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Complex analysis - Wikipedia

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class="vector-toc-numb">2</span> <span>Complex functions</span> </div> </a> <ul id="toc-Complex_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Holomorphic_functions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Holomorphic_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Holomorphic functions</span> </div> </a> <ul id="toc-Holomorphic_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conformal_map" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Conformal_map"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Conformal map</span> </div> </a> <ul id="toc-Conformal_map-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Major_results" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Major_results"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Major results</span> </div> </a> <ul id="toc-Major_results-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Complex analysis</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 57 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-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 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%AD%D9%84%D9%8A%D9%84_%D9%85%D8%B1%D9%83%D8%A8" title="تحليل مركب – Arabic" lang="ar" hreflang="ar" data-title="تحليل مركب" data-language-autonym="العربية" data-language-local-name="Arabic" 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 – Asturian" lang="ast" hreflang="ast" data-title="Analís complexu" data-language-autonym="Asturianu" data-language-local-name="Asturian" 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 – Azerbaijani" lang="az" hreflang="az" data-title="Kompleks analiz" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" 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 – Bashkir" lang="ba" hreflang="ba" data-title="Комплекслы анализ" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" 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 – Belarusian" lang="be" hreflang="be" data-title="Камплексны аналіз" data-language-autonym="Беларуская" data-language-local-name="Belarusian" 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 – Bulgarian" lang="bg" hreflang="bg" data-title="Комплексен анализ" data-language-autonym="Български" data-language-local-name="Bulgarian" 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 – Catalan" lang="ca" hreflang="ca" data-title="Anàlisi complexa" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%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 – Chuvash" lang="cv" hreflang="cv" data-title="Комплекслă анализ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</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 – Czech" lang="cs" hreflang="cs" data-title="Komplexní analýza" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Dadansoddi_cymhlyg" title="Dadansoddi cymhlyg – Welsh" lang="cy" hreflang="cy" data-title="Dadansoddi cymhlyg" data-language-autonym="Cymraeg" data-language-local-name="Welsh" 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 – Danish" lang="da" hreflang="da" data-title="Kompleks analyse" data-language-autonym="Dansk" data-language-local-name="Danish" 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 – German" lang="de" hreflang="de" data-title="Funktionentheorie" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</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 – Estonian" lang="et" hreflang="et" data-title="Kompleksmuutuja funktsioonide teooria" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</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 – Greek" lang="el" hreflang="el" data-title="Μιγαδική ανάλυση" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</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 – Spanish" lang="es" hreflang="es" data-title="Análisis complejo" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Kompleksa_analitiko" title="Kompleksa analitiko – Esperanto" lang="eo" hreflang="eo" data-title="Kompleksa analitiko" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Analisi_konplexu" title="Analisi konplexu – Basque" lang="eu" hreflang="eu" data-title="Analisi konplexu" data-language-autonym="Euskara" data-language-local-name="Basque" 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 – Persian" lang="fa" hreflang="fa" data-title="آنالیز مختلط" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Analyse_complexe" title="Analyse complexe – French" lang="fr" hreflang="fr" data-title="Analyse complexe" data-language-autonym="Français" data-language-local-name="French" 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 – Galician" lang="gl" hreflang="gl" data-title="Análise complexa" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</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 – Korean" lang="ko" hreflang="ko" data-title="복소해석학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</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 – Armenian" lang="hy" hreflang="hy" data-title="Կոմպլեքս անալիզ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</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-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Analisis_kompleks" title="Analisis kompleks – Indonesian" lang="id" hreflang="id" data-title="Analisis kompleks" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</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 – Ossetic" lang="os" hreflang="os" data-title="Комплексон анализ" data-language-autonym="Ирон" data-language-local-name="Ossetic" class="interlanguage-link-target"><span>Ирон</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Tvinnfallagreining" title="Tvinnfallagreining – Icelandic" lang="is" hreflang="is" data-title="Tvinnfallagreining" data-language-autonym="Íslenska" data-language-local-name="Icelandic" 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 – Italian" lang="it" hreflang="it" data-title="Analisi complessa" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%9E%D7%A8%D7%95%D7%9B%D7%91%D7%AA" title="אנליזה מרוכבת – Hebrew" lang="he" hreflang="he" data-title="אנליזה מרוכבת" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</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 – Georgian" lang="ka" hreflang="ka" data-title="კომპლექსური ანალიზი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</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-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Komplex_anal%C3%ADzis" title="Komplex analízis – Hungarian" lang="hu" hreflang="hu" data-title="Komplex analízis" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</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 – Macedonian" lang="mk" hreflang="mk" data-title="Комплексна анализа" data-language-autonym="Македонски" data-language-local-name="Macedonian" 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 – Maltese" lang="mt" hreflang="mt" data-title="Analisi komplessa" data-language-autonym="Malti" data-language-local-name="Maltese" class="interlanguage-link-target"><span>Malti</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-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Functietheorie" title="Functietheorie – Dutch" lang="nl" hreflang="nl" data-title="Functietheorie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%A4%87%E7%B4%A0%E8%A7%A3%E6%9E%90" title="複素解析 – Japanese" lang="ja" hreflang="ja" data-title="複素解析" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Kompleks_analyse" title="Kompleks analyse – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Kompleks analyse" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Analiza_zespolona" title="Analiza zespolona – Polish" lang="pl" hreflang="pl" data-title="Analiza zespolona" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/An%C3%A1lise_complexa" title="Análise complexa – Portuguese" lang="pt" hreflang="pt" data-title="Análise complexa" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Analiz%C4%83_complex%C4%83" title="Analiză complexă – Romanian" lang="ro" hreflang="ro" data-title="Analiză complexă" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%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 – Russian" lang="ru" hreflang="ru" data-title="Комплексный анализ" data-language-autonym="Русский" data-language-local-name="Russian" 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 – Scots" lang="sco" hreflang="sco" data-title="Complex analysis" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Analiza_komplekse" title="Analiza komplekse – Albanian" lang="sq" hreflang="sq" data-title="Analiza komplekse" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a 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data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Complexity_theory_(disambiguation)" class="mw-redirect mw-disambig" title="Complexity theory (disambiguation)">Complexity theory</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output 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.sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle"><span style="font-size: 8pt; font-weight: none"><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Mathematical analysis</a> → <b>Complex analysis</b></span></td></tr><tr><th class="sidebar-title-with-pretitle"><a class="mw-selflink selflink">Complex analysis</a></th></tr><tr><td class="sidebar-image"><span typeof="mw:File"><a href="/wiki/File:Gamma_abs_3D.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Gamma_abs_3D.png/200px-Gamma_abs_3D.png" decoding="async" width="200" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Gamma_abs_3D.png/300px-Gamma_abs_3D.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Gamma_abs_3D.png/400px-Gamma_abs_3D.png 2x" data-file-width="1280" data-file-height="1000" /></a></span></td></tr><tr><th class="sidebar-heading"> <a href="/wiki/Complex_number" title="Complex number">Complex numbers</a></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Real_number" title="Real number">Real number</a></li> <li><a href="/wiki/Imaginary_number" title="Imaginary number">Imaginary number</a></li> <li><a href="/wiki/Complex_plane" title="Complex plane">Complex plane</a></li> <li><a href="/wiki/Complex_conjugate" title="Complex conjugate">Complex conjugate</a></li> <li><a href="/wiki/Unit_complex_number" class="mw-redirect" title="Unit complex number">Unit complex number</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a class="mw-selflink-fragment" href="#Complex_functions">Complex functions</a></th></tr><tr><td class="sidebar-content"> <ul><li><a class="mw-selflink-fragment" href="#Complex_functions">Complex-valued function</a></li> <li><a href="/wiki/Analytic_function" title="Analytic function">Analytic function</a></li> <li><a href="/wiki/Holomorphic_function" title="Holomorphic function">Holomorphic function</a></li> <li><a href="/wiki/Cauchy%E2%80%93Riemann_equations" title="Cauchy–Riemann equations">Cauchy–Riemann equations</a></li> <li><a href="/wiki/Formal_power_series" title="Formal power series">Formal power series</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a class="mw-selflink-fragment" href="#Major_results">Basic theory</a></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Zeros_and_poles" title="Zeros and poles">Zeros and poles</a></li> <li><a href="/wiki/Cauchy%27s_integral_theorem" title="Cauchy&#39;s integral theorem">Cauchy's integral theorem</a></li> <li><a href="/wiki/Antiderivative_(complex_analysis)" title="Antiderivative (complex analysis)">Local primitive</a></li> <li><a href="/wiki/Cauchy%27s_integral_formula" title="Cauchy&#39;s