bigger in the second one.</li></ul> <ul><li>"sin" and "cos" are not italicized in the second one. The preceeding backslash not only prevents the letters from being italicized as if they were variables, but also provides proper spacing before and after "sin" and "cos" in some circumstances.</li></ul> <p><a href="/wiki/User:Michael_Hardy" title="User:Michael Hardy">Michael Hardy</a> <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Michael_Hardy-2006-07-03T00:05:00.000Z-Still_very_convoluted" class="ext-discussiontools-init-timestamplink">00:05, 3 July 2006 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Michael_Hardy-2006-07-03T00:05:00.000Z-Still_very_convoluted"></span><span data-mw-comment-end="c-Michael_Hardy-2006-07-03T00:05:00.000Z-Still_very_convoluted"></span> </p><p><span data-mw-comment-start="" id="c-Chrislewis.au-2006-08-13T07:44:00.000Z-Still_very_convoluted"></span>My apologies - I didnt notice the differences first time. (<a href="/wiki/User:Chrislewis.au" title="User:Chrislewis.au">Chris</a> <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Chrislewis.au-2006-08-13T07:44:00.000Z-Still_very_convoluted" class="ext-discussiontools-init-timestamplink">07:44, 13 August 2006 (UTC)</a>)<span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Chrislewis.au-2006-08-13T07:44:00.000Z-Still_very_convoluted"></span><span data-mw-comment-end="c-Chrislewis.au-2006-08-13T07:44:00.000Z-Still_very_convoluted"></span> </p><p><span data-mw-comment-start="" id="c-LovaAndriamanjay-2006-11-28T03:00:00.000Z-Still_very_convoluted"></span>I think the original poster's idea is good because now it's really unclear why you can integrate on half of the function if it's odd or even and multiply this by 2. As it is now, there are no explanations and it creates confusion even with the editors. I'll try to come up with something. (<a href="/w/index.php?title=User:LovaAndriamanjay&action=edit&redlink=1" class="new" title="User:LovaAndriamanjay (page does not exist)">LovaAndriamanjay</a> <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-LovaAndriamanjay-2006-11-28T03:00:00.000Z-Still_very_convoluted" class="ext-discussiontools-init-timestamplink">03:00, 28 November 2006 (UTC)</a>)<span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-LovaAndriamanjay-2006-11-28T03:00:00.000Z-Still_very_convoluted"></span><span data-mw-comment-end="c-LovaAndriamanjay-2006-11-28T03:00:00.000Z-Still_very_convoluted"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Error" data-mw-thread-id="h-Error-2007-05-13T22:53:00.000Z"><span data-mw-comment-start="" id="h-Error-2007-05-13T22:53:00.000Z"></span>Error<span data-mw-comment-end="h-Error-2007-05-13T22:53:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-18.244.5.107-2007-05-13T22:53:00.000Z-Error"></span>if we say that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)={\frac {1}{2}}a_{0}+\sum _{n=1}^{\infty }[a_{n}\cos(nt)+b_{n}\sin(nt)]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)={\frac {1}{2}}a_{0}+\sum _{n=1}^{\infty }[a_{n}\cos(nt)+b_{n}\sin(nt)]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b246c83e8c9ee1e070eeb915db9baea41f6bbe1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.131ex; height:6.843ex;" alt="{\displaystyle f(t)={\frac {1}{2}}a_{0}+\sum _{n=1}^{\infty }[a_{n}\cos(nt)+b_{n}\sin(nt)]}"/></span>, then defining <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(t)\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(t)\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b7cd4755d9425d5e5dd800a465adbcca9d5ba04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.603ex; height:6.009ex;" alt="{\displaystyle a_{0}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(t)\,dt}"/></span> is incorrect, as we're trying to double correct for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/693ad9f934775838bd72406b41ada4a59785d7ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{0}}"/></span> being twice as big. </p><p>It should be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(t)\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(t)\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c7d2a105f11775d26a144253fef89bac98e8ead" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.44ex; height:6.009ex;" alt="{\displaystyle a_{0}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(t)\,dt}"/></span>. I reference Strauss <i>Partial Differential Equations.</i> <a href="/w/index.php?title=User:18.244.5.107&action=edit&redlink=1" class="new" title="User:18.244.5.107 (page does not exist)">18.244.5.107</a> <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-18.244.5.107-2007-05-13T22:53:00.000Z-Error" class="ext-discussiontools-init-timestamplink">22:53, 13 May 2007 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-18.244.5.107-2007-05-13T22:53:00.000Z-Error"></span><span data-mw-comment-end="c-18.244.5.107-2007-05-13T22:53:00.000Z-Error"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Oleg_Alexandrov-2007-05-14T01:05:00.000Z-18.244.5.107-2007-05-13T22:53:00.000Z"></span>Thanks, you are right. <a href="/wiki/User:Oleg_Alexandrov" title="User:Oleg Alexandrov">Oleg Alexandrov</a> (<a href="/wiki/User_talk:Oleg_Alexandrov" title="User talk:Oleg Alexandrov">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Oleg_Alexandrov-2007-05-14T01:05:00.000Z-18.244.5.107-2007-05-13T22:53:00.000Z" class="ext-discussiontools-init-timestamplink">01:05, 14 May 2007 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Oleg_Alexandrov-2007-05-14T01:05:00.000Z-18.244.5.107-2007-05-13T22:53:00.000Z"></span><span data-mw-comment-end="c-Oleg_Alexandrov-2007-05-14T01:05:00.000Z-18.244.5.107-2007-05-13T22:53:00.000Z"></span></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Trigonometric_series_redirect?" data-mw-thread-id="h-Trigonometric_series_redirect?-2007-12-01T13:29:00.000Z"><span id="Trigonometric_series_redirect.3F"></span><span data-mw-comment-start="" id="h-Trigonometric_series_redirect?-2007-12-01T13:29:00.000Z"></span>Trigonometric series redirect?<span data-mw-comment-end="h-Trigonometric_series_redirect?-2007-12-01T13:29:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Futurebird-2007-12-01T13:29:00.000Z-Trigonometric_series_redirect?"></span>Not every Trigonometric series is a Fourier series. A trigonometric series has the form: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}A_{o}+\displaystyle \sum _{n=1}^{\infty }(A_{n}cosnx+B_{n}sinnx)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mo>+</mo> <mstyle displaystyle="true" scriptlevel="0"> <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> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>n</mi> <mi>x</mi> <mo>+</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}A_{o}+\displaystyle \sum _{n=1}^{\infty }(A_{n}cosnx+B_{n}sinnx)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbbd2567cddedde6dee0c3f2afaef92e99b2fe88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.522ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{2}}A_{o}+\displaystyle \sum _{n=1}^{\infty }(A_{n}cosnx+B_{n}sinnx)}"/></span> </p><p>It's called a Fourier series when the An and Bn terms have the form: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n}={\frac {1}{\pi }}\displaystyle \int _{0}^{2\pi }f(x)cosnxdx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>n</mi> <mi>x</mi> <mi>d</mi> <mi>x</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{n}={\frac {1}{\pi }}\displaystyle \int _{0}^{2\pi }f(x)cosnxdx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f890ebc4edaf1232d9aae2e82fd7e492162677b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.375ex; height:6.176ex;" alt="{\displaystyle A_{n}={\frac {1}{\pi }}\displaystyle \int _{0}^{2\pi }f(x)cosnxdx}"/></span> 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=0,1,2,...)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n=0,1,2,...)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5e9729d86e1de8606c57c96de53f618f2b86963" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.993ex; height:2.843ex;" alt="{\displaystyle (n=0,1,2,...)}"/></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}={\frac {1}{\pi }}\displaystyle \int _{0}^{2\pi }f(x)sinnxdx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>n</mi> <mi>x</mi> <mi>d</mi> <mi>x</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{n}={\frac {1}{\pi }}\displaystyle \int _{0}^{2\pi }f(x)sinnxdx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/288892475f9caf206e4ecd963f63869c3984be77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.459ex; height:6.176ex;" alt="{\displaystyle B_{n}={\frac {1}{\pi }}\displaystyle \int _{0}^{2\pi }f(x)sinnxdx}"/></span> 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=1,2,3,...)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n=1,2,3,...)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2201eafa4ece4cb6357a585b89e6bbbf1bce04d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.993ex; height:2.843ex;" alt="{\displaystyle (n=1,2,3,...)}"/></span> </p><p>So, there are a whole class of trig series that are not Fourier series. Shall I change it? <a href="/wiki/User:Futurebird" title="User:Futurebird">futurebird</a> <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Futurebird-2007-12-01T13:29:00.000Z-Trigonometric_series_redirect?" class="ext-discussiontools-init-timestamplink">13:29, 1 December 2007 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Futurebird-2007-12-01T13:29:00.000Z-Trigonometric_series_redirect?"></span><span data-mw-comment-end="c-Futurebird-2007-12-01T13:29:00.000Z-Trigonometric_series_redirect?"></span> </p><p><br/> </p> <dl><dd><span data-mw-comment-start="" id="c-Bob_K-2007-12-01T15:12:00.000Z-Futurebird-2007-12-01T13:29:00.000Z"></span>I can't find where it says that "every Trigonometric series is a Fourier series".</dd> <dd>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2007-12-01T15:12:00.000Z-Futurebird-2007-12-01T13:29:00.000Z" class="ext-discussiontools-init-timestamplink">15:12, 1 December 2007 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2007-12-01T15:12:00.000Z-Futurebird-2007-12-01T13:29:00.000Z"></span><span data-mw-comment-end="c-Bob_K-2007-12-01T15:12:00.000Z-Futurebird-2007-12-01T13:29:00.000Z"></span></dd></dl> <p><span data-mw-comment-start="" id="c-Futurebird-2007-12-01T15:29:00.000Z-Trigonometric_series_redirect?"></span>to me the redirect implied that this was true, I've fixed it and made a new page.<a href="/wiki/User:Futurebird" title="User:Futurebird">futurebird</a> <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Futurebird-2007-12-01T15:29:00.000Z-Trigonometric_series_redirect?" class="ext-discussiontools-init-timestamplink">15:29, 1 December 2007 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Futurebird-2007-12-01T15:29:00.000Z-Trigonometric_series_redirect?"></span><span data-mw-comment-end="c-Futurebird-2007-12-01T15:29:00.000Z-Trigonometric_series_redirect?"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id=""real_Fourier_coefficients"" data-mw-thread-id="h-"real_Fourier_coefficients"-2007-12-03T23:17:00.000Z"><span id=".22real_Fourier_coefficients.22"></span><span data-mw-comment-start="" id="h-"real_Fourier_coefficients"-2007-12-03T23:17:00.000Z"></span>"real Fourier coefficients"<span data-mw-comment-end="h-"real_Fourier_coefficients"-2007-12-03T23:17:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Bob_K-2007-12-03T23:17:00.000Z-"real_Fourier_coefficients""></span>In section: <a href="/wiki/Fourier_series#Real_Fourier_coefficients" title="Fourier series">Fourier_series#Real_Fourier_coefficients</a> it is misleading (at best) and incorrect (at worst) to refer to a_n and b_n as <b>real Fourier coefficients</b>, since they are not real-valued in general. They are the coefficients of real-valued functions. </p><p>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2007-12-03T23:17:00.000Z-"real_Fourier_coefficients"" class="ext-discussiontools-init-timestamplink">23:17, 3 December 2007 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2007-12-03T23:17:00.000Z-"real_Fourier_coefficients""></span><span data-mw-comment-end="c-Bob_K-2007-12-03T23:17:00.000Z-"real_Fourier_coefficients""></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Making_a_mountain_out_of_a_molehill_is_not_a_good_example." data-mw-thread-id="h-Making_a_mountain_out_of_a_molehill_is_not_a_good_example.-2007-12-04T11:54:00.000Z"><span data-mw-comment-start="" id="h-Making_a_mountain_out_of_a_molehill_is_not_a_good_example.-2007-12-04T11:54:00.000Z"></span>Making a mountain out of a molehill is not a good example.<span data-mw-comment-end="h-Making_a_mountain_out_of_a_molehill_is_not_a_good_example.-2007-12-04T11:54:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Bob_K-2007-12-04T11:54:00.000Z-Making_a_mountain_out_of_a_molehill_is_not_a_good_example."></span>The article says<b>:</b> </p> <blockquote> <p>One application of this Fourier series is to compute the value of the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a> at <i>s</i> = 2; by <a href="/wiki/Parseval%27s_theorem" title="Parseval's theorem">Parseval's theorem</a>, we have: </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 {1}{2\pi }}\int _{-\pi }^{\pi }x^{2}dx={\frac {1}{2}}\sum _{n>0}\left[2{\frac {(-1)^{n}}{n}}\right]^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>></mo> <mn>0</mn> </mrow> </munder> <msup> <mrow> <mo>[</mo> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2\pi }}\int _{-\pi }^{\pi }x^{2}dx={\frac {1}{2}}\sum _{n>0}\left[2{\frac {(-1)^{n}}{n}}\right]^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5535fa7c94d4c9849e931b02664ac40bb0b5d26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.512ex; height:7.176ex;" alt="{\displaystyle {\frac {1}{2\pi }}\int _{-\pi }^{\pi }x^{2}dx={\frac {1}{2}}\sum _{n>0}\left[2{\frac {(-1)^{n}}{n}}\right]^{2}}"/></span></dd></dl> <p>which yields: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n>0}{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>></mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>6</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n>0}{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/905de885f2288cdac19195187c25b999d760faf9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.35ex; height:6.843ex;" alt="{\displaystyle \sum _{n>0}{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}}"/></span>. </p> </blockquote> <p>Several points<b>:</b> </p> <ol><li>The answer is incorrect. (My guess is that the writer overlooked the factor of 2 in front of the Re operator in Parseval's formula.)</li> <li>The answer is much more easily obtained by direct integration<b>:</b> <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 {1}{2\pi }}\left[{\frac {x^{3}}{3}}\right]_{-\pi }^{\pi }={\frac {\pi ^{2}}{3}}.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>.</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2\pi }}\left[{\frac {x^{3}}{3}}\right]_{-\pi }^{\pi }={\frac {\pi ^{2}}{3}}.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0472a73d5c833658e2a4bfc7191c1340bdb681b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.815ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{2\pi }}\left[{\frac {x^{3}}{3}}\right]_{-\pi }^{\pi }={\frac {\pi ^{2}}{3}}.\,}"/></span></li> <li>I don't think this example is worth keeping, especially with all the missing steps.</li></ol> <p>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2007-12-04T11:54:00.000Z-Making_a_mountain_out_of_a_molehill_is_not_a_good_example." class="ext-discussiontools-init-timestamplink">11:54, 4 December 2007 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2007-12-04T11:54:00.000Z-Making_a_mountain_out_of_a_molehill_is_not_a_good_example."></span><span data-mw-comment-end="c-Bob_K-2007-12-04T11:54:00.000Z-Making_a_mountain_out_of_a_molehill_is_not_a_good_example."></span> </p><p><span data-mw-comment-start="" id="c-Loisel-2008-01-08T19:57:00.000Z-Making_a_mountain_out_of_a_molehill_is_not_a_good_example."></span>Dear Bob, </p><p>You are incorrect. Zeta(2) is indeed pi^2/6, which you can see in the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a> article. This specific case is also known as the <a href="/wiki/Basel_problem" title="Basel problem">Basel problem</a> and I apparently drove the math help desk at Temple U nuts in 2006 by giving this problem as a bonus question to Cal II students. </p><p>Although there are other methods for computing Zeta(2), using Parseval's identity is one of the only methods I can follow from beginning to end without going cross-eyed. </p><p>Sincerely, </p><p><a href="/wiki/User:Loisel" title="User:Loisel">Loisel</a> (<a href="/wiki/User_talk:Loisel" title="User talk:Loisel">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Loisel-2008-01-08T19:57:00.000Z-Making_a_mountain_out_of_a_molehill_is_not_a_good_example." class="ext-discussiontools-init-timestamplink">19:57, 8 January 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Loisel-2008-01-08T19:57:00.000Z-Making_a_mountain_out_of_a_molehill_is_not_a_good_example."></span><span data-mw-comment-end="c-Loisel-2008-01-08T19:57:00.000Z-Making_a_mountain_out_of_a_molehill_is_not_a_good_example."></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="inconsistent_use_of_"L"" data-mw-thread-id="h-inconsistent_use_of_"L"-2007-12-04T12:16:00.000Z"><span id="inconsistent_use_of_.22L.22"></span><span data-mw-comment-start="" id="h-inconsistent_use_of_"L"-2007-12-04T12:16:00.000Z"></span>inconsistent use of "L"<span data-mw-comment-end="h-inconsistent_use_of_"L"-2007-12-04T12:16:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><a href="/wiki/Fourier_series#The_wave_equation" title="Fourier series"><span data-mw-comment-start="" id="c-Bob_K-2007-12-04T12:16:00.000Z-inconsistent_use_of_"L""></span>Fourier_series#The_wave_equation</a> says<b>:</b> </p> <blockquote> <div class="mw-heading mw-heading3"><h3 id="The_wave_equation">The wave equation</h3></div> <p>The <a href="/wiki/Wave_equation" title="Wave equation">wave equation</a> governs the motion of a vibrating string, which may be fastened down at its endpoints. The solution of this problem requires the trigonometric expansion of a general function <i>f</i> that vanishes at the endpoints of an interval <i>x</i>=0 and <i>x</i>=<i>L</i>. The Fourier series for such a function takes the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\sum _{n=1}^{\infty }b_{n}\sin \left({\frac {n\pi }{L}}x\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <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> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>π</mi> </mrow> <mi>L</mi> </mfrac> </mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\sum _{n=1}^{\infty }b_{n}\sin \left({\frac {n\pi }{L}}x\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c1018ff9d7d67822c15ef7934ca76a9cdd07136" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.385ex; height:6.843ex;" alt="{\displaystyle f(x)=\sum _{n=1}^{\infty }b_{n}\sin \left({\frac {n\pi }{L}}x\right)}"/></span></dd></dl> <p>where </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 b_{n}={\frac {2}{L}}\int _{0}^{L}f(x)\sin \left({\frac {n\pi }{L}}x\right)\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>L</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>π</mi> </mrow> <mi>L</mi> </mfrac> </mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}={\frac {2}{L}}\int _{0}^{L}f(x)\sin \left({\frac {n\pi }{L}}x\right)\,dx.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed30bb3842c47f9b4e0c04fd930a8361c9de0a7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.231ex; height:6.176ex;" alt="{\displaystyle b_{n}={\frac {2}{L}}\int _{0}^{L}f(x)\sin \left({\frac {n\pi }{L}}x\right)\,dx.}"/></span></dd></dl> </blockquote> <p><br/> I believe the formulas should be<b>:</b> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\sum _{n=1}^{\infty }b_{n}\sin \left({\frac {n2\pi }{L}}x\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <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> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mn>2</mn> <mi>π</mi> </mrow> <mi>L</mi> </mfrac> </mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\sum _{n=1}^{\infty }b_{n}\sin \left({\frac {n2\pi }{L}}x\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c7bd2537e789782bedd049fae509240efab542f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.193ex; height:6.843ex;" alt="{\displaystyle f(x)=\sum _{n=1}^{\infty }b_{n}\sin \left({\frac {n2\pi }{L}}x\right)}"/></span></dd></dl> <p>where </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 b_{n}={\frac {2}{L}}\int _{0}^{L}f(x)\sin \left({\frac {n2\pi }{L}}x\right)\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>L</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mn>2</mn> <mi>π</mi> </mrow> <mi>L</mi> </mfrac> </mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}={\frac {2}{L}}\int _{0}^{L}f(x)\sin \left({\frac {n2\pi }{L}}x\right)\,dx.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23844ba64e565a50bbad16e5a741c0db8a96ddd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.039ex; height:6.343ex;" alt="{\displaystyle b_{n}={\frac {2}{L}}\int _{0}^{L}f(x)\sin \left({\frac {n2\pi }{L}}x\right)\,dx.}"/></span></dd></dl> <p><br/> And I suspect the subsequent paragraph is also wrong. This might be the correct version<b>:</b> </p> <blockquote> <p>Vibrations of air in a pipe that is open at one end and closed at the other are also described by the wave equation. Its solution requires expansion of a function that vanishes at <i>x</i> = 0 and whose derivative vanishes at <i>x</i>=<i>L</i>. The Fourier series for such a function takes the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\sum _{n=1}^{\infty }b_{n}\sin \left({\frac {(2n+1)\pi }{L}}x\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <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> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <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> <mi>π</mi> </mrow> <mi>L</mi> </mfrac> </mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\sum _{n=1}^{\infty }b_{n}\sin \left({\frac {(2n+1)\pi }{L}}x\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1350d676d40c4d302ab12e6f32f7b474efd3ea3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.005ex; height:6.843ex;" alt="{\displaystyle f(x)=\sum _{n=1}^{\infty }b_{n}\sin \left({\frac {(2n+1)\pi }{L}}x\right)}"/></span></dd></dl> <p>where </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 b_{n}={\frac {2}{L}}\int _{0}^{L}f(x)\sin \left({\frac {(2n+1)\pi }{L}}x\right)\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>L</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <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> <mi>π</mi> </mrow> <mi>L</mi> </mfrac> </mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}={\frac {2}{L}}\int _{0}^{L}f(x)\sin \left({\frac {(2n+1)\pi }{L}}x\right)\,dx.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b6490405dde2ba812678b89f0c772deaeb130e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.851ex; height:6.343ex;" alt="{\displaystyle b_{n}={\frac {2}{L}}\int _{0}^{L}f(x)\sin \left({\frac {(2n+1)\pi }{L}}x\right)\,dx.}"/></span></dd></dl> </blockquote> <p><small><span class="autosigned">—Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a> • <a href="/wiki/Special:Contributions/Bob_K" title="Special:Contributions/Bob K">contribs</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2007-12-04T12:16:00.000Z-inconsistent_use_of_"L"" class="ext-discussiontools-init-timestamplink">12:16, 4 December 2007 (UTC)</a></span></small><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2007-12-04T12:16:00.000Z-inconsistent_use_of_"L""></span><span data-mw-comment-end="c-Bob_K-2007-12-04T12:16:00.000Z-inconsistent_use_of_"L""></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Interpretation?" data-mw-thread-id="h-Interpretation?-2008-01-08T00:41:00.000Z"><span id="Interpretation.3F"></span><span data-mw-comment-start="" id="h-Interpretation?-2008-01-08T00:41:00.000Z"></span>Interpretation?<span data-mw-comment-end="h-Interpretation?-2008-01-08T00:41:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Bob_K-2008-01-08T00:41:00.000Z-Interpretation?"></span>I don't think the section <a href="/wiki/Fourier_series#Interpretation:_decomposing_a_movement_in_rotations" title="Fourier series">Fourier_series#Interpretation:_decomposing_a_movement_in_rotations</a> is a helpful "interpretation" of Fourier series. Perhaps it is misplaced. </p><p>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2008-01-08T00:41:00.000Z-Interpretation?" class="ext-discussiontools-init-timestamplink">00:41, 8 January 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2008-01-08T00:41:00.000Z-Interpretation?"></span><span data-mw-comment-end="c-Bob_K-2008-01-08T00:41:00.000Z-Interpretation?"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="General_case" data-mw-thread-id="h-General_case-2008-01-08T15:25:00.000Z"><span data-mw-comment-start="" id="h-General_case-2008-01-08T15:25:00.000Z"></span>General case<span data-mw-comment-end="h-General_case-2008-01-08T15:25:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Bob_K-2008-01-08T15:25:00.000Z-General_case"></span>Regarding: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{G}f(g){\overline {\chi (g)}}\,dg={\frac {1}{2\pi }}\int _{2\pi }^{}f(t)e^{-int}\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>g</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>n</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{G}f(g){\overline {\chi (g)}}\,dg={\frac {1}{2\pi }}\int _{2\pi }^{}f(t)e^{-int}\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc37f04f8a93657ca616a052a723612679dca0e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.225ex; height:5.676ex;" alt="{\displaystyle \int _{G}f(g){\overline {\chi (g)}}\,dg={\frac {1}{2\pi }}\int _{2\pi }^{}f(t)e^{-int}\,dt}"/></span></dd></dl> <p>in <a href="/wiki/Fourier_series#General_case" title="Fourier series">Fourier_series#General_case</a>, where does 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 {\frac {1}{2\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2\pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/129204d50704b07e6a4223870954242b21170354" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.331ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2\pi }}}"/></span> factor come from? </p><p>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2008-01-08T15:25:00.000Z-General_case" class="ext-discussiontools-init-timestamplink">15:25, 8 January 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2008-01-08T15:25:00.000Z-General_case"></span><span data-mw-comment-end="c-Bob_K-2008-01-08T15:25:00.000Z-General_case"></span> </p><p><br/><span data-mw-comment-start="" id="c-Bob_K-2008-01-08T18:25:00.000Z-General_case"></span> I don't really get what <i>G</i> = <b>R</b>/2π<b>Z</b> means. But evidently it's our clue that<b>:</b> </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 dg={\frac {1}{2\pi }}dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>g</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dg={\frac {1}{2\pi }}dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e234170b01471dcd53da263aa334bcfa5e20f8af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.816ex; height:5.176ex;" alt="{\displaystyle dg={\frac {1}{2\pi }}dt}"/></span></dd></dl> <p>Is it just me, or is that too much of a stretch for this article? </p><p>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2008-01-08T18:25:00.000Z-General_case" class="ext-discussiontools-init-timestamplink">18:25, 8 January 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2008-01-08T18:25:00.000Z-General_case"></span><span data-mw-comment-end="c-Bob_K-2008-01-08T18:25:00.000Z-General_case"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Loisel-2008-01-08T20:04:00.000Z-Bob_K-2008-01-08T18:25:00.000Z"></span>Dear Bob,</dd></dl> <dl><dd>There is always a normalizing coefficient that appears either in the computation of the Fourier coefficients, or in the summation formula (the Fourier series itself.) The easiest way to understand where it comes from is to think in terms of <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert spaces</a>. The functions exp(ikx) form an orthogonal basis for L^2([0,2pi]), but in order for these basis functions to have unit norm (so that they form an ortho<i>normal</i> basis), you have to do something. You either have to change the "measure" of [0,2pi] so that it measures 1 (in which case, you get 1/2pi, as in the text), or you have to put a coefficient in front of the exp(ikx), it then becomes 1/sqrt(2pi)exp(ikx), and the term 1/sqrt(2pi) also appears in the Fourier series. There are another zillion ways of doing this.</dd></dl> <dl><dd>Sincerely,</dd></dl> <dl><dd><a href="/wiki/User:Loisel" title="User:Loisel">Loisel</a> (<a href="/wiki/User_talk:Loisel" title="User talk:Loisel">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Loisel-2008-01-08T20:04:00.000Z-Bob_K-2008-01-08T18:25:00.000Z" class="ext-discussiontools-init-timestamplink">20:04, 8 January 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Loisel-2008-01-08T20:04:00.000Z-Bob_K-2008-01-08T18:25:00.000Z"></span><span data-mw-comment-end="c-Loisel-2008-01-08T20:04:00.000Z-Bob_K-2008-01-08T18:25:00.000Z"></span></dd></dl> <p><span data-mw-comment-start="" id="c-Bob_K-2008-01-08T21:48:00.000Z-General_case"></span>Thanks, but you answered the wrong question. I was specifically questioning the sufficiency of "The General Case". The case for 1/2π has to made all over again. </p><p>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2008-01-08T21:48:00.000Z-General_case" class="ext-discussiontools-init-timestamplink">21:48, 8 January 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2008-01-08T21:48:00.000Z-General_case"></span><span data-mw-comment-end="c-Bob_K-2008-01-08T21:48:00.000Z-General_case"></span> </p><p><span data-mw-comment-start="" id="c-Loisel-2008-01-10T00:02:00.000Z-General_case"></span>Dear Bob, </p><p>I also had problems with that section. Please let me know what you think now. The characters are no longer discussed, because they are discussed in details in the <a href="/wiki/Pontryagin_duality" title="Pontryagin duality">Pontryagin duality</a> article. </p><p>Cheers, </p><p><a href="/wiki/User:Loisel" title="User:Loisel">Loisel</a> (<a href="/wiki/User_talk:Loisel" title="User talk:Loisel">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Loisel-2008-01-10T00:02:00.000Z-General_case" class="ext-discussiontools-init-timestamplink">00:02, 10 January 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Loisel-2008-01-10T00:02:00.000Z-General_case"></span><span data-mw-comment-end="c-Loisel-2008-01-10T00:02:00.000Z-General_case"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Article_is_of_very_low_quality." data-mw-thread-id="h-Article_is_of_very_low_quality.-2008-01-08T21:20:00.000Z"><span data-mw-comment-start="" id="h-Article_is_of_very_low_quality.-2008-01-08T21:20:00.000Z"></span>Article is of very low quality.<span data-mw-comment-end="h-Article_is_of_very_low_quality.-2008-01-08T21:20:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Loisel-2008-01-08T21:20:00.000Z-Article_is_of_very_low_quality."></span>After reading Bob's comments, I started trying to clean up the article. In so doing, I found a large number of errors and lots of nonsense. The job is not finished, but I have other things to do right now, so I am downgrading the quality of the article. This article is not to be trusted right now. </p><p><a href="/wiki/User:Loisel" title="User:Loisel">Loisel</a> (<a href="/wiki/User_talk:Loisel" title="User talk:Loisel">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Loisel-2008-01-08T21:20:00.000Z-Article_is_of_very_low_quality." class="ext-discussiontools-init-timestamplink">21:20, 8 January 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Loisel-2008-01-08T21:20:00.000Z-Article_is_of_very_low_quality."></span><span data-mw-comment-end="c-Loisel-2008-01-08T21:20:00.000Z-Article_is_of_very_low_quality."></span> </p><p><br/><span data-mw-comment-start="" id="c-Bob_K-2008-01-08T22:45:00.000Z-Article_is_of_very_low_quality."></span> For instance, you wrote<b>:</b> </p> <dl><dd>Given a <a href="/wiki/Square-integrable" class="mw-redirect" title="Square-integrable">square-integrable</a> function <i>f</i>(<i>t</i>) of the parameter <i>t</i> (sometimes referred to as <i>time</i>), the Fourier series of <i>f</i>(<i>t</i>) is</dd></dl> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }[a_{n}\cos(nt)+b_{n}\sin(nt)]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }[a_{n}\cos(nt)+b_{n}\sin(nt)]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0311913624a865e1fa9386e5f9b8ccca589ce33f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:39.968ex; height:6.843ex;" alt="{\displaystyle f(t)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }[a_{n}\cos(nt)+b_{n}\sin(nt)]}"/></span></dd></dl></dd></dl> <p>That's only true for functions that are 2π-periodic. Fourier series is more general than that. </p><p>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2008-01-08T22:45:00.000Z-Article_is_of_very_low_quality." class="ext-discussiontools-init-timestamplink">22:45, 8 January 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2008-01-08T22:45:00.000Z-Article_is_of_very_low_quality."></span><span data-mw-comment-end="c-Bob_K-2008-01-08T22:45:00.000Z-Article_is_of_very_low_quality."></span> </p><p><span data-mw-comment-start="" id="c-Loisel-2008-01-09T22:11:00.000Z-Article_is_of_very_low_quality."></span>Dear Bob, </p><p>As per your suggestion, I have emphasized that that particular section is talking about functions of the interval [0,2π]. Note that this is not exactly the same as being periodic. </p><p>If your function is periodic with period T, what you are interested in is "discussed" in the section <b>Fourier series on a general interval [a,b]</b>, using, e.g., a=0 and b=T. </p><p>The question of which domain to use is a very interesting question, but using T-periodic functions instead of 2π-periodic functions is not the answer either because there are Fourier series for much more interesting domains, like <a href="/wiki/Lie_group" title="Lie group">Lie groups</a> and differential <a href="/wiki/Manifold" title="Manifold">manifolds</a>. The basis functions are then no longer exponentials. See <a href="/wiki/Spherical_harmonic" class="mw-redirect" title="Spherical harmonic">spherical harmonic</a> for the spherical case. </p><p>In any case, I don't think the T-periodic case is unimportant, and I have a section for it. In a minute, I will add the necessary formulae to that section. Like I said yesterday, I haven't had time to do a thorough job of the article. </p><p><a href="/wiki/User:Loisel" title="User:Loisel">Loisel</a> (<a href="/wiki/User_talk:Loisel" title="User talk:Loisel">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Loisel-2008-01-09T22:11:00.000Z-Article_is_of_very_low_quality." class="ext-discussiontools-init-timestamplink">22:11, 9 January 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Loisel-2008-01-09T22:11:00.000Z-Article_is_of_very_low_quality."></span><span data-mw-comment-end="c-Loisel-2008-01-09T22:11:00.000Z-Article_is_of_very_low_quality."></span> </p><p><span data-mw-comment-start="" id="c-Loisel-2008-01-10T00:00:00.000Z-Article_is_of_very_low_quality."></span>Dear all, </p><p>I have just finished a first pass over the article. It's not perfect, but I hope that it has fewer errors and a better flow than before. Because there were so many topics, I have elected to relegate many of them to the daughter articles that are linked with the notation <i>Main article: ...</i>. This is according to <a href="/wiki/Wikipedia:Summary_style" title="Wikipedia:Summary style">Wikipedia:Summary style</a>. </p><p>That said, I am sure there is room for improvement, so please make comments and edits. </p><p>Sincerely, </p><p><a href="/wiki/User:Loisel" title="User:Loisel">Loisel</a> (<a href="/wiki/User_talk:Loisel" title="User talk:Loisel">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Loisel-2008-01-10T00:00:00.000Z-Article_is_of_very_low_quality." class="ext-discussiontools-init-timestamplink">00:00, 10 January 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Loisel-2008-01-10T00:00:00.000Z-Article_is_of_very_low_quality."></span><span data-mw-comment-end="c-Loisel-2008-01-10T00:00:00.000Z-Article_is_of_very_low_quality."></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Bob_K-2008-01-12T13:06:00.000Z-Loisel-2008-01-10T00:00:00.000Z"></span>Here's what I think. The "general interval [a,b] and on a square" section should be split into two sections. And in the section "general interval [a,b]", the simplified formulas for [0,T] should be given in the form of an example. And we could also point out that setting T=2п produces the simplest-looking form (from your previous section).</dd> <dd>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2008-01-12T13:06:00.000Z-Loisel-2008-01-10T00:00:00.000Z" class="ext-discussiontools-init-timestamplink">13:06, 12 January 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2008-01-12T13:06:00.000Z-Loisel-2008-01-10T00:00:00.000Z"></span><span data-mw-comment-end="c-Bob_K-2008-01-12T13:06:00.000Z-Loisel-2008-01-10T00:00:00.000Z"></span></dd></dl> <p><br/> </p> <dl><dd><span data-mw-comment-start="" id="c-Bob_K-2008-01-12T13:22:00.000Z-Loisel-2008-01-10T00:00:00.000Z"></span>I also think we're missing a fairly important point made by this 8-Jan excerpt<b>:</b></dd></dl> <dl><dd><dl><dd><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 c_{n}={\frac {1}{T}}\int _{t_{0}}^{t_{0}+T}s(t)\cdot e^{-i\left(n{\frac {2\pi }{T}}\right)t}\,dt}"> <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"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>T</mi> </mrow> </msubsup> <mi>s</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>T</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{n}={\frac {1}{T}}\int _{t_{0}}^{t_{0}+T}s(t)\cdot e^{-i\left(n{\frac {2\pi }{T}}\right)t}\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1899903ac5c6b4c8600b65593779eda010a6a2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.961ex; height:6.509ex;" alt="{\displaystyle c_{n}={\frac {1}{T}}\int _{t_{0}}^{t_{0}+T}s(t)\cdot e^{-i\left(n{\frac {2\pi }{T}}\right)t}\,dt}"/></span> for all integer values of <b>n</b>,</dd></dl></dd></dl></dd></dl> <dl><dd><dl><dd>where<b>:</b> <ul><li><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 s(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c484de351ba40ccb9a5ad522c29c1aac5686c0df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.739ex; height:2.843ex;" alt="{\displaystyle s(t)}"/></span> is periodic, with 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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"/></span></li> <li><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 t_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02d3006c4190b1939b04d9b9bb21006fb4e6fa4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{0}}"/></span> is an <u>arbitrary</u> instance of real argument <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 t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3e6cc375ac6123d2342be53eba87b92fbbacf07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.486ex; height:2.009ex;" alt="{\displaystyle t.