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class="vector-toc-text">Bevezető</div> </a> </li> <li id="toc-Komplex_függvény" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Komplex_függvény"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Komplex függvény</span> </div> </a> <ul id="toc-Komplex_függvény-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Differenciálhatóság" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Differenciálhatóság"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Differenciálhatóság</span> </div> </a> <button aria-controls="toc-Differenciálhatóság-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>A(z) Differenciálhatóság alszakasz kinyitása/becsukása</span> </button> <ul id="toc-Differenciálhatóság-sublist" class="vector-toc-list"> <li id="toc-A_derivált" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#A_derivált"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>A derivált</span> </div> </a> <ul id="toc-A_derivált-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-A_Cauchy–Riemann_egyenletek" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#A_Cauchy–Riemann_egyenletek"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>A Cauchy–Riemann egyenletek</span> </div> </a> <ul id="toc-A_Cauchy–Riemann_egyenletek-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Minden_differenciálható_komplex_függvény_analitikus" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Minden_differenciálható_komplex_függvény_analitikus"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Minden differenciálható komplex függvény analitikus</span> </div> </a> <ul id="toc-Minden_differenciálható_komplex_függvény_analitikus-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Integrálás" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Integrálás"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Integrálás</span> </div> </a> <ul id="toc-Integrálás-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Holomorf_függvények" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Holomorf_függvények"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Holomorf függvények</span> </div> </a> <ul id="toc-Holomorf_függvények-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Meromorf_függvények" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Meromorf_függvények"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Meromorf függvények</span> </div> </a> <ul id="toc-Meromorf_függvények-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Jegyzetek" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Jegyzetek"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Jegyzetek</span> </div> </a> <ul id="toc-Jegyzetek-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-További_információk" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#További_információk"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>További információk</span> </div> </a> <ul id="toc-További_információk-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="Tartalomjegyzék" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Tartalomjegyzék kinyitása/becsukása" > <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">Tartalomjegyzék kinyitása/becsukása</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">Komplex analízis</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="Ugrás egy más nyelvű szócikkre. Elérhető 57 nyelven" > <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-57" 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">57 nyelv</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Complex_analysis" title="Complex analysis – angol" lang="en" hreflang="en" data-title="Complex analysis" data-language-autonym="English" data-language-local-name="angol" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%AD%D9%84%D9%8A%D9%84_%D9%85%D8%B1%D9%83%D8%A8" title="تحليل مركب – arab" lang="ar" hreflang="ar" data-title="تحليل مركب" data-language-autonym="العربية" data-language-local-name="arab" 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/Anal%C3%ADs_complexu" title="Analís complexu – asztúr" lang="ast" hreflang="ast" data-title="Analís complexu" data-language-autonym="Asturianu" data-language-local-name="asztúr" 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/Kompleks_analiz" title="Kompleks analiz – azerbajdzsáni" lang="az" hreflang="az" data-title="Kompleks analiz" data-language-autonym="Azərbaycanca" data-language-local-name="azerbajdzsáni" 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%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BB%D1%8B_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Комплекслы анализ – baskír" lang="ba" hreflang="ba" data-title="Комплекслы анализ" data-language-autonym="Башҡортса" data-language-local-name="baskír" 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%9A%D0%B0%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D1%8B_%D0%B0%D0%BD%D0%B0%D0%BB%D1%96%D0%B7" title="Камплексны аналіз – belarusz" lang="be" hreflang="be" data-title="Камплексны аналіз" data-language-autonym="Беларуская" data-language-local-name="belarusz" 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%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%B5%D0%BD_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Комплексен