오일러 공식 - 위키백과, 우리 모두의 백과사전

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[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>로그인</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"> 로그아웃한 편집자를 위한 문서 <a href="/wiki/%EB%8F%84%EC%9B%80%EB%A7%90:%EC%86%8C%EA%B0%9C" aria-label="편집에 관해 더 알아보기"><span>더 알아보기</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/%ED%8A%B9%EC%88%98:%EB%82%B4%EA%B8%B0%EC%97%AC" title="이 IP 주소의 편집 목록 [y]" accesskey="y"><span>기여</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%82%B4%EC%82%AC%EC%9A%A9%EC%9E%90%ED%86%A0%EB%A1%A0" title="현재 사용하는 IP 주소에 대한 토론 문서 [n]" accesskey="n"><span>토론</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="사이트"> <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="목차" 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">목차</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">사이드바로 이동</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">숨기기</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">처음 위치</div> </a> </li> <li id="toc-역사" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#역사"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>역사</span> </div> </a> <ul id="toc-역사-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-복소수_지수_정의와_오일러_공식" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#복소수_지수_정의와_오일러_공식"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>복소수 지수 정의와 오일러 공식</span> </div> </a> <button aria-controls="toc-복소수_지수_정의와_오일러_공식-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>복소수 지수 정의와 오일러 공식 하위섹션 토글하기</span> </button> <ul id="toc-복소수_지수_정의와_오일러_공식-sublist" class="vector-toc-list"> <li id="toc-테일러_급수를_이용한_방법" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#테일러_급수를_이용한_방법"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>테일러 급수를 이용한 방법</span> </div> </a> <ul id="toc-테일러_급수를_이용한_방법-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-미분_계산을_이용한_증명방법" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#미분_계산을_이용한_증명방법"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>미분 계산을 이용한 증명방법</span> </div> </a> <ul id="toc-미분_계산을_이용한_증명방법-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-미적분을_이용한_방법" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#미적분을_이용한_방법"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>미적분을 이용한 방법</span> </div> </a> <ul id="toc-미적분을_이용한_방법-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-미분방정식을_이용한_방법" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#미분방정식을_이용한_방법"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>미분방정식을 이용한 방법</span> </div> </a> <ul id="toc-미분방정식을_이용한_방법-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-박사가_사랑한_수식" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#박사가_사랑한_수식"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>박사가 사랑한 수식</span> </div> </a> <ul id="toc-박사가_사랑한_수식-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-쉬운_설명" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#쉬운_설명"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>쉬운 설명</span> </div> </a> <ul id="toc-쉬운_설명-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-각주" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#각주"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>각주</span> </div> </a> <ul id="toc-각주-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-같이_보기" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#같이_보기"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>같이 보기</span> </div> </a> <ul id="toc-같이_보기-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="목차" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="목차" > <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="목차 토글" > <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">목차 토글</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">오일러 공식</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="다른 언어로 문서를 방문합니다. 66개 언어로 읽을 수 있습니다" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-66" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">66개 언어</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Euler_se_formule" title="Euler se formule – 아프리칸스어" lang="af" hreflang="af" data-title="Euler se formule" data-language-autonym="Afrikaans" data-language-local-name="아프리칸스어" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B5%D9%8A%D8%BA%D8%A9_%D8%A3%D9%88%D9%8A%D9%84%D8%B1" title="صيغة أويلر – 아랍어" lang="ar" hreflang="ar" data-title="صيغة أويلر" data-language-autonym="العربية" data-language-local-name="아랍어" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/F%C3%B3rmula_d%27Euler" title="Fórmula d'Euler – 아스투리아어" lang="ast" hreflang="ast" data-title="Fórmula d'Euler" data-language-autonym="Asturianu" data-language-local-name="아스투리아어" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Eyler_d%C3%BCsturu" title="Eyler düsturu – 아제르바이잔어" lang="az" hreflang="az" data-title="Eyler düsturu" data-language-autonym="Azərbaycanca" data-language-local-name="아제르바이잔어" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%AD%D0%B9%D0%BB%D0%B5%D1%80_%D1%84%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0%D2%BB%D1%8B" title="Эйлер формулаһы – 바슈키르어" lang="ba" hreflang="ba" data-title="Эйлер формулаһы" data-language-autonym="Башҡортса" data-language-local-name="바슈키르어" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0_%D0%AD%D0%B9%D0%BB%D0%B5%D1%80%D0%B0" title="Формула Эйлера – 벨라루스어" lang="be" hreflang="be" data-title="Формула Эйлера" data-language-autonym="Беларуская" data-language-local-name="벨라루스어" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0_%D0%BD%D0%B0_%D0%9E%D0%B9%D0%BB%D0%B5%D1%80" title="Формула на Ойлер – 불가리아어" lang="bg" hreflang="bg" data-title="Формула на Ойлер" data-language-autonym="Български" data-language-local-name="불가리아어" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%85%E0%A6%AF%E0%A6%BC%E0%A6%B2%E0%A6%BE%E0%A6%B0%E0%A7%87%E0%A6%B0_%E0%A6%B8%E0%A7%82%E0%A6%A4%E0%A7%8D%E0%A6%B0" title="অয়লারের সূত্র – 벵골어" lang="bn" hreflang="bn" data-title="অয়লারের সূত্র" data-language-autonym="বাংলা" data-language-local-name="벵골어" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Eulerova_formula" title="Eulerova formula – 보스니아어" lang="bs" hreflang="bs" data-title="Eulerova formula" data-language-autonym="Bosanski" data-language-local-name="보스니아어" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/F%C3%B3rmula_d%27Euler" title="Fórmula d'Euler – 카탈로니아어" lang="ca" hreflang="ca" data-title="Fórmula d'Euler" data-language-autonym="Català" data-language-local-name="카탈로니아어" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%81%D9%88%D8%B1%D9%85%D9%88%D9%88%DA%B5%DB%8C_%D8%A6%DB%86%DB%8C%D9%84%DB%95%D8%B1" title="فورمووڵی ئۆیلەر – 소라니 쿠르드어" lang="ckb" hreflang="ckb" data-title="فورمووڵی ئۆیلەر" data-language-autonym="کوردی" data-language-local-name="소라니 쿠르드어" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Euler%C5%AFv_vzorec" title="Eulerův vzorec – 체코어" lang="cs" hreflang="cs" data-title="Eulerův vzorec" data-language-autonym="Čeština" data-language-local-name="체코어" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%AD%D0%B9%D0%BB%D0%B5%D1%80_%D1%84%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B8" title="Эйлер формули – 추바시어" lang="cv" hreflang="cv" data-title="Эйлер формули" data-language-autonym="Чӑвашла" data-language-local-name="추바시어" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Fformiwla_Euler" title="Fformiwla