}"/></span></li></ul></dd></dl></dd></dl> <dl><dd>I.e., <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{n}\,}"> <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> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{n}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7633cd0b2eed5f32eadcbca6b177776c9d6ca7c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.612ex; height:2.009ex;" alt="{\displaystyle c_{n}\,}"/></span> is invariant with respect to <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 t_{0}.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>.</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{0}.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa159915ec398b44b4416ce87ad3e30c51f67857" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.928ex; height:2.343ex;" alt="{\displaystyle t_{0}.\,}"/></span></dd> <dd>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2008-01-12T13:22:00.000Z-Loisel-2008-01-10T00:00:00.000Z" class="ext-discussiontools-init-timestamplink">13:22, 12 January 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2008-01-12T13:22:00.000Z-Loisel-2008-01-10T00:00:00.000Z"></span><span data-mw-comment-end="c-Bob_K-2008-01-12T13:22:00.000Z-Loisel-2008-01-10T00:00:00.000Z"></span></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Looking_for_an_image_of_a_vibrating_drum" data-mw-thread-id="h-Looking_for_an_image_of_a_vibrating_drum-2008-01-10T21:53:00.000Z"><span data-mw-comment-start="" id="h-Looking_for_an_image_of_a_vibrating_drum-2008-01-10T21:53:00.000Z"></span>Looking for an image of a vibrating drum<span data-mw-comment-end="h-Looking_for_an_image_of_a_vibrating_drum-2008-01-10T21:53:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Loisel-2008-01-10T21:53:00.000Z-Looking_for_an_image_of_a_vibrating_drum"></span>I am trying to improve the article with some images, and I would like to illustrate the 2d Fourier series with an image of a vibrating drum, like <a rel="nofollow" class="external autonumber" href="http://ocw.mit.edu/OcwWeb/Mathematics/18-075Fall-2004/CourseHome/">[1]</a> or like <a rel="nofollow" class="external autonumber" href="http://www.kettering.edu/~drussell/Demos/MembraneCircle/Circle.html">[2]</a> or like <a rel="nofollow" class="external autonumber" href="http://www.meta-synthesis.com/webbook/34_qn/qn_to_pt.html">[3]</a>. I haven't found any free images though. </p><p><a href="/wiki/User:Loisel" title="User:Loisel">Loisel</a> (<a href="/wiki/User_talk:Loisel" title="User talk:Loisel">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Loisel-2008-01-10T21:53:00.000Z-Looking_for_an_image_of_a_vibrating_drum" class="ext-discussiontools-init-timestamplink">21:53, 10 January 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Loisel-2008-01-10T21:53:00.000Z-Looking_for_an_image_of_a_vibrating_drum"></span><span data-mw-comment-end="c-Loisel-2008-01-10T21:53:00.000Z-Looking_for_an_image_of_a_vibrating_drum"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Oleg_Alexandrov-2008-01-11T05:28:00.000Z-Loisel-2008-01-10T21:53:00.000Z"></span>I can make the images, but it is not completely clear the me the connection to the Fourier series. The modes of a vibrating membrane are solutions of the <a href="/wiki/Wave_equation" title="Wave equation">wave equation</a> with zero <a href="/wiki/Dirichlet_boundary_condition" title="Dirichlet boundary condition">Dirichlet boundary conditions</a>. They modes are, if you wish, individual Fourier terms, but I don't think you can easily explain how you get a Fourier series out of there (do you add them up?) <a href="/wiki/User:Oleg_Alexandrov" title="User:Oleg Alexandrov">Oleg Alexandrov</a> (<a href="/wiki/User_talk:Oleg_Alexandrov" title="User talk:Oleg Alexandrov">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Oleg_Alexandrov-2008-01-11T05:28:00.000Z-Loisel-2008-01-10T21:53:00.000Z" class="ext-discussiontools-init-timestamplink">05:28, 11 January 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Oleg_Alexandrov-2008-01-11T05:28:00.000Z-Loisel-2008-01-10T21:53:00.000Z"></span><span data-mw-comment-end="c-Oleg_Alexandrov-2008-01-11T05:28:00.000Z-Loisel-2008-01-10T21:53:00.000Z"></span> <dl><dd><span data-mw-comment-start="" id="c-Oleg_Alexandrov-2008-01-12T06:37:00.000Z-Oleg_Alexandrov-2008-01-11T05:28:00.000Z"></span>See <a href="https://commons.wikimedia.org/wiki/Category:Drum_vibration_animations" class="extiw" title="commons:Category:Drum vibration animations">commons:Category:Drum vibration animations</a>, although I think they would be more appropriate at other articles, e.g., at <a href="/wiki/Vibration" title="Vibration">vibration</a>. <a href="/wiki/User:Oleg_Alexandrov" title="User:Oleg Alexandrov">Oleg Alexandrov</a> (<a href="/wiki/User_talk:Oleg_Alexandrov" title="User talk:Oleg Alexandrov">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Oleg_Alexandrov-2008-01-12T06:37:00.000Z-Oleg_Alexandrov-2008-01-11T05:28:00.000Z" class="ext-discussiontools-init-timestamplink">06:37, 12 January 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Oleg_Alexandrov-2008-01-12T06:37:00.000Z-Oleg_Alexandrov-2008-01-11T05:28:00.000Z"></span><span data-mw-comment-end="c-Oleg_Alexandrov-2008-01-12T06:37:00.000Z-Oleg_Alexandrov-2008-01-11T05:28:00.000Z"></span></dd></dl></dd></dl> <p><span data-mw-comment-start="" id="c-Loisel-2008-01-13T14:01:00.000Z-Looking_for_an_image_of_a_vibrating_drum"></span>Thanks a lot. I'm traveling right now, but I will look at it when I come back next week. <a href="/wiki/User:Loisel" title="User:Loisel">Loisel</a> (<a href="/wiki/User_talk:Loisel" title="User talk:Loisel">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Loisel-2008-01-13T14:01:00.000Z-Looking_for_an_image_of_a_vibrating_drum" class="ext-discussiontools-init-timestamplink">14:01, 13 January 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Loisel-2008-01-13T14:01:00.000Z-Looking_for_an_image_of_a_vibrating_drum"></span><span data-mw-comment-end="c-Loisel-2008-01-13T14:01:00.000Z-Looking_for_an_image_of_a_vibrating_drum"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Contradiction" data-mw-thread-id="h-Contradiction-2008-01-12T23:46:00.000Z"><span data-mw-comment-start="" id="h-Contradiction-2008-01-12T23:46:00.000Z"></span>Contradiction<span data-mw-comment-end="h-Contradiction-2008-01-12T23:46:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Bob_K-2008-01-12T23:46:00.000Z-Contradiction"></span>Here is the excerpt in question: </p> <dl><dd>===Fourier's formula for 2π-periodic functions using sines and cosines===</dd></dl> <dl><dd>Given a <a href="/wiki/Square-integrable" class="mw-redirect" title="Square-integrable">square-integrable</a> function <i>f</i>(<i>x</i>) of the variable <i>x</i> whose domain is the interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,2\pi ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,2\pi ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/348d40bf3f8b7e1c00c4346440d7e2e4f0cc9b91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.985ex; height:2.843ex;" alt="{\displaystyle [0,2\pi ]}"/></span>, the Fourier series of <i>f</i>(<i>x</i>) is</dd></dl> <p><br/> It's either periodic <b>or</b> the domain 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 [0,2\pi ].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,2\pi ].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff4354c1471c11d5701613ddb056258f5b7b8d19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle [0,2\pi ].}"/></span> But not both. </p><p>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2008-01-12T23:46:00.000Z-Contradiction" class="ext-discussiontools-init-timestamplink">23:46, 12 January 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2008-01-12T23:46:00.000Z-Contradiction"></span><span data-mw-comment-end="c-Bob_K-2008-01-12T23:46:00.000Z-Contradiction"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="The_new_figure_is_not_as_good_as_the_old_one" data-mw-thread-id="h-The_new_figure_is_not_as_good_as_the_old_one-2008-01-13T00:01:00.000Z"><span data-mw-comment-start="" id="h-The_new_figure_is_not_as_good_as_the_old_one-2008-01-13T00:01:00.000Z"></span>The new figure is not as good as the old one<span data-mw-comment-end="h-The_new_figure_is_not_as_good_as_the_old_one-2008-01-13T00:01:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Bob_K-2008-01-13T00:01:00.000Z-The_new_figure_is_not_as_good_as_the_old_one"></span>That's my opinion. Judge for yourselves<b>:</b> </p><p>new: <a class="external free" href="https://en.wikipedia.org/wiki/Image:Fourier-partial-sums-of-x.png">http://en.wikipedia.org/wiki/Image:Fourier-partial-sums-of-x.png</a> </p><p>old: <a class="external free" href="https://en.wikipedia.org/wiki/Image:Periodic_identity_function.gif">http://en.wikipedia.org/wiki/Image:Periodic_identity_function.gif</a> </p><p>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2008-01-13T00:01:00.000Z-The_new_figure_is_not_as_good_as_the_old_one" class="ext-discussiontools-init-timestamplink">00:01, 13 January 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2008-01-13T00:01:00.000Z-The_new_figure_is_not_as_good_as_the_old_one"></span><span data-mw-comment-end="c-Bob_K-2008-01-13T00:01:00.000Z-The_new_figure_is_not_as_good_as_the_old_one"></span> </p><p><span data-mw-comment-start="" id="c-Loisel-2008-01-22T15:10:00.000Z-The_new_figure_is_not_as_good_as_the_old_one"></span>Dear Bob, </p><p>I have no problem with the animated image (although I personally prefer static images, they print better.) I have made the image box smaller, which is why I had changed the animated image in the first place. Please feel free to make further changes, but ideally I would like the images to be of this standard size. On a narrow screen, wide images are not good for the flow of the text, and even on a wide screen, the images were a bit too wide before. </p><p>Sincerely, </p><p><a href="/wiki/User:Loisel" title="User:Loisel">Loisel</a> (<a href="/wiki/User_talk:Loisel" title="User talk:Loisel">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Loisel-2008-01-22T15:10:00.000Z-The_new_figure_is_not_as_good_as_the_old_one" class="ext-discussiontools-init-timestamplink">15:10, 22 January 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Loisel-2008-01-22T15:10:00.000Z-The_new_figure_is_not_as_good_as_the_old_one"></span><span data-mw-comment-end="c-Loisel-2008-01-22T15:10:00.000Z-The_new_figure_is_not_as_good_as_the_old_one"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="incorrect_use_of_"Fourier_transform"" data-mw-thread-id="h-incorrect_use_of_"Fourier_transform"-2008-01-24T15:58:00.000Z"><span id="incorrect_use_of_.22Fourier_transform.22"></span><span data-mw-comment-start="" id="h-incorrect_use_of_"Fourier_transform"-2008-01-24T15:58:00.000Z"></span>incorrect use of "Fourier transform"<span data-mw-comment-end="h-incorrect_use_of_"Fourier_transform"-2008-01-24T15:58:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Bob_K-2008-01-24T15:58:00.000Z-incorrect_use_of_"Fourier_transform""></span>Regrading this excerpt: </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}f(x)&=\sum _{n=-\infty }^{\infty }{\hat {f}}(n)\cdot e^{inx}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> <mi>x</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f(x)&=\sum _{n=-\infty }^{\infty }{\hat {f}}(n)\cdot e^{inx}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/056c8cd9857e517bf91e56bc82796ed70879c59b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:24.233ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}f(x)&=\sum _{n=-\infty }^{\infty }{\hat {f}}(n)\cdot e^{inx}\end{aligned}}}"/></span></dd></dl></dd></dl> <dl><dd>The set of coefficients <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 {\hat {f}}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd046562e1921e4ef04b0ebef548b2de59c6293" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.903ex; height:3.343ex;" alt="{\displaystyle {\hat {f}}(n)}"/></span> is called the <b>Fourier transform</b> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"/></span>. In various fields of science, it has different names and is also denoted differently, for example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F[n]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ffefdbddd63e319908aa7e0825bdf9f50f3e6f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.429ex; height:2.843ex;" alt="{\displaystyle F[n]}"/></span> and the <a href="/wiki/Frequency_domain" title="Frequency domain">frequency domain</a> in engineering, or the <a href="/wiki/Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">characteristic function</a> in probability theory.</dd></dl> <p><br/> No. The Fourier transform is: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\int _{-\infty }^{\infty }\left[\sum _{n=-\infty }^{\infty }{\hat {f}}(n)\cdot e^{inx}\right]\cdot e^{-i2\pi fx}dx&=\sum _{n=-\infty }^{\infty }{\hat {f}}(n)\cdot \int _{-\infty }^{\infty }e^{i(n-2\pi f)x}dx\\&=\sum _{n=-\infty }^{\infty }{\hat {f}}(n)\cdot \delta \left(f-{\frac {n}{2\pi }}\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="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow> <mo>[</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> <mi>x</mi> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>x</mi> </mrow> </msup> <mi>d</mi> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mo stretchy="false">)</mo> <mi>x</mi> </mrow> </msup> <mi>d</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>δ</mi> <mrow> <mo>(</mo> <mrow> <mi>f</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int _{-\infty }^{\infty }\left[\sum _{n=-\infty }^{\infty }{\hat {f}}(n)\cdot e^{inx}\right]\cdot e^{-i2\pi fx}dx&=\sum _{n=-\infty }^{\infty }{\hat {f}}(n)\cdot \int _{-\infty }^{\infty }e^{i(n-2\pi f)x}dx\\&=\sum _{n=-\infty }^{\infty }{\hat {f}}(n)\cdot \delta \left(f-{\frac {n}{2\pi }}\right)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bc748c2f75d83aed0f248604b732a9c82e25dec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:66.924ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}\int _{-\infty }^{\infty }\left[\sum _{n=-\infty }^{\infty }{\hat {f}}(n)\cdot e^{inx}\right]\cdot e^{-i2\pi fx}dx&=\sum _{n=-\infty }^{\infty }{\hat {f}}(n)\cdot \int _{-\infty }^{\infty }e^{i(n-2\pi f)x}dx\\&=\sum _{n=-\infty }^{\infty }{\hat {f}}(n)\cdot \delta \left(f-{\frac {n}{2\pi }}\right)\end{aligned}}}"/></span> </p><p>It is a <a href="/wiki/Dirac_comb" title="Dirac comb">Dirac comb</a> <u>modulated by</u> the Fourier series coefficients. </p><p>Also, this shows why f(x) is not a particularly good choice of function names. </p><p>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2008-01-24T15:58:00.000Z-incorrect_use_of_"Fourier_transform"" class="ext-discussiontools-init-timestamplink">15:58, 24 January 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2008-01-24T15:58:00.000Z-incorrect_use_of_"Fourier_transform""></span><span data-mw-comment-end="c-Bob_K-2008-01-24T15:58:00.000Z-incorrect_use_of_"Fourier_transform""></span> </p><p><span data-mw-comment-start="" id="c-Loisel-2008-01-24T19:56:00.000Z-incorrect_use_of_"Fourier_transform""></span>Dear Bob, </p><p>It turns out that in <a href="/wiki/Harmonic_analysis" title="Harmonic analysis">Harmonic analysis</a>, whenever you map a function (either of R, or of the interval, or of some manifold or group) to its Fourier coefficients, that is called the Fourier transform. The reconstruction formula is then called the inverse Fourier transform, and takes the form of either a <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a> or a <a href="/wiki/Fourier_integral" class="mw-redirect" title="Fourier integral">Fourier integral</a>. For instance, if you read <a href="/wiki/Pontryagin_duality" title="Pontryagin duality">Pontryagin duality</a>, you could be forgiven for thinking that the Fourier transform only applies to integrals, because that's all you see in that article. However, the dual group of a compact group is discrete, and so that article hides some Fourier series under the guise of what looks like a Fourier integral! </p><p>This is somewhat frustrating for some real-world users like engineers, because often one thinks of the Fourier transform as the specific integral that occurs on R. </p><p>Sincerely, </p><p><a href="/wiki/User:Loisel" title="User:Loisel">Loisel</a> (<a href="/wiki/User_talk:Loisel" title="User talk:Loisel">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Loisel-2008-01-24T19:56:00.000Z-incorrect_use_of_"Fourier_transform"" class="ext-discussiontools-init-timestamplink">19:56, 24 January 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Loisel-2008-01-24T19:56:00.000Z-incorrect_use_of_"Fourier_transform""></span><span data-mw-comment-end="c-Loisel-2008-01-24T19:56:00.000Z-incorrect_use_of_"Fourier_transform""></span> </p><p><br/><span data-mw-comment-start="" id="c-Bob_K-2008-01-24T20:44:00.000Z-incorrect_use_of_"Fourier_transform""></span> I can't say I'm surprised. We may need to disambiguate the term <i>Fourier transform</i>, because your usage conflicts with other articles in Wikipedia. </p><p>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2008-01-24T20:44:00.000Z-incorrect_use_of_"Fourier_transform"" class="ext-discussiontools-init-timestamplink">20:44, 24 January 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2008-01-24T20:44:00.000Z-incorrect_use_of_"Fourier_transform""></span><span data-mw-comment-end="c-Bob_K-2008-01-24T20:44:00.000Z-incorrect_use_of_"Fourier_transform""></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Stevenj-2008-01-26T22:05:00.000Z-Bob_K-2008-01-24T20:44:00.000Z"></span>I've also seen harmonic analysis books call the relationship between a function and its Fourier series coefficients the <a href="/wiki/Finite_Fourier_transform" title="Finite Fourier transform">finite Fourier transform</a> (hence the disambiguations on that page) (somewhat confusingly, since the number of coefficients is not finite, although the domain of f(x) is). I don't think this usage is very widespread among most users of Fourier transformation and Fourier series, however (among whom pure mathematicians are a small minority). I would say that this isn't a task for a dedicated disambig. page per se, since the meanings are closely related...rather, just a note on the corresponding pages ("this is sometimes called the 'Fourier transform' in harmonic analysis, but in common usage the latter refers to ....") would be better, while the bulk of our articles sticks with the most widespread usage. <a href="/wiki/User:Stevenj" title="User:Stevenj">—Steven G. Johnson</a> (<a href="/wiki/User_talk:Stevenj" title="User talk:Stevenj">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Stevenj-2008-01-26T22:05:00.000Z-Bob_K-2008-01-24T20:44:00.000Z" class="ext-discussiontools-init-timestamplink">22:05, 26 January 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Stevenj-2008-01-26T22:05:00.000Z-Bob_K-2008-01-24T20:44:00.000Z"></span><span data-mw-comment-end="c-Stevenj-2008-01-26T22:05:00.000Z-Bob_K-2008-01-24T20:44:00.000Z"></span></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Simplified_example" data-mw-thread-id="h-Simplified_example-2008-02-02T03:44:00.000Z"><span data-mw-comment-start="" id="h-Simplified_example-2008-02-02T03:44:00.000Z"></span>Simplified example<span data-mw-comment-end="h-Simplified_example-2008-02-02T03:44:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Loisel-2008-02-02T03:44:00.000Z-Simplified_example"></span>The way I see it, either you know how to integrate x sin nx, in which case you don't need the extra verbiage, or you don't, in which case the extra verbiage is useless anyway. </p><p>In any case, I like it much better where it is now. The text now reads: this is the Fourier series, and this is an example. </p><p><a href="/wiki/User:Loisel" title="User:Loisel">Loisel</a> (<a href="/wiki/User_talk:Loisel" title="User talk:Loisel">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Loisel-2008-02-02T03:44:00.000Z-Simplified_example" class="ext-discussiontools-init-timestamplink">03:44, 2 February 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Loisel-2008-02-02T03:44:00.000Z-Simplified_example"></span><span data-mw-comment-end="c-Loisel-2008-02-02T03:44:00.000Z-Simplified_example"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Now_sections_are_in_the_wrong_order" data-mw-thread-id="h-Now_sections_are_in_the_wrong_order-2008-02-05T19:24:00.000Z"><span data-mw-comment-start="" id="h-Now_sections_are_in_the_wrong_order-2008-02-05T19:24:00.000Z"></span>Now sections are in the wrong order<span data-mw-comment-end="h-Now_sections_are_in_the_wrong_order-2008-02-05T19:24:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Bob_K-2008-02-05T19:24:00.000Z-Now_sections_are_in_the_wrong_order"></span>The section <a href="/wiki/Fourier_series#Example:_a_simple_Fourier_series" title="Fourier series">Fourier_series#Example:_a_simple_Fourier_series</a> uses the "general interval" concept. So it should follow section <a href="/wiki/Fourier_series#Fourier_series_on_a_general_interval_.5Ba.2Cb.5D" title="Fourier series">Fourier series on a general interval [a,b] </a>. </p><p>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2008-02-05T19:24:00.000Z-Now_sections_are_in_the_wrong_order" class="ext-discussiontools-init-timestamplink">19:24, 5 February 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2008-02-05T19:24:00.000Z-Now_sections_are_in_the_wrong_order"></span><span data-mw-comment-end="c-Bob_K-2008-02-05T19:24:00.000Z-Now_sections_are_in_the_wrong_order"></span> </p><p><span data-mw-comment-start="" id="c-Bob_K-2008-02-05T19:30:00.000Z-Now_sections_are_in_the_wrong_order"></span>An easy solution is to forget about the fixed interval. Just start with a general interval, like the article was just a few weeks ago (e.g. <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Fourier_series&oldid=182959451">8-Jan</a>). </p><p>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2008-02-05T19:30:00.000Z-Now_sections_are_in_the_wrong_order" class="ext-discussiontools-init-timestamplink">19:30, 5 February 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2008-02-05T19:30:00.000Z-Now_sections_are_in_the_wrong_order"></span><span data-mw-comment-end="c-Bob_K-2008-02-05T19:30:00.000Z-Now_sections_are_in_the_wrong_order"></span> </p><p><span data-mw-comment-start="" id="c-Loisel-2008-02-05T21:59:00.000Z-Now_sections_are_in_the_wrong_order"></span>Dear Bob, </p><p>I thought you might bring that up, and one could go to arbitrary intervals, however, the article of 8 Jan had many errors in it. </p><p>What I'm going to do now instead is switch the rest of the article to [-pi,pi]. Please take a look (when I'm done) and tell me what you think. </p><p>Sincerely, </p><p><a href="/wiki/User:Loisel" title="User:Loisel">Loisel</a> (<a href="/wiki/User_talk:Loisel" title="User talk:Loisel">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Loisel-2008-02-05T21:59:00.000Z-Now_sections_are_in_the_wrong_order" class="ext-discussiontools-init-timestamplink">21:59, 5 February 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Loisel-2008-02-05T21:59:00.000Z-Now_sections_are_in_the_wrong_order"></span><span data-mw-comment-end="c-Loisel-2008-02-05T21:59:00.000Z-Now_sections_are_in_the_wrong_order"></span> </p><p><br/><span data-mw-comment-start="" id="c-Bob_K-2008-02-06T01:00:00.000Z-Now_sections_are_in_the_wrong_order"></span> Sure, I will wait and see. But before you go to that trouble, think about it... you are changing the "definition" to match one example. Wouldn't it make more sense to change the example to match the definition (which still isn't so great)? Better yet, solve the example as is, using the definition, as is. But best of all, just generalize the definition. </p><p>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2008-02-06T01:00:00.000Z-Now_sections_are_in_the_wrong_order" class="ext-discussiontools-init-timestamplink">01:00, 6 February 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2008-02-06T01:00:00.000Z-Now_sections_are_in_the_wrong_order"></span><span data-mw-comment-end="c-Bob_K-2008-02-06T01:00:00.000Z-Now_sections_are_in_the_wrong_order"></span> </p><p><span data-mw-comment-start="" id="c-Loisel-2008-02-06T07:33:00.000Z-Now_sections_are_in_the_wrong_order"></span>Dear Bob, </p><p>Well, you wrote that comment after I had changed the article. </p><p>Sincerely, </p><p><a href="/wiki/User:Loisel" title="User:Loisel">Loisel</a> (<a href="/wiki/User_talk:Loisel" title="User talk:Loisel">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Loisel-2008-02-06T07:33:00.000Z-Now_sections_are_in_the_wrong_order" class="ext-discussiontools-init-timestamplink">07:33, 6 February 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Loisel-2008-02-06T07:33:00.000Z-Now_sections_are_in_the_wrong_order"></span><span data-mw-comment-end="c-Loisel-2008-02-06T07:33:00.000Z-Now_sections_are_in_the_wrong_order"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="This_article_lacks_basic_explanation" data-mw-thread-id="h-This_article_lacks_basic_explanation-2008-02-12T12:29:00.000Z"><span data-mw-comment-start="" id="h-This_article_lacks_basic_explanation-2008-02-12T12:29:00.000Z"></span>This article lacks basic explanation<span data-mw-comment-end="h-This_article_lacks_basic_explanation-2008-02-12T12:29:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-192.35.17.15-2008-02-12T12:29:00.000Z-This_article_lacks_basic_explanation"></span>The math formulas are nice to be here, but one should put a chapter about what really Fourier series are and why do you need them. After that, you can put the formulas and other things. Do not forget tha this is Wikipedia not a math course for university. <small>—Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/Special:Contributions/192.35.17.15" title="Special:Contributions/192.35.17.15">192.35.17.15</a> (<a href="/wiki/User_talk:192.35.17.15" title="User talk:192.35.17.15">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-192.35.17.15-2008-02-12T12:29:00.000Z-This_article_lacks_basic_explanation" class="ext-discussiontools-init-timestamplink">12:29, 12 February 2008 (UTC)</a></small><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-192.35.17.15-2008-02-12T12:29:00.000Z-This_article_lacks_basic_explanation"></span><span data-mw-comment-end="c-192.35.17.15-2008-02-12T12:29:00.000Z-This_article_lacks_basic_explanation"></span> </p><p><span data-mw-comment-start="" id="c-Loisel-2008-02-13T04:40:00.000Z-This_article_lacks_basic_explanation"></span>Dear Anonymous, </p><p>Thank you for your helpful suggestion. Could you be more precise as to what information you would like to see? Maybe an example, "section XXX should start by explaining YYY". </p><p>We have been trying to make our article more useful to laypeople, as well as people who need to use the Fourier series. The introduction and the historical section are meant to be understandable to laypeople, except for the quote from Fourier, but that quote is historically significant. </p><p>I must also warn that there is no <a href="/wiki/Royal_road" class="mw-redirect" title="Royal road">royal road</a> to mathematics! (I love that quote.) There is bound to be some difficult material in a mathematical article. </p><p><a href="/wiki/User:Loisel" title="User:Loisel">Loisel</a> (<a href="/wiki/User_talk:Loisel" title="User talk:Loisel">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Loisel-2008-02-13T04:40:00.000Z-This_article_lacks_basic_explanation" class="ext-discussiontools-init-timestamplink">04:40, 13 February 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Loisel-2008-02-13T04:40:00.000Z-This_article_lacks_basic_explanation"></span><span data-mw-comment-end="c-Loisel-2008-02-13T04:40:00.000Z-This_article_lacks_basic_explanation"></span> </p><p><span data-mw-comment-start="" id="c-Bob_K-2008-02-15T14:47:00.000Z-This_article_lacks_basic_explanation"></span>Dear Loisel, </p><p>My suggestion is not malicious, but my english is bad. Just to present you a quote, one of my math teacher, a bad one said once: "Math is beautiful, but people are making it look ugly." I don't know if he meant to say teachers by people. </p><p>There is a straight road in mathematics, but to show it to others you must have the abilities to see the road and to show it to others. If I say c * (a+b) = c * a + c * b and I ask what is this you can say, probably that I just wrote the expanded formula for common factor multiply. OK, but I can say that all I wrote are some letters from alphabet and a few extra signs :-) So, formula without enough explanations is just painting. </p> <dl><dd>Encyclopedias do not replace textbooks and homework problems, hard as we might try. An encyclopedia is more like a reference book. Often, some basic background has to be assumed just to be efficient. The best thing about Wikipedia, in my opinion, is that one can often find the background they are missing by following the internal links. (Although that process may be hampered by inconsistent conventions & notations.) That doesn't mean I think this article can't still be improved or that I discourage anyone from trying. But I do think the introductory material is not too bad.</dd> <dd>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2008-02-15T14:47:00.000Z-This_article_lacks_basic_explanation" class="ext-discussiontools-init-timestamplink">14:47, 15 February 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2008-02-15T14:47:00.000Z-This_article_lacks_basic_explanation"></span><span data-mw-comment-end="c-Bob_K-2008-02-15T14:47:00.000Z-This_article_lacks_basic_explanation"></span></dd></dl> <p><span data-mw-comment-start="" id="c-Loisel-2008-02-15T22:39:00.000Z-This_article_lacks_basic_explanation"></span>Dear Anonymous commenter, </p><p>I have tried to expand the article a little bit by adding expanding on the example and providing further motivation. I am guessing that this won't be enough to satisfy you, but perhaps it helps, or perhaps you can make some specific suggestions for improvements. </p><p>Sincerely, </p><p><a href="/wiki/User:Loisel" title="User:Loisel">Loisel</a> (<a href="/wiki/User_talk:Loisel" title="User talk:Loisel">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Loisel-2008-02-15T22:39:00.000Z-This_article_lacks_basic_explanation" class="ext-discussiontools-init-timestamplink">22:39, 15 February 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Loisel-2008-02-15T22:39:00.000Z-This_article_lacks_basic_explanation"></span><span data-mw-comment-end="c-Loisel-2008-02-15T22:39:00.000Z-This_article_lacks_basic_explanation"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Dead_link" data-mw-thread-id="h-Dead_link-2008-04-24T15:11:00.000Z"><span data-mw-comment-start="" id="h-Dead_link-2008-04-24T15:11:00.000Z"></span>Dead link<span data-mw-comment-end="h-Dead_link-2008-04-24T15:11:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-129.241.138.61-2008-04-24T15:11:00.000Z-Dead_link"></span>The link "Une série de Fourier-Lebesgue divergente presque partout" does not seem to be working. <small>—Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/Special:Contributions/129.241.138.61" title="Special:Contributions/129.241.138.61">129.241.138.61</a> (<a href="/w/index.php?title=User_talk:129.241.138.61&action=edit&redlink=1" class="new" title="User talk:129.241.138.61 (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-129.241.138.61-2008-04-24T15:11:00.000Z-Dead_link" class="ext-discussiontools-init-timestamplink">15:11, 24 April 2008 (UTC)</a></small><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-129.241.138.61-2008-04-24T15:11:00.000Z-Dead_link"></span><span data-mw-comment-end="c-129.241.138.61-2008-04-24T15:11:00.000Z-Dead_link"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Introductory_Sentence." data-mw-thread-id="h-Introductory_Sentence.-2008-04-24T20:12:00.000Z"><span data-mw-comment-start="" id="h-Introductory_Sentence.-2008-04-24T20:12:00.000Z"></span>Introductory Sentence.<span data-mw-comment-end="h-Introductory_Sentence.-2008-04-24T20:12:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Thenub314-2008-04-24T20:12:00.000Z-Introductory_Sentence."></span>The article begins "In mathematics, the Fourier series is a type of Fourier analysis, which is used on functions that might otherwise be difficult or impossible to analyze." </p><p>Would be much happier about a beginning like... "Fourier series decompose a periodic function into a sum of simple oscillating functions, namely sines and cosines. The subject of Fourier series is part of the general subject of Fourer analysis. Fourier series were introduced..." </p><p>I found the phrase "a type of Fourier Analysis" strange. I had encountered this usages like this when I studied signal processing, not not since switching to mathematics. Even in signal processing literature I would not have described it as very common. So I stoped and asked several graduate students in mathematics if they understood the first sentance. And they objected to the same phrase. </p><p>I have notice the phrase "type of Fourier Analysis" has become standard since some of the pages were merged a while back, and I think it might be nice open a discussion about it. </p><p>More importantly I find the phrase " ... which is used on functions that might otherwise be difficult or impossible to analyze" a bit misleading. Fourier series are used extensively on even very simple functions for various reasons. What exactly is meant by analyze in this part of the sentence? </p><p>Lastly, and perhaps least important, would be to move any discussion of complex exponentials a bit deeper into the article. I think it would be helpful to improve readability for non-technical audiences. </p><p>I would love some feedback on these ideas. <a href="/wiki/User:Thenub314" title="User:Thenub314">Thenub314</a> (<a href="/wiki/User_talk:Thenub314" title="User talk:Thenub314">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Thenub314-2008-04-24T20:12:00.000Z-Introductory_Sentence." class="ext-discussiontools-init-timestamplink">20:12, 24 April 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Thenub314-2008-04-24T20:12:00.000Z-Introductory_Sentence."></span><span data-mw-comment-end="c-Thenub314-2008-04-24T20:12:00.000Z-Introductory_Sentence."></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="periodic_functions_as_tempered_distriutions." data-mw-thread-id="h-periodic_functions_as_tempered_distriutions.-2008-05-12T02:30:00.000Z"><span data-mw-comment-start="" id="h-periodic_functions_as_tempered_distriutions.-2008-05-12T02:30:00.000Z"></span>periodic functions as tempered distriutions.<span data-mw-comment-end="h-periodic_functions_as_tempered_distriutions.-2008-05-12T02:30:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Thenub314-2008-05-12T02:30:00.000Z-periodic_functions_as_tempered_distriutions."></span>In response to an edit comment let me explain my previous edit. As with any locally integrable function you can define a distribution by integrate against your function. Let <i>f</i> be a periodic function on <b>R</b>. For any Schwartz function φ the map </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 \varphi \mapsto \int _{\mathbb {R} }f(x)\varphi (x)\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">↦</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \mapsto \int _{\mathbb {R} }f(x)\varphi (x)\,dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28d00840f0d321f86fda2d19cad7bee5f662554e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.242ex; height:5.676ex;" alt="{\displaystyle \varphi \mapsto \int _{\mathbb {R} }f(x)\varphi (x)\,dx}"/></span></dd></dl> <p>defines a bounded linear functional. Thus defines a distribution, since it defines a tempered distribution we can discuss it's Fourier transform on <b>R</b>. <a href="/wiki/User:Thenub314" title="User:Thenub314">Thenub314</a> (<a href="/wiki/User_talk:Thenub314" title="User talk:Thenub314">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Thenub314-2008-05-12T02:30:00.000Z-periodic_functions_as_tempered_distriutions." class="ext-discussiontools-init-timestamplink">02:30, 12 May 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Thenub314-2008-05-12T02:30:00.000Z-periodic_functions_as_tempered_distriutions."></span><span data-mw-comment-end="c-Thenub314-2008-05-12T02:30:00.000Z-periodic_functions_as_tempered_distriutions."></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Dicklyon-2008-05-12T02:37:00.000Z-Thenub314-2008-05-12T02:30:00.000Z"></span>OK, I'll not be challenging that, since I don't know what a Schwartz function is, or why that makes a bounded linear functional, or what a tempered distribution is. But maybe someone else will understand and tell me it's OK. <a href="/wiki/User:Dicklyon" title="User:Dicklyon">Dicklyon</a> (<a href="/wiki/User_talk:Dicklyon" title="User talk:Dicklyon">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Dicklyon-2008-05-12T02:37:00.000Z-Thenub314-2008-05-12T02:30:00.000Z" class="ext-discussiontools-init-timestamplink">02:37, 12 May 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Dicklyon-2008-05-12T02:37:00.000Z-Thenub314-2008-05-12T02:30:00.000Z"></span><span data-mw-comment-end="c-Dicklyon-2008-05-12T02:37:00.000Z-Thenub314-2008-05-12T02:30:00.000Z"></span></dd></dl> <p><span data-mw-comment-start="" id="c-Loisel-2008-05-12T16:40:00.000Z-periodic_functions_as_tempered_distriutions."