анализ – bolgár" lang="bg" hreflang="bg" data-title="Комплексен анализ" data-language-autonym="Български" data-language-local-name="bolgár" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/An%C3%A0lisi_complexa" title="Anàlisi complexa – katalán" lang="ca" hreflang="ca" data-title="Anàlisi complexa" data-language-autonym="Català" data-language-local-name="katalán" 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/%D8%B4%DB%8C%DA%A9%D8%A7%D8%B1%DB%8C%DB%8C_%D8%A6%D8%A7%D9%88%DB%8E%D8%AA%DB%95" title="شیکاریی ئاوێتە – közép-ázsiai kurd" lang="ckb" hreflang="ckb" data-title="شیکاریی ئاوێتە" data-language-autonym="کوردی" data-language-local-name="közép-ázsiai kurd" 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/Komplexn%C3%AD_anal%C3%BDza" title="Komplexní analýza – cseh" lang="cs" hreflang="cs" data-title="Komplexní analýza" data-language-autonym="Čeština" data-language-local-name="cseh" 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%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BB%C4%83_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Комплекслă анализ – csuvas" lang="cv" hreflang="cv" data-title="Комплекслă анализ" data-language-autonym="Чӑвашла" data-language-local-name="csuvas" 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/Dadansoddi_cymhlyg" title="Dadansoddi cymhlyg – walesi" lang="cy" hreflang="cy" data-title="Dadansoddi cymhlyg" data-language-autonym="Cymraeg" data-language-local-name="walesi" 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/Kompleks_analyse" title="Kompleks analyse – dán" lang="da" hreflang="da" data-title="Kompleks analyse" data-language-autonym="Dansk" data-language-local-name="dán" 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/Funktionentheorie" title="Funktionentheorie – német" lang="de" hreflang="de" data-title="Funktionentheorie" data-language-autonym="Deutsch" data-language-local-name="német" 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%9C%CE%B9%CE%B3%CE%B1%CE%B4%CE%B9%CE%BA%CE%AE_%CE%B1%CE%BD%CE%AC%CE%BB%CF%85%CF%83%CE%B7" title="Μιγαδική ανάλυση – görög" lang="el" hreflang="el" data-title="Μιγαδική ανάλυση" data-language-autonym="Ελληνικά" data-language-local-name="görög" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Kompleksa_analitiko" title="Kompleksa analitiko – eszperantó" lang="eo" hreflang="eo" data-title="Kompleksa analitiko" data-language-autonym="Esperanto" data-language-local-name="eszperantó" 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/An%C3%A1lisis_complejo" title="Análisis complejo – spanyol" lang="es" hreflang="es" data-title="Análisis complejo" data-language-autonym="Español" data-language-local-name="spanyol" 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/Kompleksmuutuja_funktsioonide_teooria" title="Kompleksmuutuja funktsioonide teooria – észt" lang="et" hreflang="et" data-title="Kompleksmuutuja funktsioonide teooria" data-language-autonym="Eesti" data-language-local-name="észt" 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/Analisi_konplexu" title="Analisi konplexu – baszk" lang="eu" hreflang="eu" data-title="Analisi konplexu" data-language-autonym="Euskara" data-language-local-name="baszk" 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/%D8%A2%D9%86%D8%A7%D9%84%DB%8C%D8%B2_%D9%85%D8%AE%D8%AA%D9%84%D8%B7" title="آنالیز مختلط – perzsa" lang="fa" hreflang="fa" data-title="آنالیز مختلط" data-language-autonym="فارسی" data-language-local-name="perzsa" 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/Funktioteoria" title="Funktioteoria – finn" lang="fi" hreflang="fi" data-title="Funktioteoria" data-language-autonym="Suomi" data-language-local-name="finn" 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/Analyse_complexe" title="Analyse complexe – francia" lang="fr" hreflang="fr" data-title="Analyse complexe" data-language-autonym="Français" data-language-local-name="francia" 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/An%C3%A1lise_complexa" title="Análise complexa – gallego" lang="gl" hreflang="gl" data-title="Análise complexa" data-language-autonym="Galego" data-language-local-name="gallego" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%9E%D7%A8%D7%95%D7%9B%D7%91%D7%AA" title="אנליזה מרוכבת – héber" lang="he" hreflang="he" data-title="אנליזה מרוכבת" data-language-autonym="עברית" data-language-local-name="héber" 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%B8%E0%A4%AE%E0%A5%8D%E0%A4%AE%E0%A4%BF%E0%A4%B6%E0%A5%8D%E0%A4%B0_%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%8D%E0%A4%B2%E0%A5%87%E0%A4%B7%E0%A4%A3" title="सम्मिश्र विश्लेषण – hindi" lang="hi" hreflang="hi" data-title="सम्मिश्र विश्लेषण" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BF%D5%B8%D5%B4%D5%BA%D5%AC%D5%A5%D6%84%D5%BD_%D5%A1%D5%B6%D5%A1%D5%AC%D5%AB%D5%A6" title="Կոմպլեքս անալիզ – örmény" lang="hy" hreflang="hy" data-title="Կոմպլեքս անալիզ" data-language-autonym="Հայերեն" data-language-local-name="örmény" 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/Analisis_kompleks" title="Analisis kompleks – indonéz" lang="id" hreflang="id" data-title="Analisis kompleks" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonéz" 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/Tvinnfallagreining" title="Tvinnfallagreining – izlandi" lang="is" hreflang="is" data-title="Tvinnfallagreining" data-language-autonym="Íslenska" data-language-local-name="izlandi" 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/Analisi_complessa" title="Analisi complessa – olasz" lang="it" hreflang="it" data-title="Analisi complessa" data-language-autonym="Italiano" data-language-local-name="olasz" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%A4%87%E7%B4%A0%E8%A7%A3%E6%9E%90" title="複素解析 – japán" lang="ja" hreflang="ja" data-title="複素解析" data-language-autonym="日本語" data-language-local-name="japán" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%99%E1%83%9D%E1%83%9B%E1%83%9E%E1%83%9A%E1%83%94%E1%83%A5%E1%83%A1%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%90%E1%83%9C%E1%83%90%E1%83%9A%E1%83%98%E1%83%96%E1%83%98" title="კომპლექსური ანალიზი – grúz" lang="ka" hreflang="ka" data-title="კომპლექსური ანალიზი" data-language-autonym="ქართული" data-language-local-name="grúz" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B3%B5%EC%86%8C%ED%95%B4%EC%84%9D%ED%95%99" title="복소해석학 – koreai" lang="ko" hreflang="ko" data-title="복소해석학" data-language-autonym="한국어" data-language-local-name="koreai" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/An%C3%A0lisi_cumplessa" title="Anàlisi cumplessa – lombard" lang="lmo" hreflang="lmo" data-title="Anàlisi cumplessa" data-language-autonym="Lombard" data-language-local-name="lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D0%B0_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7%D0%B0" title="Комплексна анализа – macedón" lang="mk" hreflang="mk" data-title="Комплексна анализа" data-language-autonym="Македонски" data-language-local-name="macedón" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Analisi_komplessa" title="Analisi komplessa – máltai" lang="mt" hreflang="mt" data-title="Analisi komplessa" data-language-autonym="Malti" data-language-local-name="máltai" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Functietheorie" title="Functietheorie – holland" lang="nl" hreflang="nl" data-title="Functietheorie" data-language-autonym="Nederlands" data-language-local-name="holland" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Kompleks_analyse" title="Kompleks analyse – norvég (bokmål)" lang="nb" hreflang="nb" data-title="Kompleks analyse" data-language-autonym="Norsk bokmål" data-language-local-name="norvég (bokmål)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-os mw-list-item"><a href="https://os.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BE%D0%BD_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Комплексон анализ – oszét" lang="os" hreflang="os" data-title="Комплексон анализ" data-language-autonym="Ирон" data-language-local-name="oszét" class="interlanguage-link-target"><span>Ирон</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Analiza_zespolona" title="Analiza zespolona – lengyel" lang="pl" hreflang="pl" data-title="Analiza zespolona" data-language-autonym="Polski" data-language-local-name="lengyel" 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/An%C3%A1lise_complexa" title="Análise complexa – portugál" lang="pt" hreflang="pt" data-title="Análise complexa" data-language-autonym="Português" data-language-local-name="portugál" 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/Analiz%C4%83_complex%C4%83" title="Analiză complexă – román" lang="ro" hreflang="ro" data-title="Analiză complexă" data-language-autonym="Română" data-language-local-name="román" 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%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D1%8B%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Комплексный анализ – orosz" lang="ru" hreflang="ru" data-title="Комплексный анализ" data-language-autonym="Русский" data-language-local-name="orosz" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Complex_analysis" title="Complex analysis – skót" lang="sco" hreflang="sco" data-title="Complex analysis" data-language-autonym="Scots" data-language-local-name="skót" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B7%83%E0%B6%82%E0%B6%9A%E0%B7%93%E0%B6%BB%E0%B7%8A%E0%B6%AB_%E0%B7%80%E0%B7%92%E0%B7%81%E0%B7%8A%E0%B6%BD%E0%B7%9A%E0%B7%82%E0%B6%AB%E0%B6%BA" title="සංකීර්ණ විශ්ලේෂණය – szingaléz" lang="si" hreflang="si" data-title="සංකීර්ණ විශ්ලේෂණය" data-language-autonym="සිංහල" data-language-local-name="szingaléz" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Complex_analysis" title="Complex analysis – Simple English" lang="en-simple" hreflang="en-simple" data-title="Complex analysis" 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-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Komplexn%C3%A1_anal%C3%BDza" title="Komplexná analýza – szlovák" lang="sk" hreflang="sk" data-title="Komplexná analýza" data-language-autonym="Slovenčina" data-language-local-name="szlovák" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Analiza_komplekse" title="Analiza komplekse – albán" lang="sq" hreflang="sq" data-title="Analiza komplekse" data-language-autonym="Shqip" data-language-local-name="albán" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D0%B0_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7%D0%B0" title="Комплексна анализа – szerb" lang="sr" hreflang="sr" data-title="Комплексна