integral formula">Cauchy's integral formula</a></li> <li><a href="/wiki/Winding_number" title="Winding number">Winding number</a></li> <li><a href="/wiki/Laurent_series" title="Laurent series">Laurent series</a></li> <li><a href="/wiki/Isolated_singularity" title="Isolated singularity">Isolated singularity</a></li> <li><a href="/wiki/Residue_theorem" title="Residue theorem">Residue theorem</a></li> <li><a href="/wiki/Argument_principle" title="Argument principle">Argument principle</a></li> <li><a href="/wiki/Conformal_map" title="Conformal map">Conformal map</a></li> <li><a href="/wiki/Schwarz_lemma" title="Schwarz lemma">Schwarz lemma</a></li> <li><a href="/wiki/Harmonic_function" title="Harmonic function">Harmonic function</a></li> <li><a href="/wiki/Laplace%27s_equation" title="Laplace&#39;s equation">Laplace's equation</a></li> <li><a href="/wiki/Liouville%27s_theorem_(complex_analysis)" title="Liouville&#39;s theorem (complex analysis)">Liouville's theorem</a></li> <li><a href="/wiki/Picard_theorem" title="Picard theorem">Picard theorem</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Geometric_function_theory" title="Geometric function theory">Geometric function theory</a></th></tr><tr><th class="sidebar-heading"> <a class="mw-selflink-fragment" href="#History">People</a></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a></li> <li><a href="/wiki/Jacques_Hadamard" title="Jacques Hadamard">Jacques Hadamard</a></li> <li><a href="/wiki/Kiyoshi_Oka" title="Kiyoshi Oka">Kiyoshi Oka</a></li> <li><a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a></li> <li><a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Karl Weierstrass</a></li></ul></td> </tr><tr><td class="sidebar-below"> <ul><li><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/32px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> </span><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics&#32;portal</a></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Complex_analysis_sidebar" title="Template:Complex analysis sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Complex_analysis_sidebar" title="Template talk:Complex analysis sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Complex_analysis_sidebar" title="Special:EditPage/Template:Complex analysis sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Complex analysis</b>, traditionally known as the <b>theory of functions of a complex variable</b>, is the branch of <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a> that investigates <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> of <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a>. It is helpful in many branches of mathematics, including <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, <a href="/wiki/Number_theory" title="Number theory">number theory</a>, <a href="/wiki/Analytic_combinatorics" title="Analytic combinatorics">analytic combinatorics</a>, and <a href="/wiki/Applied_mathematics" title="Applied mathematics">applied mathematics</a>, as well as in <a href="/wiki/Physics" title="Physics">physics</a>, including the branches of <a href="/wiki/Hydrodynamics" class="mw-redirect" title="Hydrodynamics">hydrodynamics</a>, <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a>, <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, and <a href="/wiki/Twistor_theory" title="Twistor theory">twistor theory</a>. By extension, use of complex analysis also has applications in engineering fields such as <a href="/wiki/Nuclear_engineering" title="Nuclear engineering">nuclear</a>, <a href="/wiki/Aerospace_engineering" title="Aerospace engineering">aerospace</a>, <a href="/wiki/Mechanical_engineering" title="Mechanical engineering">mechanical</a> and <a href="/wiki/Electrical_engineering" title="Electrical engineering">electrical engineering</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">&#91;</span>1<span class="cite-bracket">&#93;</span></a></sup> </p><p>As a <a href="/wiki/Differentiable_function" title="Differentiable function">differentiable function</a> of a complex variable is equal to the <a href="/wiki/Function_series" title="Function series">sum function</a> given by its <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> (that is, it is <a href="/wiki/Analyticity_of_holomorphic_functions" title="Analyticity of holomorphic functions">analytic</a>), complex analysis is particularly concerned with <a href="/wiki/Analytic_function" title="Analytic function">analytic functions</a> of a complex variable, that is, <i><a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic functions</a></i>. The concept can be extended to <a href="/wiki/Functions_of_several_complex_variables" class="mw-redirect" title="Functions of several complex variables">functions of several complex variables</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complex_analysis&amp;action=edit&amp;section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Augustin-Louis_Cauchy_1901.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Augustin-Louis_Cauchy_1901.jpg/170px-Augustin-Louis_Cauchy_1901.jpg" decoding="async" width="170" height="236" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Augustin-Louis_Cauchy_1901.jpg/255px-Augustin-Louis_Cauchy_1901.