></span>Let <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 {\mathcal {D}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3277962e1959c3241fb1b70c7f0ac6dcefebd966" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.792ex; height:2.176ex;" alt="{\displaystyle {\mathcal {D}}}"/></span> be the set of infinitely differentiable functions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"/></span> such that, for every k>0, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{k}f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{k}f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c348198daad374da225c0dd44343fa9d73c52e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.836ex; height:3.176ex;" alt="{\displaystyle x^{k}f(x)}"/></span> tends to zero as x goes to infinity (the functions are <i>rapidly decreasing</i>). This space can be made into a metric space, and more precisely, <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 {\mathcal {D}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3277962e1959c3241fb1b70c7f0ac6dcefebd966" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.792ex; height:2.176ex;" alt="{\displaystyle {\mathcal {D}}}"/></span> is an F-space (<i>espace de type F</i>), cf. <a href="/wiki/Topological_vector_space" title="Topological vector space">topological vector space</a>. The Schwarz space <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 {\mathcal {D}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3277962e1959c3241fb1b70c7f0ac6dcefebd966" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.792ex; height:2.176ex;" alt="{\displaystyle {\mathcal {D}}}"/></span> is special because the Fourier transform of an infinitely differentiable, rapidly decreasing function is also infinitely differentiable and rapidly decreasing. A linear map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"/></span> on <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 {\mathcal {D}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3277962e1959c3241fb1b70c7f0ac6dcefebd966" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.792ex; height:2.176ex;" alt="{\displaystyle {\mathcal {D}}}"/></span> which is also continuous in the metric is called a Schwarz tempered distribution. The space of all such linear maps ("functionals") is written <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 {\mathcal {D}}'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0e2f98d79586d30daa2e37f8c17ac423dec6204" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.477ex; height:2.509ex;" alt="{\displaystyle {\mathcal {D}}'}"/></span>. Although <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 {\mathcal {D}}'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0e2f98d79586d30daa2e37f8c17ac423dec6204" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.477ex; height:2.509ex;" alt="{\displaystyle {\mathcal {D}}'}"/></span> is very abstract, one can squint and realize that it contains pretty much every function you can think of, as well as things that are not functions like measures and hairier things. The main reasons why <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 {\mathcal {D}}'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0e2f98d79586d30daa2e37f8c17ac423dec6204" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.477ex; height:2.509ex;" alt="{\displaystyle {\mathcal {D}}'}"/></span> is important is because it's the largest space of function-like things of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"/></span> on which you can define differentiation and Fourier transform. </p><p>So, almost anything you can think of has a derivative and a Fourier transform. </p><p><a href="/wiki/User:Loisel" title="User:Loisel">Loisel</a> (<a href="/wiki/User_talk:Loisel" title="User talk:Loisel">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Loisel-2008-05-12T16:40:00.000Z-periodic_functions_as_tempered_distriutions." class="ext-discussiontools-init-timestamplink">16:40, 12 May 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Loisel-2008-05-12T16:40:00.000Z-periodic_functions_as_tempered_distriutions."></span><span data-mw-comment-end="c-Loisel-2008-05-12T16:40:00.000Z-periodic_functions_as_tempered_distriutions."></span> </p><p><span data-mw-comment-start="" id="c-Loisel-2008-05-12T16:44:00.000Z-periodic_functions_as_tempered_distriutions."></span>Addendum: if you ignore Fourier transform, there's a larger space of distributions (the ones that aren't necessarily tempered) where you can also differentiate. </p><p><a href="/wiki/User:Loisel" title="User:Loisel">Loisel</a> (<a href="/wiki/User_talk:Loisel" title="User talk:Loisel">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Loisel-2008-05-12T16:44:00.000Z-periodic_functions_as_tempered_distriutions." class="ext-discussiontools-init-timestamplink">16:44, 12 May 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Loisel-2008-05-12T16:44:00.000Z-periodic_functions_as_tempered_distriutions."></span><span data-mw-comment-end="c-Loisel-2008-05-12T16:44:00.000Z-periodic_functions_as_tempered_distriutions."></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Dicklyon-2008-05-12T17:42:00.000Z-Loisel-2008-05-12T16:44:00.000Z"></span>Thanks; I'll study up on <a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">Distribution (mathematics)</a>; it's actually good to know that there's some meaningful mathematics behind the handwaving engineering uses of Fourier transform that I always thought were suspect. I always thought the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> was defined only for square-integrable functions, and extended to delta functions and such only informally, but I see now in the article that it's not so. <a href="/wiki/User:Dicklyon" title="User:Dicklyon">Dicklyon</a> (<a href="/wiki/User_talk:Dicklyon" title="User talk:Dicklyon">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Dicklyon-2008-05-12T17:42:00.000Z-Loisel-2008-05-12T16:44:00.000Z" class="ext-discussiontools-init-timestamplink">17:42, 12 May 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Dicklyon-2008-05-12T17:42:00.000Z-Loisel-2008-05-12T16:44:00.000Z"></span><span data-mw-comment-end="c-Dicklyon-2008-05-12T17:42:00.000Z-Loisel-2008-05-12T16:44:00.000Z"></span></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Vibrating_Drum" data-mw-thread-id="h-Vibrating_Drum-2008-05-25T04:28:00.000Z"><span data-mw-comment-start="" id="h-Vibrating_Drum-2008-05-25T04:28:00.000Z"></span>Vibrating Drum<span data-mw-comment-end="h-Vibrating_Drum-2008-05-25T04:28:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Gmcastil-2008-05-25T04:28:00.000Z-Vibrating_Drum"></span>The solution to the wave equation on a disc produces Bessel functions, not Fourier waves. Those Bessel functions can of course be written as a Fourier series with a simple change of basis, but I don't believe that serves the pedagogical purpose for the diagram. I would suggest removing it or producing a new animation on a square. <small>—Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/w/index.php?title=User:Gmcastil&action=edit&redlink=1" class="new" title="User:Gmcastil (page does not exist)">Gmcastil</a> (<a href="/w/index.php?title=User_talk:Gmcastil&action=edit&redlink=1" class="new" title="User talk:Gmcastil (page does not exist)">talk</a> • <a href="/wiki/Special:Contributions/Gmcastil" title="Special:Contributions/Gmcastil">contribs</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Gmcastil-2008-05-25T04:28:00.000Z-Vibrating_Drum" class="ext-discussiontools-init-timestamplink">04:28, 25 May 2008 (UTC)</a></small><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Gmcastil-2008-05-25T04:28:00.000Z-Vibrating_Drum"></span><span data-mw-comment-end="c-Gmcastil-2008-05-25T04:28:00.000Z-Vibrating_Drum"></span> </p><p><span data-mw-comment-start="" id="c-Loisel-2008-05-26T15:58:00.000Z-Vibrating_Drum"></span>The Bessel functions you speak of (really, Bessel in the r variable, complex exponential in the angular variable) form an <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> for the <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> of L^2 functions. The reconstruction formula in a Hilbert space is called the Fourier series. So all the examples (vibrating drum, spherical harmonics, etc...) are Fourier series in the sense of Hilbert spaces. <a href="/wiki/User:Loisel" title="User:Loisel">Loisel</a> (<a href="/wiki/User_talk:Loisel" title="User talk:Loisel">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Loisel-2008-05-26T15:58:00.000Z-Vibrating_Drum" class="ext-discussiontools-init-timestamplink">15:58, 26 May 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Loisel-2008-05-26T15:58:00.000Z-Vibrating_Drum"></span><span data-mw-comment-end="c-Loisel-2008-05-26T15:58:00.000Z-Vibrating_Drum"></span> </p><p><br/><span data-mw-comment-start="" id="c-Mposey82-2008-05-27T21:55:00.000Z-Vibrating_Drum"></span> Loisel - Given your description of why this image is appropriate wouldn't the page <a href="/wiki/Generalized_Fourier_series" title="Generalized Fourier series">Generalized Fourier series</a> or perhaps even <a href="/wiki/Fourier%E2%80%93Bessel_series" title="Fourier–Bessel series">Fourier–Bessel series</a> make a happier home for this example. Especially since for the purposes of this page " a Fourier series decomposes a periodic function into a sum of simple oscillating functions, namely sines and cosines." --<a href="/w/index.php?title=User:Mposey82&action=edit&redlink=1" class="new" title="User:Mposey82 (page does not exist)">Mposey82</a> (<a href="/wiki/User_talk:Mposey82" title="User talk:Mposey82">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Mposey82-2008-05-27T21:55:00.000Z-Vibrating_Drum" class="ext-discussiontools-init-timestamplink">21:55, 27 May 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Mposey82-2008-05-27T21:55:00.000Z-Vibrating_Drum"></span><span data-mw-comment-end="c-Mposey82-2008-05-27T21:55:00.000Z-Vibrating_Drum"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Ping-pong_sections" data-mw-thread-id="h-Ping-pong_sections-2008-09-10T04:35:00.000Z"><span data-mw-comment-start="" id="h-Ping-pong_sections-2008-09-10T04:35:00.000Z"></span>Ping-pong sections<span data-mw-comment-end="h-Ping-pong_sections-2008-09-10T04:35:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Dicklyon-2008-09-10T04:35:00.000Z-Ping-pong_sections"></span>We've had a lot of back-and-forth between the sections "A second example: trigonometric polynomials" and "Determinining the coefficients." I did some editing to fix a few nits and remove major duplication, but I must say I don't quite get it. The former section is a somewhat interesting if narrow example that shows that if you already have the coefficients of a trigonometric polynomial, then they agree with the formula given in "Fourier's formula for 2π-periodic functions using sines and cosines." The latter section shows that if you have any periodic function and find the coefficients that make a trigonometric polynomial equal to it, then they too obey the formula given before. If we accept Fourier's formula, then both of these sections are pretty redundant. The latter could be taken as a derivation of Fourier's formula, but it's not presented that way. What's the right way to develop this stuff? <a href="/wiki/User:Dicklyon" title="User:Dicklyon">Dicklyon</a> (<a href="/wiki/User_talk:Dicklyon" title="User talk:Dicklyon">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Dicklyon-2008-09-10T04:35:00.000Z-Ping-pong_sections" class="ext-discussiontools-init-timestamplink">04:35, 10 September 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Dicklyon-2008-09-10T04:35:00.000Z-Ping-pong_sections"></span><span data-mw-comment-end="c-Dicklyon-2008-09-10T04:35:00.000Z-Ping-pong_sections"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Thenub314-2008-09-10T06:50:00.000Z-Dicklyon-2008-09-10T04:35:00.000Z"></span>The problems I have with the section on determining Fourier coefficients is that it is mathematically flawed. Most courses given point out this is what Fourier did, and that it doesn't work. If you want to precede this way you must end up at a narrow class of functions. I am happy to remove the example all together, but I thought if someone wanted to make explicit how orthogonality worked before the section on Hilbert spaces, then I would go along with it. I have removed the section on "Determining Fourier Coefficients", among other problems it gives the impression that every trigonometric series is a Fourier series, which is false. It also had explicitly stated that if the Fourier series converged to the function then this method would work, and this is also false. <a href="/wiki/User:Thenub314" title="User:Thenub314">Thenub314</a> (<a href="/wiki/User_talk:Thenub314" title="User talk:Thenub314">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Thenub314-2008-09-10T06:50:00.000Z-Dicklyon-2008-09-10T04:35:00.000Z" class="ext-discussiontools-init-timestamplink">06:50, 10 September 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Thenub314-2008-09-10T06:50:00.000Z-Dicklyon-2008-09-10T04:35:00.000Z"></span><span data-mw-comment-end="c-Thenub314-2008-09-10T06:50:00.000Z-Dicklyon-2008-09-10T04:35:00.000Z"></span></dd></dl> <dl><dd><dl><dd><span data-mw-comment-start="" id="c-Dicklyon-2008-09-10T06:59:00.000Z-Thenub314-2008-09-10T06:50:00.000Z"></span>Well, you must be a mathematician, because you've lost me. Where exactly can this method go wrong in the case of a function that the Fourier series converges to? Are you saying that dot products with sinusoids can give wrong Fourier series coefficients? Or that the reasoning to get there was flawed? <a href="/wiki/User:Dicklyon" title="User:Dicklyon">Dicklyon</a> (<a href="/wiki/User_talk:Dicklyon" title="User talk:Dicklyon">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Dicklyon-2008-09-10T06:59:00.000Z-Thenub314-2008-09-10T06:50:00.000Z" class="ext-discussiontools-init-timestamplink">06:59, 10 September 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Dicklyon-2008-09-10T06:59:00.000Z-Thenub314-2008-09-10T06:50:00.000Z"></span><span data-mw-comment-end="c-Dicklyon-2008-09-10T06:59:00.000Z-Thenub314-2008-09-10T06:50:00.000Z"></span></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd><span data-mw-comment-start="" id="c-Thenub314-2008-09-10T17:42:00.000Z-Dicklyon-2008-09-10T06:59:00.000Z"></span>Mostly the concern is that the reasoning is flawed. Most of which come into play when you want to exchange the infinite sum and the integral. <a href="/wiki/User:Thenub314" title="User:Thenub314">Thenub314</a> (<a href="/wiki/User_talk:Thenub314" title="User talk:Thenub314">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Thenub314-2008-09-10T17:42:00.000Z-Dicklyon-2008-09-10T06:59:00.000Z" class="ext-discussiontools-init-timestamplink">17:42, 10 September 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Thenub314-2008-09-10T17:42:00.000Z-Dicklyon-2008-09-10T06:59:00.000Z"></span><span data-mw-comment-end="c-Thenub314-2008-09-10T17:42:00.000Z-Dicklyon-2008-09-10T06:59:00.000Z"></span></dd></dl></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd><dl><dd><span data-mw-comment-start="" id="c-Thenub314-2008-09-19T21:13:00.000Z-Thenub314-2008-09-10T17:42:00.000Z"></span>As a PS to this conversation, I happend upon the example I was thinking of in Pinsky's book. It turns out that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=2}^{\infty }{\frac {\sin(nx)}{\log(n)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=2}^{\infty }{\frac {\sin(nx)}{\log(n)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19f8b4181780602f1f29e70991a2f4715451e921" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:11.968ex; height:6.843ex;" alt="{\displaystyle \sum _{n=2}^{\infty }{\frac {\sin(nx)}{\log(n)}}}"/></span> converges, but it is not a Fourier series. But if you replace sin(<i>nx</i>) by cos(<i>nx</i>) then it is a Fourier series. <a href="/wiki/User:Thenub314" title="User:Thenub314">Thenub314</a> (<a href="/wiki/User_talk:Thenub314" title="User talk:Thenub314">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Thenub314-2008-09-19T21:13:00.000Z-Thenub314-2008-09-10T17:42:00.000Z" class="ext-discussiontools-init-timestamplink">21:13, 19 September 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Thenub314-2008-09-19T21:13:00.000Z-Thenub314-2008-09-10T17:42:00.000Z"></span><span data-mw-comment-end="c-Thenub314-2008-09-19T21:13:00.000Z-Thenub314-2008-09-10T17:42:00.000Z"></span></dd></dl></dd></dl></dd></dl></dd></dl> <p><span data-mw-comment-start="" id="c-Thenub314-2008-09-10T17:42:00.000Z-Ping-pong_sections"></span>I've just taken a look at the article, I think we should remove Thenub's new section which relates the exponential formulation and the sine/cosine formulation. The crux of this new "example" is already a part of the subsequent sections like <b>The modern version using complex exponentials</b> and <b>Hilbert space interpretation</b>. Any new information, which is not included in those previously existing section, should rather be merged, instead of adding a new section which repeats information found elsewhere in the article and which does not flow well. </p> <dl><dd>I am happy to delete the new section. I didn't want to completely delete <a href="/wiki/User:RJFJR" title="User:RJFJR">RJFJRs</a> additions, but I didn't want to leave a mathematical flawed argument. I wasn't trying to relate exponential and sine/cosine formulations, I don't even remember mentioning exponential functions. Those calculations are easy to deduce from the stuff in this article so I am fine with leaving them out. <a href="/wiki/User:Thenub314" title="User:Thenub314">Thenub314</a> (<a href="/wiki/User_talk:Thenub314" title="User talk:Thenub314">talk</a>)</dd></dl> <p>As for Thenub's question of what is the difference between a Fourier series and a Trigonometric series, as far as I know, the two terms are used interchangeably in the literature. </p> <dl><dd>I didn't really have a question. It is a theorem that not every trigonometric series is a Fourier series. The sets of convergence for trigonometric series is a difficult subject that I had looked into some time ago. <a href="/wiki/User:Thenub314" title="User:Thenub314">Thenub314</a> (<a href="/wiki/User_talk:Thenub314" title="User talk:Thenub314">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Thenub314-2008-09-10T17:42:00.000Z-Ping-pong_sections" class="ext-discussiontools-init-timestamplink">17:42, 10 September 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Thenub314-2008-09-10T17:42:00.000Z-Ping-pong_sections"></span><span data-mw-comment-end="c-Thenub314-2008-09-10T17:42:00.000Z-Ping-pong_sections"></span></dd></dl> <p><span data-mw-comment-start="" id="c-Loisel-2008-09-10T16:39:00.000Z-Ping-pong_sections"></span>The article as I wrote it a while back imposed that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"/></span> be square-integrable, for simplicity, but <a href="/wiki/User:Bob_K" title="User:Bob K">User:Bob K</a> attempted to generalize (although he went too far, diff is here: <a class="external free" href="https://en.wikipedia.org/w/index.php?title=Fourier_series&diff=183950639&oldid=183861651">http://en.wikipedia.org/w/index.php?title=Fourier_series&diff=183950639&oldid=183861651</a>). I subsequently "fixed" it by not being too specific on the constraint on the Fourier coefficients (they must not increase too rapidly), here is the diff: <a class="external free" href="https://en.wikipedia.org/w/index.php?title=Fourier_series&diff=185638936&oldid=185638341">http://en.wikipedia.org/w/index.php?title=Fourier_series&diff=185638936&oldid=185638341</a>. I must say I would still prefer to put a restriction on <i>f</i>, but I'm betting that the next editor who comes around and who may not necessarily know what the restrictions on the Fourier coefficients must be, will simply delete any restriction we place on <i>f</i>. Still, I'm all for going back to square-integrable functions for the earlier sections of the article. </p><p><a href="/wiki/User:Loisel" title="User:Loisel">Loisel</a> (<a href="/wiki/User_talk:Loisel" title="User talk:Loisel">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Loisel-2008-09-10T16:39:00.000Z-Ping-pong_sections" class="ext-discussiontools-init-timestamplink">16:39, 10 September 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Loisel-2008-09-10T16:39:00.000Z-Ping-pong_sections"></span><span data-mw-comment-end="c-Loisel-2008-09-10T16:39:00.000Z-Ping-pong_sections"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Thenub314-2008-09-10T18:30:00.000Z-Loisel-2008-09-10T16:39:00.000Z"></span>I absolutely agree you need a restriction on <i>f</i>. To my mind the most general condition that is easy to see makes sense is to require <i>f</i> to be integrable. Then by the triangle inequality you get:</dd></dl> <dl><dd><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 |c_{n}|=\left|\int _{0}^{2\pi }f(x)e^{-inx}\,dx\right|\leq \int _{0}^{2\pi }|f(x)||e^{-inx}|\,dx=\int _{0}^{2\pi }|f(x)|\,dx<\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>n</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>x</mi> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>n</mi> <mi>x</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>x</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |c_{n}|=\left|\int _{0}^{2\pi }f(x)e^{-inx}\,dx\right|\leq \int _{0}^{2\pi }|f(x)||e^{-inx}|\,dx=\int _{0}^{2\pi }|f(x)|\,dx<\infty .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69c57d1745752f50253e3d4e31b952dfb1cdb051" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:70.827ex; height:6.509ex;" alt="{\displaystyle |c_{n}|=\left|\int _{0}^{2\pi }f(x)e^{-inx}\,dx\right|\leq \int _{0}^{2\pi }|f(x)||e^{-inx}|\,dx=\int _{0}^{2\pi }|f(x)|\,dx<\infty .}"/></span></dd></dl></dd></dl> <dl><dd>Here I used of course that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |e^{-inx}|=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>n</mi> <mi>x</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |e^{-inx}|=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6760134cde2bee51db619df515c8877fc803bede" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.643ex; height:3.176ex;" alt="{\displaystyle |e^{-inx}|=1}"/></span>. So all the Fourier coefficients are defined and finite. I'll try putting this in, we will see what happens. —Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/User:Thenub314" title="User:Thenub314">Thenub314</a> (<a href="/wiki/User_talk:Thenub314" title="User talk:Thenub314">talk</a> • <a href="/wiki/Special:Contributions/Thenub314" title="Special:Contributions/Thenub314">contribs</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Thenub314-2008-09-10T18:30:00.000Z-Loisel-2008-09-10T16:39:00.000Z" class="ext-discussiontools-init-timestamplink">18:30, 10 September 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Thenub314-2008-09-10T18:30:00.000Z-Loisel-2008-09-10T16:39:00.000Z"></span><span data-mw-comment-end="c-Thenub314-2008-09-10T18:30:00.000Z-Loisel-2008-09-10T16:39:00.000Z"></span></dd></dl> <p><br/> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Example:_a_simple_Fourier_series" data-mw-thread-id="h-Example:_a_simple_Fourier_series-2008-11-17T01:18:00.000Z"><span data-mw-comment-start="" id="h-Example:_a_simple_Fourier_series-2008-11-17T01:18:00.000Z"></span>Example: a simple Fourier series<span data-mw-comment-end="h-Example:_a_simple_Fourier_series-2008-11-17T01:18:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Squidgyhead-2008-11-17T01:18:00.000Z-Example:_a_simple_Fourier_series"></span>The coefficients are missing a factor of <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 {1}{\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6b81c585f1905cd13e59550b847006593be2e13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.168ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{\pi }}}"/></span>. Or, conversely, the function should be periodic between -1 and 1, not <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"> <mo>−</mo> <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/f2359073fe90a84a705e02f0c1e63b32df850a60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.14ex; height:2.176ex;" alt="{\displaystyle -\pi }"/></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 \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> (or normalized differently). Also,the heat-transfer example deals with <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\in \left(0,\pi \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mi>π</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \left(0,\pi \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3ffb1bbb5bfbc781dbabdf0680fd4c4289887e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.508ex; height:2.843ex;" alt="{\displaystyle x\in \left(0,\pi \right)}"/></span>, which would give different coefficients. Or am I mistaken? <a href="/w/index.php?title=User:Squidgyhead&action=edit&redlink=1" class="new" title="User:Squidgyhead (page does not exist)">Squidgyhead</a> (<a href="/wiki/User_talk:Squidgyhead" title="User talk:Squidgyhead">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Squidgyhead-2008-11-17T01:18:00.000Z-Example:_a_simple_Fourier_series" class="ext-discussiontools-init-timestamplink">01:18, 17 November 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Squidgyhead-2008-11-17T01:18:00.000Z-Example:_a_simple_Fourier_series"></span><span data-mw-comment-end="c-Squidgyhead-2008-11-17T01:18:00.000Z-Example:_a_simple_Fourier_series"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Sines_and_cosines_more_"accessible"_than_sinusoids?" data-mw-thread-id="h-Sines_and_cosines_more_"accessible"_than_sinusoids?-2008-12-17T14:16:00.000Z"><span id="Sines_and_cosines_more_.22accessible.22_than_sinusoids.3F"></span><span data-mw-comment-start="" id="h-Sines_and_cosines_more_"accessible"_than_sinusoids?-2008-12-17T14:16:00.000Z"></span>Sines and cosines more "accessible" than sinusoids?<span data-mw-comment-end="h-Sines_and_cosines_more_"accessible"_than_sinusoids?-2008-12-17T14:16:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Dicklyon-2008-12-17T14:16:00.000Z-Sines_and_cosines_more_"accessible"_than_sinusoids?"></span>I'm not sure what TheNub means in his edit summary. It always seemed to me that saying decompose into sines and cosines was a bit vague and confusing, since it really means sine waves or sinusoids. The term "sines" tends to suggest the sines of a few angles, as opposed to a few sinusoidal functions. <a href="/wiki/User:Dicklyon" title="User:Dicklyon">Dicklyon</a> (<a href="/wiki/User_talk:Dicklyon" title="User talk:Dicklyon">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Dicklyon-2008-12-17T14:16:00.000Z-Sines_and_cosines_more_"accessible"_than_sinusoids?" class="ext-discussiontools-init-timestamplink">14:16, 17 December 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Dicklyon-2008-12-17T14:16:00.000Z-Sines_and_cosines_more_"accessible"_than_sinusoids?"></span><span data-mw-comment-end="c-Dicklyon-2008-12-17T14:16:00.000Z-Sines_and_cosines_more_"accessible"_than_sinusoids?"></span> </p><p><span data-mw-comment-start="" id="c-Thenub314-2008-12-17T16:06:00.000Z-Sines_and_cosines_more_"accessible"_than_sinusoids?"></span>I just meant that sines and cosines are familiar terms from high school. Depending on the level of mathematics a student encounters before finishing high school they may (or may not) encounter the term sinusoid (or sine and cosine). So I thought keeping the lede as it was made kept it accessible to a wider group or readers. <a href="/wiki/User:Thenub314" title="User:Thenub314">Thenub314</a> (<a href="/wiki/User_talk:Thenub314" title="User talk:Thenub314">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Thenub314-2008-12-17T16:06:00.000Z-Sines_and_cosines_more_"accessible"_than_sinusoids?" class="ext-discussiontools-init-timestamplink">16:06, 17 December 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Thenub314-2008-12-17T16:06:00.000Z-Sines_and_cosines_more_"accessible"_than_sinusoids?"></span><span data-mw-comment-end="c-Thenub314-2008-12-17T16:06:00.000Z-Sines_and_cosines_more_"accessible"_than_sinusoids?"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Definition_of_Fourier_series" data-mw-thread-id="h-Definition_of_Fourier_series-2008-12-28T09:34:00.000Z"><span data-mw-comment-start="" id="h-Definition_of_Fourier_series-2008-12-28T09:34:00.000Z"></span>Definition of Fourier series<span data-mw-comment-end="h-Definition_of_Fourier_series-2008-12-28T09:34:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Bdmy-2008-12-28T09:34:00.000Z-Definition_of_Fourier_series"></span>I know that some readers are uncomfortable with the short sentence (in "Fourier's formula for 2<i>π</i>-periodic functions using sines and cosines") </p> <dl><dd>The <a href="/wiki/Infinite_sum" class="mw-redirect" title="Infinite sum">infinite sum</a></dd></dl> <dl><dd><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 {a_{0}}{2}}+\sum _{n=1}^{\infty }[a_{n}\cos(nx)+b_{n}\sin(nx)]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }[a_{n}\cos(nx)+b_{n}\sin(nx)]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f2cc64d43dae2e5cca44921e64466b70849632c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.922ex; height:6.843ex;" alt="{\displaystyle {\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }[a_{n}\cos(nx)+b_{n}\sin(nx)]}"/></span></dd></dl></dd></dl> <dl><dd>is the <b>Fourier series</b> for <i>ƒ</i> on the interval [−<i>π</i>, <i>π</i>].</dd></dl> <p>I understand that this must be interpreted as a <i>formal</i> series of <i>functions</i> of <i>x</i>, but people who learned to distinguish between a series and the sum of the series, between series of numbers and series of functions will find the text careless. Could it be possible to be more precise, without being too long? <a href="/wiki/User:Bdmy" title="User:Bdmy">Bdmy</a> (<a href="/wiki/User_talk:Bdmy" title="User talk:Bdmy">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bdmy-2008-12-28T09:34:00.000Z-Definition_of_Fourier_series" class="ext-discussiontools-init-timestamplink">09:34, 28 December 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bdmy-2008-12-28T09:34:00.000Z-Definition_of_Fourier_series"></span><span data-mw-comment-end="c-Bdmy-2008-12-28T09:34:00.000Z-Definition_of_Fourier_series"></span> </p><p><span data-mw-comment-start="" id="c-Thenub314-2008-12-28T10:48:00.000Z-Definition_of_Fourier_series"></span>Well, one simple change may be to move the link from <a href="/wiki/Infinite_sum" class="mw-redirect" title="Infinite sum">infinite sum</a> to <a href="/wiki/Infinite_sum#Formal_definition" class="mw-redirect" title="Infinite sum">infinite sum</a> (not that I am so happy with that section of the article). This way, if there is confusion we direct the reader to a point in the article where formal series are discussed and series whose elements are functions are mentioned. <a href="/wiki/User:Thenub314" title="User:Thenub314">Thenub314</a> (<a href="/wiki/User_talk:Thenub314" title="User talk:Thenub314">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Thenub314-2008-12-28T10:48:00.000Z-Definition_of_Fourier_series" class="ext-discussiontools-init-timestamplink">10:48, 28 December 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Thenub314-2008-12-28T10:48:00.000Z-Definition_of_Fourier_series"></span><span data-mw-comment-end="c-Thenub314-2008-12-28T10:48:00.000Z-Definition_of_Fourier_series"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Formula_for_a0" data-mw-thread-id="h-Formula_for_a0-2008-12-28T16:59:00.000Z"><span data-mw-comment-start="" id="h-Formula_for_a0-2008-12-28T16:59:00.000Z"></span>Formula for <i>a</i><sub>0</sub><span data-mw-comment-end="h-Formula_for_a0-2008-12-28T16:59:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Danielsimonjr-2008-12-28T16:59:00.000Z-Formula_for_a0"></span>What's the formula for <b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/693ad9f934775838bd72406b41ada4a59785d7ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{0}}"/></span></b>? <small><span class="autosigned">—Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/w/index.php?title=User:Danielsimonjr&action=edit&redlink=1" class="new" title="User:Danielsimonjr (page does not exist)">Danielsimonjr</a> (<a href="/wiki/User_talk:Danielsimonjr" title="User talk:Danielsimonjr">talk</a> • <a href="/wiki/Special:Contributions/Danielsimonjr" title="Special:Contributions/Danielsimonjr">contribs</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Danielsimonjr-2008-12-28T16:59:00.000Z-Formula_for_a0" class="ext-discussiontools-init-timestamplink">16:59, 28 December 2008 (UTC)</a></span></small><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Danielsimonjr-2008-12-28T16:59:00.000Z-Formula_for_a0"></span><span data-mw-comment-end="c-Danielsimonjr-2008-12-28T16:59:00.000Z-Formula_for_a0"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Bdmy-2008-12-28T18:24:00.000Z-Danielsimonjr-2008-12-28T16:59:00.000Z"></span>The idea is to have <i>the same formula</i> 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 a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"/></span>, namely</dd></dl> <dl><dd><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 a_{n}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(t)\cos(nt)\,\mathrm {d} t,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(t)\cos(nt)\,\mathrm {d} t,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab42ab111ef849fd21b1ca3bc06cda12c5bed720" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.87ex; height:6.009ex;" alt="{\displaystyle a_{n}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(t)\cos(nt)\,\mathrm {d} t,}"/></span></dd></dl></dd></dl> <dl><dd>so that</dd></dl> <dl><dd><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 a_{0}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(t)\,\mathrm {d} t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(t)\,\mathrm {d} t.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f0decafe8e0f6b06f57fae1f8660b1757022e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.164ex; height:6.009ex;" alt="{\displaystyle a_{0}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(t)\,\mathrm {d} t.}"/></span></dd></dl></dd></dl> <dl><dd>It is true that this formula is a little strange, but this is how mathematicians have done for almost 200 years, and we are not going to change it. It is a bit strange because <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/693ad9f934775838bd72406b41ada4a59785d7ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{0}}"/></span> is not the mean of the function, but twice the mean. This is reflected in the formula for the Fourier series, that contains <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a39e941bdfee48cf51a6221de961daa2244fefd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.609ex; height:2.843ex;" alt="{\displaystyle a_{0}/2}"/></span>. When dealing with the complex Fourier coefficients <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{n}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b7e944bcb1be88e9a6a940638f2adce0ec4211a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.225ex; height:2.009ex;" alt="{\displaystyle c_{n}}"/></span>, you have that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{0}=a_{0}/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{0}=a_{0}/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aadd94046e456802fb0b152f0f6dfbd28dc57ed3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.769ex; height:2.843ex;" alt="{\displaystyle c_{0}=a_{0}/2}"/></span>.