анализа" data-language-autonym="Српски / srpski" data-language-local-name="szerb" 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/Komplex_analys" title="Komplex analys – svéd" lang="sv" hreflang="sv" data-title="Komplex analys" data-language-autonym="Svenska" data-language-local-name="svéd" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%A7%E0%B8%B4%E0%B9%80%E0%B8%84%E0%B8%A3%E0%B8%B2%E0%B8%B0%E0%B8%AB%E0%B9%8C%E0%B9%80%E0%B8%8A%E0%B8%B4%E0%B8%87%E0%B8%8B%E0%B9%89%E0%B8%AD%E0%B8%99" title="การวิเคราะห์เชิงซ้อน – thai" lang="th" hreflang="th" data-title="การวิเคราะห์เชิงซ้อน" data-language-autonym="ไทย" data-language-local-name="thai" 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/Karma%C5%9F%C4%B1k_analiz" title="Karmaşık analiz – török" lang="tr" hreflang="tr" data-title="Karmaşık analiz" data-language-autonym="Türkçe" data-language-local-name="török" 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%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D0%B8%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D1%96%D0%B7" title="Комплексний аналіз – ukrán" lang="uk" hreflang="uk" data-title="Комплексний аналіз" data-language-autonym="Українська" data-language-local-name="ukrán" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Gi%E1%BA%A3i_t%C3%ADch_ph%E1%BB%A9c" title="Giải tích phức – vietnámi" lang="vi" hreflang="vi" data-title="Giải tích phức" data-language-autonym="Tiếng Việt" data-language-local-name="vietnámi" 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/%E5%A4%8D%E5%88%86%E6%9E%90" title="复分析 – wu kínai" lang="wuu" hreflang="wuu" data-title="复分析" data-language-autonym="吴语" data-language-local-name="wu kínai" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-xmf mw-list-item"><a href="https://xmf.wikipedia.org/wiki/%E1%83%99%E1%83%9D%E1%83%9B%E1%83%9E%E1%83%9A%E1%83%94%E1%83%A5%E1%83%A1%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%90%E1%83%9C%E1%83%90%E1%83%9A%E1%83%98%E1%83%96%E1%83%98" title="კომპლექსური ანალიზი – Mingrelian" lang="xmf" hreflang="xmf" data-title="კომპლექსური ანალიზი" data-language-autonym="მარგალური" data-language-local-name="Mingrelian" class="interlanguage-link-target"><span>მარგალური</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%A4%87%E5%88%86%E6%9E%90" title="複分析 – kínai" lang="zh" hreflang="zh" data-title="複分析" data-language-autonym="中文" data-language-local-name="kínai" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%A4%87%E5%88%86%E6%9E%90" title="複分析 – kantoni" lang="yue" hreflang="yue" data-title="複分析" data-language-autonym="粵語" data-language-local-name="kantoni" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q193756#sitelinks-wikipedia" title="Nyelvközi hivatkozások szerkesztése" class="wbc-editpage">Hivatkozások szerkesztése</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Névterek"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Komplex_anal%C3%ADzis" title="A lap megtekintése [c]" accesskey="c"><span>Szócikk</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Vita:Komplex_anal%C3%ADzis" rel="discussion" title="Az oldal tartalmának megvitatása [t]" accesskey="t"><span>Vitalap</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Nyelvvariáns váltása" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">magyar</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Nézetek"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Komplex_anal%C3%ADzis"><span>Olvasás</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Komplex_anal%C3%ADzis&action=edit" title="Az oldal forráskódjának szerkesztése [e]" accesskey="e"><span>Szerkesztés</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Komplex_anal%C3%ADzis&action=history" title="A lap korábbi változatai [h]" accesskey="h"><span>Laptörténet</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Oldal eszközök"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Eszközök" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Eszközök</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Eszközök</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">áthelyezés az oldalsávba</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">elrejtés</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="További lehetőségek" > <div class="vector-menu-heading"> Műveletek </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Komplex_anal%C3%ADzis"><span>Olvasás</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Komplex_anal%C3%ADzis&action=edit" title="Az oldal forráskódjának szerkesztése [e]" accesskey="e"><span>Szerkesztés</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Komplex_anal%C3%ADzis&action=history"><span>Laptörténet</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Általános </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Speci%C3%A1lis:Mi_hivatkozik_erre/Komplex_anal%C3%ADzis" title="Az erre a lapra hivatkozó más lapok listája [j]" accesskey="j"><span>Mi hivatkozik erre?