Euler – 웨일스어" lang="cy" hreflang="cy" data-title="Fformiwla Euler" data-language-autonym="Cymraeg" data-language-local-name="웨일스어" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Eulers_formel" title="Eulers formel – 덴마크어" lang="da" hreflang="da" data-title="Eulers formel" data-language-autonym="Dansk" data-language-local-name="덴마크어" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Eulersche_Formel" title="Eulersche Formel – 독일어" lang="de" hreflang="de" data-title="Eulersche Formel" data-language-autonym="Deutsch" data-language-local-name="독일어" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A4%CF%8D%CF%80%CE%BF%CF%82_%CF%84%CE%BF%CF%85_%CE%8C%CE%B9%CE%BB%CE%B5%CF%81" title="Τύπος του Όιλερ – 그리스어" lang="el" hreflang="el" data-title="Τύπος του Όιλερ" data-language-autonym="Ελληνικά" data-language-local-name="그리스어" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Euler%27s_formula" title="Euler's formula – 영어" lang="en" hreflang="en" data-title="Euler's formula" data-language-autonym="English" data-language-local-name="영어" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/E%C5%ADlera_formulo" title="Eŭlera formulo – 에스페란토어" lang="eo" hreflang="eo" data-title="Eŭlera formulo" data-language-autonym="Esperanto" data-language-local-name="에스페란토어" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/F%C3%B3rmula_de_Euler" title="Fórmula de Euler – 스페인어" lang="es" hreflang="es" data-title="Fórmula de Euler" data-language-autonym="Español" data-language-local-name="스페인어" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Euleri_valem" title="Euleri valem – 에스토니아어" lang="et" hreflang="et" data-title="Euleri valem" data-language-autonym="Eesti" data-language-local-name="에스토니아어" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Eulerren_formula" title="Eulerren formula – 바스크어" lang="eu" hreflang="eu" data-title="Eulerren formula" data-language-autonym="Euskara" data-language-local-name="바스크어" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D8%B1%D9%85%D9%88%D9%84_%D8%A7%D9%88%DB%8C%D9%84%D8%B1" title="فرمول اویلر – 페르시아어" lang="fa" hreflang="fa" data-title="فرمول اویلر" data-language-autonym="فارسی" data-language-local-name="페르시아어" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Eulerin_lause_(funktioteoria)" title="Eulerin lause (funktioteoria) – 핀란드어" lang="fi" hreflang="fi" data-title="Eulerin lause (funktioteoria)" data-language-autonym="Suomi" data-language-local-name="핀란드어" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Formule_d%27Euler" title="Formule d'Euler – 프랑스어" lang="fr" hreflang="fr" data-title="Formule d'Euler" data-language-autonym="Français" data-language-local-name="프랑스어" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/F%C3%B3rmula_de_Euler" title="Fórmula de Euler – 갈리시아어" lang="gl" hreflang="gl" data-title="Fórmula de Euler" data-language-autonym="Galego" data-language-local-name="갈리시아어" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A0%D7%95%D7%A1%D7%97%D7%AA_%D7%90%D7%95%D7%99%D7%9C%D7%A8_(%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%9E%D7%A8%D7%95%D7%9B%D7%91%D7%AA)" title="נוסחת אוילר (אנליזה מרוכבת) – 히브리어" lang="he" hreflang="he" data-title="נוסחת אוילר (אנליזה מרוכבת)" data-language-autonym="עברית" data-language-local-name="히브리어" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%91%E0%A4%AF%E0%A4%B2%E0%A4%B0_%E0%A4%95%E0%A4%BE_%E0%A4%B8%E0%A5%82%E0%A4%A4%E0%A5%8D%E0%A4%B0" title="ऑयलर का सूत्र – 힌디어" lang="hi" hreflang="hi" data-title="ऑयलर का सूत्र" data-language-autonym="हिन्दी" data-language-local-name="힌디어" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Eulerova_formula" title="Eulerova formula – 크로아티아어" lang="hr" hreflang="hr" data-title="Eulerova formula" data-language-autonym="Hrvatski" data-language-local-name="크로아티아어" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Euler-k%C3%A9plet" title="Euler-képlet – 헝가리어" lang="hu" hreflang="hu" data-title="Euler-képlet" data-language-autonym="Magyar" data-language-local-name="헝가리어" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B7%D5%B5%D5%AC%D5%A5%D6%80%D5%AB_%D5%A2%D5%A1%D5%B6%D5%A1%D5%B1%D6%87" title="Էյլերի բանաձև – 아르메니아어" lang="hy" hreflang="hy" data-title="Էյլերի բանաձև" data-language-autonym="Հայերեն" data-language-local-name="아르메니아어" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Rumus_Euler" title="Rumus Euler – 인도네시아어" lang="id" hreflang="id" data-title="Rumus Euler" data-language-autonym="Bahasa Indonesia" data-language-local-name="인도네시아어" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Jafna_Eulers" title="Jafna Eulers – 아이슬란드어" lang="is" hreflang="is" data-title="Jafna Eulers" data-language-autonym="Íslenska" data-language-local-name="아이슬란드어" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Formula_di_Eulero" title="Formula di Eulero – 이탈리아어" lang="it" hreflang="it" data-title="Formula di Eulero" data-language-autonym="Italiano" data-language-local-name="이탈리아어" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%AA%E3%82%A4%E3%83%A9%E3%83%BC%E3%81%AE%E5%85%AC%E5%BC%8F" title="オイラーの公式 – 일본어" lang="ja" hreflang="ja" data-title="オイラーの公式" data-language-autonym="日本語" data-language-local-name="일본어" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%AD%D0%B9%D0%BB%D0%B5%D1%80_%D1%84%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0%D1%81%D1%8B" title="Эйлер формуласы – 카자흐어" lang="kk" hreflang="kk" data-title="Эйлер формуласы" data-language-autonym="Қазақша" data-language-local-name="카자흐어" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%9A%E1%9E%BC%E1%9E%94%E1%9E%98%E1%9E%93%E1%9F%92%E1%9E%8F%E1%9E%A2%E1%9E%99%E1%9E%9B%E1%9F%90%E1%9E%9A" title="រូបមន្តអយល័រ – 크메르어" lang="km" hreflang="km" data-title="រូបមន្តអយល័រ" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="크메르어" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Euleri_formula" title="Euleri formula – 라틴어" lang="la" hreflang="la" data-title="Euleri formula" data-language-autonym="Latina" data-language-local-name="라틴어" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Oilerio_formul%C4%97" title="Oilerio formulė – 리투아니아어" lang="lt" hreflang="lt" data-title="Oilerio formulė" data-language-autonym="Lietuvių" data-language-local-name="리투아니아어" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Eilera_formula" title="Eilera formula – 라트비아어" lang="lv" hreflang="lv" data-title="Eilera formula" data-language-autonym="Latviešu" data-language-local-name="라트비아어" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9E%D1%98%D0%BB%D0%B5%D1%80%D0%BE%D0%B2%D0%B0_%D1%84%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0" title="Ојлерова формула – 마케도니아어" lang="mk" hreflang="mk" data-title="Ојлерова формула" data-language-autonym="Македонски" data-language-local-name="마케도니아어" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Formula_Euler" title="Formula Euler – 말레이어" lang="ms" hreflang="ms" data-title="Formula Euler" data-language-autonym="Bahasa Melayu" data-language-local-name="말레이어" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Formule_van_Euler" title="Formule van Euler – 네덜란드어" lang="nl" hreflang="nl" data-title="Formule van Euler" data-language-autonym="Nederlands" data-language-local-name="네덜란드어" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Eulerformelen" title="Eulerformelen – 노르웨이어(니노르스크)" lang="nn" hreflang="nn" data-title="Eulerformelen" data-language-autonym="Norsk