</dd> <dd>--<a href="/wiki/User:Bdmy" title="User:Bdmy">Bdmy</a> (<a href="/wiki/User_talk:Bdmy" title="User talk:Bdmy">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bdmy-2008-12-28T18:24:00.000Z-Danielsimonjr-2008-12-28T16:59:00.000Z" class="ext-discussiontools-init-timestamplink">18:24, 28 December 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bdmy-2008-12-28T18:24:00.000Z-Danielsimonjr-2008-12-28T16:59:00.000Z"></span><span data-mw-comment-end="c-Bdmy-2008-12-28T18:24:00.000Z-Danielsimonjr-2008-12-28T16:59:00.000Z"></span></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Merger_proposal" data-mw-thread-id="h-Merger_proposal-2009-03-11T18:33:00.000Z"><span data-mw-comment-start="" id="h-Merger_proposal-2009-03-11T18:33:00.000Z"></span>Merger proposal<span data-mw-comment-end="h-Merger_proposal-2009-03-11T18:33:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Taeshadow-2009-03-11T18:33:00.000Z-Merger_proposal"></span>Not sure if <a href="/wiki/Fourier_theorem" class="mw-redirect" title="Fourier theorem">Fourier theorem</a> has any information that this article doesn't have, but it's clearly subsumed in scope. --<a href="/wiki/User:Taeshadow" title="User:Taeshadow">Taeshadow</a> (<a href="/wiki/User_talk:Taeshadow" title="User talk:Taeshadow">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Taeshadow-2009-03-11T18:33:00.000Z-Merger_proposal" class="ext-discussiontools-init-timestamplink">18:33, 11 March 2009 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Taeshadow-2009-03-11T18:33:00.000Z-Merger_proposal"></span><span data-mw-comment-end="c-Taeshadow-2009-03-11T18:33:00.000Z-Merger_proposal"></span> </p> <ul><li><b><span data-mw-comment-start="" id="c-Dicklyon-2009-03-13T05:24:00.000Z-Taeshadow-2009-03-11T18:33:00.000Z"></span>Support</b> – <a href="/wiki/Fourier_theorem" class="mw-redirect" title="Fourier theorem">Fourier theorem</a> is little more than a stub, thinly edited, covered better at <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a>. <a href="/wiki/User:Dicklyon" title="User:Dicklyon">Dicklyon</a> (<a href="/wiki/User_talk:Dicklyon" title="User talk:Dicklyon">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Dicklyon-2009-03-13T05:24:00.000Z-Taeshadow-2009-03-11T18:33:00.000Z" class="ext-discussiontools-init-timestamplink">05:24, 13 March 2009 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Dicklyon-2009-03-13T05:24:00.000Z-Taeshadow-2009-03-11T18:33:00.000Z"></span><span data-mw-comment-end="c-Dicklyon-2009-03-13T05:24:00.000Z-Taeshadow-2009-03-11T18:33:00.000Z"></span></li></ul> <dl><dd><dl><dd><span data-mw-comment-start="" id="c-Paul_Laroque-2009-04-16T01:47:00.000Z-Dicklyon-2009-03-13T05:24:00.000Z"></span>I agree also. The <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a> article is far better and encompassed everything from the short paragraph at <a href="/wiki/Fourier_theorem" class="mw-redirect" title="Fourier theorem">Fourier theorem</a>. --<a href="/wiki/User:Paul_Laroque" title="User:Paul Laroque">Paul Laroque</a> (<a href="/wiki/User_talk:Paul_Laroque" title="User talk:Paul Laroque">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Paul_Laroque-2009-04-16T01:47:00.000Z-Dicklyon-2009-03-13T05:24:00.000Z" class="ext-discussiontools-init-timestamplink">01:47, 16 April 2009 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Paul_Laroque-2009-04-16T01:47:00.000Z-Dicklyon-2009-03-13T05:24:00.000Z"></span><span data-mw-comment-end="c-Paul_Laroque-2009-04-16T01:47:00.000Z-Dicklyon-2009-03-13T05:24:00.000Z"></span></dd></dl></dd></dl> <p><span data-mw-comment-start="" id="c-Michael_Hardy-2009-04-16T02:13:00.000Z-Merger_proposal"></span>I agree. If the article titled <a href="/wiki/Fourier_theorem" class="mw-redirect" title="Fourier theorem">Fourier theorem</a> actually stated a theorem, maybe it would have a point. <a href="/wiki/User:Michael_Hardy" title="User:Michael Hardy">Michael Hardy</a> (<a href="/wiki/User_talk:Michael_Hardy" title="User talk:Michael Hardy">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Michael_Hardy-2009-04-16T02:13:00.000Z-Merger_proposal" class="ext-discussiontools-init-timestamplink">02:13, 16 April 2009 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Michael_Hardy-2009-04-16T02:13:00.000Z-Merger_proposal"></span><span data-mw-comment-end="c-Michael_Hardy-2009-04-16T02:13:00.000Z-Merger_proposal"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Dicklyon-2009-04-16T02:17:00.000Z-Michael_Hardy-2009-04-16T02:13:00.000Z"></span>Merged. <a href="/wiki/User:Dicklyon" title="User:Dicklyon">Dicklyon</a> (<a href="/wiki/User_talk:Dicklyon" title="User talk:Dicklyon">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Dicklyon-2009-04-16T02:17:00.000Z-Michael_Hardy-2009-04-16T02:13:00.000Z" class="ext-discussiontools-init-timestamplink">02:17, 16 April 2009 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Dicklyon-2009-04-16T02:17:00.000Z-Michael_Hardy-2009-04-16T02:13:00.000Z"></span><span data-mw-comment-end="c-Dicklyon-2009-04-16T02:17:00.000Z-Michael_Hardy-2009-04-16T02:13:00.000Z"></span></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Reviewers_of_the_original_Fourier_article" data-mw-thread-id="h-Reviewers_of_the_original_Fourier_article-2009-11-28T02:25:00.000Z"><span data-mw-comment-start="" id="h-Reviewers_of_the_original_Fourier_article-2009-11-28T02:25:00.000Z"></span>Reviewers of the original Fourier article<span data-mw-comment-end="h-Reviewers_of_the_original_Fourier_article-2009-11-28T02:25:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-98.210.57.10-2009-11-28T02:25:00.000Z-Reviewers_of_the_original_Fourier_article"></span>In the section "Revolutionary Article" its is stated, "When Fourier submitted his paper in 1807, the committee (which included Lagrange, Laplace, Malus and Legendre, among others)". This statement is factually incorrect. </p><p>Fact: Four reviewers were appointed by the Academy. These were: Laplace, Lagrange, Monge and LaCroix. </p><p>T. N. Narasimhan, tnnarasimhan@LBL.gov <a href="/wiki/Special:Contributions/98.210.57.10" title="Special:Contributions/98.210.57.10">98.210.57.10</a> (<a href="/w/index.php?title=User_talk:98.210.57.10&action=edit&redlink=1" class="new" title="User talk:98.210.57.10 (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-98.210.57.10-2009-11-28T02:25:00.000Z-Reviewers_of_the_original_Fourier_article" class="ext-discussiontools-init-timestamplink">02:25, 28 November 2009 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-98.210.57.10-2009-11-28T02:25:00.000Z-Reviewers_of_the_original_Fourier_article"></span><span data-mw-comment-end="c-98.210.57.10-2009-11-28T02:25:00.000Z-Reviewers_of_the_original_Fourier_article"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-LutzL-2009-11-30T09:47:00.000Z-98.210.57.10-2009-11-28T02:25:00.000Z"></span>Any sources for that? Because replacing a badly sourced statement by a badly sourced statement will, in all likelyhood, not happen.--<a href="/wiki/User:LutzL" title="User:LutzL">LutzL</a> (<a href="/wiki/User_talk:LutzL" title="User talk:LutzL">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-LutzL-2009-11-30T09:47:00.000Z-98.210.57.10-2009-11-28T02:25:00.000Z" class="ext-discussiontools-init-timestamplink">09:47, 30 November 2009 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-LutzL-2009-11-30T09:47:00.000Z-98.210.57.10-2009-11-28T02:25:00.000Z"></span><span data-mw-comment-end="c-LutzL-2009-11-30T09:47:00.000Z-98.210.57.10-2009-11-28T02:25:00.000Z"></span></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Vibrating_Disc_doesn't_look_right" data-mw-thread-id="h-Vibrating_Disc_doesn't_look_right-2008-05-02T04:12:00.000Z"><span id="Vibrating_Disc_doesn.27t_look_right"></span><span data-mw-comment-start="" id="h-Vibrating_Disc_doesn't_look_right-2008-05-02T04:12:00.000Z"></span>Vibrating Disc doesn't look right<span data-mw-comment-end="h-Vibrating_Disc_doesn't_look_right-2008-05-02T04:12:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-117.196.34.205-2009-12-14T16:16:00.000Z-Vibrating_Disc_doesn't_look_right"></span>it states that this is a variable when integer is constant <span style="font-size: smaller;" class="autosigned">—Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/Special:Contributions/117.196.34.205" title="Special:Contributions/117.196.34.205">117.196.34.205</a> (<a href="/wiki/User_talk:117.196.34.205" title="User talk:117.196.34.205">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-117.196.34.205-2009-12-14T16:16:00.000Z-Vibrating_Disc_doesn't_look_right" class="ext-discussiontools-init-timestamplink">16:16, 14 December 2009 (UTC)</a></span><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-117.196.34.205-2009-12-14T16:16:00.000Z-Vibrating_Disc_doesn't_look_right"></span><span data-mw-comment-end="c-117.196.34.205-2009-12-14T16:16:00.000Z-Vibrating_Disc_doesn't_look_right"></span> </p><p><span data-mw-comment-start="" id="c-129.105.207.213-2008-05-02T04:12:00.000Z-Vibrating_Disc_doesn't_look_right"></span>I would be surprised if the animation of the vibrating disc is correct. If it is a circular disk, as depicted, the solution should have circular symmetry. I understand that this kind of symmetry preservation doesn't always hold, but in this fairly simple case I think it should. </p><p>I learned that vibrating disks have solutions that look like radial Bessel functions anyhow. <small>—Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/Special:Contributions/129.105.207.213" title="Special:Contributions/129.105.207.213">129.105.207.213</a> (<a href="/w/index.php?title=User_talk:129.105.207.213&action=edit&redlink=1" class="new" title="User talk:129.105.207.213 (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-129.105.207.213-2008-05-02T04:12:00.000Z-Vibrating_Disc_doesn't_look_right" class="ext-discussiontools-init-timestamplink">04:12, 2 May 2008 (UTC)</a></small><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-129.105.207.213-2008-05-02T04:12:00.000Z-Vibrating_Disc_doesn't_look_right"></span><span data-mw-comment-end="c-129.105.207.213-2008-05-02T04:12:00.000Z-Vibrating_Disc_doesn't_look_right"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Too_technical" data-mw-thread-id="h-Too_technical-2007-12-19T16:00:00.000Z"><span data-mw-comment-start="" id="h-Too_technical-2007-12-19T16:00:00.000Z"></span>Too technical<span data-mw-comment-end="h-Too_technical-2007-12-19T16:00:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-192.35.17.15-2007-12-19T16:00:00.000Z-Too_technical"></span>All this is too much math for a common user trying to understand or get a clue of what Fourier did. I know, we are not all math people, but more instroduction is necesary nad then you can start with formulas. <a href="/wiki/Special:Contributions/192.35.17.15" title="Special:Contributions/192.35.17.15">192.35.17.15</a> (<a href="/wiki/User_talk:192.35.17.15" title="User talk:192.35.17.15">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-192.35.17.15-2007-12-19T16:00:00.000Z-Too_technical" class="ext-discussiontools-init-timestamplink">16:00, 19 December 2007 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-192.35.17.15-2007-12-19T16:00:00.000Z-Too_technical"></span><span data-mw-comment-end="c-192.35.17.15-2007-12-19T16:00:00.000Z-Too_technical"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Bob_K-2007-12-20T22:47:00.000Z-192.35.17.15-2007-12-19T16:00:00.000Z"></span>The article as a whole is disjointed and probably too detailed, but the <b>intro</b> looks OK to me. If you don't understand it, the internal links, like <a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a>, provide background material which you might need. If you can't understand the background material either, you probably aren't in the target group for an article like this one. If you do understand the article, but think you can do better, you are free to show us the way.</dd> <dd>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2007-12-20T22:47:00.000Z-192.35.17.15-2007-12-19T16:00:00.000Z" class="ext-discussiontools-init-timestamplink">22:47, 20 December 2007 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2007-12-20T22:47:00.000Z-192.35.17.15-2007-12-19T16:00:00.000Z"></span><span data-mw-comment-end="c-Bob_K-2007-12-20T22:47:00.000Z-192.35.17.15-2007-12-19T16:00:00.000Z"></span></dd></dl> <p><span data-mw-comment-start="" id="c-Michael_Hardy-2007-12-21T01:09:00.000Z-Too_technical"></span>I can imagine a more layperson-oriented intro, but it would certainly take some work. <a href="/wiki/User:Michael_Hardy" title="User:Michael Hardy">Michael Hardy</a> (<a href="/wiki/User_talk:Michael_Hardy" title="User talk:Michael Hardy">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Michael_Hardy-2007-12-21T01:09:00.000Z-Too_technical" class="ext-discussiontools-init-timestamplink">01:09, 21 December 2007 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Michael_Hardy-2007-12-21T01:09:00.000Z-Too_technical"></span><span data-mw-comment-end="c-Michael_Hardy-2007-12-21T01:09:00.000Z-Too_technical"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Loisel-2008-01-08T20:32:00.000Z-Michael_Hardy-2007-12-21T01:09:00.000Z"></span>I took a stab at it. <a href="/wiki/User:Loisel" title="User:Loisel">Loisel</a> (<a href="/wiki/User_talk:Loisel" title="User talk:Loisel">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Loisel-2008-01-08T20:32:00.000Z-Michael_Hardy-2007-12-21T01:09:00.000Z" class="ext-discussiontools-init-timestamplink">20:32, 8 January 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Loisel-2008-01-08T20:32:00.000Z-Michael_Hardy-2007-12-21T01:09:00.000Z"></span><span data-mw-comment-end="c-Loisel-2008-01-08T20:32:00.000Z-Michael_Hardy-2007-12-21T01:09:00.000Z"></span></dd></dl> <dl><dd><dl><dd><span data-mw-comment-start="" id="c-137.222.187.157-2008-01-09T12:25:00.000Z-Loisel-2008-01-08T20:32:00.000Z"></span>Putting a lay person's introduction at the start of the article is fine. However you should not simplify the article <i>at the expense of the technical user</i>. Loisel: I think the edits you have made do just this. In particular the choice of a particular period and a particular integration regime instead of the more general <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 [t_{0},t_{0}+T]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [t_{0},t_{0}+T]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42861c0e088bf0086b4c7ca5bbd5c05f5b625955" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.592ex; height:2.843ex;" alt="{\displaystyle [t_{0},t_{0}+T]}"/></span> could (a) leave more technical users who don't know much about Fourier series with the impression that the integration region isn't movable by an arbitrary offset, (b) annoy those technical users who wanted the formula in the more general form and will now have to convert it themselves (which if they are intending to actually use the formulas will likely be quite a large number of such users, and (c) confuse intermediate users who wish to make the Fourier series of a function with a period of other than <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 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }"/></span> but don't understand the details well enough to change the given formula.</dd></dl></dd></dl> <dl><dd><dl><dd>If you really feel there is a class of user out there who will be unable to understand the more general form of the equations and yet will want more than a lay person's non-mathematical introduction to the topic, then maybe make a 'simple case' section where you give them in the form you have used in your edit.</dd></dl></dd></dl> <dl><dd><dl><dd>Another criticism (sorry!) is that with respect to the lay person's introduction, I would have though that a simple modern description of the technique ranks higher than a historical account of its development --- so I'm not convinced about moving the 'Historical Development' section to the top of the article.</dd></dl></dd></dl> <dl><dd><dl><dd>But really my largest gripe is just the removal of the more general form; as a technical user of Wikipedia I personally love the fact that you can get gritty detailed descriptions of mathematical techniques that you wouldn't find in other encyclopedias.</dd></dl></dd></dl> <dl><dd><dl><dd><a href="/wiki/Special:Contributions/137.222.187.157" title="Special:Contributions/137.222.187.157">137.222.187.157</a> (<a href="/w/index.php?title=User_talk:137.222.187.157&action=edit&redlink=1" class="new" title="User talk:137.222.187.157 (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-137.222.187.157-2008-01-09T12:25:00.000Z-Loisel-2008-01-08T20:32:00.000Z" class="ext-discussiontools-init-timestamplink">12:25, 9 January 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-137.222.187.157-2008-01-09T12:25:00.000Z-Loisel-2008-01-08T20:32:00.000Z"></span><span data-mw-comment-end="c-137.222.187.157-2008-01-09T12:25:00.000Z-Loisel-2008-01-08T20:32:00.000Z"></span></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd><span data-mw-comment-start="" id="c-Loisel-2008-01-09T22:29:00.000Z-137.222.187.157-2008-01-09T12:25:00.000Z"></span>Dear Anonymous,</dd></dl></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd>You are very interested in the T-periodic case. I have made some notes to that effect in the T-periodic section, giving explicit formulae. However, please note that the T-periodic cse is not "the general" case. The general case is the <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> orthonormal basis approach. However, without going to an abstract Hilbert space, there are interesting Fourier series which are not on intervals, see <a href="/wiki/Spherical_harmonic" class="mw-redirect" title="Spherical harmonic">spherical harmonic</a>.</dd></dl></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd>Sincerely,</dd></dl></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd><a href="/wiki/User:Loisel" title="User:Loisel">Loisel</a> (<a href="/wiki/User_talk:Loisel" title="User talk:Loisel">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Loisel-2008-01-09T22:29:00.000Z-137.222.187.157-2008-01-09T12:25:00.000Z" class="ext-discussiontools-init-timestamplink">22:29, 9 January 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Loisel-2008-01-09T22:29:00.000Z-137.222.187.157-2008-01-09T12:25:00.000Z"></span><span data-mw-comment-end="c-Loisel-2008-01-09T22:29:00.000Z-137.222.187.157-2008-01-09T12:25:00.000Z"></span></dd></dl></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd><span data-mw-comment-start="" id="c-97.123.87.61-2009-10-17T18:16:00.000Z-137.222.187.157-2008-01-09T12:25:00.000Z"></span>I agree, the 2pi period constraint threw me off quite a bit. It's the <i>reason</i> for the equations, the explanation of what's being done, that's missing. Simplifying shouldn't even be necessary, except to show what happens when you plug in certain circumstances. In other words, it's not the math that's wrong, it's the article. <a href="/wiki/Special:Contributions/97.123.87.61" title="Special:Contributions/97.123.87.61">97.123.87.61</a> (<a href="/w/index.php?title=User_talk:97.123.87.61&action=edit&redlink=1" class="new" title="User talk:97.123.87.61 (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-97.123.87.61-2009-10-17T18:16:00.000Z-137.222.187.157-2008-01-09T12:25:00.000Z" class="ext-discussiontools-init-timestamplink">18:16, 17 October 2009 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-97.123.87.61-2009-10-17T18:16:00.000Z-137.222.187.157-2008-01-09T12:25:00.000Z"></span><span data-mw-comment-end="c-97.123.87.61-2009-10-17T18:16:00.000Z-137.222.187.157-2008-01-09T12:25:00.000Z"></span></dd></dl></dd></dl></dd></dl> <p><span data-mw-comment-start="" id="c-PrBeacon-2010-10-06T07:46:00.000Z-Too_technical"></span>In regards to a comment above, awhile back: <i>you should not simplify the article at the expense of the technical user</i> ... I understand your general points about article quality, but an encyclopedia can't be all things to everyone. (There's an old chestnut about "Try to make everyone happy, you make no one happy.") Besides, the more technical readers can always skim/skip the intro. Wikilinks are fine for further detail or branching topics, but an article should stand on its own. I think the intro reads well at the moment but could use some tweaking. For instance, what general field of math does this fall under? I haven't studied diffy-q's since college, but I always thought this was the domain of <b>calculus</b> -- yet that term does not appear in the article. -<small><a href="/wiki/User:PrBeacon" title="User:PrBeacon">PrBeacon</a> <a href="/wiki/User_talk:PrBeacon" title="User talk:PrBeacon">(talk)</a></small> <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-PrBeacon-2010-10-06T07:46:00.000Z-Too_technical" class="ext-discussiontools-init-timestamplink">07:46, 6 October 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-PrBeacon-2010-10-06T07:46:00.000Z-Too_technical"></span><span data-mw-comment-end="c-PrBeacon-2010-10-06T07:46:00.000Z-Too_technical"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-PrBeacon-2010-10-06T07:56:00.000Z-PrBeacon-2010-10-06T07:46:00.000Z"></span>Would it be appropriate to add CAT:Multivariable Calculus, Differential equations and/or Partial differential equations? -<small><a href="/wiki/User:PrBeacon" title="User:PrBeacon">PrBeacon</a> <a href="/wiki/User_talk:PrBeacon" title="User talk:PrBeacon">(talk)</a></small> <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-PrBeacon-2010-10-06T07:56:00.000Z-PrBeacon-2010-10-06T07:46:00.000Z" class="ext-discussiontools-init-timestamplink">07:56, 6 October 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-PrBeacon-2010-10-06T07:56:00.000Z-PrBeacon-2010-10-06T07:46:00.000Z"></span><span data-mw-comment-end="c-PrBeacon-2010-10-06T07:56:00.000Z-PrBeacon-2010-10-06T07:46:00.000Z"></span> <dl><dd><span data-mw-comment-start="" id="c-LutzL-2010-10-06T13:33:00.000Z-PrBeacon-2010-10-06T07:56:00.000Z"></span>No, No and No. Fourier series are in one variable, ODEs and PDEs are analyzed using the Lagrange tranform or the Fourier transform, not the series. There is a connection from the series to Sturm-Liouville, but that does not deserve a category.--<a href="/wiki/User:LutzL" title="User:LutzL">LutzL</a> (<a href="/wiki/User_talk:LutzL" title="User talk:LutzL">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-LutzL-2010-10-06T13:33:00.000Z-PrBeacon-2010-10-06T07:56:00.000Z" class="ext-discussiontools-init-timestamplink">13:33, 6 October 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-LutzL-2010-10-06T13:33:00.000Z-PrBeacon-2010-10-06T07:56:00.000Z"></span><span data-mw-comment-end="c-LutzL-2010-10-06T13:33:00.000Z-PrBeacon-2010-10-06T07:56:00.000Z"></span> <dl><dd><span data-mw-comment-start="" id="c-PrBeacon-2010-10-07T01:03:00.000Z-LutzL-2010-10-06T13:33:00.000Z"></span>Ok, what branch of mathematics would you suggest? -<small><a href="/wiki/User:PrBeacon" title="User:PrBeacon">PrBeacon</a> <a href="/wiki/User_talk:PrBeacon" title="User talk:PrBeacon">(talk)</a></small> <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-PrBeacon-2010-10-07T01:03:00.000Z-LutzL-2010-10-06T13:33:00.000Z" class="ext-discussiontools-init-timestamplink">01:03, 7 October 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-PrBeacon-2010-10-07T01:03:00.000Z-LutzL-2010-10-06T13:33:00.000Z"></span><span data-mw-comment-end="c-PrBeacon-2010-10-07T01:03:00.000Z-LutzL-2010-10-06T13:33:00.000Z"></span></dd></dl></dd></dl></dd></dl> <p><span data-mw-comment-start="" id="c-72.225.204.182-2010-12-09T06:55:00.000Z-Too_technical"></span>A great many of the scientific articles in Wikipedia seem to be addressed primarily to fellow specialists, or at any rate students already well-versed in the subject matter. That's great for professionals, but why cannot the articles of an enyclopedia for the general public unpack the technical terms as it goes along? I don't see why such unpacking would have to come at the expense of technical precision or comprehensiveness. <span style="font-size: smaller;" class="autosigned">—Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/Special:Contributions/72.225.204.182" title="Special:Contributions/72.225.204.182">72.225.204.182</a> (<a href="/w/index.php?title=User_talk:72.225.204.182&action=edit&redlink=1" class="new" title="User talk:72.225.204.182 (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-72.225.204.182-2010-12-09T06:55:00.000Z-Too_technical" class="ext-discussiontools-init-timestamplink">06:55, 9 December 2010 (UTC)</a></span><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-72.225.204.182-2010-12-09T06:55:00.000Z-Too_technical"></span><span data-mw-comment-end="c-72.225.204.182-2010-12-09T06:55:00.000Z-Too_technical"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Dicklyon-2010-12-09T07:50:00.000Z-72.225.204.182-2010-12-09T06:55:00.000Z"></span>I think that's what wiki-links are for. If you find terms that you don't understand, and they're not linked to an article that explains them, let us know here. <a href="/wiki/User:Dicklyon" title="User:Dicklyon">Dicklyon</a> (<a href="/wiki/User_talk:Dicklyon" title="User talk:Dicklyon">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Dicklyon-2010-12-09T07:50:00.000Z-72.225.204.182-2010-12-09T06:55:00.000Z" class="ext-discussiontools-init-timestamplink">07:50, 9 December 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Dicklyon-2010-12-09T07:50:00.000Z-72.225.204.182-2010-12-09T06:55:00.000Z"></span><span data-mw-comment-end="c-Dicklyon-2010-12-09T07:50:00.000Z-72.225.204.182-2010-12-09T06:55:00.000Z"></span></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Very_first_example" data-mw-thread-id="h-Very_first_example-2010-05-24T04:14:00.000Z"><span data-mw-comment-start="" id="h-Very_first_example-2010-05-24T04:14:00.000Z"></span>Very first example<span data-mw-comment-end="h-Very_first_example-2010-05-24T04:14:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-70.119.103.136-2010-05-24T04:14:00.000Z-Very_first_example"></span>There will be no <i>π</i> in the denominator in the final result since it's canceled out.<br/> look at <a rel="nofollow" class="external free" href="https://planetmath.org/?op=getobj&from=objects&id=4718">http://planetmath.org/?op=getobj&from=objects&id=4718</a> Thanks. <span style="font-size: smaller;" class="autosigned">—Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/Special:Contributions/70.119.103.136" title="Special:Contributions/70.119.103.136">70.119.103.136</a> (<a href="/w/index.php?title=User_talk:70.119.103.136&action=edit&redlink=1" class="new" title="User talk:70.119.103.136 (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-70.119.103.136-2010-05-24T04:14:00.000Z-Very_first_example" class="ext-discussiontools-init-timestamplink">04:14, 24 May 2010 (UTC)</a></span><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-70.119.103.136-2010-05-24T04:14:00.000Z-Very_first_example"></span><span data-mw-comment-end="c-70.119.103.136-2010-05-24T04:14:00.000Z-Very_first_example"></span> </p><p><span data-mw-comment-start="" id="c-128.95.41.29-2010-05-26T02:51:00.000Z-Very_first_example"></span>NOTE now fixed; this error was introduced by a very recent, incorrect edit. <a href="/wiki/Special:Contributions/128.95.41.29" title="Special:Contributions/128.95.41.29">128.95.41.29</a> (<a href="/w/index.php?title=User_talk:128.95.41.29&action=edit&redlink=1" class="new" title="User talk:128.95.41.29 (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-128.95.41.29-2010-05-26T02:51:00.000Z-Very_first_example" class="ext-discussiontools-init-timestamplink">02:51, 26 May 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-128.95.41.29-2010-05-26T02:51:00.000Z-Very_first_example"></span><span data-mw-comment-end="c-128.95.41.29-2010-05-26T02:51:00.000Z-Very_first_example"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Quote_isn't_actually_Fourier's?" data-mw-thread-id="h-Quote_isn't_actually_Fourier's?-2010-06-09T15:56:00.000Z"><span id="Quote_isn.27t_actually_Fourier.27s.3F"></span><span data-mw-comment-start="" id="h-Quote_isn't_actually_Fourier's?-2010-06-09T15:56:00.000Z"></span>Quote isn't actually Fourier's?<span data-mw-comment-end="h-Quote_isn't_actually_Fourier's?-2010-06-09T15:56:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Tcnuk-2010-06-09T15:56:00.000Z-Quote_isn't_actually_Fourier's?"></span>We have a quote under the heading Revolutionary Article attributed to Fourier. However, if you check up the reference with the link provided, you'll see that the words aren't actually his. Despite the article being listed as "par M. Fourier", it's written in the third person with constant references to "M. Fourier" did this/that etc. On the first page, there is a footnote: </p><p><i>Cet Article, que nous avons déjà signalé dans l'Avant-Propos du Tome I, n'est pas de Fourier. Signé de l'initialo P, il a été écrit par Poisson, qui était un des rédacteurs du Bulletin des Sciences pour la partie mathématique. A raison de l'intérêt historique qu'il présente comme étant le premier écrit où l'on ait fait connaître la théorie de Fourier, nous avons eru devoir le reproduire intégralement.</i> </p><p>In my own loose, and possibly mildly inaccurate translation: </p><p><i>This article, as we have already indicated in the "Avant-Propos" of Book 1, is not by Fourier. Signed with the initial P, it has been written by Poisson, who was one of the editors of the Bulletin des Sciences for the mathematical part. For reason of historical interest, it is presented as though it were the first writing where the theory of Fourier was made known, [we have had to reproduce it in full?]</i> (is "eru" supposed to be "eu"?) </p><p>Not quite sure what to make of this - sounds like a rather bizarre practice, but I believe that the original memoire may have been lost. Anybody got any ideas on how this should be correctly attributed? <small><span class="autosigned">—Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/User:Tcnuk" title="User:Tcnuk">Tcnuk</a> (<a href="/wiki/User_talk:Tcnuk" title="User talk:Tcnuk">talk</a> • <a href="/wiki/Special:Contributions/Tcnuk" title="Special:Contributions/Tcnuk">contribs</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Tcnuk-2010-06-09T15:56:00.000Z-Quote_isn't_actually_Fourier's?" class="ext-discussiontools-init-timestamplink">15:56, 9 June 2010 (UTC)</a></span></small><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tcnuk-2010-06-09T15:56:00.000Z-Quote_isn't_actually_Fourier's?"></span><span data-mw-comment-end="c-Tcnuk-2010-06-09T15:56:00.000Z-Quote_isn't_actually_Fourier's?"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Tcnuk-2010-06-10T12:03:00.000Z-Tcnuk-2010-06-09T15:56:00.000Z"></span>I've put a footnote to this effect into the article. If anybody can shine any light on this mystery, I'd be delighted to hear from them... <a href="/wiki/User:Tcnuk" title="User:Tcnuk">Tcnuk</a> (<a href="/wiki/User_talk:Tcnuk" title="User talk:Tcnuk">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Tcnuk-2010-06-10T12:03:00.000Z-Tcnuk-2010-06-09T15:56:00.000Z" class="ext-discussiontools-init-timestamplink">12:03, 10 June 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tcnuk-2010-06-10T12:03:00.000Z-Tcnuk-2010-06-09T15:56:00.000Z"></span><span data-mw-comment-end="c-Tcnuk-2010-06-10T12:03:00.000Z-Tcnuk-2010-06-09T15:56:00.000Z"></span></dd></dl> <dl><dd><span data-mw-comment-start="" id="c-Tamfang-2010-12-09T16:22:00.000Z-Tcnuk-2010-06-09T15:56:00.000Z"></span>The last sentence: <i>Because of the historic interest <b>that it presents</b> as being the first ....</i> —<a href="/wiki/User:Tamfang" title="User:Tamfang">Tamfang</a> (<a href="/wiki/User_talk:Tamfang" title="User talk:Tamfang">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Tamfang-2010-12-09T16:22:00.000Z-Tcnuk-2010-06-09T15:56:00.000Z" class="ext-discussiontools-init-timestamplink">16:22, 9 December 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tamfang-2010-12-09T16:22:00.000Z-Tcnuk-2010-06-09T15:56:00.000Z"></span><span data-mw-comment-end="c-Tamfang-2010-12-09T16:22:00.000Z-Tcnuk-2010-06-09T15:56:00.000Z"></span></dd></dl> <dl><dd><span data-mw-comment-start="" id="c-LutzL-2010-12-09T19:20:00.000Z-Tcnuk-2010-06-09T15:56:00.000Z"></span>Not "eru, not "eu", it's <i>"nous avons cru devoir le reproduire intégralement"</i>, <i>"we believed it to be our duty to reproduce it in full"</i>.--<a href="/wiki/User:LutzL" title="User:LutzL">LutzL</a> (<a href="/wiki/User_talk:LutzL" title="User talk:LutzL">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-LutzL-2010-12-09T19:20:00.000Z-Tcnuk-2010-06-09T15:56:00.000Z" class="ext-discussiontools-init-timestamplink">19:20, 9 December 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-LutzL-2010-12-09T19:20:00.000Z-Tcnuk-2010-06-09T15:56:00.000Z"></span><span data-mw-comment-end="c-LutzL-2010-12-09T19:20:00.000Z-Tcnuk-2010-06-09T15:56:00.000Z"></span></dd></dl> <dl><dd><span data-mw-comment-start="" id="c-LutzL-2010-12-09T19:31:00.000Z-Tcnuk-2010-06-09T15:56:00.000Z"></span>And the translation of the last sentence is slightly wrong: "For reasons of historical interest, as it appears to be the first text to publicate the theory of Fourier, we deemed it indispensable to publish it in full."--<a href="/wiki/User:LutzL" title="User:LutzL">LutzL</a> (<a href="/wiki/User_talk:LutzL" title="User talk:LutzL">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-LutzL-2010-12-09T19:31:00.000Z-Tcnuk-2010-06-09T15:56:00.000Z" class="ext-discussiontools-init-timestamplink">19:31, 9 December 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-LutzL-2010-12-09T19:31:00.000Z-Tcnuk-2010-06-09T15:56:00.000Z"></span><span data-mw-comment-end="c-LutzL-2010-12-09T19:31:00.000Z-Tcnuk-2010-06-09T15:56:00.000Z"></span> <dl><dd><span data-mw-comment-start="" id="c-LutzL-2010-12-09T20:26:00.000Z-LutzL-2010-12-09T19:31:00.000Z"></span>And perhaps it should be "in its original form" or similar for "integralement".--<a href="/wiki/User:LutzL" title="User:LutzL">LutzL</a> (<a href="/wiki/User_talk:LutzL" title="User talk:LutzL">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-LutzL-2010-12-09T20:26:00.000Z-LutzL-2010-12-09T19:31:00.000Z" class="ext-discussiontools-init-timestamplink">20:26, 9 December 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-LutzL-2010-12-09T20:26:00.000Z-LutzL-2010-12-09T19:31:00.000Z"></span><span data-mw-comment-end="c-LutzL-2010-12-09T20:26:00.000Z-LutzL-2010-12-09T19:31:00.000Z"></span></dd></dl></dd> <dd><span data-mw-comment-start="" id="c-LutzL-2011-01-14T21:18:00.000Z-Tcnuk-2010-06-09T15:56:00.000Z"></span>On decembre 21st 1807, Fourier presented his current work in something like a seminar at the "Academie des Sciences". The base of this presentation was a "memoire", perhaps hand-written, that was probably given to some mathematicians, but not published. Obviously, Poisson found it interesting enough to publish his notes of this memoire in 1808 on 4 pages in the "Nouveau Bulletin des Sciences, par la Societe Philomathique", <a href="<a rel="nofollow" class="external free" href="http://www.archive.org/stream/nouveaubulletind11807soci#page/112/mode/2up">http://www.archive.org/stream/nouveaubulletind11807soci#page/112/mode/2up</a>"> thanks to google available online</a>. This was reprinted as the earliest published occurence of the computation of (Fourier)trigonometric series coefficients by integrals and the notorious claim that the series converges to the given function. The computation of coefficients of trigonometric polynomials by something like the discrete Fourier transform was known some decades earlier.--<a href="/wiki/User:LutzL" title="User:LutzL">LutzL</a> (<a href="/wiki/User_talk:LutzL" title="User talk:LutzL">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-LutzL-2011-01-14T21:18:00.000Z-Tcnuk-2010-06-09T15:56:00.000Z" class="ext-discussiontools-init-timestamplink">21:18, 14 January 2011 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-LutzL-2011-01-14T21:18:00.000Z-Tcnuk-2010-06-09T15:56:00.000Z"></span><span data-mw-comment-end="c-LutzL-2011-01-14T21:18:00.000Z-Tcnuk-2010-06-09T15:56:00.000Z"></span></dd></dl> <p>Poisson's 1808 publication of a 4-page brief summary and comment on Fourier's memoir/presentation to the Institut doesn't qualify as a first publication of the memoir, although it did help protect Fourier from anyone who might have wished to publish before Fourier was able to, and thereby compete for priority. Although Fourier included a version of his earlier 1807 and 1811 work in his 1922 book, the first real publication of the 1807 memoir manuscript is given below, and yes, the 1807 manuscript did go missing for several decades. Fourier, Jean-Baptiste Joseph. "1811 mathematics prize paper." In Joseph Fourier, 1768-1830; a survey of his life and work, based on a critical edition of his monograph on the propagation of heat, presented to the Institut de France i 1807., by Ivor Grattan-Guinness, 30-440. Cambridge, MA: The MIT Press, 1972. Fourier did turn in a 234-page hand-written copy of his 1807 memoir to the Institute. However, Grattan-Guinness thinks that Fourier may have presented without the aid of the usual extract, since none survives <b>[For new information and discussion on this point, please see note below by AJR]</b>. To the previous contributor, the notoriousness of Fourier's claim faded as and when it became known that all practical data satisfies the requirements for convergence. Even Fourier himself, in using the words "arbitrary function" also said that the function had to be finite in extent, thereby presaging Dirichlet's conditions. In effect Fourier was saying that the function could be arbitrary, as long as it was integrable. Fourier's daring gave mathematics the necessary kick in the pants, but it took over 100 years for the bruise to fully go away. ;-) </p><p><br/> <b>AJR:</b> John Herivel's book, <i>Joseph Fourier: The man and the physicist</i>, published in 1975, in other words 3 years after I. Grattan-Guinness' landmark publication of Fourier's 1807 memoir, has this: </p><p><b>J Herivel, Page 318: Chapt XXI Fourier to an unknown correspondent, around 1808-9</b> </p><p>"I have the honour to send you two notes concerning the memoir on heat. The first(Note 1) is the one that was read at the Institut in place of the reading of the memoir... [presumably due to the excessive length of the actual memoir, comprising 234 sides!, in other words Fourier is referring to the extract. Therefore Fourier *did* read an extract, rather than presenting without using one]. </p><p><b>J Herivel, Page 320:</b> Note 1. This was the abstract (extrait) of Fourier's 1807 memoir which has been retained in the MS. 1851 of the library of the Nationale École des Ponts et Chaussées, Paris. It must therefore be dated 1807, and not 1809 as suggested by Grattan-Guiness (3), p. 497. </p><p>When I looked in Grattan-Guinness, I immediately saw that this would not be a simple correction, since discussion of the extrait in q uestion appears in multiple locations, and is quite embedded in the discussion. For example, see </p><p><b>Grattan-Guinness,</b> page 24 (lines 14-18, 22-24); page 26 (lines 6 and 24). </p><p><b>Grattan-Guinness, Page xii Abbreviations of Titles of Key Works by Fourier</b> </p><p>4. "Extrait," for the paper sent by Fourier to the Institut de France in October 1808. [I had long wondered about this, since as far as I knew there was only the 1807 extract -AJR] </p><p>5. "Notes," for the set of footnotes to the Extrait. </p><p>I have not seen a correction published by Ivor Grattan-Guinness, nor any acknowledgement of the problem. I also have yet to find any further discussion by John Herivel. One of them must be more correct than the other in this matter. </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Formula_incorrect" data-mw-thread-id="h-Formula_incorrect-2011-01-13T16:09:00.000Z"><span data-mw-comment-start="" id="h-Formula_incorrect-2011-01-13T16:09:00.000Z"></span>Formula incorrect<span data-mw-comment-end="h-Formula_incorrect-2011-01-13T16:09:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-99.52.193.237-2011-01-13T16:09:00.000Z-Formula_incorrect"></span>I believe the formula giving the conversion of coefficients into exponential form is incorrect (and the equivalent problem exists in the other direction as well). it suggests that c<sub>0</sub> = a<sub>0</sub>/2. This should be c<sub>0</sub> = a<sub>0</sub>. since a<sub>0</sub> cos(0x) + b<sub>0</sub> sin(0x) = a<sub>0</sub>, and c<sub>0</sub> e<sup>i0x</sup> = c<sub>0</sub>. </p><p><a href="/wiki/Special:Contributions/99.52.193.237" title="Special:Contributions/99.52.193.237">99.52.193.237</a> (<a href="/w/index.php?title=User_talk:99.52.193.237&action=edit&redlink=1" class="new" title="User talk:99.52.193.237 (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-99.52.193.237-2011-01-13T16:09:00.000Z-Formula_incorrect" class="ext-discussiontools-init-timestamplink">16:09, 13 January 2011 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-99.52.193.237-2011-01-13T16:09:00.000Z-Formula_incorrect"></span><span data-mw-comment-end="c-99.52.193.237-2011-01-13T16:09:00.000Z-Formula_incorrect"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Bob_K-2011-01-14T18:39:00.000Z-99.52.193.237-2011-01-13T16:09:00.000Z"></span>You appear to be assuming something like a<sub>n</sub> cos(nx) + b<sub>n</sub> sin(nx) = c<sub>n</sub> e<sup>inx</sup>. Try proving it.