</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Speci%C3%A1lis:Kapcsol%C3%B3d%C3%B3_v%C3%A1ltoztat%C3%A1sok/Komplex_anal%C3%ADzis" rel="nofollow" title="Az erről a 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background-color:#ccccff;"><b>Általános</b></td></tr><tr><td class="hlist" style="text-align:center;"> <ul><li><a href="/wiki/Matematikus" title="Matematikus">Matematikusok</a></li> <li><a href="/wiki/A_matematika_t%C3%B6rt%C3%A9nete" title="A matematika története">Matematikatörténet</a></li> <li><a href="/wiki/Matematikafiloz%C3%B3fia" title="Matematikafilozófia">Matematikafilozófia</a></li> <li><small><a href="/wiki/Port%C3%A1l:Matematika" title="Portál:Matematika">Portál</a></small></li></ul> </td></tr><tr><td style="text-align:center; background-color:#ccccff;"><div class="navbar noprint hlist plainlinks mini" style="display:inline;font-size:xx-small"><style data-mw-deduplicate="TemplateStyles:r26593303">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><span style="display:none"><a href="/wiki/Sablon:Matematika" title="Sablon:Matematika">Sablon:Matematika</a></span><ul style="display:inline"><li class="nv-view"><a class="external text" href="https://hu.wikipedia.org/wiki/Sablon:Matematika"><span title="Mutasd ezt a sablont">m</span></a></li> <li class="nv-talk"><a class="external text" href="https://hu.wikipedia.org/wiki/Sablonvita:Matematika"><span title="A sablon vitalapja">v</span></a></li> <li class="nv-edit"><a class="external text" href="https://hu.wikipedia.org/w/index.php?title=Sablon:Matematika&action=edit"><span title="A sablon szerkesztése">sz</span></a></li></ul></div></td></tr></tbody></table> <p>A <b>komplex analízis</b> vagy <b>komplexfüggvény-tan</b> a matematika azon ága, amely a komplex változós komplex értékű függvényekkel foglalkozik. Alkalmazzák kétdimenziós fizikai problémák modellezésében és a <a href="/wiki/Sz%C3%A1melm%C3%A9let" title="Számelmélet">számelméletben</a> is. </p><p>A komplex analízisben központi szerep jut a függvények differenciálhatóságának, s konkrétan a <a href="/wiki/Holomorf_f%C3%BCggv%C3%A9nyek" title="Holomorf függvények">holomorf</a> illetve a <a href="/wiki/Meromorf_f%C3%BCggv%C3%A9nyek" title="Meromorf függvények">meromorf függvények</a> vizsgálatának. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Komplex_függvény"><span id="Komplex_f.C3.BCggv.C3.A9ny"></span>Komplex függvény</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Komplex_anal%C3%ADzis&action=edit&section=1" title="Szakasz szerkesztése: Komplex függvény"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="dablink noprint noviewer" style="padding-left: 2em; vertical-align: middle;" cellpadding="0" cellspacing="0"><tbody><tr><td style="padding-right:.25em;"><span typeof="mw:File"><a href="/wiki/F%C3%A1jl:Searchtool_right.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Searchtool_right.svg/14px-Searchtool_right.svg.png" decoding="async" width="14" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Searchtool_right.svg/21px-Searchtool_right.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/32/Searchtool_right.svg/28px-Searchtool_right.svg.png 2x" data-file-width="60" data-file-height="60" /></a></span></td><td><i>Bővebben: <a href="/wiki/Komplex_f%C3%BCggv%C3%A9ny" title="Komplex függvény">Komplex függvény</a></i></td></tr></tbody></table> <p>Komplex függvény alatt olyan függvényeket értünk, melyeknek az értelmezési tartománya és az értékkészlete egyaránt a komplex sík részhalmaza. </p> <div class="mw-heading mw-heading2"><h2 id="Differenciálhatóság"><span id="Differenci.C3.A1lhat.C3.B3s.C3.A1g"></span>Differenciálhatóság</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Komplex_anal%C3%ADzis&action=edit&section=2" title="Szakasz szerkesztése: Differenciálhatóság"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="A_derivált"><span id="A_deriv.C3.A1lt"></span>A derivált</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Komplex_anal%C3%ADzis&action=edit&section=3" title="Szakasz szerkesztése: A derivált"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Valamely <span class="mwe-math-element"><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\in \mathbb {C} \to \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in \mathbb {C} \to \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05ed4af84a4bd455742015391a34f83171005f56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.089ex; height:2.509ex;" alt="{\displaystyle f\in \mathbb {C} \to \mathbb {C} }"></span> függvény deriváltja a z helyen a valós esethez hasonlóan értelmezhető. Ha az alábbi határérték létezik, akkor f a z helyen differenciálható, s a határértéket az f függvény z pontban vett deriváltjának nevezzük: </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'(z)=\lim _{h\to 0}{\frac {f(z+h)-f(z)}{h}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'(z)=\lim _{h\to 0}{\frac {f(z+h)-f(z)}{h}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/908a9f57664a288b570b0bbb73458a5b9d8a4670" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.395ex; height:5.843ex;" alt="{\displaystyle f'(z)=\lim _{h\to 0}{\frac {f(z+h)-f(z)}{h}}\,}"></span></dd></dl> <p>Ha egy f függvény valamely Ω halmaz minden pontján differenciálható, akkor definiálható a derivált függvény is: </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':\Omega \to \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>:</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f':\Omega \to \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3ff241391a881e0a011152b29d3b15d08a18d1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.913ex; height:2.843ex;" alt="{\displaystyle f':\Omega \to \mathbb {C} }"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="A_Cauchy–Riemann_egyenletek"><span id="A_Cauchy.E2.80.93Riemann_egyenletek"></span>A Cauchy–Riemann egyenletek</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Komplex_anal%C3%ADzis&action=edit&section=4" title="Szakasz szerkesztése: A Cauchy–Riemann egyenletek"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A komplex függvények differenciálhatóságra adnak ekvivalens feltételt a Cauchy–Riemann egyenletek.