nynorsk" data-language-local-name="노르웨이어(니노르스크)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Eulers_formel" title="Eulers formel – 노르웨이어(보크말)" lang="nb" hreflang="nb" data-title="Eulers formel" data-language-autonym="Norsk bokmål" data-language-local-name="노르웨이어(보크말)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Formula_d%27Euler" title="Formula d'Euler – 오크어" lang="oc" hreflang="oc" data-title="Formula d'Euler" data-language-autonym="Occitan" data-language-local-name="오크어" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wz%C3%B3r_Eulera" title="Wzór Eulera – 폴란드어" lang="pl" hreflang="pl" data-title="Wzór Eulera" data-language-autonym="Polski" data-language-local-name="폴란드어" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/F%C3%B3rmula_de_Euler" title="Fórmula de Euler – 포르투갈어" lang="pt" hreflang="pt" data-title="Fórmula de Euler" data-language-autonym="Português" data-language-local-name="포르투갈어" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Formula_lui_Euler" title="Formula lui Euler – 루마니아어" lang="ro" hreflang="ro" data-title="Formula lui Euler" data-language-autonym="Română" data-language-local-name="루마니아어" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0_%D0%AD%D0%B9%D0%BB%D0%B5%D1%80%D0%B0" title="Формула Эйлера – 러시아어" lang="ru" hreflang="ru" data-title="Формула Эйлера" data-language-autonym="Русский" data-language-local-name="러시아어" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Euler%27s_formula" title="Euler's formula – Simple English" lang="en-simple" hreflang="en-simple" data-title="Euler's formula" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Eulerjeva_formula" title="Eulerjeva formula – 슬로베니아어" lang="sl" hreflang="sl" data-title="Eulerjeva formula" data-language-autonym="Slovenščina" data-language-local-name="슬로베니아어" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9E%D1%98%D0%BB%D0%B5%D1%80%D0%BE%D0%B2%D0%B0_%D1%84%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0" title="Ојлерова формула – 세르비아어" lang="sr" hreflang="sr" data-title="Ојлерова формула" data-language-autonym="Српски / srpski" data-language-local-name="세르비아어" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Eulers_formel" title="Eulers formel – 스웨덴어" lang="sv" hreflang="sv" data-title="Eulers formel" data-language-autonym="Svenska" data-language-local-name="스웨덴어" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%86%E0%AE%AF%E0%AF%8D%E0%AE%B2%E0%AE%B0%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%B5%E0%AE%BE%E0%AE%AF%E0%AF%8D%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AE%BE%E0%AE%9F%E0%AF%81" title="ஆய்லரின் வாய்ப்பாடு – 타밀어" lang="ta" hreflang="ta" data-title="ஆய்லரின் வாய்ப்பாடு" data-language-autonym="தமிழ்" data-language-local-name="타밀어" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0%D2%B3%D0%BE%D0%B8_%D0%AD%D0%B9%D0%BB%D0%B5%D1%80" title="Формулаҳои Эйлер – 타지크어" lang="tg" hreflang="tg" data-title="Формулаҳои Эйлер" data-language-autonym="Тоҷикӣ" data-language-local-name="타지크어" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AA%E0%B8%B9%E0%B8%95%E0%B8%A3%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B8%AD%E0%B9%87%E0%B8%AD%E0%B8%A2%E0%B9%80%E0%B8%A5%E0%B8%AD%E0%B8%A3%E0%B9%8C" title="สูตรของอ็อยเลอร์ – 태국어" lang="th" hreflang="th" data-title="สูตรของอ็อยเลอร์" data-language-autonym="ไทย" data-language-local-name="태국어" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Euler_form%C3%BCl%C3%BC" title="Euler formülü – 튀르키예어" lang="tr" hreflang="tr" data-title="Euler formülü" data-language-autonym="Türkçe" data-language-local-name="튀르키예어" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0_%D0%95%D0%B9%D0%BB%D0%B5%D1%80%D0%B0" title="Формула Ейлера – 우크라이나어" lang="uk" hreflang="uk" data-title="Формула Ейлера" data-language-autonym="Українська" data-language-local-name="우크라이나어" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B9%D8%A7%D8%A6%D9%84%D8%B1%DB%8C_%DA%A9%D9%84%DB%8C%DB%81" title="عائلری کلیہ – 우르두어" lang="ur" hreflang="ur" data-title="عائلری کلیہ" data-language-autonym="اردو" data-language-local-name="우르두어" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Euler_formulalari" title="Euler formulalari – 우즈베크어" lang="uz" hreflang="uz" data-title="Euler formulalari" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="우즈베크어" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/C%C3%B4ng_th%E1%BB%A9c_Euler" title="Công thức Euler – 베트남어" lang="vi" hreflang="vi" data-title="Công thức Euler" data-language-autonym="Tiếng Việt" data-language-local-name="베트남어" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%AC%A7%E6%8B%89%E5%85%AC%E5%BC%8F" title="欧拉公式 – 우어" lang="wuu" hreflang="wuu" data-title="欧拉公式" data-language-autonym="吴语" data-language-local-name="우어" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a 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class="vector-pinnable-header-label">보이기</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">사이드바로 이동</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">숨기기</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">위키백과, 우리 모두의 백과사전.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ko" dir="ltr"><p><span class="nowrap"></span> </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/%ED%8C%8C%EC%9D%BC:Euler%27s_formula.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/360px-Euler%27s_formula.svg.png" decoding="async" width="360" height="370" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/540px-Euler%27s_formula.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/720px-Euler%27s_formula.svg.png 2x" data-file-width="760" data-file-height="782" /></a><figcaption><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=\cos x+i\sin x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=\cos x+i\sin x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04f4f3cc7216a30e77f0a6a4151adab1ca7a668e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.617ex; height:2.343ex;" alt="{\displaystyle z=\cos x+i\sin x}"></span>는 복소평면에서 단위원을 뜻한다.</figcaption></figure> <p><b>오일러 공식</b>(Euler's formula)은 수학자 <a href="/wiki/%EB%A0%88%EC%98%A8%ED%95%98%EB%A5%B4%ED%8A%B8_%EC%98%A4%EC%9D%BC%EB%9F%AC" title="레온하르트 오일러">레온하르트 오일러</a>의 이름이 붙은 공식이다. </p><p>사용되는 경우로는 복소수 지수를 정의하는 데에 출발점이 되며, <a href="/wiki/%EC%82%BC%EA%B0%81%ED%95%A8%EC%88%98" class="mw-redirect" title="삼각함수">삼각함수</a>와 <a href="/wiki/%EC%A7%80%EC%88%98%ED%95%A8%EC%88%98" class="mw-redirect" title="지수함수">지수함수</a>에 대한 관계를 나타낸다. <a href="/wiki/%EC%98%A4%EC%9D%BC%EB%9F%AC_%ED%95%AD%EB%93%B1%EC%8B%9D" title="오일러 항등식">오일러 항등식</a>은 이 공식의 특수한 경우이다. </p><p>오일러 공식은 다음과 같다. <a href="/wiki/%EC%8B%A4%EC%88%98" title="실수">실수</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mtext> </mtext> </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/88d7c2a9294f2d337df5ace5a38a631c19dad665" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.91ex; height:1.676ex;" alt="{\displaystyle x~}"></span>에 대해, <a href="/wiki/%ED%97%88%EC%88%98" title="허수">허수</a> 지수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ix~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi>x</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ix~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e05e27aea9f2199b59510a31dcbb88bdc1331a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.713ex; height:2.176ex;" alt="{\displaystyle ix~}"></span>를 다음과 같이 정의한다. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{ix}\,=\,\cos x+i\sin x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{ix}\,=\,\cos x+i\sin x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdbbe899b8f3a08cc2d8f6562a2a6216ac3fe010" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:20.126ex; height:2.843ex;" alt="{\displaystyle e^{ix}\,=\,\cos x+i\sin x}"></span></dd></dl> <p>여기서, e는 <a href="/wiki/%EC%9E%90%EC%97%B0%EB%A1%9C%EA%B7%B8%EC%9D%98_%EB%B0%91" title="자연로그의 밑">자연로그의 밑</a>이고, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f48f89dad66f900b53ceaa01c58578e50eed9a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.383ex; height:2.176ex;" alt="{\displaystyle i~}"></span>는 제곱하여 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa70a68a9cb7496c5e9cb2ef4a9cabe61f684f92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.551ex; height:2.343ex;" alt="{\displaystyle -1~}"></span>이 되는(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{2}=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i^{2}=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88e98a401d352e5037d5043028e2d7f449e83fa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.926ex; height:2.843ex;" alt="{\displaystyle i^{2}=-1}"></span>) <a href="/wiki/%ED%97%88%EC%88%98%EB%8B%A8%EC%9C%84" class="mw-redirect" title="허수단위">허수단위</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee55beec18afd710e7ab767964b915b020c65093" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.856ex; height:2.176ex;" alt="{\displaystyle \sin }"></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 \cos }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e473a3de151d75296f141f9f482fe59d582a7509" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.111ex; height:1.676ex;" alt="{\displaystyle \cos }"></span>은 <a href="/wiki/%EC%82%BC%EA%B0%81%ED%95%A8%EC%88%98" class="mw-redirect" title="삼각함수">삼각함수</a>의 사인과 코사인 함수이다. </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 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>에 <span class="mwe-math-element"><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>를 대입하여, <a href="/wiki/%EC%98%A4%EC%9D%BC%EB%9F%AC_%ED%95%AD%EB%93%B1%EC%8B%9D" title="오일러 항등식">오일러 항등식</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\pi }=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>π</mi> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\pi }=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00ba628c5431f37a0be483c26111b1078c036f6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.894ex; height:2.843ex;" alt="{\displaystyle e^{i\pi }=-1}"></span>을 구할 수 있다. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="역사"><span id=".EC.97.AD.EC.82.AC"></span>역사</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%98%A4%EC%9D%BC%EB%9F%AC_%EA%B3%B5%EC%8B%9D&action=edit&section=1" title="부분 편집: 역사"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>오일러 공식은 <a href="/wiki/1714%EB%85%84" title="1714년">1714년</a> <a href="/wiki/%EB%A1%9C%EC%A0%80_%EC%BD%94%EC%B8%A0" title="로저 코츠">로저 코츠</a>가 다음과 같은 형태로 처음 발견하였다. </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 \ln(\cos x+i\sin x)\,=\,ix}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mi>i</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(\cos x+i\sin x)\,=\,ix}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/718e99eb8c15c1193b98fcc9b8a4b7d2f76c62a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.184ex; height:2.843ex;" alt="{\displaystyle \ln(\cos x+i\sin x)\,=\,ix}"></span></dd></dl> <p>지금과 같은 모양의 오일러 공식은 1748년 오일러가 <a href="/wiki/%EB%AC%B4%ED%95%9C%EA%B8%89%EC%88%98" class="mw-redirect" title="무한급수">무한급수</a>의 좌우 극한값이 같음을 증명하면서 발표되었다. 그러나 로저와 오일러 모두 이 공식이 지닌 '<a href="/wiki/%EB%B3%B5%EC%86%8C%EC%88%98" title="복소수">복소수</a>를 <a href="/wiki/%EB%B3%B5%EC%86%8C%ED%8F%89%EB%A9%B4" title="복소평면">복소평면</a> 위의 하나의 점으로 볼 수 있다'는 기하학적 의미를 눈치채지는 못하였고, 이것은 약 50년이 지난 후에나 발견되었다. 오일러는 현재의 교육과정에서 보다 훨씬 이른 시기에 학생들에게 복소수를 가르쳤다. 그의 기초 대수학 교재인 <<a href="/wiki/%EB%8C%80%EC%88%98%ED%95%99_%EC%9B%90%EB%A1%A0" title="대수학 원론">대수학 원론</a>>(Elements of Algebra)에 보면 교재의 거의 맨 앞부분부터 복소수를 도입하고 있고 교재 전체를 통틀어 자연스럽게 사용하고 있다.<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> </p> <div class="mw-heading mw-heading2"><h2 id="복소수_지수_정의와_오일러_공식"><span id=".EB.B3.B5.EC.86.8C.EC.88.98_.EC.A7.80.EC.88.98_.EC.A0.95.EC.9D.98.EC.99.80_.EC.98.A4.EC.9D.BC.EB.9F.AC_.EA.B3.B5.EC.8B.9D"></span>복소수 지수 정의와 오일러 공식</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%98%A4%EC%9D%BC%EB%9F%AC_%EA%B3%B5%EC%8B%9D&action=edit&section=2" title="부분 편집: 복소수 지수 정의와 오일러 공식"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="테일러_급수를_이용한_방법"><span id=".ED.85.8C.EC.9D.BC.EB.9F.AC_.EA.B8.89.EC.88.98.EB.A5.BC_.EC.9D.B4.EC.9A.A9.ED.95.9C_.EB.B0.A9.EB.B2.95"></span>테일러 급수를 이용한 방법</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%98%A4%EC%9D%BC%EB%9F%AC_%EA%B3%B5%EC%8B%9D&action=edit&section=3" title="부분 편집: 테일러 급수를 이용한 방법"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%ED%85%8C%EC%9D%BC%EB%9F%AC_%EA%B8%89%EC%88%98" title="테일러 급수">테일러 급수</a>에 따라 실수 범위에서 다음의 식이 성립한다. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x}={{x^{0}} \over {0!}}+{{x^{1}} \over {1!}}+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{x}={{x^{0}} \over {0!}}+{{x^{1}} \over {1!}}+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eefc582176ab0b4f85664d9d66f1b23cb335b954" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.544ex; height:6.843ex;" alt="{\displaystyle e^{x}={{x^{0}} \over {0!}}+{{x^{1}} \over {1!}}+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}}"></span></dd></dl> <p>따라서 </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{array}{rll}e^{x}&{}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\cdots &{}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}\\\cos x&{}=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots &{}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}\\\sin x&{}=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots &{}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\\\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mrow> <mn>6</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <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> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mrow> <mn>5</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mrow> <mn>7</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <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> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rll}e^{x}&{}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\cdots &{}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}\\\cos x&{}=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots &{}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}\\\sin x&{}=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots &{}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\\\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdb4840642b6fb261230625123eceb7fd226ce38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:57.772ex; height:14.509ex;" alt="{\displaystyle {\begin{array}{rll}e^{x}&{}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\cdots &{}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}\\\cos x&{}=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots &{}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}\\\sin x&{}=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots &{}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\\\end{array}}}"></span></dd></dl> <p>이때 <span class="mwe-math-element"><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>가 <a href="/wiki/%EB%B3%B5%EC%86%8C%EC%88%98" title="복소수">복소수</a>일 때에 앞의 <a href="/wiki/%EB%AC%B4%ED%95%9C%EA%B8%89%EC%88%98" class="mw-redirect" title="무한급수">무한급수</a>를 각각의 함수로 <b>정의</b>한다. 