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/d/d3/Augustin-Louis_Cauchy_1901.jpg 2x" data-file-width="280" data-file-height="388" /></a><figcaption><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a>, one of the founders of complex analysis</figcaption></figure> <p>Complex analysis is one of the classical branches in mathematics, with roots in the 18th century and just prior. Important mathematicians associated with complex numbers include <a href="/wiki/Euler" class="mw-redirect" title="Euler">Euler</a>, <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a>, <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Riemann</a>, <a href="/wiki/Cauchy" class="mw-redirect" title="Cauchy">Cauchy</a>, <a href="/wiki/G%C3%B6sta_Mittag-Leffler" title="Gösta Mittag-Leffler">Gösta Mittag-Leffler</a>, <a href="/wiki/Weierstrass" class="mw-redirect" title="Weierstrass">Weierstrass</a>, and many more in the 20th century. Complex analysis, in particular the theory of <a href="/wiki/Conformal_mapping" class="mw-redirect" title="Conformal mapping">conformal mappings</a>, has many physical applications and is also used throughout <a href="/wiki/Analytic_number_theory" title="Analytic number theory">analytic number theory</a>. In modern times, it has become very popular through a new boost from <a href="/wiki/Complex_dynamics" title="Complex dynamics">complex dynamics</a> and the pictures of <a href="/wiki/Fractal" title="Fractal">fractals</a> produced by iterating <a href="/wiki/Holomorphic_functions" class="mw-redirect" title="Holomorphic functions">holomorphic functions</a>. Another important application of complex analysis is in <a href="/wiki/String_theory" title="String theory">string theory</a> which examines conformal invariants in <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Complex_functions">Complex functions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complex_analysis&amp;action=edit&amp;section=2" title="Edit section: Complex functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Exponentials_of_complex_number_within_unit_circle-2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Exponentials_of_complex_number_within_unit_circle-2.svg/320px-Exponentials_of_complex_number_within_unit_circle-2.svg.png" decoding="async" width="320" height="319" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Exponentials_of_complex_number_within_unit_circle-2.svg/480px-Exponentials_of_complex_number_within_unit_circle-2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Exponentials_of_complex_number_within_unit_circle-2.svg/640px-Exponentials_of_complex_number_within_unit_circle-2.svg.png 2x" data-file-width="591" data-file-height="590" /></a><figcaption>An <a href="/wiki/Exponentiation" title="Exponentiation">exponential</a> function <span class="texhtml"><i>A</i><sup><i>n</i></sup></span> of a discrete (<a href="/wiki/Integer" title="Integer">integer</a>) variable <span class="texhtml mvar" style="font-style:italic;">n</span>, similar to <a href="/wiki/Geometric_progression" title="Geometric progression">geometric progression</a></figcaption></figure> <p>A complex function is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> from <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> to complex numbers. In other words, it is a function that has a (not necessarily proper) subset of the complex numbers as a <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> and the complex numbers as a <a href="/wiki/Codomain" title="Codomain">codomain</a>. Complex functions are generally assumed to have a domain that contains a nonempty <a href="/wiki/Open_subset" class="mw-redirect" title="Open subset">open subset</a> of the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>. </p><p>For any complex function, the values <span class="mwe-math-element"><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> from the domain and their images <span class="mwe-math-element"><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> in the range may be separated into <a href="/wiki/Real_number" title="Real number">real</a> and <a href="/wiki/Imaginary_number" title="Imaginary number">imaginary</a> parts: </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=x+iy\quad {\text{ and }}\quad f(z)=f(x+iy)=u(x,y)+iv(x,y),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>&#xA0;and&#xA0;</mtext> </mrow> <mspace width="1em" /> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>v</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=x+iy\quad {\text{ and }}\quad f(z)=f(x+iy)=u(x,y)+iv(x,y),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86025940b06aa6ca9aa3289d7a320106bd2af2af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.861ex; height:2.843ex;" alt="{\displaystyle z=x+iy\quad {\text{ and }}\quad f(z)=f(x+iy)=u(x,y)+iv(x,y),}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y,u(x,y),v(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,u(x,y),v(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e14e1e37e7cb9bcb6968ea7026f66b5d659f77fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.701ex; height:2.843ex;" alt="{\displaystyle x,y,u(x,y),v(x,y)}"></span> are all real-valued. </p><p>In other words, a complex function <span class="mwe-math-element"><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 {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">&#x2192;<!-- → --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {C} \to \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61c4e3a3b342f10ee3f8556c374a36eafa559d0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {C} \to \mathbb {C} }"></span> may be decomposed into </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u:\mathbb {R} ^{2}\to \mathbb {R} \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</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">&#x2192;<!-- → --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u:\mathbb {R} ^{2}\to \mathbb {R} \quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e815a1e0691cc33c8369581a4ee81ef5fc7e20b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.614ex; height:2.676ex;" alt="{\displaystyle u:\mathbb {R} ^{2}\to \mathbb {R} \quad }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad v:\mathbb {R} ^{2}\to \mathbb {R} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mi>v</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">&#x2192;<!