<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> Ezek mögött az van, hogy a határértéknek az adott pontban a komplex sík minden irányából közelítve azonosnak kell lennie. Mivel a komplex sík izomorf a kétdimenziós valós vektortérrel, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> komplex változós függvény felírható ekvivalens módon <span class="mwe-math-element"><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:\mathbb {R} ^{2}\to \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25c44be73c93794e647a2a66e3fc18f527395064" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.294ex; height:3.009ex;" alt="{\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} ^{2}}"></span> alakban a következőképpen: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y)={\begin{bmatrix}f_{1}(x,y)\\f_{2}(x,y)\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)={\begin{bmatrix}f_{1}(x,y)\\f_{2}(x,y)\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a82cf74e77b62ff3b222c04b0c9b533939667c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.434ex; height:6.176ex;" alt="{\displaystyle f(x,y)={\begin{bmatrix}f_{1}(x,y)\\f_{2}(x,y)\end{bmatrix}}}"></span></dd></dl> <p>Pontosan akkor differenciálható <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> valamely <span class="mwe-math-element"><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=x+yi}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=x+yi}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/639f77c05613faa61f43ee28a4d5ca7c35fffa38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.315ex; height:2.509ex;" alt="{\displaystyle z=x+yi}"></span> pontban, ha teljesülnek az úgynevezett Cauchy–Riemann egyenletek: </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 \partial _{1}f_{1}(x,y)=\partial _{2}f_{2}(x,y)\qquad \partial _{1}f_{2}(x,y)=-\partial _{2}f_{1}(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="2em" /> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{1}f_{1}(x,y)=\partial _{2}f_{2}(x,y)\qquad \partial _{1}f_{2}(x,y)=-\partial _{2}f_{1}(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18e4c0ed792a2f207c43dd323d8d15d6f6318fb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.893ex; height:2.843ex;" alt="{\displaystyle \partial _{1}f_{1}(x,y)=\partial _{2}f_{2}(x,y)\qquad \partial _{1}f_{2}(x,y)=-\partial _{2}f_{1}(x,y)}"></span></dd></dl> <p>Ekkor a derivált értéke a következő: </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'(z)=\partial _{1}f_{1}(x,y)+\partial _{1}f_{2}(x,y)i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'(z)=\partial _{1}f_{1}(x,y)+\partial _{1}f_{2}(x,y)i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eefbd81377b1307a6b665a84a9efb39f507e5836" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.265ex; height:3.009ex;" alt="{\displaystyle f'(z)=\partial _{1}f_{1}(x,y)+\partial _{1}f_{2}(x,y)i}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Minden_differenciálható_komplex_függvény_analitikus"><span id="Minden_differenci.C3.A1lhat.C3.B3_komplex_f.C3.BCggv.C3.A9ny_analitikus"></span>Minden differenciálható komplex függvény analitikus</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Komplex_anal%C3%ADzis&action=edit&section=5" title="Szakasz szerkesztése: Minden differenciálható komplex függvény analitikus"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Megmutatható, hogy minden differenciálható komplex függvény <a href="/wiki/Val%C3%B3s_analitikus_f%C3%BCggv%C3%A9ny" title="Valós analitikus függvény">analitikus</a>, azaz az adott pont egy környezetében a függvény <a href="/wiki/Taylor-sor" title="Taylor-sor">Taylor-sora</a> létezik és előállítja a függvényt. </p> <div class="mw-heading mw-heading2"><h2 id="Integrálás"><span id="Integr.C3.A1l.C3.A1s"></span>Integrálás</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Komplex_anal%C3%ADzis&action=edit&section=6" title="Szakasz szerkesztése: Integrálás"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mivel mind a változónak, mind a függvény értékének lehet valós és képzetes része is, az integrálás a vektorfüggvényekéhez hasonló. Legelterjedtebb a komplex síkon végigfutó görbe menti <b>vonalintegrál</b>. <b>Cauchy alaptétel</b>: bármely analitikus függvényt egy zárt görbén integrálva az eredmény nulla. </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 \oint f(z)dz=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∮</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint f(z)dz=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f7284f6ce4be9abe38fe0847d62e47e54cc92f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.322ex; height:5.676ex;" alt="{\displaystyle \oint f(z)dz=0}"></span></dd></dl> <p>A vonalintegrált