그러면 </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}e^{iz}&{}=1+iz+{\frac {(iz)^{2}}{2!}}+{\frac {(iz)^{3}}{3!}}+{\frac {(iz)^{4}}{4!}}+{\frac {(iz)^{5}}{5!}}+{\frac {(iz)^{6}}{6!}}+{\frac {(iz)^{7}}{7!}}+{\frac {(iz)^{8}}{8!}}+\cdots \\&{}=1+iz-{\frac {z^{2}}{2!}}-{\frac {iz^{3}}{3!}}+{\frac {z^{4}}{4!}}+{\frac {iz^{5}}{5!}}-{\frac {z^{6}}{6!}}-{\frac {iz^{7}}{7!}}+{\frac {z^{8}}{8!}}+\cdots \\&{}=\left(1-{\frac {z^{2}}{2!}}+{\frac {z^{4}}{4!}}-{\frac {z^{6}}{6!}}+{\frac {z^{8}}{8!}}-\cdots \right)+i\left(z-{\frac {z^{3}}{3!}}+{\frac {z^{5}}{5!}}-{\frac {z^{7}}{7!}}+\cdots \right)\\&{}=\cos z+i\sin z\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> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>z</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mi>z</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mrow> <mn>5</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mrow> <mrow> <mn>6</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> </mrow> <mrow> <mn>7</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> </mrow> <mrow> <mn>8</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mi>z</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mrow> <mn>5</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mrow> <mn>6</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> </mrow> <mrow> <mn>7</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mrow> <mn>8</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mrow> <mn>6</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mrow> <mn>8</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mo>⋯</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>i</mi> <mrow> <mo>(</mo> <mrow> <mi>z</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mrow> <mn>5</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mrow> <mn>7</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>z</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>z</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}e^{iz}&{}=1+iz+{\frac {(iz)^{2}}{2!}}+{\frac {(iz)^{3}}{3!}}+{\frac {(iz)^{4}}{4!}}+{\frac {(iz)^{5}}{5!}}+{\frac {(iz)^{6}}{6!}}+{\frac {(iz)^{7}}{7!}}+{\frac {(iz)^{8}}{8!}}+\cdots \\&{}=1+iz-{\frac {z^{2}}{2!}}-{\frac {iz^{3}}{3!}}+{\frac {z^{4}}{4!}}+{\frac {iz^{5}}{5!}}-{\frac {z^{6}}{6!}}-{\frac {iz^{7}}{7!}}+{\frac {z^{8}}{8!}}+\cdots \\&{}=\left(1-{\frac {z^{2}}{2!}}+{\frac {z^{4}}{4!}}-{\frac {z^{6}}{6!}}+{\frac {z^{8}}{8!}}-\cdots \right)+i\left(z-{\frac {z^{3}}{3!}}+{\frac {z^{5}}{5!}}-{\frac {z^{7}}{7!}}+\cdots \right)\\&{}=\cos z+i\sin z\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14cc6e94d4cb9db103322d9cff63dc8ba42242b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.171ex; width:76.974ex; height:21.509ex;" alt="{\displaystyle {\begin{aligned}e^{iz}&{}=1+iz+{\frac {(iz)^{2}}{2!}}+{\frac {(iz)^{3}}{3!}}+{\frac {(iz)^{4}}{4!}}+{\frac {(iz)^{5}}{5!}}+{\frac {(iz)^{6}}{6!}}+{\frac {(iz)^{7}}{7!}}+{\frac {(iz)^{8}}{8!}}+\cdots \\&{}=1+iz-{\frac {z^{2}}{2!}}-{\frac {iz^{3}}{3!}}+{\frac {z^{4}}{4!}}+{\frac {iz^{5}}{5!}}-{\frac {z^{6}}{6!}}-{\frac {iz^{7}}{7!}}+{\frac {z^{8}}{8!}}+\cdots \\&{}=\left(1-{\frac {z^{2}}{2!}}+{\frac {z^{4}}{4!}}-{\frac {z^{6}}{6!}}+{\frac {z^{8}}{8!}}-\cdots \right)+i\left(z-{\frac {z^{3}}{3!}}+{\frac {z^{5}}{5!}}-{\frac {z^{7}}{7!}}+\cdots \right)\\&{}=\cos z+i\sin z\end{aligned}}}"></span></dd></dl> <p>가 된다. </p> <div class="mw-heading mw-heading3"><h3 id="미분_계산을_이용한_증명방법"><span id=".EB.AF.B8.EB.B6.84_.EA.B3.84.EC.82.B0.EC.9D.84_.EC.9D.B4.EC.9A.A9.ED.95.9C_.EC.A6.9D.EB.AA.85.EB.B0.A9.EB.B2.95"></span>미분 계산을 이용한 증명방법</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%98%A4%EC%9D%BC%EB%9F%AC_%EA%B3%B5%EC%8B%9D&action=edit&section=4" title="부분 편집: 미분 계산을 이용한 증명방법"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=e^{-ix}(\cos x+i\sin x)\cdots (1)}"> <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> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=e^{-ix}(\cos x+i\sin x)\cdots (1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61797a01517e95537ed08166198e30e7b5172309" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.327ex; height:3.176ex;" alt="{\displaystyle f(x)=e^{-ix}(\cos x+i\sin x)\cdots (1)}"></span> 라면,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dx}}f(x)=-ie^{-ix}(\cos x+i\sin x)+e^{-ix}(-\sin x+i\cos x)=e^{-ix}(-i\cos x+\sin x-\sin x+i\cos x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>i</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}f(x)=-ie^{-ix}(\cos x+i\sin x)+e^{-ix}(-\sin x+i\cos x)=e^{-ix}(-i\cos x+\sin x-\sin x+i\cos x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ecce289e448e37803fc73435fd72acba4969757" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:102.007ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dx}}f(x)=-ie^{-ix}(\cos x+i\sin x)+e^{-ix}(-\sin x+i\cos x)=e^{-ix}(-i\cos x+\sin x-\sin x+i\cos x)=0}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \therefore f(x)=C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∴</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \therefore f(x)=C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fe072cf50cf316db906e81825a2aea66da0eb2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.478ex; height:2.843ex;" alt="{\displaystyle \therefore f(x)=C}"></span> (단, <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>는 상수)</dd></dl> <p>(1)에 <span class="mwe-math-element"><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=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}"></span>을 대입하면, </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(0)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(0)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00884d9c3ef0dbc18fe185a4444f1da931c8ba0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.511ex; height:2.843ex;" alt="{\displaystyle f(0)=1}"></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 \therefore C=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∴</mo> <mi>C</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \therefore C=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4ace1f8905f4144ccbbee49db1da0b85ebc5e3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.223ex; height:2.343ex;" alt="{\displaystyle \therefore C=1}"></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 e^{-ix}(\cos x+i\sin x)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-ix}(\cos x+i\sin x)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17640a81e7dfa7fbf39809d72abe405bd867e4bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.602ex; height:3.176ex;" alt="{\displaystyle e^{-ix}(\cos x+i\sin x)=1}"></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 e^{ix}=(\cos x+i\sin x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{ix}=(\cos x+i\sin x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/289bdc7a26f9ef9448972e5783a582a8c53b1c7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.161ex; height:3.176ex;" alt="{\displaystyle e^{ix}=(\cos x+i\sin x)}"></span></dd></dl> <p><i><a href="/wiki/Q.E.D." title="Q.E.D.">Q.E.D.