</dd> <dd>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2011-01-14T18:39:00.000Z-99.52.193.237-2011-01-13T16:09:00.000Z" class="ext-discussiontools-init-timestamplink">18:39, 14 January 2011 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2011-01-14T18:39:00.000Z-99.52.193.237-2011-01-13T16:09:00.000Z"></span><span data-mw-comment-end="c-Bob_K-2011-01-14T18:39:00.000Z-99.52.193.237-2011-01-13T16:09:00.000Z"></span></dd></dl> <p><span data-mw-comment-start="" id="c-99.52.193.237-2011-01-16T17:04:00.000Z-Formula_incorrect"></span>What I'm suggesting (in more detail) is that a<sub>0</sub> cos(0x) + b<sub>0</sub> sin(0x) = c<sub>0</sub> cos(0x) + c<sub>0</sub> i sin(0x), by Euler's formula. The sine terms drop out so: a<sub>0</sub> cos(0x) = c<sub>0</sub> cos(0x) and c<sub>0</sub> = a<sub>0</sub>. <span style="font-size: smaller;" class="autosigned">—Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/Special:Contributions/99.52.193.237" title="Special:Contributions/99.52.193.237">99.52.193.237</a> (<a href="/w/index.php?title=User_talk:99.52.193.237&action=edit&redlink=1" class="new" title="User talk:99.52.193.237 (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-99.52.193.237-2011-01-16T17:04:00.000Z-Formula_incorrect" class="ext-discussiontools-init-timestamplink">17:04, 16 January 2011 (UTC)</a></span><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-99.52.193.237-2011-01-16T17:04:00.000Z-Formula_incorrect"></span><span data-mw-comment-end="c-99.52.193.237-2011-01-16T17:04:00.000Z-Formula_incorrect"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-LutzL-2011-01-16T17:12:00.000Z-99.52.193.237-2011-01-16T17:04:00.000Z"></span>And that is totally correct, except that the constant term of the real form of the series is given as <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 {a_{0}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a_{0}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e9329744d373e38354b7a13001fed84c56bd28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.12ex; height:4.676ex;" alt="{\displaystyle {\frac {a_{0}}{2}}}"/></span>. This is done to have a unified integral formula for the real coefficients. And to be consistent with the conversion in the opposite direction given above the discussed one. Write that down explicitely for the case n=0.--<a href="/wiki/User:LutzL" title="User:LutzL">LutzL</a> (<a href="/wiki/User_talk:LutzL" title="User talk:LutzL">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-LutzL-2011-01-16T17:12:00.000Z-99.52.193.237-2011-01-16T17:04:00.000Z" class="ext-discussiontools-init-timestamplink">17:12, 16 January 2011 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-LutzL-2011-01-16T17:12:00.000Z-99.52.193.237-2011-01-16T17:04:00.000Z"></span><span data-mw-comment-end="c-LutzL-2011-01-16T17:12:00.000Z-99.52.193.237-2011-01-16T17:04:00.000Z"></span></dd></dl> <p><span data-mw-comment-start="" id="c-Brvman-2011-05-04T23:49:00.000Z-Formula_incorrect"></span>Might it be useful to include this formula: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(x)dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(x)dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7559046d13a66d0a3e2e7cdff050cd51acce2bae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.033ex; height:6.009ex;" alt="{\displaystyle a_{0}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(x)dx}"/></span> in the article as the general form for a<sub>0</sub>? <small><span class="autosigned">—Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/User:Brvman" title="User:Brvman">Brvman</a> (<a href="/wiki/User_talk:Brvman" title="User talk:Brvman">talk</a> • <a href="/wiki/Special:Contributions/Brvman" title="Special:Contributions/Brvman">contribs</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Brvman-2011-05-04T23:49:00.000Z-Formula_incorrect" class="ext-discussiontools-init-timestamplink">23:49, 4 May 2011 (UTC)</a></span></small><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Brvman-2011-05-04T23:49:00.000Z-Formula_incorrect"></span><span data-mw-comment-end="c-Brvman-2011-05-04T23:49:00.000Z-Formula_incorrect"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Formula_and_graph_axis_labelling" data-mw-thread-id="h-Formula_and_graph_axis_labelling-2011-12-07T14:52:00.000Z"><span data-mw-comment-start="" id="h-Formula_and_graph_axis_labelling-2011-12-07T14:52:00.000Z"></span>Formula and graph axis labelling<span data-mw-comment-end="h-Formula_and_graph_axis_labelling-2011-12-07T14:52:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Cephas_Borg-2011-12-07T14:52:00.000Z-Formula_and_graph_axis_labelling"></span>Sorry to ask such a stupid question, but shouldn't the graph axes in the "Plot of a periodic identity function" associated with Example 1 be the same as the formula states? I.e. instead of x(t) and t (which appear nowhere in the formula or the coefficient calcs), shouldn't it be labelled f(x) and x? Or am I missing something obvious? (I'm no mathematician, as you can probably tell) <a href="/wiki/User:Cephas_Borg" title="User:Cephas Borg">Cephas Borg</a> (<a href="/wiki/User_talk:Cephas_Borg" title="User talk:Cephas Borg">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Cephas_Borg-2011-12-07T14:52:00.000Z-Formula_and_graph_axis_labelling" class="ext-discussiontools-init-timestamplink">14:52, 7 December 2011 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Cephas_Borg-2011-12-07T14:52:00.000Z-Formula_and_graph_axis_labelling"></span><span data-mw-comment-end="c-Cephas_Borg-2011-12-07T14:52:00.000Z-Formula_and_graph_axis_labelling"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-172.190.77.188-2013-02-10T02:11:00.000Z-Formula_and_graph_axis_labelling"></span>You're not missing anything. I'd guess the figure was originally created for a different purpose or different text... not an unusual occurrence at Wikipedia. It's not a perfect fit, but it's close enough that nobody has bothered to fix it. I don't think anyone would complain if you want to fix it yourself.</dd></dl> <p>- I would if I could but I do not know how to fix a fig. It is confusing enough that it should be fixed <a href="/wiki/Special:Contributions/172.190.77.188" title="Special:Contributions/172.190.77.188">172.190.77.188</a> (<a href="/w/index.php?title=User_talk:172.190.77.188&action=edit&redlink=1" class="new" title="User talk:172.190.77.188 (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-172.190.77.188-2013-02-10T02:11:00.000Z-Formula_and_graph_axis_labelling" class="ext-discussiontools-init-timestamplink">02:11, 10 February 2013 (UTC)</a> aburr@aol.com<span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-172.190.77.188-2013-02-10T02:11:00.000Z-Formula_and_graph_axis_labelling"></span><span data-mw-comment-end="c-172.190.77.188-2013-02-10T02:11:00.000Z-Formula_and_graph_axis_labelling"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Bob_K-2011-12-07T18:16:00.000Z-172.190.77.188-2013-02-10T02:11:00.000Z"></span>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2011-12-07T18:16:00.000Z-172.190.77.188-2013-02-10T02:11:00.000Z" class="ext-discussiontools-init-timestamplink">18:16, 7 December 2011 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2011-12-07T18:16:00.000Z-172.190.77.188-2013-02-10T02:11:00.000Z"></span><span data-mw-comment-end="c-Bob_K-2011-12-07T18:16:00.000Z-172.190.77.188-2013-02-10T02:11:00.000Z"></span></dd></dl> <p><span data-mw-comment-start="" id="c-Cephas_Borg-2011-12-08T02:49:00.000Z-Formula_and_graph_axis_labelling"></span>Thank Fourier it wasn't just me! I'm currently learning to modify the original source. Thanks for the encouragement, Bob. (This is my first real Wikipedia contribution, woo hoo! :). <a href="/wiki/User:Cephas_Borg" title="User:Cephas Borg">Cephas Borg</a> (<a href="/wiki/User_talk:Cephas_Borg" title="User talk:Cephas Borg">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Cephas_Borg-2011-12-08T02:49:00.000Z-Formula_and_graph_axis_labelling" class="ext-discussiontools-init-timestamplink">02:49, 8 December 2011 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Cephas_Borg-2011-12-08T02:49:00.000Z-Formula_and_graph_axis_labelling"></span><span data-mw-comment-end="c-Cephas_Borg-2011-12-08T02:49:00.000Z-Formula_and_graph_axis_labelling"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Bob_K-2011-12-09T01:20:00.000Z-Cephas_Borg-2011-12-08T02:49:00.000Z"></span>One thing I should have checked and mentioned is the section <i>File usage on other wikis</i> at <a class="external free" href="https://commons.wikimedia.org/wiki/File:Sawtooth_pi.svg">http://commons.wikimedia.org/wiki/File:Sawtooth_pi.svg</a>. Luckily, all it lists is this article. (Whew!) Thanks for your help, and good luck with your first "adventure".</dd> <dd>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2011-12-09T01:20:00.000Z-Cephas_Borg-2011-12-08T02:49:00.000Z" class="ext-discussiontools-init-timestamplink">01:20, 9 December 2011 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2011-12-09T01:20:00.000Z-Cephas_Borg-2011-12-08T02:49:00.000Z"></span><span data-mw-comment-end="c-Bob_K-2011-12-09T01:20:00.000Z-Cephas_Borg-2011-12-08T02:49:00.000Z"></span> <dl><dd><span data-mw-comment-start="" id="c-Olli_Niemitalo-2013-02-10T06:42:00.000Z-Bob_K-2011-12-09T01:20:00.000Z"></span>I changed the axis titles in <a href="/wiki/Inkscape" title="Inkscape">Inkscape</a> to <i>f</i>(<i>x</i>) and <i>x</i> which are OK for all Wikipedia articles where the image is used. <a href="/wiki/User:Olli_Niemitalo" title="User:Olli Niemitalo">Olli Niemitalo</a> (<a href="/wiki/User_talk:Olli_Niemitalo" title="User talk:Olli Niemitalo">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Olli_Niemitalo-2013-02-10T06:42:00.000Z-Bob_K-2011-12-09T01:20:00.000Z" class="ext-discussiontools-init-timestamplink">06:42, 10 February 2013 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Olli_Niemitalo-2013-02-10T06:42:00.000Z-Bob_K-2011-12-09T01:20:00.000Z"></span><span data-mw-comment-end="c-Olli_Niemitalo-2013-02-10T06:42:00.000Z-Bob_K-2011-12-09T01:20:00.000Z"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Request_for_comment_by_readers_of_this_article" data-mw-thread-id="h-Request_for_comment_by_readers_of_this_article-2012-04-21T17:07:00.000Z"><span data-mw-comment-start="" id="h-Request_for_comment_by_readers_of_this_article-2012-04-21T17:07:00.000Z"></span>Request for comment by readers of this article<span data-mw-comment-end="h-Request_for_comment_by_readers_of_this_article-2012-04-21T17:07:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Brews_ohare-2012-04-21T17:07:00.000Z-Request_for_comment_by_readers_of_this_article"></span>A <a href="/wiki/Talk:Wavelength#rfc_DDA12B5" title="Talk:Wavelength"> request for comment</a> regarding a reference to this article is open for discussion. Readers of this article are invited to contribute. <a href="/wiki/User:Brews_ohare" title="User:Brews ohare">Brews ohare</a> (<a href="/wiki/User_talk:Brews_ohare" title="User talk:Brews ohare">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Brews_ohare-2012-04-21T17:07:00.000Z-Request_for_comment_by_readers_of_this_article" class="ext-discussiontools-init-timestamplink">17:07, 21 April 2012 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Brews_ohare-2012-04-21T17:07:00.000Z-Request_for_comment_by_readers_of_this_article"></span><span data-mw-comment-end="c-Brews_ohare-2012-04-21T17:07:00.000Z-Request_for_comment_by_readers_of_this_article"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Riemann_paper_and_1811_essay_competition" data-mw-thread-id="h-Riemann_paper_and_1811_essay_competition-2012-11-10T04:10:00.000Z"><span data-mw-comment-start="" id="h-Riemann_paper_and_1811_essay_competition-2012-11-10T04:10:00.000Z"></span>Riemann paper and 1811 essay competition<span data-mw-comment-end="h-Riemann_paper_and_1811_essay_competition-2012-11-10T04:10:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-5.151.82.6-2012-11-10T04:10:00.000Z-Riemann_paper_and_1811_essay_competition"></span>I have an idea for a new reference for <a href="/wiki/Riemann" class="mw-redirect" title="Riemann">Riemann</a>. I think it might be this : Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe. (Part of the) <a href="/wiki/Habilitationsschrift" class="mw-redirect" title="Habilitationsschrift">Habilitationsschrift</a>, 1854. </p><p>Help from German speakers, please? Also what was the name of the prize competition that Fourier entered in 1811? <span style="font-size: smaller;" class="autosigned">— Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/Special:Contributions/5.151.82.6" title="Special:Contributions/5.151.82.6">5.151.82.6</a> (<a href="/wiki/User_talk:5.151.82.6" title="User talk:5.151.82.6">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-5.151.82.6-2012-11-10T04:10:00.000Z-Riemann_paper_and_1811_essay_competition" class="ext-discussiontools-init-timestamplink">04:10, 10 November 2012 (UTC)</a></span><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-5.151.82.6-2012-11-10T04:10:00.000Z-Riemann_paper_and_1811_essay_competition"></span><span data-mw-comment-end="c-5.151.82.6-2012-11-10T04:10:00.000Z-Riemann_paper_and_1811_essay_competition"></span> </p> <ul><li><span data-mw-comment-start="" id="c-5.151.82.6-2012-11-10T04:45:00.000Z-5.151.82.6-2012-11-10T04:10:00.000Z"></span>Here are two important potential reference sources that I think could help with the issues I raised. <a rel="nofollow" class="external autonumber" href="https://docs.google.com/viewer?a=v&q=cache:pw2WZGGnt-kJ:iws.collin.edu/ebock/Scientists/More%2520Scientists%2520page%25201/Fourier-bio.pdf+&hl=en&gl=uk&pid=bl&srcid=ADGEESifaRpB4KtSGVM547KJ_Lxg1_rM5149OoJmSFyTXN1gpFotYzfn9zJ-_bljaWMFT3Oa7CNP9sspAO1TXXIV7gmqLW9g83pJJzeTH8Q-WHo5e65CChnURtqoLXXfHf39CLy3kuEF&sig=AHIEtbQZk8kXogMOPEXH0LIa1JM7kp965g">[4]</a> and Connections in Mathematical Analysis: The Case of Fourier Series, Enrique A. Gonzalez-Velasco. American Mathematical Monthly, Volume 99, Issue 5 (May, 1992), 427-441. <a rel="nofollow" class="external autonumber" href="https://docs.google.com/viewer?a=v&q=cache:LA_45nkhf8gJ:www.math.ucdavis.edu/~saito/courses/121/gonzalez.pdf+&hl=en&gl=uk&pid=bl&srcid=ADGEESj35PLNarwPuicjnNrICrcCRkbkDz5Kc_J_h7-rA3Hjt9TRbtcs07b5-6piDSr6MH9S-UtCTeXpXffl0STjlPi6_Gz4UtePcgshJZUej4g3VwY6K5rn9r7rq58rPjUxIDqTHTZV&sig=AHIEtbQb-4m7TQ8HbT9slCoysSviYS04rQ">[5]</a> <span style="font-size: smaller;" class="autosigned">— Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/Special:Contributions/5.151.82.6" title="Special:Contributions/5.151.82.6">5.151.82.6</a> (<a href="/wiki/User_talk:5.151.82.6" title="User talk:5.151.82.6">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-5.151.82.6-2012-11-10T04:45:00.000Z-5.151.82.6-2012-11-10T04:10:00.000Z" class="ext-discussiontools-init-timestamplink">04:45, 10 November 2012 (UTC)</a><span data-mw-comment-end="c-5.151.82.6-2012-11-10T04:45:00.000Z-5.151.82.6-2012-11-10T04:10:00.000Z"></span></span><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-5.151.82.6-2012-11-10T04:45:00.000Z-5.151.82.6-2012-11-10T04:10:00.000Z"></span></li></ul> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Fourier's_theorem" data-mw-thread-id="h-Fourier's_theorem-2013-03-11T14:22:00.000Z"><span id="Fourier.27s_theorem"></span><span data-mw-comment-start="" id="h-Fourier's_theorem-2013-03-11T14:22:00.000Z"></span>Fourier's theorem<span data-mw-comment-end="h-Fourier's_theorem-2013-03-11T14:22:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Arbitrarily0-2013-03-11T14:22:00.000Z-Fourier's_theorem"></span>It has been said </p> <table class="cquote pullquote" role="presentation" style="margin:auto; border-collapse: collapse; border: none; width: auto;"> <tbody><tr> <td style="width: 20px; vertical-align: top; border: none; color: #B2B7F2; font-size: 40px; font-family: 'Times New Roman', Times, serif; font-weight: bold; line-height: .6em; text-align: left; padding: 10px 10px;">“ </td> <td style="vertical-align: top; border: none; padding: 4px 10px;">Fourier’s theorem is not only one of the most beautiful results of modern analysis, but it is said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics </td> <td style="width: 20px; vertical-align: bottom; border: none; color: #B2B7F2; font-size: 40px; font-family: 'Times New Roman', Times, serif; font-weight: bold; line-height: .6em; text-align: right; padding: 10px 10px;">” </td></tr> <tr> <td colspan="3" class="cquotecite" style="border: none; padding-right: 4%; font-size: smaller; line-height:normal; text-align: right;"><cite>— <a href="/wiki/Lord_Kelvin" title="Lord Kelvin">Lord Kelvin</a> <a rel="nofollow" class="external autonumber" href="http://www.nonoscience.info/fourier-series.html">[6]</a></cite> </td></tr></tbody></table> <p>but this article does explicitly mention "<a href="/w/index.php?title=Fourier%E2%80%99s_theorem&action=edit&redlink=1" class="new" title="Fourier’s theorem (page does not exist)">Fourier’s theorem</a>". Does it go by another name? Several dictionaries define it as the theorem that states that under suitable conditions any periodic function can be represented by a Fourier series. <a href="/wiki/User:Arbitrarily0" title="User:Arbitrarily0"><span style="color:black"><b><u><i><big>A</big>rbitrarily<big>0</big></i></u></b></span></a> <sup><b>(<a href="/wiki/User_talk:Arbitrarily0" title="User talk:Arbitrarily0"><span style="font-variant: small-caps; color:#FF4500;">talk</span></a>)</b></sup> <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Arbitrarily0-2013-03-11T14:22:00.000Z-Fourier's_theorem" class="ext-discussiontools-init-timestamplink">14:22, 11 March 2013 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Arbitrarily0-2013-03-11T14:22:00.000Z-Fourier's_theorem"></span><span data-mw-comment-end="c-Arbitrarily0-2013-03-11T14:22:00.000Z-Fourier's_theorem"></span> </p> <span data-mw-comment-start="" id="c-Bob_K-2013-03-12T21:00:00.000Z-Fourier's_theorem"></span><style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>:"'The Fourier theorem' consists not of one single theorem, but in several theorems all on a common theme."</p><div class="templatequotecite">— <cite>D.C.Champeney, <i><cite>A Handbook of Fourier Theorems</cite></i>, 1989, p 2, <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><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0521366887" title="Special:BookSources/0521366887">0521366887</a></cite></div></blockquote> <dl><dd><dl><dd><dl><dd>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2013-03-12T21:00:00.000Z-Fourier's_theorem" class="ext-discussiontools-init-timestamplink">21:00, 12 March 2013 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2013-03-12T21:00:00.000Z-Fourier's_theorem"></span><span data-mw-comment-end="c-Bob_K-2013-03-12T21:00:00.000Z-Fourier's_theorem"></span></dd></dl></dd></dl></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Fourier_series_on_a_general_interval_[a,_a_+_τ]" data-mw-thread-id="h-Fourier_series_on_a_general_interval_[a,_a_+_τ]-2013-08-11T20:40:00.000Z"><span id="Fourier_series_on_a_general_interval_.5Ba.2C_a_.2B_.CF.84.5D"></span><span data-mw-comment-start="" id="h-Fourier_series_on_a_general_interval_[a,_a_+_τ]-2013-08-11T20:40:00.000Z"></span>Fourier series on a general interval [a, a + τ]<span data-mw-comment-end="h-Fourier_series_on_a_general_interval_[a,_a_+_τ]-2013-08-11T20:40:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Bdmy-2013-08-11T20:40:00.000Z-Fourier_series_on_a_general_interval_[a,_a_+_τ]"></span>This section needs some attention. It introduces in vague terms (the words "square integrable" are mentioned) two functions <i>g</i> and <i>h</i>, and claims that </p> <dl><dd><dl><dd><ul><li><i>g</i>(<i>x</i>) and <i>h</i>(<i>x</i>) are equal everywhere, except possibly at discontinuities</li></ul></dd></dl></dd></dl> <p>This contrasts with the more careful treatment in the section "Fourier's formula for 2π-periodic functions". <a href="/wiki/User:Bdmy" title="User:Bdmy">Bdmy</a> (<a href="/wiki/User_talk:Bdmy" title="User talk:Bdmy">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bdmy-2013-08-11T20:40:00.000Z-Fourier_series_on_a_general_interval_[a,_a_+_τ]" class="ext-discussiontools-init-timestamplink">20:40, 11 August 2013 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bdmy-2013-08-11T20:40:00.000Z-Fourier_series_on_a_general_interval_[a,_a_+_τ]"></span><span data-mw-comment-end="c-Bdmy-2013-08-11T20:40:00.000Z-Fourier_series_on_a_general_interval_[a,_a_+_τ]"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Bob_K-2013-08-13T13:58:00.000Z-Bdmy-2013-08-11T20:40:00.000Z"></span>Rather than repeat a lot of the same verbiage, perhaps we should simply generalize the "2π-periodic functions" section. --<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2013-08-13T13:58:00.000Z-Bdmy-2013-08-11T20:40:00.000Z" class="ext-discussiontools-init-timestamplink">13:58, 13 August 2013 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2013-08-13T13:58:00.000Z-Bdmy-2013-08-11T20:40:00.000Z"></span><span data-mw-comment-end="c-Bob_K-2013-08-13T13:58:00.000Z-Bdmy-2013-08-11T20:40:00.000Z"></span></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Fourier's_formulae_for_T-periodic_functions" data-mw-thread-id="h-Fourier's_formulae_for_T-periodic_functions-2013-08-22T23:35:00.000Z"><span id="Fourier.27s_formulae_for_T-periodic_functions"></span><span data-mw-comment-start="" id="h-Fourier's_formulae_for_T-periodic_functions-2013-08-22T23:35:00.000Z"></span>Fourier's formulae for T-periodic functions<span data-mw-comment-end="h-Fourier's_formulae_for_T-periodic_functions-2013-08-22T23:35:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-A_Thousand_Doors-2013-08-22T23:35:00.000Z-Fourier's_formulae_for_T-periodic_functions"></span>Unless I'm missing something, I'm a little confused as to why the formulae for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"/></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 b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28e2d72f6dd9375c8f1f59f1effd9b4e5492ac97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.509ex;" alt="{\displaystyle b_{n}}"/></span> are given only for the specific case of 2π-periodic functions. Wouldn't it be better to state the more general T-periodic cases (e.g. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}={\frac {2}{T}}\int _{\tfrac {-T}{2}}^{\tfrac {T}{2}}f(x)\cos({\tfrac {2{\pi }nx}{T}})\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>T</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mo>−</mo> <mi>T</mi> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>T</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> <mi>n</mi> <mi>x</mi> </mrow> <mi>T</mi> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}={\frac {2}{T}}\int _{\tfrac {-T}{2}}^{\tfrac {T}{2}}f(x)\cos({\tfrac {2{\pi }nx}{T}})\,dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca1f0522acdbda1d24655f1bd32bf0618d38b871" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:30.774ex; height:8.343ex;" alt="{\displaystyle a_{n}={\frac {2}{T}}\int _{\tfrac {-T}{2}}^{\tfrac {T}{2}}f(x)\cos({\tfrac {2{\pi }nx}{T}})\,dx}"/></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}={\frac {2}{T}}\int _{0}^{T}f(x)\cos({\tfrac {2{\pi }nx}{T}})\,dx,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>T</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> <mi>n</mi> <mi>x</mi> </mrow> <mi>T</mi> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}={\frac {2}{T}}\int _{0}^{T}f(x)\cos({\tfrac {2{\pi }nx}{T}})\,dx,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97829dce78bd0670ef6de522c8a5b3bca6b11f1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.478ex; height:6.176ex;" alt="{\displaystyle a_{n}={\frac {2}{T}}\int _{0}^{T}f(x)\cos({\tfrac {2{\pi }nx}{T}})\,dx,}"/></span> from which the T=2π case would follow almost immediately?<br/> <a href="/wiki/User:A_Thousand_Doors" class="mw-redirect" title="User:A Thousand Doors">A Thousand Doors</a> (<a href="/wiki/User_talk:A_Thousand_Doors" title="User talk:A Thousand Doors">talk</a> | <a href="/wiki/Special:Contributions/A_Thousand_Doors" title="Special:Contributions/A Thousand Doors">contribs</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-A_Thousand_Doors-2013-08-22T23:35:00.000Z-Fourier's_formulae_for_T-periodic_functions" class="ext-discussiontools-init-timestamplink">23:35, 22 August 2013 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-A_Thousand_Doors-2013-08-22T23:35:00.000Z-Fourier's_formulae_for_T-periodic_functions"></span><span data-mw-comment-end="c-A_Thousand_Doors-2013-08-22T23:35:00.000Z-Fourier's_formulae_for_T-periodic_functions"></span> </p><p><br/> </p> <dl><dd><span data-mw-comment-start="" id="c-Bob_K-2013-08-23T03:39:00.000Z-A_Thousand_Doors-2013-08-22T23:35:00.000Z"></span>Yes, of course (in my humble opinion). But why stop there? Why not: <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 a_{n}={\frac {2}{T}}\int _{\alpha }^{\alpha +T}f(x)\cos({\tfrac {2{\pi }nx}{T}})\,dx,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>T</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>+</mo> <mi>T</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> <mi>n</mi> <mi>x</mi> </mrow> <mi>T</mi> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}={\frac {2}{T}}\int _{\alpha }^{\alpha +T}f(x)\cos({\tfrac {2{\pi }nx}{T}})\,dx,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42e43ed3aebee7077e53c14d0e8b4945dc1691aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.808ex; height:6.176ex;" alt="{\displaystyle a_{n}={\frac {2}{T}}\int _{\alpha }^{\alpha +T}f(x)\cos({\tfrac {2{\pi }nx}{T}})\,dx,}"/></span></dd></dl></dd> <dd>where α is any arbitrary number?</dd> <dd>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2013-08-23T03:39:00.000Z-A_Thousand_Doors-2013-08-22T23:35:00.000Z" class="ext-discussiontools-init-timestamplink">03:39, 23 August 2013 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2013-08-23T03:39:00.000Z-A_Thousand_Doors-2013-08-22T23:35:00.000Z"></span><span data-mw-comment-end="c-Bob_K-2013-08-23T03:39:00.000Z-A_Thousand_Doors-2013-08-22T23:35:00.000Z"></span></dd></dl> <dl><dd><dl><dd><span data-mw-comment-start="" id="c-A_Thousand_Doors-2013-08-23T10:42:00.000Z-Bob_K-2013-08-23T03:39:00.000Z"></span>Yep, I'd agree with that, it's probably a better idea. Happy editing, <a href="/wiki/User:A_Thousand_Doors" class="mw-redirect" title="User:A Thousand Doors">A Thousand Doors</a> (<a href="/wiki/User_talk:A_Thousand_Doors" title="User talk:A Thousand Doors">talk</a> | <a href="/wiki/Special:Contributions/A_Thousand_Doors" title="Special:Contributions/A Thousand Doors">contribs</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-A_Thousand_Doors-2013-08-23T10:42:00.000Z-Bob_K-2013-08-23T03:39:00.000Z" class="ext-discussiontools-init-timestamplink">10:42, 23 August 2013 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-A_Thousand_Doors-2013-08-23T10:42:00.000Z-Bob_K-2013-08-23T03:39:00.000Z"></span><span data-mw-comment-end="c-A_Thousand_Doors-2013-08-23T10:42:00.000Z-Bob_K-2013-08-23T03:39:00.000Z"></span></dd></dl></dd></dl> <dl><dd><span data-mw-comment-start="" id="c-Bob_K-2013-08-23T13:02:00.000Z-A_Thousand_Doors-2013-08-22T23:35:00.000Z"></span>Our answer (I assume) is that Fourier didn't do it that way. The "compromise" I settled for long ago is section <a href="/wiki/Fourier_series#Fourier_series_on_a_general_interval_.5Ba.2C.C2.A0a_.2B_.CF.84.29" title="Fourier series">Fourier series on a general interval [a, a + τ)</a>.</dd> <dd>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2013-08-23T13:02:00.000Z-A_Thousand_Doors-2013-08-22T23:35:00.000Z" class="ext-discussiontools-init-timestamplink">13:02, 23 August 2013 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2013-08-23T13:02:00.000Z-A_Thousand_Doors-2013-08-22T23:35:00.000Z"></span><span data-mw-comment-end="c-Bob_K-2013-08-23T13:02:00.000Z-A_Thousand_Doors-2013-08-22T23:35:00.000Z"></span></dd></dl> <dl><dd><dl><dd><span data-mw-comment-start="" id="c-A_Thousand_Doors-2013-08-27T11:08:00.000Z-Bob_K-2013-08-23T13:02:00.000Z"></span>It just seems to me that we're missing a step when we're going from defining <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"/></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 b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28e2d72f6dd9375c8f1f59f1effd9b4e5492ac97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.509ex;" alt="{\displaystyle b_{n}}"/></span> for <a href="/wiki/Fourier_series#Fourier's_formula_for_2π-periodic_functions_using_sines_and_cosines" title="Fourier series">2π-periodic functions with real-valued coefficients</a> to defining <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"/></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 b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28e2d72f6dd9375c8f1f59f1effd9b4e5492ac97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.509ex;" alt="{\displaystyle b_{n}}"/></span> for <a href="/wiki/Fourier_series#Fourier_series_on_a_general_interval_.5Ba.2C.C2.A0a_.2B_.CF.84.29" title="Fourier series">τ-periodic functions with <i>complex</i>-valued coefficients</a>. The equations that we've listed above at the sorts of things that I would expect to see in this article, as that was how I was always taught about Fourier series. I think it would be best to include them in the article somewhere, even if it's just something like <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Fourier_series&diff=570379709&oldid=569315968">this</a>. <a href="/wiki/User:A_Thousand_Doors" class="mw-redirect" title="User:A Thousand Doors">A Thousand Doors</a> (<a href="/wiki/User_talk:A_Thousand_Doors" title="User talk:A Thousand Doors">talk</a> | <a href="/wiki/Special:Contributions/A_Thousand_Doors" title="Special:Contributions/A Thousand Doors">contribs</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-A_Thousand_Doors-2013-08-27T11:08:00.000Z-Bob_K-2013-08-23T13:02:00.000Z" class="ext-discussiontools-init-timestamplink">11:08, 27 August 2013 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-A_Thousand_Doors-2013-08-27T11:08:00.000Z-Bob_K-2013-08-23T13:02:00.000Z"></span><span data-mw-comment-end="c-A_Thousand_Doors-2013-08-27T11:08:00.000Z-Bob_K-2013-08-23T13:02:00.000Z"></span></dd></dl></dd></dl> <dl><dd><span data-mw-comment-start="" id="c-Bob_K-2013-08-27T12:18:00.000Z-A_Thousand_Doors-2013-08-22T23:35:00.000Z"></span>My preference is to keep that section as it was, even if we have to add a similar section for real-valued coefficients. However, there is no rule that says this article must preserve the chronology of historical events. I.e. we are free to begin with the general interval approach for both real and complex coefficients. Then simply point out that the special case τ=2π, and a=-π was the historical starting point for Fourier. Simple, clean, and effective.</dd> <dd>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2013-08-27T12:18:00.000Z-A_Thousand_Doors-2013-08-22T23:35:00.000Z" class="ext-discussiontools-init-timestamplink">12:18, 27 August 2013 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2013-08-27T12:18:00.000Z-A_Thousand_Doors-2013-08-22T23:35:00.000Z"></span><span data-mw-comment-end="c-Bob_K-2013-08-27T12:18:00.000Z-A_Thousand_Doors-2013-08-22T23:35:00.000Z"></span> <dl><dd><span data-mw-comment-start="" id="c-A_Thousand_Doors-2013-08-27T12:27:00.000Z-Bob_K-2013-08-27T12:18:00.000Z"></span>Sounds like a good idea to me, I'd be happy with that. I just think that it's important to list those definitions for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"/></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 b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28e2d72f6dd9375c8f1f59f1effd9b4e5492ac97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.509ex;" alt="{\displaystyle b_{n}}"/></span> somewhere in the article. Beginning with the general interval approach and then describing Fourier's special case would be my preference too. <a href="/wiki/User:A_Thousand_Doors" class="mw-redirect" title="User:A Thousand Doors">A Thousand Doors</a> (<a href="/wiki/User_talk:A_Thousand_Doors" title="User talk:A Thousand Doors">talk</a> | <a href="/wiki/Special:Contributions/A_Thousand_Doors" title="Special:Contributions/A Thousand Doors">contribs</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-A_Thousand_Doors-2013-08-27T12:27:00.000Z-Bob_K-2013-08-27T12:18:00.000Z" class="ext-discussiontools-init-timestamplink">12:27, 27 August 2013 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-A_Thousand_Doors-2013-08-27T12:27:00.000Z-Bob_K-2013-08-27T12:18:00.000Z"></span><span data-mw-comment-end="c-A_Thousand_Doors-2013-08-27T12:27:00.000Z-Bob_K-2013-08-27T12:18:00.000Z"></span></dd> <dd><span data-mw-comment-start="" id="c-A_Thousand_Doors-2013-08-30T22:16:00.000Z-Bob_K-2013-08-27T12:18:00.000Z"></span>Quite like the new structure of the article, nice work. I think it's a smart idea to introduce the concepts of Fourier series and Fourier coefficients as early as possible. <a href="/wiki/User:A_Thousand_Doors" class="mw-redirect" title="User:A Thousand Doors">A Thousand Doors</a> (<a href="/wiki/User_talk:A_Thousand_Doors" title="User talk:A Thousand Doors">talk</a> | <a href="/wiki/Special:Contributions/A_Thousand_Doors" title="Special:Contributions/A Thousand Doors">contribs</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-A_Thousand_Doors-2013-08-30T22:16:00.000Z-Bob_K-2013-08-27T12:18:00.000Z" class="ext-discussiontools-init-timestamplink">22:16, 30 August 2013 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-A_Thousand_Doors-2013-08-30T22:16:00.000Z-Bob_K-2013-08-27T12:18:00.000Z"></span><span data-mw-comment-end="c-A_Thousand_Doors-2013-08-30T22:16:00.000Z-Bob_K-2013-08-27T12:18:00.000Z"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="translation_of_paper's_title?" data-mw-thread-id="h-translation_of_paper's_title?-2013-09-10T19:42:00.000Z"><span id="translation_of_paper.27s_title.3F"></span><span data-mw-comment-start="" id="h-translation_of_paper's_title?-2013-09-10T19:42:00.000Z"></span>translation of paper's title?<span data-mw-comment-end="h-translation_of_paper's_title?-2013-09-10T19:42:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-RJFJR-2013-09-10T19:42:00.000Z-translation_of_paper's_title?"></span>Under divergence we have "Une série de Fourier-Lebesgue divergente presque partout". Should we include the translation of this title into English? <a href="/wiki/User:RJFJR" title="User:RJFJR">RJFJR</a> (<a href="/wiki/User_talk:RJFJR" title="User talk:RJFJR">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-RJFJR-2013-09-10T19:42:00.000Z-translation_of_paper's_title?" class="ext-discussiontools-init-timestamplink">19:42, 10 September 2013 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-RJFJR-2013-09-10T19:42:00.000Z-translation_of_paper's_title?"></span><span data-mw-comment-end="c-RJFJR-2013-09-10T19:42:00.000Z-translation_of_paper's_title?"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="The_coefficient_in_"2.1_Example_1:_a_simple_Fourier_series"" data-mw-thread-id="h-The_coefficient_in_"2.1_Example_1:_a_simple_Fourier_series"-2013-11-06T16:42:00.000Z"><span id="The_coefficient_in_.222.1_Example_1:_a_simple_Fourier_series.22"></span><span data-mw-comment-start="" id="h-The_coefficient_in_"2.1_Example_1:_a_simple_Fourier_series"-2013-11-06T16:42:00.000Z"></span>The coefficient in "2.1 Example 1: a simple Fourier series"<span data-mw-comment-end="h-The_coefficient_in_"2.1_Example_1:_a_simple_Fourier_series"-2013-11-06T16:42:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Wangguansong-2013-11-06T16:42:00.000Z-The_coefficient_in_"2.1_Example_1:_a_simple_Fourier_series""></span>I think there is an error or typo in the Fourier coefficient <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 b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28e2d72f6dd9375c8f1f59f1effd9b4e5492ac97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.509ex;" alt="{\displaystyle b_{n}}"/></span> of Example 1. It should be: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}b_{n}&{}={\frac {1}{\pi }}\int _{-\pi }^{\pi }s(x)\sin(nx)\,dx\\&=-{\frac {2}{n}}\cos(n\pi )+{\frac {2}{\pi n^{2}}}\sin(n\pi )\\&={\frac {2\,(-1)^{n+1}}{n}},\quad n\geq 1.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>n</mi> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>π</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mi>π</mi> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>π</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="1em"></mspace> <mi>n</mi> <mo>≥</mo> <mn>1.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}b_{n}&{}={\frac {1}{\pi }}\int _{-\pi }^{\pi }s(x)\sin(nx)\,dx\\&=-{\frac {2}{n}}\cos(n\pi )+{\frac {2}{\pi n^{2}}}\sin(n\pi )\\&={\frac {2\,(-1)^{n+1}}{n}},\quad n\geq 1.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ad2bd5406c336dcefee1328ffa2691853bf7f59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.001ex; margin-bottom: -0.337ex; width:33.375ex; height:17.676ex;" alt="{\displaystyle {\begin{aligned}b_{n}&{}={\frac {1}{\pi }}\int _{-\pi }^{\pi }s(x)\sin(nx)\,dx\\&=-{\frac {2}{n}}\cos(n\pi )+{\frac {2}{\pi n^{2}}}\sin(n\pi )\\&={\frac {2\,(-1)^{n+1}}{n}},\quad n\geq 1.\end{aligned}}}"/></span></dd></dl> <p>That is, there is an extra <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> in the denominator of the current article. </p><p>Here is one reference: <a rel="nofollow" class="external free" href="http://watkins.cs.queensu.ca/~jstewart/861/sampling.pdf">http://watkins.cs.queensu.ca/~jstewart/861/sampling.pdf</a> </p><p>Can someone confirm that this is an error? <small><span class="autosigned">— Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/w/index.php?title=User:Wangguansong&action=edit&redlink=1" class="new" title="User:Wangguansong (page does not exist)">Wangguansong</a> (<a href="/w/index.php?title=User_talk:Wangguansong&action=edit&redlink=1" class="new" title="User talk:Wangguansong (page does not exist)">talk</a> • <a href="/wiki/Special:Contributions/Wangguansong" title="Special:Contributions/Wangguansong">contribs</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Wangguansong-2013-11-06T16:42:00.000Z-The_coefficient_in_"2.1_Example_1:_a_simple_Fourier_series"" class="ext-discussiontools-init-timestamplink">16:42, 6 November 2013 (UTC)</a></span></small><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Wangguansong-2013-11-06T16:42:00.000Z-The_coefficient_in_"2.1_Example_1:_a_simple_Fourier_series""></span><span data-mw-comment-end="c-Wangguansong-2013-11-06T16:42:00.000Z-The_coefficient_in_"2.1_Example_1:_a_simple_Fourier_series""></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Bob_K-2013-11-06T22:44:00.000Z-Wangguansong-2013-11-06T16:42:00.000Z"></span>Perhaps you forgot that s(x) = x/π (not just x). Otherwise see <a rel="nofollow" class="external text" href="http://www.wolframalpha.com/input/?i=%5Cfrac%7B1%7D%7B%5Cpi%7D%5Cint_%7B-%5Cpi%7D%5E%7B%5Cpi%7Dx%2F%5Cpi+%5Csin%28nx%29%5C%2C+dx">this link</a>.