sokszor akkor is tudjuk értelmezni, ha a függvény nem analitikus, azaz a a görbén belül szakadása, <b>pólusa</b> van. Példaként az <span class="mwe-math-element"><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(z)=1/z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z)=1/z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19a8605a3b5b1f951367e5792f3fa49e3042ef33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.688ex; height:2.843ex;" alt="{\displaystyle f(z)=1/z}"></span> függvényt az origó körüli körön integrálva (kihasználva, hogy <span class="mwe-math-element"><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=|z|e^{i\phi })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ϕ</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=|z|e^{i\phi })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5db98164339afc551a8b59d54e9ddf5b73a6852" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.336ex; height:3.176ex;" alt="{\displaystyle z=|z|e^{i\phi })}"></span> </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 \oint {\frac {1}{z}}dz=i\int _{0}^{2\pi }d\phi =2\pi i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> <mi>d</mi> <mi>z</mi> <mo>=</mo> <mi>i</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <mi>d</mi> <mi>ϕ</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint {\frac {1}{z}}dz=i\int _{0}^{2\pi }d\phi =2\pi i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3bd36f98a9b104dab02b8deb70acf346f7daf75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.015ex; height:6.176ex;" alt="{\displaystyle \oint {\frac {1}{z}}dz=i\int _{0}^{2\pi }d\phi =2\pi i}"></span></dd></dl> <p>Ebből megkapható, hogy egy <span class="mwe-math-element"><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 {f(z)}{z-z_{\circ }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>z</mi> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {f(z)}{z-z_{\circ }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2df1237155f871a278829b8d2394f001f1f0d2ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.853ex; margin-bottom: -0.318ex; width:6.9ex; height:6.009ex;" alt="{\displaystyle {\frac {f(z)}{z-z_{\circ }}}}"></span> alakú függvény, ahol <span class="mwe-math-element"><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(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8dd568d570b390c337c0a911f0a1c5c214e8240" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.176ex; height:2.843ex;" alt="{\displaystyle f(z)}"></span> tetszőleges, analitkus függvény, <span class="mwe-math-element"><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_{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0c17d18dfc9ccc41a8515ff1ccf742eb464902f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.26ex; margin-bottom: -0.412ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{\circ }}"></span> pólust tartalmazó zárt görbére vett integrálja az analitikus függvény <span class="mwe-math-element"><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_{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0c17d18dfc9ccc41a8515ff1ccf742eb464902f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.26ex; margin-bottom: -0.412ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{\circ }}"></span> pont-beli értékét adja. </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 \oint {\frac {f(z)}{z-z_{\circ }}}dz=f(z_{\circ })2\pi i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>z</mi> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>z</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msub> <mo stretchy="false">)</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint {\frac {f(z)}{z-z_{\circ }}}dz=f(z_{\circ })2\pi i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4de9f940264d83ff71199fb497baee261438fa64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.404ex; height:6.176ex;" alt="{\displaystyle \oint {\frac {f(z)}{z-z_{\circ }}}dz=f(z_{\circ })2\pi i}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Holomorf_függvények"><span id="Holomorf_f.C3.BCggv.C3.A9nyek"></span>Holomorf függvények</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Komplex_anal%C3%ADzis&action=edit&section=7" title="Szakasz szerkesztése: Holomorf függvények"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="dablink noprint noviewer" style="padding-left: 2em; vertical-align: middle;" cellpadding="0" cellspacing="0"><tbody><tr><td style="padding-right:.25em;"><span typeof="mw:File"><a href="/wiki/F%C3%A1jl:Searchtool_right.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Searchtool_right.svg/14px-Searchtool_right.svg.png" decoding="async" width="14" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Searchtool_right.svg/21px-Searchtool_right.