</a></i> </p> <div class="mw-heading mw-heading3"><h3 id="미적분을_이용한_방법"><span id=".EB.AF.B8.EC.A0.81.EB.B6.84.EC.9D.84_.EC.9D.B4.EC.9A.A9.ED.95.9C_.EB.B0.A9.EB.B2.95"></span>미적분을 이용한 방법</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%98%A4%EC%9D%BC%EB%9F%AC_%EA%B3%B5%EC%8B%9D&action=edit&section=5" title="부분 편집: 미적분을 이용한 방법"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>다음과 같은 복소수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>를 생각하자: </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 z=\cos x+i\sin x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=\cos x+i\sin x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05535d0a5ca528219e00012aed0255969d5bda54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.004ex; height:2.343ex;" alt="{\displaystyle z=\cos x+i\sin x\,}"></span></dd></dl> <p>양변을 <span class="mwe-math-element"><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>에 대해 미분하면: </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 {dz}{dx}}=-\sin x+i\cos x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dz}{dx}}=-\sin x+i\cos x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b87ba9620ff51e242abca44425450cbf37c9840b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.106ex; height:5.509ex;" alt="{\displaystyle {\frac {dz}{dx}}=-\sin x+i\cos x}"></span></dd></dl> <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 i^{2}=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i^{2}=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88e98a401d352e5037d5043028e2d7f449e83fa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.926ex; height:2.843ex;" alt="{\displaystyle i^{2}=-1}"></span>이므로: </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 {dz}{dx}}=i^{2}\sin x+i\cos x=i(\cos x+i\sin x)=iz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>i</mi> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>i</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dz}{dx}}=i^{2}\sin x+i\cos x=i(\cos x+i\sin x)=iz}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4acb9985884359fcd79b71cc251371cf55f2773d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:46.284ex; height:5.509ex;" alt="{\displaystyle {\frac {dz}{dx}}=i^{2}\sin x+i\cos x=i(\cos x+i\sin x)=iz}"></span></dd></dl> <p>z를 이항한 후 양변을 적분하면: </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}{\frac {1}{z}}\,{\frac {dz}{dx}}&=i\\\int {\frac {1}{z}}\,dz&=\int i\,dx\\\ln z&=ix+C\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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>i</mi> </mtd> </mtr> <mtr> <mtd> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>∫</mo> <mi>i</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>ln</mi> <mo>⁡</mo> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>i</mi> <mi>x</mi> <mo>+</mo> <mi>C</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {1}{z}}\,{\frac {dz}{dx}}&=i\\\int {\frac {1}{z}}\,dz&=\int i\,dx\\\ln z&=ix+C\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ace8a76accb089aa7dc98e8095d99d9397f7e60c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:17.859ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {1}{z}}\,{\frac {dz}{dx}}&=i\\\int {\frac {1}{z}}\,dz&=\int i\,dx\\\ln z&=ix+C\end{aligned}}}"></span></dd></dl> <p>(여기에서 <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>는 적분 상수이다.) </p><p>이제 <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f750a7094a396d89a81974cdf35783db2bb287b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.027ex; height:2.176ex;" alt="{\displaystyle C=0}"></span>이라는 것을 증명한다. <span class="mwe-math-element"><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=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}"></span>일 경우를 계산해보면 </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}\ln z&=C\\z&=\cos x+i\sin x=\cos 0+i\sin 0=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> <mi>ln</mi> <mo>⁡</mo> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>C</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mn>0</mn> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mn>0</mn> <mo>=</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ln z&=C\\z&=\cos x+i\sin x=\cos 0+i\sin 0=1\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5d83c05370259a52b5b4048b14f626e1fed5899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:41.15ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\ln z&=C\\z&=\cos x+i\sin x=\cos 0+i\sin 0=1\end{aligned}}}"></span></dd></dl> <p>따라서 </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}\ln 1&=C\\C&=0\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>ln</mi> <mo>⁡</mo> <mn>1</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>C</mi> </mtd> </mtr> <mtr> <mtd> <mi>C</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ln 1&=C\\C&=0\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc7b3c25ea20609de5139f9b5c318ab8f9769bc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.105ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\ln 1&=C\\C&=0\end{aligned}}}"></span></dd></dl> <p>따라서 다음과 같은 식이 성립한다: </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}\ln z&=ix\\z&=e^{ix}\\e^{ix}&=\cos x+i\sin x\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>ln</mi> <mo>⁡</mo> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>i</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ln z&=ix\\z&=e^{ix}\\e^{ix}&=\cos x+i\sin x\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f150b040ac72a5c373499346ca45a3aff0a45947" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:20.695ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}\ln z&=ix\\z&=e^{ix}\\e^{ix}&=\cos x+i\sin x\end{aligned}}}"></span></dd></dl> <p><i><a href="/wiki/Q.E.D." title="Q.E.D.">Q.E.D.</a></i> </p> <div class="mw-heading mw-heading3"><h3 id="미분방정식을_이용한_방법"><span id=".EB.AF.B8.EB.B6.84.EB.B0.A9.EC.A0.95.EC.8B.9D.EC.9D.84_.EC.9D.B4.EC.9A.A9.ED.95.9C_.EB.B0.A9.EB.B2.95"></span>미분방정식을 이용한 방법</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%98%A4%EC%9D%BC%EB%9F%AC_%EA%B3%B5%EC%8B%9D&action=edit&section=6" title="부분 편집: 미분방정식을 이용한 방법"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>함수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}"></span>를 다음과 같이 정의한다. </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 g(x)=e^{ix}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)=e^{ix}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a03d49d5645c023caa33e9b612abca25b02a9002" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.177ex; height:3.176ex;" alt="{\displaystyle g(x)=e^{ix}}"></span></dd></dl> <p>허수단위 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>는 상수이므로 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}"></span>의 <a href="/wiki/%EB%8F%84%ED%95%A8%EC%88%98" class="mw-redirect" title="도함수">도함수</a>와 <a href="/wiki/%EC%9D%B4%EA%B3%84%EB%8F%84%ED%95%A8%EC%88%98" title="이계도함수">이계도함수</a>는 다음과 