</dd> <dd>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2013-11-06T22:44:00.000Z-Wangguansong-2013-11-06T16:42:00.000Z" class="ext-discussiontools-init-timestamplink">22:44, 6 November 2013 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2013-11-06T22:44:00.000Z-Wangguansong-2013-11-06T16:42:00.000Z"></span><span data-mw-comment-end="c-Bob_K-2013-11-06T22:44:00.000Z-Wangguansong-2013-11-06T16:42:00.000Z"></span></dd></dl> <dl><dd><dl><dd><span data-mw-comment-start="" id="c-Wangguansong-2013-11-07T14:35:00.000Z-Bob_K-2013-11-06T22:44:00.000Z"></span>Thank you for clearing that for me! I missed that pi. <a href="/w/index.php?title=User:Wangguansong&action=edit&redlink=1" class="new" title="User:Wangguansong (page does not exist)">Wangguansong</a> (<a href="/w/index.php?title=User_talk:Wangguansong&action=edit&redlink=1" class="new" title="User talk:Wangguansong (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Wangguansong-2013-11-07T14:35:00.000Z-Bob_K-2013-11-06T22:44:00.000Z" class="ext-discussiontools-init-timestamplink">14:35, 7 November 2013 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Wangguansong-2013-11-07T14:35:00.000Z-Bob_K-2013-11-06T22:44:00.000Z"></span><span data-mw-comment-end="c-Wangguansong-2013-11-07T14:35:00.000Z-Bob_K-2013-11-06T22:44:00.000Z"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Why_are_we_using_a_finite_series_in_the_definition?" data-mw-thread-id="h-Why_are_we_using_a_finite_series_in_the_definition?-2014-02-10T02:15:00.000Z"><span id="Why_are_we_using_a_finite_series_in_the_definition.3F"></span><span data-mw-comment-start="" id="h-Why_are_we_using_a_finite_series_in_the_definition?-2014-02-10T02:15:00.000Z"></span>Why are we using a finite series in the definition?<span data-mw-comment-end="h-Why_are_we_using_a_finite_series_in_the_definition?-2014-02-10T02:15:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-174.3.243.185-2014-02-10T02:15:00.000Z-Why_are_we_using_a_finite_series_in_the_definition?"></span>The definition section uses a summation from n=1 to N, and then takes the limit as N approaches infinity. I don't see the point of introducing the variable N. If you're trying to learn fourier series, you presumably already understand the idea of infinite sums being a limit. <span style="font-size: smaller;" class="autosigned">— Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/Special:Contributions/174.3.243.185" title="Special:Contributions/174.3.243.185">174.3.243.185</a> (<a href="/w/index.php?title=User_talk:174.3.243.185&action=edit&redlink=1" class="new" title="User talk:174.3.243.185 (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-174.3.243.185-2014-02-10T02:15:00.000Z-Why_are_we_using_a_finite_series_in_the_definition?" class="ext-discussiontools-init-timestamplink">02:15, 10 February 2014 (UTC)</a></span><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-174.3.243.185-2014-02-10T02:15:00.000Z-Why_are_we_using_a_finite_series_in_the_definition?"></span><span data-mw-comment-end="c-174.3.243.185-2014-02-10T02:15:00.000Z-Why_are_we_using_a_finite_series_in_the_definition?"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Bob_K-2014-02-10T05:14:00.000Z-174.3.243.185-2014-02-10T02:15:00.000Z"></span>I've never seen it done that way either. But three of the figures reflect that approach. And it leads nicely into the discussion of convergence. The term "partial sum", seen in one of the figure captions, used to be in the prose as well. It got dropped (by me) when I did some work on the Definition section a while ago. It just wasn't a good fit anywhere. But perhaps that was a mistake.(?)</dd> <dd>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2014-02-10T05:14:00.000Z-174.3.243.185-2014-02-10T02:15:00.000Z" class="ext-discussiontools-init-timestamplink">05:14, 10 February 2014 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2014-02-10T05:14:00.000Z-174.3.243.185-2014-02-10T02:15:00.000Z"></span><span data-mw-comment-end="c-Bob_K-2014-02-10T05:14:00.000Z-174.3.243.185-2014-02-10T02:15:00.000Z"></span></dd></dl> <p><span data-mw-comment-start="" id="c-198.73.178.11-2014-02-10T17:40:00.000Z-Why_are_we_using_a_finite_series_in_the_definition?"></span>Maybe we can make the definition an infinite sum, and then include something akin to "the infinite sum can often be approximated with a finite sum to a high degree of accuracy, sometimes called a partial sum of the Fourier series." This can lead into the approximation and convergence section.<a href="/wiki/Special:Contributions/198.73.178.11" title="Special:Contributions/198.73.178.11">198.73.178.11</a> (<a href="/wiki/User_talk:198.73.178.11" title="User talk:198.73.178.11">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-198.73.178.11-2014-02-10T17:40:00.000Z-Why_are_we_using_a_finite_series_in_the_definition?" class="ext-discussiontools-init-timestamplink">17:40, 10 February 2014 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-198.73.178.11-2014-02-10T17:40:00.000Z-Why_are_we_using_a_finite_series_in_the_definition?"></span><span data-mw-comment-end="c-198.73.178.11-2014-02-10T17:40:00.000Z-Why_are_we_using_a_finite_series_in_the_definition?"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Stevenj-2014-02-10T18:33:00.000Z-198.73.178.11-2014-02-10T17:40:00.000Z"></span>Replace "can often be approximated" with "can <i>always</i> be approximated". The ability to approximate with arbitrary accuracy by a (sufficiently large) finite partial sum is the definition of convergence. <a href="/wiki/User:Stevenj" title="User:Stevenj">— Steven G. Johnson</a> (<a href="/wiki/User_talk:Stevenj" title="User talk:Stevenj">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Stevenj-2014-02-10T18:33:00.000Z-198.73.178.11-2014-02-10T17:40:00.000Z" class="ext-discussiontools-init-timestamplink">18:33, 10 February 2014 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Stevenj-2014-02-10T18:33:00.000Z-198.73.178.11-2014-02-10T17:40:00.000Z"></span><span data-mw-comment-end="c-Stevenj-2014-02-10T18:33:00.000Z-198.73.178.11-2014-02-10T17:40:00.000Z"></span></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="layperson_readability" data-mw-thread-id="h-layperson_readability-2013-09-17T19:58:00.000Z"><span data-mw-comment-start="" id="h-layperson_readability-2013-09-17T19:58:00.000Z"></span>layperson readability<span data-mw-comment-end="h-layperson_readability-2013-09-17T19:58:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Bajatmerc-2013-09-17T19:58:00.000Z-layperson_readability"></span>First we must ask who is the audience. In this case it is the general public in my opinion, or the layperson. </p><p>This page is a long way from a layperson finding a Fourier series coefficient of y(t). It isn't above the ability of someone who passed high school to do. It is too obscure though. </p><p>The subscripts and symbols are hurdles for laypeople in my opinion. People could be referred to half a dozen other pages to learn the symbols. I suspect most would give up. </p><p>Somewhere along the line in mathematics, someone's shorthand became standard, and mathematics became another language. <a href="/w/index.php?title=User:Bajatmerc&action=edit&redlink=1" class="new" title="User:Bajatmerc (page does not exist)">Bajatmerc</a> (<a href="/wiki/User_talk:Bajatmerc" title="User talk:Bajatmerc">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bajatmerc-2013-09-17T19:58:00.000Z-layperson_readability" class="ext-discussiontools-init-timestamplink">19:58, 17 September 2013 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bajatmerc-2013-09-17T19:58:00.000Z-layperson_readability"></span><span data-mw-comment-end="c-Bajatmerc-2013-09-17T19:58:00.000Z-layperson_readability"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Bob_K-2013-09-17T23:26:00.000Z-Bajatmerc-2013-09-17T19:58:00.000Z"></span>It is not the purview of every Wikipedia article that relies on mathematics to re-teach standard concepts and notation to the general public. Yes, mathematics has its own language... no way around that.</dd> <dd>FWIW, the missing concept, IMO, is that in my 45 years of experience with "Fourier analysis", I have never knowingly "found" a Fourier series coefficient, and I don't know anyone who has. What we actually do is analyze "data" with tools such as DFTs. And our ability to interpret those DFTs depends on our understanding of how they are related to the underlying continuous transforms and inverse transforms.</dd> <dd>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2013-09-17T23:26:00.000Z-Bajatmerc-2013-09-17T19:58:00.000Z" class="ext-discussiontools-init-timestamplink">23:26, 17 September 2013 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2013-09-17T23:26:00.000Z-Bajatmerc-2013-09-17T19:58:00.000Z"></span><span data-mw-comment-end="c-Bob_K-2013-09-17T23:26:00.000Z-Bajatmerc-2013-09-17T19:58:00.000Z"></span> <dl><dd><span data-mw-comment-start="" id="c-Trlkly-2014-05-13T22:26:00.000Z-Bob_K-2013-09-17T23:26:00.000Z"></span>You don't have to teach concepts, but you should, if possible, give a lay understanding in the lead section so that the lay person can get a general understanding, even if it's incomplete. Right now, the image does a better job than the lead. — <a href="/wiki/User_talk:Trlkly" title="User talk:Trlkly">trlkly</a> <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Trlkly-2014-05-13T22:26:00.000Z-Bob_K-2013-09-17T23:26:00.000Z" class="ext-discussiontools-init-timestamplink">22:26, 13 May 2014 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Trlkly-2014-05-13T22:26:00.000Z-Bob_K-2013-09-17T23:26:00.000Z"></span><span data-mw-comment-end="c-Trlkly-2014-05-13T22:26:00.000Z-Bob_K-2013-09-17T23:26:00.000Z"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="1/2_[f(x0+)+f(x0-)]" data-mw-thread-id="h-1/2_[f(x0+)+f(x0-)]-2014-02-02T08:38:00.000Z"><span id="1.2F2_.5Bf.28x0.2B.29.2Bf.28x0-.29.5D"></span><span data-mw-comment-start="" id="h-1/2_[f(x0+)+f(x0-)]-2014-02-02T08:38:00.000Z"></span>1/2 [f(x0+)+f(x0-)]<span data-mw-comment-end="h-1/2_[f(x0+)+f(x0-)]-2014-02-02T08:38:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <blockquote> <p><span data-mw-comment-start="" id="c-دالبا-2014-02-02T08:38:00.000Z-1/2_[f(x0+)+f(x0-)]"></span>If f were continuous at x0, f(x0)=f(x0+)=f(x0-). At a jump however, there is no prior relation between f(x0) and f(x0+-), but it is fairly common for the value of f at the jump x0 to be precisely at the midpoint of the jump. That is f(x0)= 1/2 [f(x0+)+f(x0-)]. </p> </blockquote> <p><a rel="nofollow" class="external autonumber" href="http://www.math.ubc.ca/~feldman/math/fourier.pdf">[7]</a> <span style="border:solid 2px #1E90FF;border-radius:3px"><a href="/wiki/User:%D8%AF%D8%A7%D9%84%D8%A8%D8%A7" class="mw-redirect" title="User:دالبا">Dalba</a></span> <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-دالبا-2014-02-02T08:38:00.000Z-1/2_[f(x0+)+f(x0-)]" class="ext-discussiontools-init-timestamplink">08:38, 2 February 2014 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-دالبا-2014-02-02T08:38:00.000Z-1/2_[f(x0+)+f(x0-)]"></span><span data-mw-comment-end="c-دالبا-2014-02-02T08:38:00.000Z-1/2_[f(x0+)+f(x0-)]"></span> </p><dl><dd><span data-mw-comment-start="" id="c-LutzL-2014-05-14T10:52:00.000Z-دالبا-2014-02-02T08:38:00.000Z"></span>This is a rather strange quote. Under the assumptions for pointwise convergence, the value of the Fourier series at a jump is the midpoint of the one-sided limits. Independent of the value of the function at that point. This gives a condition for pointwise convergence towards the original function, but it is not a common occurrence. It is at least as popular to have the function value be one of the one-sided limits.--<a href="/wiki/User:LutzL" title="User:LutzL">LutzL</a> (<a href="/wiki/User_talk:LutzL" title="User talk:LutzL">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-LutzL-2014-05-14T10:52:00.000Z-دالبا-2014-02-02T08:38:00.000Z" class="ext-discussiontools-init-timestamplink">10:52, 14 May 2014 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-LutzL-2014-05-14T10:52:00.000Z-دالبا-2014-02-02T08:38:00.000Z"></span><span data-mw-comment-end="c-LutzL-2014-05-14T10:52:00.000Z-دالبا-2014-02-02T08:38:00.000Z"></span></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Eye-catching_GIF" data-mw-thread-id="h-Eye-catching_GIF-2014-03-07T13:48:00.000Z"><span data-mw-comment-start="" id="h-Eye-catching_GIF-2014-03-07T13:48:00.000Z"></span>Eye-catching GIF<span data-mw-comment-end="h-Eye-catching_GIF-2014-03-07T13:48:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <span data-mw-comment-start="" id="c-Bob_K-2014-03-07T13:48:00.000Z-Eye-catching_GIF"></span><figure typeof="mw:File/Thumb"><a href="/wiki/File:SquareWaveFourierArrows.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/6b/SquareWaveFourierArrows.gif" decoding="async" width="300" height="90" class="mw-file-element" data-file-width="300" data-file-height="90"/></a><figcaption>A Fourier series enables any <a href="/wiki/Periodic_function" title="Periodic function">periodic motion</a>, such as the <a href="/wiki/Square_wave" title="Square wave">square wave</a> motion of the blue circle at the bottom, to be represented as the sum of many (often infinitely many) <a href="/wiki/Sine_wave" title="Sine wave">sinusoidally-oscillating</a> terms with different <a href="/wiki/Frequency" title="Frequency">frequencies</a>. The arrows show the six largest terms in the Fourier series, which sum to the purple circle.</figcaption></figure> <p>It's cool, but IMO it's sideways. Amplitude is customarily vertical. Partly for that reason, I think the only people who will actually comprehend it are those who have already internalized File:Fourier_Series.svg and are able to perform the mental (or laptop) rotation. For them, it is just a novelty. For others, it might serve the purpose of attracting attention, like a blinking light. But hopefully they won't get "stuck" there. My 2 cents. </p><p>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2014-03-07T13:48:00.000Z-Eye-catching_GIF" class="ext-discussiontools-init-timestamplink">13:48, 7 March 2014 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2014-03-07T13:48:00.000Z-Eye-catching_GIF"></span><span data-mw-comment-end="c-Bob_K-2014-03-07T13:48:00.000Z-Eye-catching_GIF"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Sbyrnes321-2014-03-07T16:53:00.000Z-Bob_K-2014-03-07T13:48:00.000Z"></span>It's not vertical because tall and narrow figures tend to make wikipedia page layout very difficult, compared to short and wide figures. (Or is there a trick I'm not aware of?)</dd></dl> <dl><dd>If you think the figure is doing more harm than good, you are entitled to delete it! --<a href="/wiki/User:Sbyrnes321" title="User:Sbyrnes321">Steve</a> (<a href="/wiki/User_talk:Sbyrnes321" title="User talk:Sbyrnes321">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Sbyrnes321-2014-03-07T16:53:00.000Z-Bob_K-2014-03-07T13:48:00.000Z" class="ext-discussiontools-init-timestamplink">16:53, 7 March 2014 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Sbyrnes321-2014-03-07T16:53:00.000Z-Bob_K-2014-03-07T13:48:00.000Z"></span><span data-mw-comment-end="c-Sbyrnes321-2014-03-07T16:53:00.000Z-Bob_K-2014-03-07T13:48:00.000Z"></span></dd></dl> <dl><dd><span data-mw-comment-start="" id="c-Sbyrnes321-2014-03-08T02:04:00.000Z-Bob_K-2014-03-07T13:48:00.000Z"></span>Incidentally, even disregarding page layout issues, I'm not convinced that vertical is really better. Fourier series the concept can appear in many superficially different settings: Vertical position as a function of horizontal position (your favorite), horizontal position as a function of time (my animation), vertical position as a function of time (your preference for my animation), color as a function of position (image processing), and on and on. Presenting the same concept in more than one superficial setting has a pedagogical value: It helps readers construct a more mature and refined conceptual understanding.</dd></dl> <dl><dd>(Yes, I acknowledge that this philosophy should not be taken to an extreme, where the settings are so bizarre that they are distracting and frustrating. But I think that in this particular situation, horizontal is OK.)</dd></dl> <dl><dd>Also, if it stays at the top of the article, it may be the first thing that people ever see about Fourier series, so they would not necessarily have the same preconception that you have, i.e. "x(t) is weird and y(t) is normal". --<a href="/wiki/User:Sbyrnes321" title="User:Sbyrnes321">Steve</a> (<a href="/wiki/User_talk:Sbyrnes321" title="User talk:Sbyrnes321">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Sbyrnes321-2014-03-08T02:04:00.000Z-Bob_K-2014-03-07T13:48:00.000Z" class="ext-discussiontools-init-timestamplink">02:04, 8 March 2014 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Sbyrnes321-2014-03-08T02:04:00.000Z-Bob_K-2014-03-07T13:48:00.000Z"></span><span data-mw-comment-end="c-Sbyrnes321-2014-03-08T02:04:00.000Z-Bob_K-2014-03-07T13:48:00.000Z"></span></dd></dl> <p><span data-mw-comment-start="" id="c-2601:703:0:851E:91E5:CA2D:218:296C-2016-02-25T16:27:00.000Z-Eye-catching_GIF"></span>I should like to differ on the description of the Fourier series. First this definition of a Fourier series is applicable only to scientists and engineers. Hence is not really mathematics but engineering. In mathematics a Fourier series is defined within any inner product space within a maximal orthonormal set within that inner product space. The engineering version set out in this article is valid only for the inner product space L2[a,b]. <small class="autosigned">— Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/Special:Contributions/2601:703:0:851E:91E5:CA2D:218:296C" title="Special:Contributions/2601:703:0:851E:91E5:CA2D:218:296C">2601:703:0:851E:91E5:CA2D:218:296C</a> (<a href="/w/index.php?title=User_talk:2601:703:0:851E:91E5:CA2D:218:296C&action=edit&redlink=1" class="new" title="User talk:2601:703:0:851E:91E5:CA2D:218:296C (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-2601:703:0:851E:91E5:CA2D:218:296C-2016-02-25T16:27:00.000Z-Eye-catching_GIF" class="ext-discussiontools-init-timestamplink">16:27, 25 February 2016 (UTC)</a></small><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-2601:703:0:851E:91E5:CA2D:218:296C-2016-02-25T16:27:00.000Z-Eye-catching_GIF"></span><span data-mw-comment-end="c-2601:703:0:851E:91E5:CA2D:218:296C-2016-02-25T16:27:00.000Z-Eye-catching_GIF"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="there_is_an_asterisk_(*)._What_does_it_mean?" data-mw-thread-id="h-there_is_an_asterisk_(*)._What_does_it_mean?-2016-06-16T08:59:00.000Z"><span id="there_is_an_asterisk_.28.2A.29._What_does_it_mean.3F"></span><span data-mw-comment-start="" id="h-there_is_an_asterisk_(*)._What_does_it_mean?-2016-06-16T08:59:00.000Z"></span>there is an asterisk (*). What does it mean?<span data-mw-comment-end="h-there_is_an_asterisk_(*)._What_does_it_mean?-2016-06-16T08:59:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-88.188.110.51-2016-06-16T08:59:00.000Z-there_is_an_asterisk_(*)._What_does_it_mean?"></span>Dear Author, please accept my many thanks for your helpful article, and permit me a question: in "Definition" above "we can also write the function in these equivalent forms:" at the end of formula, there is an asterisk (*). What does it mean? Regards, Georges Theodosiou, <a href="/wiki/Special:Contributions/88.188.110.51" title="Special:Contributions/88.188.110.51">88.188.110.51</a> (<a href="/wiki/User_talk:88.188.110.51" title="User talk:88.188.110.51">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-88.188.110.51-2016-06-16T08:59:00.000Z-there_is_an_asterisk_(*)._What_does_it_mean?" class="ext-discussiontools-init-timestamplink">08:59, 16 June 2016 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-88.188.110.51-2016-06-16T08:59:00.000Z-there_is_an_asterisk_(*)._What_does_it_mean?"></span><span data-mw-comment-end="c-88.188.110.51-2016-06-16T08:59:00.000Z-there_is_an_asterisk_(*)._What_does_it_mean?"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Bob_K-2016-06-16T12:56:00.000Z-88.188.110.51-2016-06-16T08:59:00.000Z"></span>It is a common notation for the <a href="/wiki/Complex_conjugate#Notation" title="Complex conjugate">Complex_conjugate#Notation</a> operation. --<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2016-06-16T12:56:00.000Z-88.188.110.51-2016-06-16T08:59:00.000Z" class="ext-discussiontools-init-timestamplink">12:56, 16 June 2016 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2016-06-16T12:56:00.000Z-88.188.110.51-2016-06-16T08:59:00.000Z"></span><span data-mw-comment-end="c-Bob_K-2016-06-16T12:56:00.000Z-88.188.110.51-2016-06-16T08:59:00.000Z"></span></dd></dl> <dl><dd><dl><dd><span data-mw-comment-start="" id="c-88.188.110.51-2016-06-16T13:48:00.000Z-Bob_K-2016-06-16T12:56:00.000Z"></span>Mr Author, please let me express you my sincere thanks for your helpful answer. Georges Theodosiou <a href="/wiki/Special:Contributions/88.188.110.51" title="Special:Contributions/88.188.110.51">88.188.110.51</a> (<a href="/wiki/User_talk:88.188.110.51" title="User talk:88.188.110.51">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-88.188.110.51-2016-06-16T13:48:00.000Z-Bob_K-2016-06-16T12:56:00.000Z" class="ext-discussiontools-init-timestamplink">13:48, 16 June 2016 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-88.188.110.51-2016-06-16T13:48:00.000Z-Bob_K-2016-06-16T12:56:00.000Z"></span><span data-mw-comment-end="c-88.188.110.51-2016-06-16T13:48:00.000Z-Bob_K-2016-06-16T12:56:00.000Z"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Fourier_Series_definition_and_other_problems" data-mw-thread-id="h-Fourier_Series_definition_and_other_problems-2017-10-17T20:33:00.000Z"><span data-mw-comment-start="" id="h-Fourier_Series_definition_and_other_problems-2017-10-17T20:33:00.000Z"></span>Fourier Series definition and other problems<span data-mw-comment-end="h-Fourier_Series_definition_and_other_problems-2017-10-17T20:33:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Bob_K-2017-10-17T20:33:00.000Z-Fourier_Series_definition_and_other_problems"></span>There is a problem with the first equation on the page. The Fourier Series is defined as a series with amplitude and phase angle for each harmonic then the cosine function should be used. This is consistent with the idea of "phasor". The consistent Fourier Series definition follows as: </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 s_{N}(x)={\frac {A_{0}}{2}}+\sum _{n=1}^{N}A_{n}\times \cos \left({\tfrac {2\pi nx}{P}}+\phi _{n}\right),\quad {\text{for integer}}\ N\ \geq \ 1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>×</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> </mrow> <mi>P</mi> </mfrac> </mstyle> </mrow> <mo>+</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>for integer</mtext> </mrow> <mtext> </mtext> <mi>N</mi> <mtext> </mtext> <mo>≥</mo> <mtext> </mtext> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{N}(x)={\frac {A_{0}}{2}}+\sum _{n=1}^{N}A_{n}\times \cos \left({\tfrac {2\pi nx}{P}}+\phi _{n}\right),\quad {\text{for integer}}\ N\ \geq \ 1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d40cec2b1f57f5e1063ec1d3a2b37a58d2102323" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:63.314ex; height:7.343ex;" alt="{\displaystyle s_{N}(x)={\frac {A_{0}}{2}}+\sum _{n=1}^{N}A_{n}\times \cos \left({\tfrac {2\pi nx}{P}}+\phi _{n}\right),\quad {\text{for integer}}\ N\ \geq \ 1.}"/></span></dd></dl> <p>Note, I have also used the multiplication symbol <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 \times }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>×</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \times }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffafff1ad26cbe49045f19a67ce532116a32703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.019ex; margin-bottom: -0.19ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \times }"/></span> instead of the dot product symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }"/></span>. From an aesthetic perspective, I would even prefer to remove the multiplication symbol completely. </p><p>Another problem is in the definition of the equivalent forms. The identity only holds if the sine part is complex, as shown in the following equation: </p> <table class="wikitable" style="text-align:left"> <tbody><tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}s_{N}(x)&=\overbrace {a_{0}} ^{A_{0}}/2+\sum _{n=1}^{N}\left(\overbrace {a_{n}} ^{A_{n}\sin(\phi _{n})}\cos \left({\tfrac {2\pi nx}{P}}\right)+\overbrace {b_{n}} ^{A_{n}\cos(\phi _{n})}i\,\sin \left({\tfrac {2\pi nx}{P}}\right)\right)\\&=\sum _{n=-N}^{N}c_{n}\times e^{i{\tfrac {2\pi nx}{P}}},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mover> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mover> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> </mrow> <mi>P</mi> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mover> <mi>i</mi> <mspace width="thinmathspace"></mspace> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> </mrow> <mi>P</mi> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>×</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> </mrow> <mi>P</mi> </mfrac> </mstyle> </mrow> </mrow> </msup> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}s_{N}(x)&=\overbrace {a_{0}} ^{A_{0}}/2+\sum _{n=1}^{N}\left(\overbrace {a_{n}} ^{A_{n}\sin(\phi _{n})}\cos \left({\tfrac {2\pi nx}{P}}\right)+\overbrace {b_{n}} ^{A_{n}\cos(\phi _{n})}i\,\sin \left({\tfrac {2\pi nx}{P}}\right)\right)\\&=\sum _{n=-N}^{N}c_{n}\times e^{i{\tfrac {2\pi nx}{P}}},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1635a515222a732d0f5f178e63df2bb37faad838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.171ex; width:67.807ex; height:17.509ex;" alt="{\displaystyle {\begin{aligned}s_{N}(x)&=\overbrace {a_{0}} ^{A_{0}}/2+\sum _{n=1}^{N}\left(\overbrace {a_{n}} ^{A_{n}\sin(\phi _{n})}\cos \left({\tfrac {2\pi nx}{P}}\right)+\overbrace {b_{n}} ^{A_{n}\cos(\phi _{n})}i\,\sin \left({\tfrac {2\pi nx}{P}}\right)\right)\\&=\sum _{n=-N}^{N}c_{n}\times e^{i{\tfrac {2\pi nx}{P}}},\end{aligned}}}"/></span> </td></tr></tbody></table> <p>Please discuss and comment. </p> <dl><dd>The generalization to complex-valued s(x) happens in section <a href="/wiki/Fourier_series#Complex-valued_functions" title="Fourier series">Fourier_series#Complex-valued_functions</a>.</dd></dl> <dl><dd>As clearly stated in the article, the identity you don't understand is just basic trigonometry:</dd></dl> <dl><dd><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 \sin \left({\tfrac {2\pi nx}{P}}+\phi _{n}\right)\equiv \sin(\phi _{n})\cos \left({\tfrac {2\pi nx}{P}}\right)+\cos(\phi _{n})\sin \left({\tfrac {2\pi nx}{P}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> </mrow> <mi>P</mi> </mfrac> </mstyle> </mrow> <mo>+</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>≡</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> </mrow> <mi>P</mi> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> </mrow> <mi>P</mi> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \left({\tfrac {2\pi nx}{P}}+\phi _{n}\right)\equiv \sin(\phi _{n})\cos \left({\tfrac {2\pi nx}{P}}\right)+\cos(\phi _{n})\sin \left({\tfrac {2\pi nx}{P}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3749dbab6378387aa48f5155fa5c7f7d51a35bcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:58.325ex; height:4.843ex;" alt="{\displaystyle \sin \left({\tfrac {2\pi nx}{P}}+\phi _{n}\right)\equiv \sin(\phi _{n})\cos \left({\tfrac {2\pi nx}{P}}\right)+\cos(\phi _{n})\sin \left({\tfrac {2\pi nx}{P}}\right).}"/></span></dd></dl></dd></dl> <dl><dd>Therefore the formula:</dd></dl> <dl><dd><table class="wikitable" style="text-align:left"> <tbody><tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}s_{N}(x)&=\overbrace {a_{0}} ^{A_{0}}/2+\sum _{n=1}^{N}\left(\overbrace {a_{n}} ^{A_{n}\sin(\phi _{n})}\cos \left({\tfrac {2\pi nx}{P}}\right)+\overbrace {b_{n}} ^{A_{n}\cos(\phi _{n})}\sin \left({\tfrac {2\pi nx}{P}}\right)\right)\\&=\sum _{n=-N}^{N}c_{n}\cdot e^{i{\tfrac {2\pi nx}{P}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mover> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mover> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> </mrow> <mi>P</mi> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mover> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> </mrow> <mi>P</mi> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> </mrow> <mi>P</mi> </mfrac> </mstyle> </mrow> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}s_{N}(x)&=\overbrace {a_{0}} ^{A_{0}}/2+\sum _{n=1}^{N}\left(\overbrace {a_{n}} ^{A_{n}\sin(\phi _{n})}\cos \left({\tfrac {2\pi nx}{P}}\right)+\overbrace {b_{n}} ^{A_{n}\cos(\phi _{n})}\sin \left({\tfrac {2\pi nx}{P}}\right)\right)\\&=\sum _{n=-N}^{N}c_{n}\cdot e^{i{\tfrac {2\pi nx}{P}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2773b2d39ad9bf5b197a2b326ac5edda98146ed9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.171ex; width:66.617ex; height:17.509ex;" alt="{\displaystyle {\begin{aligned}s_{N}(x)&=\overbrace {a_{0}} ^{A_{0}}/2+\sum _{n=1}^{N}\left(\overbrace {a_{n}} ^{A_{n}\sin(\phi _{n})}\cos \left({\tfrac {2\pi nx}{P}}\right)+\overbrace {b_{n}} ^{A_{n}\cos(\phi _{n})}\sin \left({\tfrac {2\pi nx}{P}}\right)\right)\\&=\sum _{n=-N}^{N}c_{n}\cdot e^{i{\tfrac {2\pi nx}{P}}}\end{aligned}}}"/></span> </td></tr></tbody></table></dd></dl> <dl><dd>is intentionally real-valued. And in case you missed it<b>:</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{n}\cdot e^{i{\tfrac {2\pi nx}{P}}}}"> <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> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> </mrow> <mi>P</mi> </mfrac> </mstyle> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{n}\cdot e^{i{\tfrac {2\pi nx}{P}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d16bbb7a335b6e2d83bbd05056c6f4a7f0784e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.314ex; height:4.343ex;" alt="{\displaystyle c_{n}\cdot e^{i{\tfrac {2\pi nx}{P}}}}"/></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 c_{-n}\cdot e^{i{\tfrac {2\pi (-n)x}{P}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msub> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mi>x</mi> </mrow> <mi>P</mi> </mfrac> </mstyle> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{-n}\cdot e^{i{\tfrac {2\pi (-n)x}{P}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e2e075b2bdad9798f726a5eeacb10929f90b7d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.15ex; height:4.843ex;" alt="{\displaystyle c_{-n}\cdot e^{i{\tfrac {2\pi (-n)x}{P}}}}"/></span> are complex conjugates. Their sum is real.</dd></dl> <dl><dd>The • operator is commonly used to represent the product of 2 operands, which technically <u>is</u> a one-dimensional <u>dot product</u>. So, at worst, you can call it overkill, if you insist on thinking in those terms. But I'm confident that most readers just see it as an alternative to ×, just as / is an alternative to ÷.</dd> <dd>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2017-10-17T20:33:00.000Z-Fourier_Series_definition_and_other_problems" class="ext-discussiontools-init-timestamplink">20:33, 17 October 2017 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2017-10-17T20:33:00.000Z-Fourier_Series_definition_and_other_problems"></span><span data-mw-comment-end="c-Bob_K-2017-10-17T20:33:00.000Z-Fourier_Series_definition_and_other_problems"></span></dd></dl> <dl><dd><dl><dd><span data-mw-comment-start="" id="c-Dicklyon-2019-04-25T05:24:00.000Z-Bob_K-2017-10-17T20:33:00.000Z"></span>I think you're both right. That is, the derivation seems correct, but starting with the sine instead of the cosine for defining the sinusoid with a phase angle seems a bit unconventional, and makes it a little messier, like needing to divide by i to get c_n. <a href="/wiki/User:Dicklyon" title="User:Dicklyon">Dicklyon</a> (<a href="/wiki/User_talk:Dicklyon" title="User talk:Dicklyon">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Dicklyon-2019-04-25T05:24:00.000Z-Bob_K-2017-10-17T20:33:00.000Z" class="ext-discussiontools-init-timestamplink">05:24, 25 April 2019 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Dicklyon-2019-04-25T05:24:00.000Z-Bob_K-2017-10-17T20:33:00.000Z"></span><span data-mw-comment-end="c-Dicklyon-2019-04-25T05:24:00.000Z-Bob_K-2017-10-17T20:33:00.000Z"></span></dd></dl></dd></dl> <dl><dd><span data-mw-comment-start="" id="c-Bob_K-2019-04-26T23:33:00.000Z-Bob_K-2017-10-17T20:33:00.000Z"></span>"Conventional" is the sine-cosine form<b>:</b></dd></dl> <dl><dd><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 {1}{2}}a_{0}+\sum _{n=1}^{\infty }[a_{n}\cos(2\pi \ n\ x)+b_{n}\sin(2\pi \ n\ x)].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mtext> </mtext> <mi>n</mi> <mtext> </mtext> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mtext> </mtext> <mi>n</mi> <mtext> </mtext> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}a_{0}+\sum _{n=1}^{\infty }[a_{n}\cos(2\pi \ n\ x)+b_{n}\sin(2\pi \ n\ x)].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5974c3d6e83eb3fd04b7d61e3871f32113cc23a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.043ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{2}}a_{0}+\sum _{n=1}^{\infty }[a_{n}\cos(2\pi \ n\ x)+b_{n}\sin(2\pi \ n\ x)].}"/></span></dd></dl></dd></dl> <dl><dd>Unconventional would be<b>:</b></dd></dl> <dl><dd><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 {1}{2}}a_{0}+\sum _{n=1}^{\infty }[a_{n}\cos(2\pi \ n\ x)-b_{n}\sin(2\pi \ n\ x)].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mtext> </mtext> <mi>n</mi> <mtext> </mtext> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mtext> </mtext> <mi>n</mi> <mtext> </mtext> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}a_{0}+\sum _{n=1}^{\infty }[a_{n}\cos(2\pi \ n\ x)-b_{n}\sin(2\pi \ n\ x)].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20e7032d931bc1fa7769cd637eb929335ed4482a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.043ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{2}}a_{0}+\sum _{n=1}^{\infty }[a_{n}\cos(2\pi \ n\ x)-b_{n}\sin(2\pi \ n\ x)].}"/></span></dd></dl></dd></dl> <dl><dd>But the former suggests<b>:</b></dd></dl> <dl><dd><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 \sin(2\pi nx+\varphi _{n})\equiv \sin(\varphi _{n})\cos(2\pi nx)+\cos(\varphi _{n})\sin(2\pi nx),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> <mo>+</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≡</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(2\pi nx+\varphi _{n})\equiv \sin(\varphi _{n})\cos(2\pi nx)+\cos(\varphi _{n})\sin(2\pi nx),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2d58ebdd83e668555a3f4db4171af8bfc3bb116" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.908ex; height:2.843ex;" alt="{\displaystyle \sin(2\pi nx+\varphi _{n})\equiv \sin(\varphi _{n})\cos(2\pi nx)+\cos(\varphi _{n})\sin(2\pi nx),}"/></span></dd></dl></dd></dl> <dl><dd>because of consistently "+" signs. For sign consistency, the latter suggests<b>:</b></dd></dl> <dl><dd><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 \cos(2\pi nx+\varphi _{n})\equiv \cos(\varphi _{n})\cos(2\pi nx)-\sin(\varphi _{n})\sin(2\pi nx),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> <mo>+</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≡</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(2\pi nx+\varphi _{n})\equiv \cos(\varphi _{n})\cos(2\pi nx)-\sin(\varphi _{n})\sin(2\pi nx),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5d02ef7b262b2497254a8a5fff9cd645878bd82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.164ex; height:2.843ex;" alt="{\displaystyle \cos(2\pi nx+\varphi _{n})\equiv \cos(\varphi _{n})\cos(2\pi nx)-\sin(\varphi _{n})\sin(2\pi nx),}"/></span></dd></dl></dd></dl> <dl><dd>which is also OK with me.</dd> <dd>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2019-04-26T23:33:00.000Z-Bob_K-2017-10-17T20:33:00.000Z" class="ext-discussiontools-init-timestamplink">23:33, 26 April 2019 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2019-04-26T23:33:00.000Z-Bob_K-2017-10-17T20:33:00.000Z"></span><span data-mw-comment-end="c-Bob_K-2019-04-26T23:33:00.000Z-Bob_K-2017-10-17T20:33:00.000Z"></span></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Label_for_the_cool_gif_is_unclear." data-mw-thread-id="h-Label_for_the_cool_gif_is_unclear.-2019-08-25T08:23:00.000Z"><span data-mw-comment-start="" id="h-Label_for_the_cool_gif_is_unclear.-2019-08-25T08:23:00.000Z"></span>Label for the cool gif is unclear.<span data-mw-comment-end="h-Label_for_the_cool_gif_is_unclear.-2019-08-25T08:23:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-2001:8003:E448:D401:7D27:BA42:5CB1:E259-2019-08-25T08:23:00.000Z-Label_for_the_cool_gif_is_unclear."></span>The gif showing the synthesis is fabulous, but perhaps it should be stated explicitly that it is the SUM of the four sine waves that creates it. <small class="autosigned">— Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/Special:Contributions/2001:8003:E448:D401:7D27:BA42:5CB1:E259" title="Special:Contributions/2001:8003:E448:D401:7D27:BA42:5CB1:E259">2001:8003:E448:D401:7D27:BA42:5CB1:E259</a> (<a href="/w/index.php?title=User_talk:2001:8003:E448:D401:7D27:BA42:5CB1:E259&action=edit&redlink=1" class="new" title="User talk:2001:8003:E448:D401:7D27:BA42:5CB1:E259 (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-2001:8003:E448:D401:7D27:BA42:5CB1:E259-2019-08-25T08:23:00.000Z-Label_for_the_cool_gif_is_unclear." class="ext-discussiontools-init-timestamplink">08:23, 25 August 2019 (UTC)</a></small><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-2001:8003:E448:D401:7D27:BA42:5CB1:E259-2019-08-25T08:23:00.000Z-Label_for_the_cool_gif_is_unclear."></span><span data-mw-comment-end="c-2001:8003:E448:D401:7D27:BA42:5CB1:E259-2019-08-25T08:23:00.000Z-Label_for_the_cool_gif_is_unclear."