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/32/Searchtool_right.svg/28px-Searchtool_right.svg.png 2x" data-file-width="60" data-file-height="60" /></a></span></td><td><i>Bővebben: <a href="/wiki/Holomorf_f%C3%BCggv%C3%A9nyek" title="Holomorf függvények">Holomorf függvények</a></i></td></tr></tbody></table> <p>A komplex sík valamely nyílt részhalmazán értelmezett függvényt holomorfnak nevezzük, ha differenciálható. </p><p>A terminológia az ógörög <b>holos</b> (<a href="https://hu.wiktionary.org/wiki/%E1%BD%85%CE%BB%CE%BF%CF%82" class="extiw" title="wikt:ὅλος">ὅλος</a>) szóból származik, amely azt jelenti <b>egész</b>, s arra utal, hogy a függvény az egész értelmezési tartományán differenciálható. </p> <div class="mw-heading mw-heading2"><h2 id="Meromorf_függvények"><span id="Meromorf_f.C3.BCggv.C3.A9nyek"></span>Meromorf függvények</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Komplex_anal%C3%ADzis&action=edit&section=8" title="Szakasz szerkesztése: Meromorf függvények"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="dablink noprint noviewer" style="padding-left: 2em; vertical-align: middle;" cellpadding="0" cellspacing="0"><tbody><tr><td style="padding-right:.25em;"><span typeof="mw:File"><a href="/wiki/F%C3%A1jl:Searchtool_right.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Searchtool_right.svg/14px-Searchtool_right.svg.png" decoding="async" width="14" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Searchtool_right.svg/21px-Searchtool_right.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/32/Searchtool_right.svg/28px-Searchtool_right.svg.png 2x" data-file-width="60" data-file-height="60" /></a></span></td><td><i>Bővebben: <a href="/wiki/Meromorf_f%C3%BCggv%C3%A9nyek" title="Meromorf függvények">Meromorf függvények</a></i></td></tr></tbody></table> <p>A komplex sík valamely nyílt részhalmazán értelmezett függvényt meromorfnak nevezzük, ha legfeljebb izolált pontokban nem differenciálható. </p><p>A szó az ógörög <b>meros</b> (<a href="https://hu.wiktionary.org/wiki/%CE%BC%CE%AD%CF%81%CE%BF%CF%82" class="extiw" title="wikt:μέρος">μέρος</a>) szóból ered, mely azt jelenti <b>rész</b>, utalva arra, hogy a függvény csak az értelmezési tartományának egy részén differenciálható. </p> <div class="mw-heading mw-heading2"><h2 id="Jegyzetek">Jegyzetek</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Komplex_anal%C3%ADzis&action=edit&section=9" title="Szakasz szerkesztése: Jegyzetek"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="ref-1col"><div style="-moz-column-count:2; -webkit-column-count:2; column-count:2; -webkit-column-gap: 3em; -moz-column-gap: 3em; column-gap: 3em;"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Simonovits András: Válogatott fejezetek a matematika történetéből. 105. old. Typotex Kiadó, 2009. <a href="/wiki/Speci%C3%A1lis:K%C3%B6nyvforr%C3%A1sok/9789632790268" title="Speciális:Könyvforrások/9789632790268">ISBN 978-963-279-026-8</a></span> </li> </ol></div></div><div class="ref-1col"><div style="-moz-column-count:2; -webkit-column-count:2; column-count:2; -webkit-column-gap: 3em; -moz-column-gap: 3em; column-gap: 3em;"></div></div> <div class="mw-heading mw-heading2"><h2 id="További_információk"><span id="Tov.C3.A1bbi_inform.C3.A1ci.C3.B3k"></span>További információk</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Komplex_anal%C3%ADzis&action=edit&section=10" title="Szakasz szerkesztése: További információk"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><i>Komplex analízis, 1-2.</i>; Erdélyi Tankönyvtanács, Kolozsvár, 2004–2007 <ul><li>Teodor Bulboacă–Németh Sándor; <i>1.</i>; 2004</li> <li>Teodor Bulboacă–Salamon Júlia: <i>2. Feladatok és megoldások</i>; 2007</li></ul></li></ul> <div class="noprint noviewer" style="overflow: hidden; clear: both;"><div style="margin-left:0; margin-right:2px;"><ul style="display:block; list-style-image:none; list-style-type:none; width:100%; vertical-align:middle; margin:0; padding:0; min-height: 27px;"><li style="float:left; min-height: 27px; line-height:25px; width:100%; margin:0; margin-top:.5em; margin-left:0; margin-right:0; padding:0; border:1px solid #CCF; background-color:#F0EEFF"><span typeof="mw:File"><a href="/wiki/F%C3%A1jl:P_cartesian_graph.svg" class="mw-file-description" title="matematika"><img alt="matematika" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/P_cartesian_graph.svg/25px-P_cartesian_graph.svg.png" decoding="async" width="25" height="23" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/P_cartesian_graph.svg/38px-P_cartesian_graph.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/23/P_cartesian_graph.svg/50px-P_cartesian_graph.svg.png 2x" data-file-width="400" data-file-height="360" /></a></span> <b><a href="/wiki/Port%C3%A1l:Matematika" title="Portál:Matematika">Matematikaportál</a></b> • összefoglaló, színes tartalomajánló lap</li></ul></div></div></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">A lap eredeti címe: „<a dir="ltr" href="https://hu.wikipedia.org/w/index.php?title=Komplex_analízis&oldid=27532980">https://hu.wikipedia.org/w/index.php?title=Komplex_analízis&oldid=27532980</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikip%C3%A9dia:Kateg%C3%B3ri%C3%A1k" title="Wikipédia:Kategóriák">Kategória</a>: <ul><li><a href="/wiki/Kateg%C3%B3ria:Komplex_anal%C3%ADzis" title="Kategória:Komplex analízis">Komplex analízis</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> A lap utolsó módosítása: 2024. október 20., 18:31</li> <li id="footer-info-copyright">A lap szövege <a rel="nofollow" class="external text" href="http://creativecommons.org/licenses/by-sa/4.0/deed.hu">Creative Commons Nevezd meg! – Így add tovább! 4.0</a> licenc alatt van; egyes esetekben más módon is felhasználható. 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