같다. </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}g'(x)&=ie^{ix}\\g''(x)&=i^{2}e^{ix}=-e^{ix}\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> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>i</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>g</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}g'(x)&=ie^{ix}\\g''(x)&=i^{2}e^{ix}=-e^{ix}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25a4f168055b517ac48d9abb989ab6bdd294323c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.654ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}g'(x)&=ie^{ix}\\g''(x)&=i^{2}e^{ix}=-e^{ix}\end{aligned}}}"></span></dd></dl> <p>이로부터 </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 g''(x)=-g(x)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g''(x)=-g(x)\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1a75c177a3f0dbc6da683f4e407891aef83f830" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.137ex; height:3.009ex;" alt="{\displaystyle g''(x)=-g(x)\ }"></span> 또는</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 g''(x)+g(x)=0\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g''(x)+g(x)=0\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8218b2fda35a5e69600cb6b8bfa6db919ac76c27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.331ex; height:3.009ex;" alt="{\displaystyle g''(x)+g(x)=0\ }"></span> 라는 2차 선형 <a href="/wiki/%EB%AF%B8%EB%B6%84%EB%B0%A9%EC%A0%95%EC%8B%9D" title="미분방정식">미분방정식</a>이 만들어지고,</dd></dl> <p><a href="/wiki/%EC%9D%BC%EC%B0%A8_%EB%8F%85%EB%A6%BD" class="mw-redirect" title="일차 독립">일차 독립</a>인 두 해가 발생한다. </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}g_{1}(x)&=\cos x\\g_{2}(x)&=\sin x\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}g_{1}(x)&=\cos x\\g_{2}(x)&=\sin x\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/020be456cf972b09e82121e6497ccc4a603eb75f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.98ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}g_{1}(x)&=\cos x\\g_{2}(x)&=\sin x\end{aligned}}}"></span></dd></dl> <p>한편, 차수가 같은 미분방정식의 어떤 <a href="/wiki/%EC%84%A0%ED%98%95_%EA%B2%B0%ED%95%A9" title="선형 결합">선형 결합</a>도 해가 될 수 있으므로 위의 미분방정식의 일반적인 해는 다음과 같다. </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}g(x)&=Ag_{1}(x)+Bg_{2}(x)\\&=A\cos x+B\sin x\\g'(x)&=-A\sin x+B\cos x\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>A</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>B</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>A</mi> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>B</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>A</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>B</mi> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}g(x)&=Ag_{1}(x)+Bg_{2}(x)\\&=A\cos x+B\sin x\\g'(x)&=-A\sin x+B\cos x\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48207ff53e7879e84361c9766b37f13983b91fb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.73ex; margin-bottom: -0.275ex; width:27.122ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}g(x)&=Ag_{1}(x)+Bg_{2}(x)\\&=A\cos x+B\sin x\\g'(x)&=-A\sin x+B\cos x\end{aligned}}}"></span> <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>와 <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>는 상수)</dd></dl></dd></dl> <p>그리고 여기에 함수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}"></span> 의 초기 조건 </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}g(0)&=e^{i0}&=1\\g'(0)&=ie^{i0}&=i\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>g</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>0</mn> </mrow> </msup> </mtd> <mtd> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>i</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>0</mn> </mrow> </msup> </mtd> <mtd> <mo>=</mo> <mi>i</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}g(0)&=e^{i0}&=1\\g'(0)&=ie^{i0}&=i\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09e63d24a79172a23bd4f8314363b3940cb3a1ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.393ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}g(0)&=e^{i0}&=1\\g'(0)&=ie^{i0}&=i\end{aligned}}}"></span> 을 대입하면,</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 {\begin{aligned}g(0)&={\color {White}-}A\cos 0+B\sin 0&=A\\g'(0)&=-A\sin 0+B\cos 0&=B\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>g</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#FFFFFF"> <mo>−</mo> </mstyle> </mrow> <mi>A</mi> <mi>cos</mi> <mo>⁡</mo> <mn>0</mn> <mo>+</mo> <mi>B</mi> <mi>sin</mi> <mo>⁡</mo> <mn>0</mn> </mtd> <mtd> <mo>=</mo> <mi>A</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>A</mi> <mi>sin</mi> <mo>⁡</mo> <mn>0</mn> <mo>+</mo> <mi>B</mi> <mi>cos</mi> <mo>⁡</mo> <mn>0</mn> </mtd> <mtd> <mo>=</mo> <mi>B</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}g(0)&={\color {White}-}A\cos 0+B\sin 0&=A\\g'(0)&=-A\sin 0+B\cos 0&=B\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01002464380f4acb8eeb828944c7e8f4cb9b4a4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.483ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}g(0)&={\color {White}-}A\cos 0+B\sin 0&=A\\g'(0)&=-A\sin 0+B\cos 0&=B\end{aligned}}}"></span></dd></dl> <p><br /> 곧, </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}g(0)&=A&=1\\g'(0)&=B&=i\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>g</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>A</mi> </mtd> <mtd> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>B</mi> </mtd> <mtd> <mo>=</mo> <mi>i</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}g(0)&=A&=1\\g'(0)&=B&=i\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08ef943312dd6c2d0200dc4041ec0344b131dddf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.65ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}g(0)&=A&=1\\g'(0)&=B&=i\end{aligned}}}"></span></dd></dl> <p>이므로 </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 g(x)\,=\,e^{ix}=\cos x+i\sin x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)\,=\,e^{ix}=\cos x+i\sin x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdbd73e3c601413b8604cdb3f210287021a128fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.48ex; height:3.176ex;" alt="{\displaystyle g(x)\,=\,e^{ix}=\cos x+i\sin x}"></span>이다.</dd></dl> <p><a href="/wiki/Q.E.D." title="Q.E.D.">Q.E.D.</a> </p> <div class="mw-heading mw-heading2"><h2 id="박사가_사랑한_수식"><span id=".EB.B0.95.EC.82.AC.EA.B0.80_.EC.82.AC.EB.9E.91.ED.95.9C_.EC.88.98.EC.8B.9D"></span>박사가 사랑한 수식</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%98%A4%EC%9D%BC%EB%9F%AC_%EA%B3%B5%EC%8B%9D&action=edit&section=7" title="부분 편집: 박사가 사랑한 수식"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>오일러 공식은 위 소설의 모티브로도 사용이 되었다. 오일러 공식에 숨어 있는 뜻을 cis 함수</b> 또는 <b>복소 지수 함수</b>는 오일러 공식으로부터 바로 유도되는 함수로, 다음과 같이 정의되는 것이다. </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 \operatorname {cis} (\theta )=e^{i\theta }=\cos \theta +i\sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cis</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cis} (\theta )=e^{i\theta }=\cos \theta +i\sin \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2576620507cff98370b59cb47be9a06a8074f9bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.298ex; height:3.176ex;" alt="{\displaystyle \operatorname {cis} (\theta )=e^{i\theta }=\cos \theta +i\sin \theta }"></span></dd></dl> <p>이 함수는 <a href="/wiki/%ED%91%B8%EB%A6%AC%EC%97%90_%EB%B3%80%ED%99%98" title="푸리에 변환">푸리에 변환</a>이나 <a href="/wiki/%ED%8E%98%EC%9D%B4%EC%A0%80_(%EC%A0%84%EC%9E%90)" title="페이저 (전자)">페이저</a> 등에서 복소수와 관련된 연산을 할 때 흔히 사용되는 것이다. </p> <div class="mw-heading mw-heading2"><h2 id="쉬운_설명"><span id=".EC.89.AC.EC.9A.B4_.EC.84.A4.EB.AA.85"></span>쉬운 설명</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%98%A4%EC%9D%BC%EB%9F%AC_%EA%B3%B5%EC%8B%9D&action=edit&section=8" title="부분 편집: 쉬운 설명"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>우선 이 공식을 이해하기 위해서는 <a href="/wiki/%EB%A9%B1%EA%B8%89%EC%88%98" title="멱급수">멱급수</a>가 무엇인지 알고 있어야 한다. 