></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Coefficient_of_'"`UNIQ--postMath-00000070-QINU`"'" data-mw-thread-id="h-Coefficient_of_'"`UNIQ--postMath-00000070-QINU`"'-2020-05-31T19:16:00.000Z"><span id="Coefficient_of_.7F.27.22.60UNIQ--postMath-00000070-QINU.60.22.27.7F"></span><span data-mw-comment-start="" id="h-Coefficient_of_'"`UNIQ--postMath-00000070-QINU`"'-2020-05-31T19:16:00.000Z"></span>Coefficient of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/693ad9f934775838bd72406b41ada4a59785d7ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{0}}"/></span><span data-mw-comment-end="h-Coefficient_of_'"`UNIQ--postMath-00000070-QINU`"'-2020-05-31T19:16:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Thellamabotherer-2020-05-31T19:16:00.000Z-Coefficient_of_'"`UNIQ--postMath-00000070-QINU`"'"></span>Sorry if I'm just being stupid here, which is probably the case. </p><p>Shouldn't the coefficent of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/693ad9f934775838bd72406b41ada4a59785d7ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{0}}"/></span> (underneath the first equation box in the definition section) be <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 {2}{P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>P</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcc5410727492841b0fb59a144d17c6d7aae244b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.582ex; height:5.176ex;" alt="{\displaystyle {\frac {2}{P}}}"/></span> rather than <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 {1}{P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>P</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/decd68a69fcb9a2a5c159842ef98df1f9c0d1b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.582ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{P}}}"/></span>. This feels like an easy mistake to make when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"/></span> is normally going to be <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 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }"/></span>. <small class="autosigned">— Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/w/index.php?title=User:Thellamabotherer&action=edit&redlink=1" class="new" title="User:Thellamabotherer (page does not exist)">Thellamabotherer</a> (<a href="/w/index.php?title=User_talk:Thellamabotherer&action=edit&redlink=1" class="new" title="User talk:Thellamabotherer (page does not exist)">talk</a> • <a href="/wiki/Special:Contributions/Thellamabotherer" title="Special:Contributions/Thellamabotherer">contribs</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Thellamabotherer-2020-05-31T19:16:00.000Z-Coefficient_of_'"`UNIQ--postMath-00000070-QINU`"'" class="ext-discussiontools-init-timestamplink">19:16, 31 May 2020 (UTC)</a></small><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Thellamabotherer-2020-05-31T19:16:00.000Z-Coefficient_of_'"`UNIQ--postMath-00000070-QINU`"'"></span><span data-mw-comment-end="c-Thellamabotherer-2020-05-31T19:16:00.000Z-Coefficient_of_'"`UNIQ--postMath-00000070-QINU`"'"></span> </p><p><br/> </p><p><span data-mw-comment-start="" id="c-Hqurve-2020-06-05T15:56:00.000Z-Coefficient_of_'"`UNIQ--postMath-00000070-QINU`"'"></span>I believe you are correct. If you substitute n=0 into the defintion of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"/></span> you arrive at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}={\frac {2}{P}}\int _{P}s(x)dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>P</mi> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}={\frac {2}{P}}\int _{P}s(x)dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70e2134a65bbbad21a9d99fd8f875342c2d3407c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.272ex; height:5.676ex;" alt="{\displaystyle a_{0}={\frac {2}{P}}\int _{P}s(x)dx}"/></span>. </p><p>In fact, there are two equivalent ways to define <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/693ad9f934775838bd72406b41ada4a59785d7ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{0}}"/></span> and your formula for <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 s_{N}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{N}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae692e7ee7e30e7b4153eb6c21e7906dfd52c64d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.921ex; height:2.843ex;" alt="{\displaystyle s_{N}(x)}"/></span> is affected by this choice. If you choose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/693ad9f934775838bd72406b41ada4a59785d7ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{0}}"/></span> to have a consistent formula with that of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"/></span> (ie. the formula with <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 {2}{P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>P</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcc5410727492841b0fb59a144d17c6d7aae244b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.582ex; height:5.176ex;" alt="{\displaystyle {\frac {2}{P}}}"/></span>), your <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 s_{N}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{N}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae692e7ee7e30e7b4153eb6c21e7906dfd52c64d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.921ex; height:2.843ex;" alt="{\displaystyle s_{N}(x)}"/></span> is defined as follows </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 s_{N}(x)={\frac {a_{0}}{2}}+\sum _{n=1}^{N}a_{n}\cos \left({\frac {2\pi nx}{P}}\right)+b_{n}\sin \left({\frac {2\pi nx}{P}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> </mrow> <mi>P</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> </mrow> <mi>P</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{N}(x)={\frac {a_{0}}{2}}+\sum _{n=1}^{N}a_{n}\cos \left({\frac {2\pi nx}{P}}\right)+b_{n}\sin \left({\frac {2\pi nx}{P}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/716806ffad08501988038b296155082ca19daed0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:51.92ex; height:7.343ex;" alt="{\displaystyle s_{N}(x)={\frac {a_{0}}{2}}+\sum _{n=1}^{N}a_{n}\cos \left({\frac {2\pi nx}{P}}\right)+b_{n}\sin \left({\frac {2\pi nx}{P}}\right)}"/></span> (Equation 2 in the article)</dd></dl> <p>However you can choose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/693ad9f934775838bd72406b41ada4a59785d7ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{0}}"/></span> to be half of this value (ie. the formula with <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 {1}{P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>P</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/decd68a69fcb9a2a5c159842ef98df1f9c0d1b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.582ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{P}}}"/></span>) so that you don't have to later divide by two in the formula for <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 s_{N}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{N}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae692e7ee7e30e7b4153eb6c21e7906dfd52c64d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.921ex; height:2.843ex;" alt="{\displaystyle s_{N}(x)}"/></span>. </p><p>Whoever made this edit was unaware of that difference in formula for <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 s_{N}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{N}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae692e7ee7e30e7b4153eb6c21e7906dfd52c64d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.921ex; height:2.843ex;" alt="{\displaystyle s_{N}(x)}"/></span> and made a bad edit. Moreover, the rest of the article makes use of the consistent formula for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/693ad9f934775838bd72406b41ada4a59785d7ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{0}}"/></span>. So in order to ensure the consistency of the article, it was best that I undid his edit. The formulas should now be correct. </p><p>--<a href="/w/index.php?title=User:Hqurve&action=edit&redlink=1" class="new" title="User:Hqurve (page does not exist)">Hqurve</a> (<a href="/w/index.php?title=User_talk:Hqurve&action=edit&redlink=1" class="new" title="User talk:Hqurve (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Hqurve-2020-06-05T15:56:00.000Z-Coefficient_of_'"`UNIQ--postMath-00000070-QINU`"'" class="ext-discussiontools-init-timestamplink">15:56, 5 June 2020 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Hqurve-2020-06-05T15:56:00.000Z-Coefficient_of_'"`UNIQ--postMath-00000070-QINU`"'"></span><span data-mw-comment-end="c-Hqurve-2020-06-05T15:56:00.000Z-Coefficient_of_'"`UNIQ--postMath-00000070-QINU`"'"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="No_such_topic_yet" data-mw-thread-id="h-No_such_topic_yet-2020-10-29T23:15:00.000Z"><span data-mw-comment-start="" id="h-No_such_topic_yet-2020-10-29T23:15:00.000Z"></span>No such topic yet<span data-mw-comment-end="h-No_such_topic_yet-2020-10-29T23:15:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-104.162.239.6-2020-10-29T23:15:00.000Z-No_such_topic_yet"></span>See also: Fourier analysis § History <a href="/wiki/Special:Contributions/104.162.239.6" title="Special:Contributions/104.162.239.6">104.162.239.6</a> (<a href="/w/index.php?title=User_talk:104.162.239.6&action=edit&redlink=1" class="new" title="User talk:104.162.239.6 (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-104.162.239.6-2020-10-29T23:15:00.000Z-No_such_topic_yet" class="ext-discussiontools-init-timestamplink">23:15, 29 October 2020 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-104.162.239.6-2020-10-29T23:15:00.000Z-No_such_topic_yet"></span><span data-mw-comment-end="c-104.162.239.6-2020-10-29T23:15:00.000Z-No_such_topic_yet"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="The_first_few_paragraphs_should_be_moved" data-mw-thread-id="h-The_first_few_paragraphs_should_be_moved-2021-10-28T00:09:00.000Z"><span data-mw-comment-start="" id="h-The_first_few_paragraphs_should_be_moved-2021-10-28T00:09:00.000Z"></span>The first few paragraphs should be moved<span data-mw-comment-end="h-The_first_few_paragraphs_should_be_moved-2021-10-28T00:09:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Cahmad25-2021-10-28T00:09:00.000Z-The_first_few_paragraphs_should_be_moved"></span>From the engineering and the maths perspective, those 3 paragraphs aren't helpful for the reader where they are placed : </p> <pre> 1 History 2 Definition 2.1 Complex-valued functions </pre> <p>We need to start with the definition of the series and the coefficients, the visual gif plus the problem of the convergence, the whole in less than 10 lines. Take a look at <a href="/wiki/Convergence_of_Fourier_series" title="Convergence of Fourier series">Convergence of Fourier series</a>. Only after that, we can go into disgression, the intuitive ideas, the theorems, the history, the real vs complex notation. Assuming the reader doesn't know what is a series and a complex number is ridiculous, as well as beginning the article with the partial sums. <a href="/w/index.php?title=User:Reuns&action=edit&redlink=1" class="new" title="User:Reuns (page does not exist)">Reuns</a> (<a href="/wiki/User_talk:Reuns" title="User talk:Reuns">talk</a>) </p> <dl><dd>From the perspective of a high school student who's interested in this, I agree with Reuns. The history should be lower, then the definition. <a href="/wiki/User:Cahmad25" title="User:Cahmad25">Cahmad25</a> (<a href="/wiki/User_talk:Cahmad25" title="User talk:Cahmad25">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Cahmad25-2021-10-28T00:09:00.000Z-The_first_few_paragraphs_should_be_moved" class="ext-discussiontools-init-timestamplink">00:09, 28 October 2021 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Cahmad25-2021-10-28T00:09:00.000Z-The_first_few_paragraphs_should_be_moved"></span><span data-mw-comment-end="c-Cahmad25-2021-10-28T00:09:00.000Z-The_first_few_paragraphs_should_be_moved"></span></dd></dl> <dl><dd><dl><dd><span data-mw-comment-start="" id="c-Bob_K-2021-10-28T13:52:00.000Z-Cahmad25-2021-10-28T00:09:00.000Z"></span>Thank you, and I am inclined to agree. So I would propose a reorganization something like this:</dd></dl></dd></dl> <ul><li>Definition (excluding complex-valued functions, for now)</li> <li>Convergence & Example 1 (A simple Fourier series)</li> <li>History, Beginnings, Birth of harmonic analysis, Example 2 (Fourier's motivation)</li> <li>Table of common Fourier series</li> <li>Complex-valued functions & Other common notations</li> <li>Table of basic properties (section 7.1)</li> <li>Other properties (sections 7.2 - 7.10)</li> <li>Extensions & Example 3 (complex Fourier series animation)</li> <li>Approximation and convergence of Fourier series</li></ul> <dl><dd><dl><dd>Note that this leaves two sections on the topic of convergence, but they are too dissimilar to be merged.</dd> <dd>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2021-10-28T13:52:00.000Z-Cahmad25-2021-10-28T00:09:00.000Z" class="ext-discussiontools-init-timestamplink">13:52, 28 October 2021 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2021-10-28T13:52:00.000Z-Cahmad25-2021-10-28T00:09:00.000Z"></span><span data-mw-comment-end="c-Bob_K-2021-10-28T13:52:00.000Z-Cahmad25-2021-10-28T00:09:00.000Z"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Should_this_article_definition_start_with_a_scary_correlation_function?" data-mw-thread-id="h-Should_this_article_definition_start_with_a_scary_correlation_function?-2022-03-15T07:36:00.000Z"><span id="Should_this_article_definition_start_with_a_scary_correlation_function.3F"></span><span data-mw-comment-start="" id="h-Should_this_article_definition_start_with_a_scary_correlation_function?-2022-03-15T07:36:00.000Z"></span>Should this article definition start with a scary correlation function?<span data-mw-comment-end="h-Should_this_article_definition_start_with_a_scary_correlation_function?-2022-03-15T07:36:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Em3rgent0rdr-2022-03-15T07:36:00.000Z-Should_this_article_definition_start_with_a_scary_correlation_function?"></span>I have honestly never seen correlation functions be used as an introductory starting point to Fourier series. Is invoking correlation even necessary in the definition? This article doesn't even have a clear transition from the correlation function into the Fourier series amplitude-phase form <span class="autosigned" style="font-size:85%;">— Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/User:Em3rgent0rdr" title="User:Em3rgent0rdr">Em3rgent0rdr</a> (<a href="/wiki/User_talk:Em3rgent0rdr#top" title="User talk:Em3rgent0rdr">talk</a> • <a href="/wiki/Special:Contributions/Em3rgent0rdr" title="Special:Contributions/Em3rgent0rdr">contribs</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Em3rgent0rdr-2022-03-15T07:36:00.000Z-Should_this_article_definition_start_with_a_scary_correlation_function?" class="ext-discussiontools-init-timestamplink">07:36, 15 March 2022 (UTC)</a></span><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Em3rgent0rdr-2022-03-15T07:36:00.000Z-Should_this_article_definition_start_with_a_scary_correlation_function?"></span><span data-mw-comment-end="c-Em3rgent0rdr-2022-03-15T07:36:00.000Z-Should_this_article_definition_start_with_a_scary_correlation_function?"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Em3rgent0rdr-2022-03-15T12:12:00.000Z-Em3rgent0rdr-2022-03-15T07:36:00.000Z"></span>well I moved the correlation function down a bit so it wasn't the first thing in definition. I'm still not entirely sure it even belongs in definition. <a href="/wiki/User:Em3rgent0rdr" title="User:Em3rgent0rdr">Em3rgent0rdr</a> (<a href="/wiki/User_talk:Em3rgent0rdr" title="User talk:Em3rgent0rdr">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Em3rgent0rdr-2022-03-15T12:12:00.000Z-Em3rgent0rdr-2022-03-15T07:36:00.000Z" class="ext-discussiontools-init-timestamplink">12:12, 15 March 2022 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Em3rgent0rdr-2022-03-15T12:12:00.000Z-Em3rgent0rdr-2022-03-15T07:36:00.000Z"></span><span data-mw-comment-end="c-Em3rgent0rdr-2022-03-15T12:12:00.000Z-Em3rgent0rdr-2022-03-15T07:36:00.000Z"></span> <dl><dd><span data-mw-comment-start="" id="c-Em3rgent0rdr-2022-03-16T11:39:00.000Z-Em3rgent0rdr-2022-03-15T12:12:00.000Z"></span>I believe I have transitioned into talking about the correlation function better now, by saying that this correlation can be used to determine the coefficients A_n and phi_n. So I think I am ok now with this correlation function being here. <a href="/wiki/User:Em3rgent0rdr" title="User:Em3rgent0rdr">Em3rgent0rdr</a> (<a href="/wiki/User_talk:Em3rgent0rdr" title="User talk:Em3rgent0rdr">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Em3rgent0rdr-2022-03-16T11:39:00.000Z-Em3rgent0rdr-2022-03-15T12:12:00.000Z" class="ext-discussiontools-init-timestamplink">11:39, 16 March 2022 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Em3rgent0rdr-2022-03-16T11:39:00.000Z-Em3rgent0rdr-2022-03-15T12:12:00.000Z"></span><span data-mw-comment-end="c-Em3rgent0rdr-2022-03-16T11:39:00.000Z-Em3rgent0rdr-2022-03-15T12:12:00.000Z"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Fourier's_Theorem_is_necessary_but_not_in_Wikipedia" data-mw-thread-id="h-Fourier's_Theorem_is_necessary_but_not_in_Wikipedia-2022-03-15T12:17:00.000Z"><span id="Fourier.27s_Theorem_is_necessary_but_not_in_Wikipedia"></span><span data-mw-comment-start="" id="h-Fourier's_Theorem_is_necessary_but_not_in_Wikipedia-2022-03-15T12:17:00.000Z"></span>Fourier's Theorem is necessary but not in Wikipedia<span data-mw-comment-end="h-Fourier's_Theorem_is_necessary_but_not_in_Wikipedia-2022-03-15T12:17:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Em3rgent0rdr-2022-03-15T12:17:00.000Z-Fourier's_Theorem_is_necessary_but_not_in_Wikipedia"></span>Fourier's Theorem underpins the concept of Fourier Series being able to represent any reasonable periodic function. However there is no mention of this crucial theorem. searching Wikipedia for it results in a redirect to Fourier Series, which is honestly a bit of circular logic. I believe it should be somewhere. maybe the Fourier theorem deserves its own page, or at least a section in this article. <a href="/wiki/User:Em3rgent0rdr" title="User:Em3rgent0rdr">Em3rgent0rdr</a> (<a href="/wiki/User_talk:Em3rgent0rdr" title="User talk:Em3rgent0rdr">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Em3rgent0rdr-2022-03-15T12:17:00.000Z-Fourier's_Theorem_is_necessary_but_not_in_Wikipedia" class="ext-discussiontools-init-timestamplink">12:17, 15 March 2022 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Em3rgent0rdr-2022-03-15T12:17:00.000Z-Fourier's_Theorem_is_necessary_but_not_in_Wikipedia"></span><span data-mw-comment-end="c-Em3rgent0rdr-2022-03-15T12:17:00.000Z-Fourier's_Theorem_is_necessary_but_not_in_Wikipedia"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Em3rgent0rdr-2022-03-16T11:41:00.000Z-Em3rgent0rdr-2022-03-15T12:17:00.000Z"></span>I think I have resolved this by renaming a section at the bottom of the article to be called "Fourier theorem proving convergence of Fourier series" and now I can refer to that section earlier in the article to reference "the Fourier Theorem". <a href="/wiki/User:Em3rgent0rdr" title="User:Em3rgent0rdr">Em3rgent0rdr</a> (<a href="/wiki/User_talk:Em3rgent0rdr" title="User talk:Em3rgent0rdr">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Em3rgent0rdr-2022-03-16T11:41:00.000Z-Em3rgent0rdr-2022-03-15T12:17:00.000Z" class="ext-discussiontools-init-timestamplink">11:41, 16 March 2022 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Em3rgent0rdr-2022-03-16T11:41:00.000Z-Em3rgent0rdr-2022-03-15T12:17:00.000Z"></span><span data-mw-comment-end="c-Em3rgent0rdr-2022-03-16T11:41:00.000Z-Em3rgent0rdr-2022-03-15T12:17:00.000Z"></span></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="I_brought_the_animations_to_be_fixed_in_between_paragraphs_of_the_intro_(using_align_=_none)." data-mw-thread-id="h-I_brought_the_animations_to_be_fixed_in_between_paragraphs_of_the_intro_(using_a-2022-03-16T12:42:00.000Z"><span id="I_brought_the_animations_to_be_fixed_in_between_paragraphs_of_the_intro_.28using_align_.3D_none.29."></span><span data-mw-comment-start="" id="h-I_brought_the_animations_to_be_fixed_in_between_paragraphs_of_the_intro_(using_a-2022-03-16T12:42:00.000Z"></span>I brought the animations to be fixed in between paragraphs of the intro (using align = none).<span data-mw-comment-end="h-I_brought_the_animations_to_be_fixed_in_between_paragraphs_of_the_intro_(using_a-2022-03-16T12:42:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Em3rgent0rdr-2022-03-16T12:42:00.000Z-I_brought_the_animations_to_be_fixed_in_between_paragraphs_of_the_intro_(using_a"></span>I think these animations are great for understand the basics. Unfortunately, they previously would appear on the right-bottom corner of my browser and were sortof missed. I think I have reworded the introduction so that those animations can be included directly inside the intro. After invoking "Fourier synthesis" I am immediately showing the square wave being added up. And after invoking "Fourier transform" I am immediately showing the 6 sine waves being broken up into frequency domain representation. <span class="autosigned" style="font-size:85%;">— Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/User:Em3rgent0rdr" title="User:Em3rgent0rdr">Em3rgent0rdr</a> (<a href="/wiki/User_talk:Em3rgent0rdr#top" title="User talk:Em3rgent0rdr">talk</a> • <a href="/wiki/Special:Contributions/Em3rgent0rdr" title="Special:Contributions/Em3rgent0rdr">contribs</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Em3rgent0rdr-2022-03-16T12:42:00.000Z-I_brought_the_animations_to_be_fixed_in_between_paragraphs_of_the_intro_(using_a" class="ext-discussiontools-init-timestamplink">12:42, 16 March 2022 (UTC)</a></span><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Em3rgent0rdr-2022-03-16T12:42:00.000Z-I_brought_the_animations_to_be_fixed_in_between_paragraphs_of_the_intro_(using_a"></span><span data-mw-comment-end="c-Em3rgent0rdr-2022-03-16T12:42:00.000Z-I_brought_the_animations_to_be_fixed_in_between_paragraphs_of_the_intro_(using_a"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Maybe_use_'ξ'_instead_of_'f'_for_frequency_to_avoid_conflict_with_'f'_for_function" data-mw-thread-id="h-Maybe_use_'ξ'_instead_of_'f'_for_frequency_to_avoid_conflict_with_'f'_for_funct-2022-03-15T07:31:00.000Z"><span id="Maybe_use_.27.CE.BE.27_instead_of_.27f.27_for_frequency_to_avoid_conflict_with_.27f.27_for_function"></span><span data-mw-comment-start="" id="h-Maybe_use_'ξ'_instead_of_'f'_for_frequency_to_avoid_conflict_with_'f'_for_funct-2022-03-15T07:31:00.000Z"></span>Maybe use 'ξ' instead of 'f' for frequency to avoid conflict with 'f' for function<span data-mw-comment-end="h-Maybe_use_'ξ'_instead_of_'f'_for_frequency_to_avoid_conflict_with_'f'_for_funct-2022-03-15T07:31:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Em3rgent0rdr-2022-03-15T07:31:00.000Z-Maybe_use_'ξ'_instead_of_'f'_for_frequency_to_avoid_conflict_with_'f'_for_funct"></span>This would also make this article more consistent with the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> article. I guess the downside would be that people scared of the letter 'ξ' and not knowing how to pronounce it. <span class="autosigned" style="font-size:85%;">— Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/User:Em3rgent0rdr" title="User:Em3rgent0rdr">Em3rgent0rdr</a> (<a href="/wiki/User_talk:Em3rgent0rdr#top" title="User talk:Em3rgent0rdr">talk</a> • <a href="/wiki/Special:Contributions/Em3rgent0rdr" title="Special:Contributions/Em3rgent0rdr">contribs</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Em3rgent0rdr-2022-03-15T07:31:00.000Z-Maybe_use_'ξ'_instead_of_'f'_for_frequency_to_avoid_conflict_with_'f'_for_funct" class="ext-discussiontools-init-timestamplink">07:31, 15 March 2022 (UTC)</a></span><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Em3rgent0rdr-2022-03-15T07:31:00.000Z-Maybe_use_'ξ'_instead_of_'f'_for_frequency_to_avoid_conflict_with_'f'_for_funct"></span><span data-mw-comment-end="c-Em3rgent0rdr-2022-03-15T07:31:00.000Z-Maybe_use_'ξ'_instead_of_'f'_for_frequency_to_avoid_conflict_with_'f'_for_funct"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Bob_K-2022-03-15T19:52:00.000Z-Em3rgent0rdr-2022-03-15T07:31:00.000Z"></span>My preference is to keep <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"/></span> for <a href="/wiki/Frequency" title="Frequency">Ordinary frequency</a>, which is consistent with other articles. And I agree that <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 \xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b461aaf61091abd5d2c808931c48b8ff9647db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \xi }"/></span> is unnecessarily intimidating. So what we've pretty much settled on is to avoid using <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"/></span> for both things in the same article. It's the same thing in the literature, different authors, different preferences.</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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"/></span> is another one. Many articles use it instead of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"/></span>. So we have both <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(t)\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ X(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">⟺</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mi>X</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ X(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/858a3b88136ceceebe84bc5f9486003c9a41a8ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.524ex; height:4.176ex;" alt="{\displaystyle x(t)\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ X(f)}"/></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 f(x)\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ {\hat {f}}(\xi )}"> <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> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">⟺</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ {\hat {f}}(\xi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cb38d71da2c23f03006b608a37cae2fa26e6d41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.434ex; height:4.176ex;" alt="{\displaystyle f(x)\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ {\hat {f}}(\xi )}"/></span> expressing the same concept.</dd> <dd>Since I don't like using <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"/></span> for a function, and I do like using <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"/></span> for frequency, my preference is:</dd></dl> <dl><dd><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 s(t)\ ({\text{or}}\ s(x))\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ S(f),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>or</mtext> </mrow> <mtext> </mtext> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">⟺</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mi>S</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(t)\ ({\text{or}}\ s(x))\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ S(f),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f3d0f53e5ee6922a19890116ad2ba962017275a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.725ex; height:4.176ex;" alt="{\displaystyle s(t)\ ({\text{or}}\ s(x))\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ S(f),}"/></span> (<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 s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"/></span> for "signal" 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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"/></span> for "spectrum")<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></dd></dl></dd> <dd>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2022-03-15T19:52:00.000Z-Em3rgent0rdr-2022-03-15T07:31:00.000Z" class="ext-discussiontools-init-timestamplink">19:52, 15 March 2022 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2022-03-15T19:52:00.000Z-Em3rgent0rdr-2022-03-15T07:31:00.000Z"></span><span data-mw-comment-end="c-Bob_K-2022-03-15T19:52:00.000Z-Em3rgent0rdr-2022-03-15T07:31:00.000Z"></span></dd></dl> <span data-mw-comment-start="" id="c-Em3rgent0rdr-2022-03-16T11:03:00.000Z-Maybe_use_'ξ'_instead_of_'f'_for_frequency_to_avoid_conflict_with_'f'_for_funct"></span><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-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVan_Trees1968" class="citation book cs1">Van Trees, Harry L (1968). <i>Detection, Estimation, and Modulation Theory</i>. Vol. 1. New York: John Wiley. p. 680. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-09517-6" title="Special:BookSources/0-471-09517-6"><bdi>0-471-09517-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Detection%2C+Estimation%2C+and+Modulation+Theory&rft.place=New+York&rft.pages=680&rft.pub=John+Wiley&rft.date=1968&rft.isbn=0-471-09517-6&rft.aulast=Van+Trees&rft.aufirst=Harry+L&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATalk%3AFourier+series%2FArchive+2" class="Z3988"></span></span> </li> </ol></div></div> <dl><dd><dl><dd>Yeah, I understand. What I have now done is added a sentence at the start of "Table of common Fourier series" which says</dd> <dd>"Readers be aware that up until this point in the article, f had previously represented frequency, but no longer does."</dd> <dd>So I suppose I can accept this. <a href="/wiki/User:Em3rgent0rdr" title="User:Em3rgent0rdr">Em3rgent0rdr</a> (<a href="/wiki/User_talk:Em3rgent0rdr" title="User talk:Em3rgent0rdr">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Em3rgent0rdr-2022-03-16T11:03:00.000Z-Maybe_use_'ξ'_instead_of_'f'_for_frequency_to_avoid_conflict_with_'f'_for_funct" class="ext-discussiontools-init-timestamplink">11:03, 16 March 2022 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Em3rgent0rdr-2022-03-16T11:03:00.000Z-Maybe_use_'ξ'_instead_of_'f'_for_frequency_to_avoid_conflict_with_'f'_for_funct"></span><span data-mw-comment-end="c-Em3rgent0rdr-2022-03-16T11:03:00.000Z-Maybe_use_'ξ'_instead_of_'f'_for_frequency_to_avoid_conflict_with_'f'_for_funct"></span></dd></dl></dd></dl> <p><span data-mw-comment-start="" id="c-Bob_K-2022-03-16T12:04:00.000Z-Maybe_use_'ξ'_instead_of_'f'_for_frequency_to_avoid_conflict_with_'f'_for_funct"></span>Thank you. Actually, as I wrote the above I had forgotten that this article an exception to the "avoid using <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"/></span> for both things in the same article" mantra. (I knew there was at least one exception, but did not remember this is it.) Anyhow, I think your work-around is sufficient. If not, I'd be willing to replace f(x) with s(x) in the table, including the figures.<br/> --<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2022-03-16T12:04:00.000Z-Maybe_use_'ξ'_instead_of_'f'_for_frequency_to_avoid_conflict_with_'f'_for_funct" class="ext-discussiontools-init-timestamplink">12:04, 16 March 2022 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2022-03-16T12:04:00.000Z-Maybe_use_'ξ'_instead_of_'f'_for_frequency_to_avoid_conflict_with_'f'_for_funct"></span><span data-mw-comment-end="c-Bob_K-2022-03-16T12:04:00.000Z-Maybe_use_'ξ'_instead_of_'f'_for_frequency_to_avoid_conflict_with_'f'_for_funct"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Em3rgent0rdr-2022-03-16T12:37:00.000Z-Bob_K-2022-03-16T12:04:00.000Z"></span>If you could replace f(x) with s(x) in the table, that would be great and avoid any possible confusion. <a href="/wiki/User:Em3rgent0rdr" title="User:Em3rgent0rdr">Em3rgent0rdr</a> (<a href="/wiki/User_talk:Em3rgent0rdr" title="User talk:Em3rgent0rdr">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Em3rgent0rdr-2022-03-16T12:37:00.000Z-Bob_K-2022-03-16T12:04:00.000Z" class="ext-discussiontools-init-timestamplink">12:37, 16 March 2022 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Em3rgent0rdr-2022-03-16T12:37:00.000Z-Bob_K-2022-03-16T12:04:00.000Z"></span><span data-mw-comment-end="c-Em3rgent0rdr-2022-03-16T12:37:00.000Z-Bob_K-2022-03-16T12:04:00.000Z"></span></dd></dl> <p><span data-mw-comment-start="" id="c-Bob_K-2022-03-17T11:56:00.000Z-Maybe_use_'ξ'_instead_of_'f'_for_frequency_to_avoid_conflict_with_'f'_for_funct"></span>Happy to give it a try. --<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series/Archive_2#c-Bob_K-2022-03-17T11:56:00.000Z-Maybe_use_'ξ'_instead_of_'f'_for_frequency_to_avoid_conflict_with_'f'_for_funct" class="ext-discussiontools-init-timestamplink">11:56, 17 March 2022 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2022-03-17T11:56:00.000Z-Maybe_use_'ξ'_instead_of_'f'_for_frequency_to_avoid_conflict_with_'f'_for_funct"></span><span data-mw-comment-end="c-Bob_K-2022-03-17T11:56:00.000Z-Maybe_use_'ξ'_instead_of_'f'_for_frequency_to_avoid_conflict_with_'f'_for_funct"></span> </p>    </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=Talk:Fourier_series/Archive_2&oldid=1140628542">https://en.wikipedia.org/w/index.php?title=Talk:Fourier_series/Archive_2&oldid=1140628542</a>"</div></div> <div id="catlinks" class="catlinks catlinks-allhidden" data-mw="interface"></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 21 February 2023, at 00:26<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=Talk:Fourier_series/Archive_2&oldid=1140628542&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-74cc59cb9d-kfpjm","wgBackendResponseTime":185,"wgDiscussionToolsPageThreads":[{"headingLevel":2,"name":"h-87.69.5.52-2006-11-28T15:15:00.000Z","type":"heading","level":0,"id":"h-Possible_Error-2006-11-28T15:15:00.000Z","replies":[{"timestamp":"2006-11-28T15:15:00.000Z","author":"87.69.5.52","type":"comment","level":1,"id":"c-87.69.5.52-2006-11-28T15:15:00.000Z-Possible_Error","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Fresheneesz-2006-05-24T17:41:00.000Z","type":"heading","level":0,"id":"h-Format_of_equations-2006-05-24T17:41:00.000Z","replies":[{"timestamp":"2006-05-24T17:41:00.000Z","author":"Fresheneesz","type":"comment","level":1,"id":"c-Fresheneesz-2006-05-24T17:41:00.000Z-Format_of_equations","replies":[{"timestamp":"2006-05-25T01:04:00.000Z","author":"Oleg Alexandrov","type":"comment","level":2,"id":"c-Oleg_Alexandrov-2006-05-25T01:04:00.000Z-Fresheneesz-2006-05-24T17:41:00.000Z","replies":[{"timestamp":"2006-05-25T02:51:00.000Z","author":"Fresheneesz","type":"comment","level":3,"id":"c-Fresheneesz-2006-05-25T02:51:00.000Z-Oleg_Alexandrov-2006-05-25T01:04:00.000Z","replies":[{"timestamp":"2006-05-25T15:12:00.000Z","author":"Oleg Alexandrov","type":"comment","level":4,"id":"c-Oleg_Alexandrov-2006-05-25T15:12:00.000Z-Fresheneesz-2006-05-25T02:51:00.000Z","replies":[{"timestamp":"2006-05-26T04:18:00.000Z","author":"Fresheneesz","type":"comment","level":5,"id":"c-Fresheneesz-2006-05-26T04:18:00.000Z-Oleg_Alexandrov-2006-05-25T15:12:00.000Z","replies":[]}]}]},{"timestamp":"2006-11-24T22:08:00.000Z","author":"85.220.116.138","type":"comment","level":3,"id":"c-85.220.116.138-2006-11-24T22:08:00.000Z-Oleg_Alexandrov-2006-05-25T01:04:00.000Z","replies":[]}]}]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Fresheneesz-2006-05-26T21:20:00.000Z","type":"heading","level":0,"id":"h-relationship_between_real_and_complex_forms-2006-05-26T21:20:00.000Z","replies":[{"timestamp":"2006-05-26T21:20:00.000Z","author":"Fresheneesz","type":"comment","level":1,"id":"c-Fresheneesz-2006-05-26T21:20:00.000Z-relationship_between_real_and_complex_forms","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Fresheneesz-2006-07-02T19:08:00.000Z","type":"heading","level":0,"id":"h-Still_very_convoluted-2006-07-02T19:08:00.000Z","replies":[{"timestamp":"2006-07-02T19:08:00.000Z","author":"Fresheneesz","type":"comment","level":1,"id":"c-Fresheneesz-2006-07-02T19:08:00.000Z-Still_very_convoluted","replies":[]},{"timestamp":"2006-07-03T00:05:00.000Z","author":"Michael Hardy","type":"comment","level":1,"id":"c-Michael_Hardy-2006-07-03T00:05:00.000Z-Still_very_convoluted","replies":[]},{"timestamp":"2006-08-13T07:44:00.000Z","author":"Chrislewis.au","type":"comment","level":1,"id":"c-Chrislewis.au-2006-08-13T07:44:00.000Z-Still_very_convoluted","replies":[],"displayName":"Chris"},{"timestamp":"2006-11-28T03:00:00.000Z","author":"LovaAndriamanjay","type":"comment","level":1,"id":"c-LovaAndriamanjay-2006-11-28T03:00:00.000Z-Still_very_convoluted","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-18.244.5.107-2007-05-13T22:53:00.000Z","type":"heading","level":0,"id":"h-Error-2007-05-13T22:53:00.000Z","replies":[{"timestamp":"2007-05-13T22:53:00.000Z","author":"18.244.5.107","type":"comment","level":1,"id":"c-18.244.5.107-2007-05-13T22:53:00.000Z-Error","replies":[{"timestamp":"2007-05-14T01:05:00.000Z","author":"Oleg Alexandrov","type":"comment","level":2,"id":"c-Oleg_Alexandrov-2007-05-14T01:05:00.000Z-18.244.5.107-2007-05-13T22:53:00.000Z","replies":[]}]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Futurebird-2007-12-01T13:29:00.000Z","type":"heading","level":0,"id":"h-Trigonometric_series_redirect?-2007-12-01T13:29:00.000Z","replies":[{"timestamp":"2007-12-01T13:29:00.000Z","author":"Futurebird","type":"comment","level":1,"id":"c-Futurebird-2007-12-01T13:29:00.000Z-Trigonometric_series_redirect?","replies":[{"timestamp":"2007-12-01T15:12:00.000Z","author":"Bob K","type":"comment","level":2,"id":"c-Bob_K-2007-12-01T15:12:00.000Z-Futurebird-2007-12-01T13:29:00.000Z","replies":[]}]},{"timestamp":"2007-12-01T15:29:00.000Z","author":"Futurebird","type":"comment","level":1,"id":"c-Futurebird-2007-12-01T15:29:00.000Z-Trigonometric_series_redirect?","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Bob_K-2007-12-03T23:17:00.000Z","type":"heading","level":0,"id":"h-\"real_Fourier_coefficients\"-2007-12-03T23:17:00.000Z","replies":[{"timestamp":"2007-12-03T23:17:00.000Z","author":"Bob K","type":"comment","level":1,"id":"c-Bob_K-2007-12-03T23:17:00.000Z-\"real_Fourier_coefficients\"","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Bob_K-2007-12-04T11:54:00.000Z","type":"heading","level":0,"id":"h-Making_a_mountain_out_of_a_molehill_is_not_a_good_example.-2007-12-04T11:54:00.000Z","replies":[{"timestamp":"2007-12-04T11:54:00.000Z","author":"Bob K","type":"comment","level":1,"id":"c-Bob_K-2007-12-04T11:54:00.000Z-Making_a_mountain_out_of_a_molehill_is_not_a_good_example.","replies":[]},{"timestamp":"2008-01-08T19:57:00.000Z","author":"Loisel","type":"comment","level":1,"id":"c-Loisel-2008-01-08T19:57:00.000Z-Making_a_mountain_out_of_a_molehill_is_not_a_good_example.","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Bob_K-2007-12-04T12:16:00.000Z","type":"heading","level":0,"id":"h-inconsistent_use_of_\"L\"-2007-12-04T12:16:00.000Z","replies":[{"timestamp":"2007-12-04T12:16:00.000Z","author":"Bob K","type":"comment","level":1,"id":"c-Bob_K-2007-12-04T12:16:00.000Z-inconsistent_use_of_\"L\"","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Bob_K-2008-01-08T00:41:00.000Z","type":"heading","level":0,"id":"h-Interpretation?-2008-01-08T00:41:00.000Z","replies":[{"timestamp":"2008-01-08T00:41:00.000Z","author":"Bob K","type":"comment","level":1,"id":"c-Bob_K-2008-01-08T00:41:00.000Z-Interpretation?","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Bob_K-2008-01-08T15:25:00.000Z","type":"heading","level":0,"id":"h-General_case-2008-01-08T15:25:00.000Z","replies":[{"timestamp":"2008-01-08T15:25:00.000Z","author":"Bob K","type":"comment","level":1,"id":"c-Bob_K-2008-01-08T15:25:00.000Z-General_case","replies":[]},{"timestamp":"2008-01-08T18:25:00.000Z","author":"Bob K","type":"comment","level":1,"id":"c-Bob_K-2008-01-08T18:25:00.000Z-General_case","replies":[{"timestamp":"2008-01-08T20:04:00.000Z","author":"Loisel","type":"comment","level":2,"id":"c-Loisel-2008-01-08T20:04:00.000Z-Bob_K-2008-01-08T18:25:00.000Z","replies":[]}]},{"timestamp":"2008-01-08T21:48:00.000Z","author":"Bob K","type":"comment","level":1,"id":"c-Bob_K-2008-01-08T21:48:00.000Z-General_case","replies":[]},{"timestamp":"2008-01-10T00:02:00.000Z","author":"Loisel","type":"comment","level":1,"id":"c-Loisel-2008-01-10T00:02:00.000Z-General_case","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Loisel-2008-01-08T21:20:00.000Z","type":"heading","level":0,"id":"h-Article_is_of_very_low_quality.-2008-01-08T21:20:00.000Z","replies":[{"timestamp":"2008-01-08T21:20:00.000Z","author":"Loisel","type":"comment","level":1,"id":"c-Loisel-2008-01-08T21:20:00.000Z-Article_is_of_very_low_quality.","replies":[]},{"timestamp":"2008-01-08T22:45:00.000Z","author":"Bob K","type":"comment","level":1,"id":"c-Bob_K-2008-01-08T22:45:00.000Z-Article_is_of_very_low_quality.","replies":[]},{"timestamp":"2008-01-09T22:11:00.000Z","author":"Loisel","type":"comment","level":1,"id":"c-Loisel-2008-01-09T22:11:00.000Z-Article_is_of_very_low_quality.","replies":[]},{"timestamp":"2008-01-10T00:00:00.000Z","author":"Loisel","type":"comment","level":1,"id":"c-Loisel-2008-01-10T00:00:00.000Z-Article_is_of_very_low_quality.","replies":[{"timestamp":"2008-01-12T13:06:00.000Z","author":"Bob K","type":"comment","level":2,"id":"c-Bob_K-2008-01-12T13:06:00.000Z-Loisel-2008-01-10T00:00:00.000Z","replies":[]},{"timestamp":"2008-01-12T13:22:00.000Z","author":"Bob K","type":"comment","level":2,"id":"c-Bob_K-2008-01-12T13:22:00.000Z-Loisel-2008-01-10T00:00:00.000Z","replies":[]}]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Loisel-2008-01-10T21:53:00.000Z","type":"heading","level":0,"id":"h-Looking_for_an_image_of_a_vibrating_drum-2008-01-10T21:53:00.000Z","replies":[{"timestamp":"2008-01-10T21:53:00.000Z","author":"Loisel","type":"comment","level":1,"id":"c-Loisel-2008-01-10T21:53:00.000Z-Looking_for_an_image_of_a_vibrating_drum","replies":[{"timestamp":"2008-01-11T05:28:00.000Z","author":"Oleg Alexandrov","type":"comment","level":2,"id":"c-Oleg_Alexandrov-2008-01-11T05:28:00.000Z-Loisel-2008-01-10T21:53:00.000Z","replies":[{"timestamp":"2008-01-12T06:37:00.000Z","author":"Oleg Alexandrov","type":"comment","level":3,"id":"c-Oleg_Alexandrov-2008-01-12T06:37:00.000Z-Oleg_Alexandrov-2008-01-11T05:28:00.000Z","replies":[]}]}]},{"timestamp":"2008-01-13T14:01:00.000Z","author":"Loisel","type":"comment","level":1,"id":"c-Loisel-2008-01-13T14:01:00.000Z-Looking_for_an_image_of_a_vibrating_drum","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Bob_K-2008-01-12T23:46:00.000Z","type":"heading","level":0,"id":"h-Contradiction-2008-01-12T23:46:00.000Z","replies":[{"timestamp":"2008-01-12T23:46:00.000Z","author":"Bob K","type":"comment","level":1,"id":"c-Bob_K-2008-01-12T23:46:00.000Z-Contradiction","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Bob_K-2008-01-13T00:01:00.000Z","type":"heading","level":0,"id":"h-The_new_figure_is_not_as_good_as_the_old_one-2008-01-13T00:01:00.000Z","replies":[{"timestamp":"2008-01-13T00:01:00.000Z","author":"Bob K","type":"comment","level":1,"id":"c-Bob_K-2008-01-13T00:01:00.000Z-The_new_figure_is_not_as_good_as_the_old_one","replies":[]},{"timestamp":"2008-01-22T15:10:00.000Z","author":"Loisel","type":"comment","level":1,"id":"c-Loisel-2008-01-22T15:10:00.000Z-The_new_figure_is_not_as_good_as_the_old_one","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Bob_K-2008-01-24T15:58:00.000Z","type":"heading","level":0,"id":"h-incorrect_use_of_\"Fourier_transform\"-2008-01-24T15:58:00.000Z","replies":[{"timestamp":"2008-01-24T15:58:00.000Z","author":"Bob K","type":"comment","level":1,"id":"c-Bob_K-2008-01-24T15:58:00.000Z-incorrect_use_of_\"Fourier_transform\"","replies":[]},{"timestamp":"2008-01-24T19:56:00.000Z","author":"Loisel","type":"comment","level":1,"id":"c-Loisel-2008-01-24T19:56:00.000Z-incorrect_use_of_\"Fourier_transform\"","replies":[]},{"timestamp":"2008-01-24T20:44:00.000Z","author":"Bob K","type":"comment","level":1,"id":"c-Bob_K-2008-01-24T20:44:00.000Z-incorrect_use_of_\"Fourier_transform\"","replies":[{"timestamp":"2008-01-26T22:05:00.000Z","author":"Stevenj","type":"comment","level":2,"id":"c-Stevenj-2008-01-26T22:05:00.000Z-Bob_K-2008-01-24T20:44:00.000Z","replies":[],"displayName":"—Steven G. Johnson"}]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Loisel-2008-02-02T03:44:00.000Z","type":"heading","level":0,"id":"h-Simplified_example-2008-02-02T03:44:00.000Z","replies":[{"timestamp":"2008-02-02T03:44:00.000Z","author":"Loisel","type":"comment","level":1,"id":"c-Loisel-2008-02-02T03:44:00.000Z-Simplified_example","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Bob_K-2008-02-05T19:24:00.000Z","type":"heading","level":0,"id":"h-Now_sections_are_in_the_wrong_order-2008-02-05T19:24:00.000Z","replies":[{"timestamp":"2008-02-05T19:24:00.000Z","author":"Bob K","type":"comment","level":1,"id":"c-Bob_K-2008-02-05T19:24:00.000Z-Now_sections_are_in_the_wrong_order","replies":[]},{"timestamp":"2008-02-05T19:30:00.000Z","author":"Bob K","type":"comment","level":1,"id":"c-Bob_K-2008-02-05T19:30:00.000Z-Now_sections_are_in_the_wrong_order","replies":[]},{"timestamp":"2008-02-05T21:59:00.000Z","author":"Loisel","type":"comment","level":1,"id":"c-Loisel-2008-02-05T21:59:00.000Z-Now_sections_are_in_the_wrong_order","replies":[]},{"timestamp":"2008-02-06T01:00:00.000Z","author":"Bob K","type":"comment","level":1,"id":"c-Bob_K-2008-02-06T01:00:00.000Z-Now_sections_are_in_the_wrong_order","replies":[]},{"timestamp":"2008-02-06T07:33:00.000Z","author":"Loisel","type":"comment","level":1,"id":"c-Loisel-2008-02-06T07:33:00.000Z-Now_sections_are_in_the_wrong_order","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-192.35.17.15-2008-02-12T12:29:00.000Z","type":"heading","level":0,"id":"h-This_article_lacks_basic_explanation-2008-02-12T12:29:00.000Z","replies":[{"timestamp":"2008-02-12T12:29:00.000Z","author":"192.35.17.15","type":"comment","level":1,"id":"c-192.35.17.15-2008-02-12T12:29:00.000Z-This_article_lacks_basic_explanation","replies":[]},{"timestamp":"2008-02-13T04:40:00.000Z","author":"Loisel","type":"comment","level":1,"id":"c-Loisel-2008-02-13T04:40:00.000Z-This_article_lacks_basic_explanation","replies":[]},{"timestamp":"2008-02-15T14:47:00.000Z","author":"Bob K","type":"comment","level":1,"id":"c-Bob_K-2008-02-15T14:47:00.000Z-This_article_lacks_basic_explanation","replies":[]},{"timestamp":"2008-02-15T22:39:00.000Z","author":"Loisel","type":"comment","level":1,"id":"c-Loisel-2008-02-15T22:39:00.000Z-This_article_lacks_basic_explanation","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-129.241.138.61-2008-04-24T15:11:00.000Z","type":"heading","level":0,"id":"h-Dead_link-2008-04-24T15:11:00.000Z","replies":[{"timestamp":"2008-04-24T15:11:00.000Z","author":"129.241.138.61","type":"comment","level":1,"id":"c-129.241.138.61-2008-04-24T15:11:00.000Z-Dead_link","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Thenub314-2008-04-24T20:12:00.000Z","type":"heading","level":0,"id":"h-Introductory_Sentence.-2008-04-24T20:12:00.000Z","replies":[{"timestamp":"2008-04-24T20:12:00.000Z","author":"Thenub314","type":"comment","level":1,"id":"c-Thenub314-2008-04-24T20:12:00.000Z-Introductory_Sentence.","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Thenub314-2008-05-12T02:30:00.000Z","type":"heading","level":0,"id":"h-periodic_functions_as_tempered_distriutions.-2008-05-12T02:30:00.000Z","replies":[{"timestamp":"2008-05-12T02:30:00.000Z","author":"Thenub314","type":"comment","level":1,"id":"c-Thenub314-2008-05-12T02:30:00.000Z-periodic_functions_as_tempered_distriutions.","replies":[{"timestamp":"2008-05-12T02:37:00.000Z","author":"Dicklyon","type":"comment","level":2,"id":"c-Dicklyon-2008-05-12T02:37:00.000Z-Thenub314-2008-05-12T02:30:00.000Z","replies":[]}]},{"timestamp":"2008-05-12T16:40:00.000Z","author":"Loisel","type":"comment","level":1,"id":"c-Loisel-2008-05-12T16:40:00.000Z-periodic_functions_as_tempered_distriutions.","replies":[]},{"timestamp":"2008-05-12T16:44:00.000Z","author":"Loisel","type":"comment","level":1,"id":"c-Loisel-2008-05-12T16:44:00.000Z-periodic_functions_as_tempered_distriutions.","replies":[{"timestamp":"2008-05-12T17:42:00.000Z","author":"Dicklyon","type":"comment","level":2,"id":"c-Dicklyon-2008-05-12T17:42:00.000Z-Loisel-2008-05-12T16:44:00.000Z","replies":[]}]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Gmcastil-2008-05-25T04:28:00.000Z","type":"heading","level":0,"id":"h-Vibrating_Drum-2008-05-25T04:28:00.000Z","replies":[{"timestamp":"2008-05-25T04:28:00.000Z","author":"Gmcastil","type":"comment","level":1,"id":"c-Gmcastil-2008-05-25T04:28:00.000Z-Vibrating_Drum","replies":[]},{"timestamp":"2008-05-26T15:58:00.000Z","author":"Loisel","type":"comment","level":1,"id":"c-Loisel-2008-05-26T15:58:00.000Z-Vibrating_Drum","replies":[]},{"timestamp":"2008-05-27T21:55:00.000Z","author":"Mposey82","type":"comment","level":1,"id":"c-Mposey82-2008-05-27T21:55:00.000Z-Vibrating_Drum","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Dicklyon-2008-09-10T04:35:00.000Z","type":"heading","level":0,"id":"h-Ping-pong_sections-2008-09-10T04:35:00.000Z","replies":[{"timestamp":"2008-09-10T04:35:00.000Z","author":"Dicklyon","type":"comment","level":1,"id":"c-Dicklyon-2008-09-10T04:35:00.000Z-Ping-pong_sections","replies":[{"timestamp":"2008-09-10T06:50:00.000Z","author":"Thenub314","type":"comment","level":2,"id":"c-Thenub314-2008-09-10T06:50:00.000Z-Dicklyon-2008-09-10T04:35:00.000Z","replies":[{"timestamp":"2008-09-10T06:59:00.000Z","author":"Dicklyon","type":"comment","level":3,"id":"c-Dicklyon-2008-09-10T06:59:00.000Z-Thenub314-2008-09-10T06:50:00.000Z","replies":[{"timestamp":"2008-09-10T17:42:00.000Z","author":"Thenub314","type":"comment","level":4,"id":"c-Thenub314-2008-09-10T17:42:00.000Z-Dicklyon-2008-09-10T06:59:00.000Z","replies":[{"timestamp":"2008-09-19T21:13:00.000Z","author":"Thenub314","type":"comment","level":5,"id":"c-Thenub314-2008-09-19T21:13:00.000Z-Thenub314-2008-09-10T17:42:00.000Z","replies":[]}]}]}]}]},{"timestamp":"2008-09-10T17:42:00.000Z","author":"Thenub314","type":"comment","level":1,"id":"c-Thenub314-2008-09-10T17:42:00.000Z-Ping-pong_sections","replies":[]},{"timestamp":"2008-09-10T16:39:00.000Z","author":"Loisel","type":"comment","level":1,"id":"c-Loisel-2008-09-10T16:39:00.000Z-Ping-pong_sections","replies":[{"timestamp":"2008-09-10T18:30:00.000Z","author":"Thenub314","type":"comment","level":2,"id":"c-Thenub314-2008-09-10T18:30:00.000Z-Loisel-2008-09-10T16:39:00.000Z","replies":[]}]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Squidgyhead-2008-11-17T01:18:00.000Z","type":"heading","level":0,"id":"h-Example:_a_simple_Fourier_series-2008-11-17T01:18:00.000Z","replies":[{"timestamp":"2008-11-17T01:18:00.000Z","author":"Squidgyhead","type":"comment","level":1,"id":"c-Squidgyhead-2008-11-17T01:18:00.000Z-Example:_a_simple_Fourier_series","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Dicklyon-2008-12-17T14:16:00.000Z","type":"heading","level":0,"id":"h-Sines_and_cosines_more_\"accessible\"_than_sinusoids?-2008-12-17T14:16:00.000Z","replies":[{"timestamp":"2008-12-17T14:16:00.000Z","author":"Dicklyon","type":"comment","level":1,"id":"c-Dicklyon-2008-12-17T14:16:00.000Z-Sines_and_cosines_more_\"accessible\"_than_sinusoids?","replies":[]},{"timestamp":"2008-12-17T16:06:00.000Z","author":"Thenub314","type":"comment","level":1,"id":"c-Thenub314-2008-12-17T16:06:00.000Z-Sines_and_cosines_more_\"accessible\"_than_sinusoids?","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Bdmy-2008-12-28T09:34:00.000Z","type":"heading","level":0,"id":"h-Definition_of_Fourier_series-2008-12-28T09:34:00.000Z","replies":[{"timestamp":"2008-12-28T09:34:00.000Z","author":"Bdmy","type":"comment","level":1,"id":"c-Bdmy-2008-12-28T09:34:00.000Z-Definition_of_Fourier_series","replies":[]},{"timestamp":"2008-12-28T10:48:00.000Z","author":"Thenub314","type":"comment","level":1,"id":"c-Thenub314-2008-12-28T10:48:00.000Z-Definition_of_Fourier_series","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Danielsimonjr-2008-12-28T16:59:00.000Z","type":"heading","level":0,"id":"h-Formula_for_a0-2008-12-28T16:59:00.000Z","replies":[{"timestamp":"2008-12-28T16:59:00.000Z","author":"Danielsimonjr","type":"comment","level":1,"id":"c-Danielsimonjr-2008-12-28T16:59:00.000Z-Formula_for_a0","replies":[{"timestamp":"2008-12-28T18:24:00.000Z","author":"Bdmy","type":"comment","level":2,"id":"c-Bdmy-2008-12-28T18:24:00.000Z-Danielsimonjr-2008-12-28T16:59:00.000Z","replies":[]}]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Taeshadow-2009-03-11T18:33:00.000Z","type":"heading","level":0,"id":"h-Merger_proposal-2009-03-11T18:33:00.000Z","replies":[{"timestamp":"2009-03-11T18:33:00.000Z","author":"Taeshadow","type":"comment","level":1,"id":"c-Taeshadow-2009-03-11T18:33:00.000Z-Merger_proposal","replies":[{"timestamp":"2009-03-13T05:24:00.000Z","author":"Dicklyon","type":"comment","level":2,"id":"c-Dicklyon-2009-03-13T05:24:00.000Z-Taeshadow-2009-03-11T18:33:00.000Z","replies":[{"timestamp":"2009-04-16T01:47:00.000Z","author":"Paul Laroque","type":"comment","level":3,"id":"c-Paul_Laroque-2009-04-16T01:47:00.000Z-Dicklyon-2009-03-13T05:24:00.000Z","replies":[]}]}]},{"timestamp":"2009-04-16T02:13:00.000Z","author":"Michael Hardy","type":"comment","level":1,"id":"c-Michael_Hardy-2009-04-16T02:13:00.000Z-Merger_proposal","replies":[{"timestamp":"2009-04-16T02:17:00.000Z","author":"Dicklyon","type":"comment","level":2,"id":"c-Dicklyon-2009-04-16T02:17:00.000Z-Michael_Hardy-2009-04-16T02:13:00.000Z","replies":[]}]}],"uneditableSection":true},{"headingLevel":2,"name":"h-98.210.57.10-2009-11-28T02:25:00.000Z","type":"heading","level":0,"id":"h-Reviewers_of_the_original_Fourier_article-2009-11-28T02:25:00.000Z","replies":[{"timestamp":"2009-11-28T02:25:00.000Z","author":"98.210.57.10","type":"comment","level":1,"id":"c-98.210.57.10-2009-11-28T02:25:00.000Z-Reviewers_of_the_original_Fourier_article","replies":[{"timestamp":"2009-11-30T09:47:00.000Z","author":"LutzL","type":"comment","level":2,"id":"c-LutzL-2009-11-30T09:47:00.000Z-98.210.57.10-2009-11-28T02:25:00.000Z","replies":[]}]}],"uneditableSection":true},{"headingLevel":2,"name":"h-129.105.207.213-2008-05-02T04:12:00.000Z","type":"heading","level":0,"id":"h-Vibrating_Disc_doesn't_look_right-2008-05-02T04:12:00.000Z","replies":[{"timestamp":"2009-12-14T16:16:00.000Z","author":"117.196.34.205","type":"comment","level":1,"id":"c-117.196.34.205-2009-12-14T16:16:00.000Z-Vibrating_Disc_doesn't_look_right","replies":[]},{"timestamp":"2008-05-02T04:12:00.000Z","author":"129.105.207.213","type":"comment","level":1,"id":"c-129.105.207.213-2008-05-02T04:12:00.000Z-Vibrating_Disc_doesn't_look_right","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-192.35.17.15-2007-12-19T16:00:00.000Z","type":"heading","level":0,"id":"h-Too_technical-2007-12-19T16:00:00.000Z","replies":[{"timestamp":"2007-12-19T16:00:00.000Z","author":"192.35.17.15","type":"comment","level":1,"id":"c-192.35.17.15-2007-12-19T16:00:00.000Z-Too_technical","replies":[{"timestamp":"2007-12-20T22:47:00.000Z","author":"Bob K","type":"comment","level":2,"id":"c-Bob_K-2007-12-20T22:47:00.000Z-192.35.17.15-2007-12-19T16:00:00.000Z","replies":[]}]},{"timestamp":"2007-12-21T01:09:00.000Z","author":"Michael Hardy","type":"comment","level":1,"id":"c-Michael_Hardy-2007-12-21T01:09:00.000Z-Too_technical","replies":[{"timestamp":"2008-01-08T20:32:00.000Z","author":"Loisel","type":"comment","level":2,"id":"c-Loisel-2008-01-08T20:32:00.000Z-Michael_Hardy-2007-12-21T01:09:00.000Z","replies":[{"timestamp":"2008-01-09T12:25:00.000Z","author":"137.222.187.157","type":"comment","level":3,"id":"c-137.222.187.157-2008-01-09T12:25:00.000Z-Loisel-2008-01-08T20:32:00.000Z","replies":[{"timestamp":"2008-01-09T22:29:00.000Z","author":"Loisel","type":"comment","level":4,"id":"c-Loisel-2008-01-09T22:29:00.000Z-137.222.187.157-2008-01-09T12:25:00.000Z","replies":[]},{"timestamp":"2009-10-17T18:16:00.000Z","author":"97.123.87.61","type":"comment","level":4,"id":"c-97.123.87.61-2009-10-17T18:16:00.000Z-137.222.187.157-2008-01-09T12:25:00.000Z","replies":[]}]}]}]},{"timestamp":"2010-10-06T07:46:00.000Z","author":"PrBeacon","type":"comment","level":1,"id":"c-PrBeacon-2010-10-06T07:46:00.000Z-Too_technical","replies":[{"timestamp":"2010-10-06T07:56:00.000Z","author":"PrBeacon","type":"comment","level":2,"id":"c-PrBeacon-2010-10-06T07:56:00.000Z-PrBeacon-2010-10-06T07:46:00.000Z","replies":[{"timestamp":"2010-10-06T13:33:00.000Z","author":"LutzL","type":"comment","level":3,"id":"c-LutzL-2010-10-06T13:33:00.000Z-PrBeacon-2010-10-06T07:56:00.000Z","replies":[{"timestamp":"2010-10-07T01:03:00.000Z","author":"PrBeacon","type":"comment","level":4,"id":"c-PrBeacon-2010-10-07T01:03:00.000Z-LutzL-2010-10-06T13:33:00.000Z","replies":[]}]}]}]},{"timestamp":"2010-12-09T06:55:00.000Z","author":"72.225.204.182","type":"comment","level":1,"id":"c-72.225.204.182-2010-12-09T06:55:00.000Z-Too_technical","replies":[{"timestamp":"2010-12-09T07:50:00.000Z","author":"Dicklyon","type":"comment","level":2,"id":"c-Dicklyon-2010-12-09T07:50:00.000Z-72.225.204.182-2010-12-09T06:55:00.000Z","replies":[]}]}],"uneditableSection":true},{"headingLevel":2,"name":"h-70.119.103.136-2010-05-24T04:14:00.000Z","type":"heading","level":0,"id":"h-Very_first_example-2010-05-24T04:14:00.000Z","replies":[{"timestamp":"2010-05-24T04:14:00.000Z","author":"70.119.103.136","type":"comment","level":1,"id":"c-70.119.103.136-2010-05-24T04:14:00.000Z-Very_first_example","replies":[]},{"timestamp":"2010-05-26T02:51:00.000Z","author":"128.95.41.29","type":"comment","level":1,"id":"c-128.95.41.29-2010-05-26T02:51:00.000Z-Very_first_example","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Tcnuk-2010-06-09T15:56:00.000Z","type":"heading","level":0,"id":"h-Quote_isn't_actually_Fourier's?-2010-06-09T15:56:00.000Z","replies":[{"timestamp":"2010-06-09T15:56:00.000Z","author":"Tcnuk","type":"comment","level":1,"id":"c-Tcnuk-2010-06-09T15:56:00.000Z-Quote_isn't_actually_Fourier's?","replies":[{"timestamp":"2010-06-10T12:03:00.000Z","author":"Tcnuk","type":"comment","level":2,"id":"c-Tcnuk-2010-06-10T12:03:00.000Z-Tcnuk-2010-06-09T15:56:00.000Z","replies":[]},{"timestamp":"2010-12-09T16:22:00.000Z","author":"Tamfang","type":"comment","level":2,"id":"c-Tamfang-2010-12-09T16:22:00.000Z-Tcnuk-2010-06-09T15:56:00.000Z","replies":[]},{"timestamp":"2010-12-09T19:20:00.000Z","author":"LutzL","type":"comment","level":2,"id":"c-LutzL-2010-12-09T19:20:00.000Z-Tcnuk-2010-06-09T15:56:00.000Z","replies":[]},{"timestamp":"2010-12-09T19:31:00.000Z","author":"LutzL","type":"comment","level":2,"id":"c-LutzL-2010-12-09T19:31:00.000Z-Tcnuk-2010-06-09T15:56:00.000Z","replies":[{"timestamp":"2010-12-09T20:26:00.000Z","author":"LutzL","type":"comment","level":3,"id":"c-LutzL-2010-12-09T20:26:00.000Z-LutzL-2010-12-09T19:31:00.000Z","replies":[]}]},{"timestamp":"2011-01-14T21:18:00.000Z","author":"LutzL","type":"comment","level":2,"id":"c-LutzL-2011-01-14T21:18:00.000Z-Tcnuk-2010-06-09T15:56:00.000Z","replies":[]}]}],"uneditableSection":true},{"headingLevel":2,"name":"h-99.52.193.237-2011-01-13T16:09:00.000Z","type":"heading","level":0,"id":"h-Formula_incorrect-2011-01-13T16:09:00.000Z","replies":[{"timestamp":"2011-01-13T16:09:00.000Z","author":"99.52.193.237","type":"comment","level":1,"id":"c-99.52.193.237-2011-01-13T16:09:00.000Z-Formula_incorrect","replies":[{"timestamp":"2011-01-14T18:39:00.000Z","author":"Bob K","type":"comment","level":2,"id":"c-Bob_K-2011-01-14T18:39:00.000Z-99.52.193.237-2011-01-13T16:09:00.000Z","replies":[]}]},{"timestamp":"2011-01-16T17:04:00.000Z","author":"99.52.193.237","type":"comment","level":1,"id":"c-99.52.193.237-2011-01-16T17:04:00.000Z-Formula_incorrect","replies":[{"timestamp":"2011-01-16T17:12:00.000Z","author":"LutzL","type":"comment","level":2,"id":"c-LutzL-2011-01-16T17:12:00.000Z-99.52.193.237-2011-01-16T17:04:00.000Z","replies":[]}]},{"timestamp":"2011-05-04T23:49:00.000Z","author":"Brvman","type":"comment","level":1,"id":"c-Brvman-2011-05-04T23:49:00.000Z-Formula_incorrect","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Cephas_Borg-2011-12-07T14:52:00.000Z","type":"heading","level":0,"id":"h-Formula_and_graph_axis_labelling-2011-12-07T14:52:00.000Z","replies":[{"timestamp":"2011-12-07T14:52:00.000Z","author":"Cephas Borg","type":"comment","level":1,"id":"c-Cephas_Borg-2011-12-07T14:52:00.000Z-Formula_and_graph_axis_labelling","replies":[]},{"timestamp":"2013-02-10T02:11:00.000Z","author":"172.190.77.188","type":"comment","level":1,"id":"c-172.190.77.188-2013-02-10T02:11:00.000Z-Formula_and_graph_axis_labelling","replies":[{"timestamp":"2011-12-07T18:16:00.000Z","author":"Bob K","type":"comment","level":2,"id":"c-Bob_K-2011-12-07T18:16:00.000Z-172.190.77.188-2013-02-10T02:11:00.000Z","replies":[]}]},{"timestamp":"2011-12-08T02:49:00.000Z","author":"Cephas Borg","type":"comment","level":1,"id":"c-Cephas_Borg-2011-12-08T02:49:00.000Z-Formula_and_graph_axis_labelling","replies":[{"timestamp":"2011-12-09T01:20:00.000Z","author":"Bob K","type":"comment","level":2,"id":"c-Bob_K-2011-12-09T01:20:00.000Z-Cephas_Borg-2011-12-08T02:49:00.000Z","replies":[{"timestamp":"2013-02-10T06:42:00.000Z","author":"Olli Niemitalo","type":"comment","level":3,"id":"c-Olli_Niemitalo-2013-02-10T06:42:00.000Z-Bob_K-2011-12-09T01:20:00.000Z","replies":[]}]}]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Brews_ohare-2012-04-21T17:07:00.000Z","type":"heading","level":0,"id":"h-Request_for_comment_by_readers_of_this_article-2012-04-21T17:07:00.000Z","replies":[{"timestamp":"2012-04-21T17:07:00.000Z","author":"Brews ohare","type":"comment","level":1,"id":"c-Brews_ohare-2012-04-21T17:07:00.000Z-Request_for_comment_by_readers_of_this_article","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-5.151.82.6-2012-11-10T04:10:00.000Z","type":"heading","level":0,"id":"h-Riemann_paper_and_1811_essay_competition-2012-11-10T04:10:00.000Z","replies":[{"timestamp":"2012-11-10T04:10:00.000Z","author":"5.151.82.6","type":"comment","level":1,"id":"c-5.151.82.6-2012-11-10T04:10:00.000Z-Riemann_paper_and_1811_essay_competition","replies":[{"timestamp":"2012-11-10T04:45:00.000Z","author":"5.151.82.6","type":"comment","level":2,"id":"c-5.151.82.6-2012-11-10T04:45:00.000Z-5.151.82.6-2012-11-10T04:10:00.000Z","replies":[]}]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Arbitrarily0-2013-03-11T14:22:00.000Z","type":"heading","level":0,"id":"h-Fourier's_theorem-2013-03-11T14:22:00.000Z","replies":[{"timestamp":"2013-03-11T14:22:00.000Z","author":"Arbitrarily0","type":"comment","level":1,"id":"c-Arbitrarily0-2013-03-11T14:22:00.000Z-Fourier's_theorem","replies":[]},{"timestamp":"2013-03-12T21:00:00.000Z","author":"Bob K","type":"comment","level":1,"id":"c-Bob_K-2013-03-12T21:00:00.000Z-Fourier's_theorem","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Bdmy-2013-08-11T20:40:00.000Z","type":"heading","level":0,"id":"h-Fourier_series_on_a_general_interval_[a,_a_+_τ]-2013-08-11T20:40:00.000Z","replies":[{"timestamp":"2013-08-11T20:40:00.000Z","author":"Bdmy","type":"comment","level":1,"id":"c-Bdmy-2013-08-11T20:40:00.000Z-Fourier_series_on_a_general_interval_[a,_a_+_τ]","replies":[{"timestamp":"2013-08-13T13:58:00.000Z","author":"Bob K","type":"comment","level":2,"id":"c-Bob_K-2013-08-13T13:58:00.000Z-Bdmy-2013-08-11T20:40:00.000Z","replies":[]}]}],"uneditableSection":true},{"headingLevel":2,"name":"h-A_Thousand_Doors-2013-08-22T23:35:00.000Z","type":"heading","level":0,"id":"h-Fourier's_formulae_for_T-periodic_functions-2013-08-22T23:35:00.000Z","replies":[{"timestamp":"2013-08-22T23:35:00.000Z","author":"A Thousand Doors","type":"comment","level":1,"id":"c-A_Thousand_Doors-2013-08-22T23:35:00.000Z-Fourier's_formulae_for_T-periodic_functions","replies":[{"timestamp":"2013-08-23T03:39:00.000Z","author":"Bob K","type":"comment","level":2,"id":"c-Bob_K-2013-08-23T03:39:00.000Z-A_Thousand_Doors-2013-08-22T23:35:00.000Z","replies":[{"timestamp":"2013-08-23T10:42:00.000Z","author":"A Thousand Doors","type":"comment","level":3,"id":"c-A_Thousand_Doors-2013-08-23T10:42:00.000Z-Bob_K-2013-08-23T03:39:00.000Z","replies":[]}]},{"timestamp":"2013-08-23T13:02:00.000Z","author":"Bob K","type":"comment","level":2,"id":"c-Bob_K-2013-08-23T13:02:00.000Z-A_Thousand_Doors-2013-08-22T23:35:00.000Z","replies":[{"timestamp":"2013-08-27T11:08:00.000Z","author":"A Thousand Doors","type":"comment","level":3,"id":"c-A_Thousand_Doors-2013-08-27T11:08:00.000Z-Bob_K-2013-08-23T13:02:00.000Z","replies":[]}]},{"timestamp":"2013-08-27T12:18:00.000Z","author":"Bob K","type":"comment","level":2,"id":"c-Bob_K-2013-08-27T12:18:00.000Z-A_Thousand_Doors-2013-08-22T23:35:00.000Z","replies":[{"timestamp":"2013-08-27T12:27:00.000Z","author":"A Thousand Doors","type":"comment","level":3,"id":"c-A_Thousand_Doors-2013-08-27T12:27:00.000Z-Bob_K-2013-08-27T12:18:00.000Z","replies":[]},{"timestamp":"2013-08-30T22:16:00.000Z","author":"A Thousand Doors","type":"comment","level":3,"id":"c-A_Thousand_Doors-2013-08-30T22:16:00.000Z-Bob_K-2013-08-27T12:18:00.000Z","replies":[]}]}]}],"uneditableSection":true},{"headingLevel":2,"name":"h-RJFJR-2013-09-10T19:42:00.000Z","type":"heading","level":0,"id":"h-translation_of_paper's_title?-2013-09-10T19:42:00.000Z","replies":[{"timestamp":"2013-09-10T19:42:00.000Z","author":"RJFJR","type":"comment","level":1,"id":"c-RJFJR-2013-09-10T19:42:00.000Z-translation_of_paper's_title?","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Wangguansong-2013-11-06T16:42:00.000Z","type":"heading","level":0,"id":"h-The_coefficient_in_\"2.1_Example_1:_a_simple_Fourier_series\"-2013-11-06T16:42:00.000Z","replies":[{"timestamp":"2013-11-06T16:42:00.000Z","author":"Wangguansong","type":"comment","level":1,"id":"c-Wangguansong-2013-11-06T16:42:00.000Z-The_coefficient_in_\"2.1_Example_1:_a_simple_Fourier_series\"","replies":[{"timestamp":"2013-11-06T22:44:00.000Z","author":"Bob K","type":"comment","level":2,"id":"c-Bob_K-2013-11-06T22:44:00.000Z-Wangguansong-2013-11-06T16:42:00.000Z","replies":[{"timestamp":"2013-11-07T14:35:00.000Z","author":"Wangguansong","type":"comment","level":3,"id":"c-Wangguansong-2013-11-07T14:35:00.000Z-Bob_K-2013-11-06T22:44:00.000Z","replies":[]}]}]}],"uneditableSection":true},{"headingLevel":2,"name":"h-174.3.243.185-2014-02-10T02:15:00.000Z","type":"heading","level":0,"id":"h-Why_are_we_using_a_finite_series_in_the_definition?-2014-02-10T02:15:00.000Z","replies":[{"timestamp":"2014-02-10T02:15:00.000Z","author":"174.3.243.185","type":"comment","level":1,"id":"c-174.3.243.185-2014-02-10T02:15:00.000Z-Why_are_we_using_a_finite_series_in_the_definition?","replies":[{"timestamp":"2014-02-10T05:14:00.000Z","author":"Bob K","type":"comment","level":2,"id":"c-Bob_K-2014-02-10T05:14:00.000Z-174.3.243.185-2014-02-10T02:15:00.000Z","replies":[]}]},{"timestamp":"2014-02-10T17:40:00.000Z","author":"198.73.178.11","type":"comment","level":1,"id":"c-198.73.178.11-2014-02-10T17:40:00.000Z-Why_are_we_using_a_finite_series_in_the_definition?","replies":[{"timestamp":"2014-02-10T18:33:00.000Z","author":"Stevenj","type":"comment","level":2,"id":"c-Stevenj-2014-02-10T18:33:00.000Z-198.73.178.11-2014-02-10T17:40:00.000Z","replies":[],"displayName":"— Steven G. Johnson"}]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Bajatmerc-2013-09-17T19:58:00.000Z","type":"heading","level":0,"id":"h-layperson_readability-2013-09-17T19:58:00.000Z","replies":[{"timestamp":"2013-09-17T19:58:00.000Z","author":"Bajatmerc","type":"comment","level":1,"id":"c-Bajatmerc-2013-09-17T19:58:00.000Z-layperson_readability","replies":[{"timestamp":"2013-09-17T23:26:00.000Z","author":"Bob K","type":"comment","level":2,"id":"c-Bob_K-2013-09-17T23:26:00.000Z-Bajatmerc-2013-09-17T19:58:00.000Z","replies":[{"timestamp":"2014-05-13T22:26:00.000Z","author":"Trlkly","type":"comment","level":3,"id":"c-Trlkly-2014-05-13T22:26:00.000Z-Bob_K-2013-09-17T23:26:00.000Z","replies":[]}]}]}],"uneditableSection":true},{"headingLevel":2,"name":"h-دالبا-2014-02-02T08:38:00.000Z","type":"heading","level":0,"id":"h-1/2_[f(x0+)+f(x0-)]-2014-02-02T08:38:00.000Z","replies":[{"timestamp":"2014-02-02T08:38:00.000Z","author":"دالبا","type":"comment","level":1,"id":"c-دالبا-2014-02-02T08:38:00.000Z-1/2_[f(x0+)+f(x0-)]","replies":[{"timestamp":"2014-05-14T10:52:00.000Z","author":"LutzL","type":"comment","level":2,"id":"c-LutzL-2014-05-14T10:52:00.000Z-دالبا-2014-02-02T08:38:00.000Z","replies":[]}],"displayName":"Dalba"}],"uneditableSection":true},{"headingLevel":2,"name":"h-Bob_K-2014-03-07T13:48:00.000Z","type":"heading","level":0,"id":"h-Eye-catching_GIF-2014-03-07T13:48:00.000Z","replies":[{"timestamp":"2014-03-07T13:48:00.000Z","author":"Bob K","type":"comment","level":1,"id":"c-Bob_K-2014-03-07T13:48:00.000Z-Eye-catching_GIF","replies":[{"timestamp":"2014-03-07T16:53:00.000Z","author":"Sbyrnes321","type":"comment","level":2,"id":"c-Sbyrnes321-2014-03-07T16:53:00.000Z-Bob_K-2014-03-07T13:48:00.000Z","replies":[],"displayName":"Steve"},{"timestamp":"2014-03-08T02:04:00.000Z","author":"Sbyrnes321","type":"comment","level":2,"id":"c-Sbyrnes321-2014-03-08T02:04:00.000Z-Bob_K-2014-03-07T13:48:00.000Z","replies":[],"displayName":"Steve"}]},{"timestamp":"2016-02-25T16:27:00.000Z","author":"2601:703:0:851E:91E5:CA2D:218:296C","type":"comment","level":1,"id":"c-2601:703:0:851E:91E5:CA2D:218:296C-2016-02-25T16:27:00.000Z-Eye-catching_GIF","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-88.188.110.51-2016-06-16T08:59:00.000Z","type":"heading","level":0,"id":"h-there_is_an_asterisk_(*)._What_does_it_mean?-2016-06-16T08:59:00.000Z","replies":[{"timestamp":"2016-06-16T08:59:00.000Z","author":"88.188.110.51","type":"comment","level":1,"id":"c-88.188.110.51-2016-06-16T08:59:00.000Z-there_is_an_asterisk_(*)._What_does_it_mean?","replies":[{"timestamp":"2016-06-16T12:56:00.000Z","author":"Bob K","type":"comment","level":2,"id":"c-Bob_K-2016-06-16T12:56:00.000Z-88.188.110.51-2016-06-16T08:59:00.000Z","replies":[{"timestamp":"2016-06-16T13:48:00.000Z","author":"88.188.110.51","type":"comment","level":3,"id":"c-88.188.110.51-2016-06-16T13:48:00.000Z-Bob_K-2016-06-16T12:56:00.000Z","replies":[]}]}]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Bob_K-2017-10-17T20:33:00.000Z","type":"heading","level":0,"id":"h-Fourier_Series_definition_and_other_problems-2017-10-17T20:33:00.000Z","replies":[{"timestamp":"2017-10-17T20:33:00.000Z","author":"Bob K","type":"comment","level":1,"id":"c-Bob_K-2017-10-17T20:33:00.000Z-Fourier_Series_definition_and_other_problems","replies":[{"timestamp":"2019-04-25T05:24:00.000Z","author":"Dicklyon","type":"comment","level":3,"id":"c-Dicklyon-2019-04-25T05:24:00.000Z-Bob_K-2017-10-17T20:33:00.000Z","replies":[]},{"timestamp":"2019-04-26T23:33:00.000Z","author":"Bob K","type":"comment","level":2,"id":"c-Bob_K-2019-04-26T23:33:00.000Z-Bob_K-2017-10-17T20:33:00.000Z","replies":[]}]}],"uneditableSection":true},{"headingLevel":2,"name":"h-2001:8003:E448:D401:7D27:BA42:5CB1:E259-2019-08-25T08:23:00.000Z","type":"heading","level":0,"id":"h-Label_for_the_cool_gif_is_unclear.-2019-08-25T08:23:00.000Z","replies":[{"timestamp":"2019-08-25T08:23:00.000Z","author":"2001:8003:E448:D401:7D27:BA42:5CB1:E259","type":"comment","level":1,"id":"c-2001:8003:E448:D401:7D27:BA42:5CB1:E259-2019-08-25T08:23:00.000Z-Label_for_the_cool_gif_is_unclear.","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Thellamabotherer-2020-05-31T19:16:00.000Z","type":"heading","level":0,"id":"h-Coefficient_of_'\"`UNIQ--postMath-00000070-QINU`\"'-2020-05-31T19:16:00.000Z","replies":[{"timestamp":"2020-05-31T19:16:00.000Z","author":"Thellamabotherer","type":"comment","level":1,"id":"c-Thellamabotherer-2020-05-31T19:16:00.000Z-Coefficient_of_'\"`UNIQ--postMath-00000070-QINU`\"'","replies":[]},{"timestamp":"2020-06-05T15:56:00.000Z","author":"Hqurve","type":"comment","level":1,"id":"c-Hqurve-2020-06-05T15:56:00.000Z-Coefficient_of_'\"`UNIQ--postMath-00000070-QINU`\"'","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-104.162.239.6-2020-10-29T23:15:00.000Z","type":"heading","level":0,"id":"h-No_such_topic_yet-2020-10-29T23:15:00.000Z","replies":[{"timestamp":"2020-10-29T23:15:00.000Z","author":"104.162.239.6","type":"comment","level":1,"id":"c-104.162.239.6-2020-10-29T23:15:00.000Z-No_such_topic_yet","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Cahmad25-2021-10-28T00:09:00.000Z","type":"heading","level":0,"id":"h-The_first_few_paragraphs_should_be_moved-2021-10-28T00:09:00.000Z","replies":[{"timestamp":"2021-10-28T00:09:00.000Z","author":"Cahmad25","type":"comment","level":1,"id":"c-Cahmad25-2021-10-28T00:09:00.000Z-The_first_few_paragraphs_should_be_moved","replies":[{"timestamp":"2021-10-28T13:52:00.000Z","author":"Bob K","type":"comment","level":3,"id":"c-Bob_K-2021-10-28T13:52:00.000Z-Cahmad25-2021-10-28T00:09:00.000Z","replies":[]}]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Em3rgent0rdr-2022-03-15T07:36:00.000Z","type":"heading","level":0,"id":"h-Should_this_article_definition_start_with_a_scary_correlation_function?-2022-03-15T07:36:00.000Z","replies":[{"timestamp":"2022-03-15T07:36:00.000Z","author":"Em3rgent0rdr","type":"comment","level":1,"id":"c-Em3rgent0rdr-2022-03-15T07:36:00.000Z-Should_this_article_definition_start_with_a_scary_correlation_function?","replies":[{"timestamp":"2022-03-15T12:12:00.000Z","author":"Em3rgent0rdr","type":"comment","level":2,"id":"c-Em3rgent0rdr-2022-03-15T12:12:00.000Z-Em3rgent0rdr-2022-03-15T07:36:00.000Z","replies":[{"timestamp":"2022-03-16T11:39:00.000Z","author":"Em3rgent0rdr","type":"comment","level":3,"id":"c-Em3rgent0rdr-2022-03-16T11:39:00.000Z-Em3rgent0rdr-2022-03-15T12:12:00.000Z","replies":[]}]}]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Em3rgent0rdr-2022-03-15T12:17:00.000Z","type":"heading","level":0,"id":"h-Fourier's_Theorem_is_necessary_but_not_in_Wikipedia-2022-03-15T12:17:00.000Z","replies":[{"timestamp":"2022-03-15T12:17:00.000Z","author":"Em3rgent0rdr","type":"comment","level":1,"id":"c-Em3rgent0rdr-2022-03-15T12:17:00.000Z-Fourier's_Theorem_is_necessary_but_not_in_Wikipedia","replies":[{"timestamp":"2022-03-16T11:41:00.000Z","author":"Em3rgent0rdr","type":"comment","level":2,"id":"c-Em3rgent0rdr-2022-03-16T11:41:00.000Z-Em3rgent0rdr-2022-03-15T12:17:00.000Z","replies":[]}]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Em3rgent0rdr-2022-03-16T12:42:00.000Z","type":"heading","level":0,"id":"h-I_brought_the_animations_to_be_fixed_in_between_paragraphs_of_the_intro_(using_a-2022-03-16T12:42:00.000Z","replies":[{"timestamp":"2022-03-16T12:42:00.000Z","author":"Em3rgent0rdr","type":"comment","level":1,"id":"c-Em3rgent0rdr-2022-03-16T12:42:00.000Z-I_brought_the_animations_to_be_fixed_in_between_paragraphs_of_the_intro_(using_a","replies":[]}],"uneditableSection":true},{"headingLevel":2,"name":"h-Em3rgent0rdr-2022-03-15T07:31:00.000Z","type":"heading","level":0,"id":"h-Maybe_use_'ξ'_instead_of_'f'_for_frequency_to_avoid_conflict_with_'f'_for_funct-2022-03-15T07:31:00.000Z","replies":[{"timestamp":"2022-03-15T07:31:00.000Z","author":"Em3rgent0rdr","type":"comment","level":1,"id":"c-Em3rgent0rdr-2022-03-15T07:31:00.000Z-Maybe_use_'ξ'_instead_of_'f'_for_frequency_to_avoid_conflict_with_'f'_for_funct","replies":[{"timestamp":"2022-03-15T19:52:00.000Z","author":"Bob K","type":"comment","level":2,"id":"c-Bob_K-2022-03-15T19:52:00.000Z-Em3rgent0rdr-2022-03-15T07:31:00.000Z","replies":[]}]},{"timestamp":"2022-03-16T11:03:00.000Z","author":"Em3rgent0rdr","type":"comment","level":1,"id":"c-Em3rgent0rdr-2022-03-16T11:03:00.000Z-Maybe_use_'ξ'_instead_of_'f'_for_frequency_to_avoid_conflict_with_'f'_for_funct","replies":[]},{"timestamp":"2022-03-16T12:04:00.000Z","author":"Bob K","type":"comment","level":1,"id":"c-Bob_K-2022-03-16T12:04:00.000Z-Maybe_use_'ξ'_instead_of_'f'_for_frequency_to_avoid_conflict_with_'f'_for_funct","replies":[{"timestamp":"2022-03-16T12:37:00.000Z","author":"Em3rgent0rdr","type":"comment","level":2,"id":"c-Em3rgent0rdr-2022-03-16T12:37:00.000Z-Bob_K-2022-03-16T12:04:00.000Z","replies":[]}]},{"timestamp":"2022-03-17T11:56:00.000Z","author":"Bob K","type":"comment","level":1,"id":"c-Bob_K-2022-03-17T11:56:00.000Z-Maybe_use_'ξ'_instead_of_'f'_for_frequency_to_avoid_conflict_with_'f'_for_funct","replies":[]}],"uneditableSection":true}],"wgPageParseReport":{"discussiontools":{"limitreport-timeusage":"0.260"},"limitreport":{"cputime":"0.645","walltime":"0.883","ppvisitednodes":{"value":2130,"limit":1000000},"postexpandincludesize":{"value":9817,"limit":2097152},"templateargumentsize":{"value":2274,"limit":2097152},"expansiondepth":{"value":17,"limit":100},"expensivefunctioncount":{"value":7,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":13749,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 218.331 1 -total"," 38.19% 83.389 1 Template:Automatic_archive_navigator"," 29.70% 64.836 1 Template:Reflist"," 25.50% 55.678 1 Template:Bq"," 24.71% 53.953 1 Template:Cite_book"," 18.55% 40.494 1 Template:ISBN"," 14.34% 31.298 1 Template:Catalog_lookup_link"," 13.56% 29.614 2 Template:Is_redirect"," 4.33% 9.446 1 Template:Cquote"," 4.18% 9.134 1 Template:Comma_separated_entries"]},"scribunto":{"limitreport-timeusage":{"value":"0.100","limit":"10.000"},"limitreport-memusage":{"value":3672420,"limit":52428800}},"cachereport":{"origin":"mw-web.codfw.main-f69cdc8f6-4cmjj","timestamp":"20241124004939","ttl":864000,"transientcontent":true}}});});</script> </body> </html>

CINXE.COM

Talk:Fourier series/Archive 2 - Wikipedia