멱급수란 하나의 수의 지수를 증가시키며 모두 더한 값을 말하며, 지수함수의 멱급수는 다음과 같다. </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 e^{x}=1+x+{\frac {x^{2}}{2}}+{\frac {x^{3}}{3\times 2}}+{\frac {x^{4}}{4\times 3\times 2}}+\cdot \cdot \cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mn>3</mn> <mo>×</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow> <mn>4</mn> <mo>×</mo> <mn>3</mn> <mo>×</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋅</mo> <mo>⋅</mo> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{x}=1+x+{\frac {x^{2}}{2}}+{\frac {x^{3}}{3\times 2}}+{\frac {x^{4}}{4\times 3\times 2}}+\cdot \cdot \cdot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ace7f1e2720a50cef360cbe190f3234283c28ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:44.247ex; height:5.843ex;" alt="{\displaystyle e^{x}=1+x+{\frac {x^{2}}{2}}+{\frac {x^{3}}{3\times 2}}+{\frac {x^{4}}{4\times 3\times 2}}+\cdot \cdot \cdot }"></span> </p><p>이 멱급수의 지수 <span class="mwe-math-element"><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>에 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle iz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle iz}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42006d44120493f9b1963f78de623524c26872f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.891ex; height:2.176ex;" alt="{\displaystyle iz}"></span>(복소수에 임의의 수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>를 곱한 값)을 추가하면, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\sin z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\sin z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca02ab1c7af6f3d31420d2003775901f9c4dc69c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.52ex; height:2.176ex;" alt="{\displaystyle i\sin z}"></span>에 <span class="mwe-math-element"><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 z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26ac0eb3eee2aa3b07447f04fa98ff5964b2772" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.586ex; height:1.676ex;" alt="{\displaystyle \cos z}"></span>를 더한 값과 완벽하게 일치했다. </p><p>즉, <span class="mwe-math-element"><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^{iz}\,=\,\cos z+i\sin z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>z</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mi>z</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{iz}\,=\,\cos z+i\sin z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be9703af96c109826addee1cba25dff343b39fb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.472ex; height:2.843ex;" alt="{\displaystyle e^{iz}\,=\,\cos z+i\sin z}"></span>가 성립하는 것이다. </p><p>이 등식에서 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>에 <span class="mwe-math-element"><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>를 대입했을 때 나오는 특수한 등식이 바로 오일러의 등식이다. </p><p><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="각주"><span id=".EA.B0.81.EC.A3.BC"></span>각주</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%98%A4%EC%9D%BC%EB%9F%AC_%EA%B3%B5%EC%8B%9D&action=edit&section=9" title="부분 편집: 각주"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r35556958">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.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"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">[참고] (대수학 원론 레온하르트 오일러,번억 Christopher James Sangwin,School of Mathematics,University of Birmingham,United Kingdom,B15 2TT,<a href="/wiki/%ED%8A%B9%EC%88%98:%EC%B1%85%EC%B0%BE%EA%B8%B0/9781899618736" class="internal mw-magiclink-isbn">ISBN 978-1-89961-873-6</a>) <a rel="nofollow" class="external free" href="https://web.archive.org/web/20110413234352/http://web.mat.bham.ac.uk/C.J.Sangwin/euler/">https://web.archive.org/web/20110413234352/http://web.mat.bham.ac.uk/C.J.Sangwin/euler/</a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><cite class="citation book">리처드 엘위스. 《수학 100》.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%EC%88%98%ED%95%99+100&rft.au=%EB%A6%AC%EC%B2%98%EB%93%9C+%EC%97%98%EC%9C%84%EC%8A%A4&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%98%A4%EC%9D%BC%EB%9F%AC+%EA%B3%B5%EC%8B%9D" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="같이_보기"><span id=".EA.B0.99.EC.9D.B4_.EB.B3.B4.EA.B8.B0"></span>같이 보기</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%98%A4%EC%9D%BC%EB%9F%AC_%EA%B3%B5%EC%8B%9D&action=edit&section=10" title="부분 편집: 같이 보기"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=%EA%B2%94%ED%8F%B0%EB%93%9C_%EC%83%81%EC%88%98&action=edit&redlink=1" class="new" title="겔폰드 상수 (없는 문서)">겔폰트 상수</a></li> <li><a href="/wiki/%EC%98%A4%EC%9D%BC%EB%9F%AC%EC%9D%98_%EB%93%B1%EC%8B%9D" class="mw-redirect" title="오일러의 등식">오일러의 등식</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">원본 주소 "<a dir="ltr" href="https://ko.wikipedia.org/w/index.php?title=오일러_공식&oldid=37777064">https://ko.wikipedia.org/w/index.php?title=오일러_공식&oldid=37777064</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" 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</div> </div> </div> </nav> <div class="vector-sticky-header-context-bar-primary" aria-hidden="true" ><span class="mw-page-title-main">오일러 공식</span></div> </div> </div> <div class="vector-sticky-header-end" aria-hidden="true"> <div class="vector-sticky-header-icons"> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-talk-sticky-header" tabindex="-1" data-event-name="talk-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbles mw-ui-icon-wikimedia-speechBubbles"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-subject-sticky-header" tabindex="-1" data-event-name="subject-sticky-header"><span class="vector-icon mw-ui-icon-article mw-ui-icon-wikimedia-article"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-history-sticky-header" tabindex="-1" data-event-name="history-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-history mw-ui-icon-wikimedia-wikimedia-history"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only mw-watchlink" id="ca-watchstar-sticky-header" tabindex="-1" data-event-name="watch-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-star mw-ui-icon-wikimedia-wikimedia-star"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-edit-sticky-header" tabindex="-1" data-event-name="wikitext-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-wikiText mw-ui-icon-wikimedia-wikimedia-wikiText"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-ve-edit-sticky-header" tabindex="-1" data-event-name="ve-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-edit mw-ui-icon-wikimedia-wikimedia-edit"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-editLock mw-ui-icon-wikimedia-wikimedia-editLock"></span> <span></span> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-language mw-ui-icon-wikimedia-wikimedia-language"></span> <span>66개 언어</span> </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbleAdd-progressive mw-ui-icon-wikimedia-speechBubbleAdd-progressive"></span> <span>새 주제</span> </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> 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