-- → --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad v:\mathbb {R} ^{2}\to \mathbb {R} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5de9188d9bf5e66a6cf2fd2359754cc29d1a6ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.059ex; height:3.009ex;" alt="{\displaystyle \quad v:\mathbb {R} ^{2}\to \mathbb {R} ,}"></span></dd></dl> <p>i.e., into two real-valued functions (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" 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 u}"></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 v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span>) of two real variables (<span class="mwe-math-element"><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>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>). </p><p>Similarly, any complex-valued function <span class="texhtml mvar" style="font-style:italic;">f</span> on an arbitrary <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> <span class="texhtml mvar" style="font-style:italic;">X</span> (is <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> to, and therefore, in that sense, it) can be considered as an <a href="/wiki/Ordered_pair" title="Ordered pair">ordered pair</a> of two <a href="/wiki/Real-valued_function" title="Real-valued function">real-valued functions</a>: <span class="texhtml">(Re <i>f</i>, Im <i>f</i>)</span> or, alternatively, as a <a href="/wiki/Vector-valued_function" title="Vector-valued function">vector-valued function</a> from <span class="texhtml mvar" style="font-style:italic;">X</span> into <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/066b155c535a38739cc0c4b288324cbb7a4a227a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}.}"></span> </p><p>Some properties of complex-valued functions (such as <a href="/wiki/Continuous_function" title="Continuous function">continuity</a>) are nothing more than the corresponding properties of vector valued functions of two real variables. Other concepts of complex analysis, such as <a href="/wiki/Differentiability" class="mw-redirect" title="Differentiability">differentiability</a>, are direct generalizations of the similar concepts for real functions, but may have very different properties. In particular, every <a href="/wiki/Holomorphic_function" title="Holomorphic function">differentiable complex function</a> is <a href="/wiki/Analytic_function" title="Analytic function">analytic</a> (see next section), and two differentiable functions that are equal in a <a href="/wiki/Neighborhood_(mathematics)" class="mw-redirect" title="Neighborhood (mathematics)">neighborhood</a> of a point are equal on the intersection of their domain (if the domains are <a href="/wiki/Connected_space" title="Connected space">connected</a>). The latter property is the basis of the principle of <a href="/wiki/Analytic_continuation" title="Analytic continuation">analytic continuation</a> which allows extending every real <a href="/wiki/Analytic_function" title="Analytic function">analytic function</a> in a unique way for getting a complex analytic function whose domain is the whole complex plane with a finite number of <a href="/wiki/Arc_(geometry)" class="mw-redirect" title="Arc (geometry)">curve arcs</a> removed. Many basic and <a href="/wiki/Special_functions" title="Special functions">special</a> complex functions are defined in this way, including the <a href="/wiki/Exponential_function#Complex_plane" title="Exponential function">complex exponential function</a>, <a href="/wiki/Complex_logarithm" title="Complex logarithm">complex logarithm functions</a>, and <a href="/wiki/Trigonometric_functions#In_the_complex_plane" title="Trigonometric functions">trigonometric functions</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Holomorphic_functions">Holomorphic functions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complex_analysis&amp;action=edit&amp;section=3" title="Edit section: Holomorphic functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Holomorphic_function" title="Holomorphic function">Holomorphic function</a></div> <p>Complex functions that are <a href="/wiki/Differentiable" class="mw-redirect" title="Differentiable">differentiable</a> at every point of an <a href="/wiki/Open_set" title="Open set">open subset</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 \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">&#x03A9;<!-- Ω --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span> of the complex plane are said to be <i>holomorphic on</i> <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">&#x03A9;<!-- Ω --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span>.</span> In the context of complex analysis, the derivative of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e72d1d86e86355892b39b8eb32b964834e113bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{0}}"></span> is defined to be<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">&#91;</span>2<span class="cite-bracket">&#93;</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(z_{0})=\lim _{z\to z_{0}}{\frac {f(z)-f(z_{0})}{z-z_{0}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>&#x2032;</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo stretchy="false">&#x2192;<!-- → --></mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>&#x2212;<!-- − --></mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>z</mi> <mo>&#x2212;<!-- − --></mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'(z_{0})=\lim _{z\to z_{0}}{\frac {f(z)-f(z_{0})}{z-z_{0}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34df99111d0727236ade3f006e89f736e2a7222c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:27.167ex; height:6.009ex;" alt="{\displaystyle f&#039;(z_{0})=\lim _{z\to z_{0}}{\frac {f(z)-f(z_{0})}{z-z_{0}}}.}"></span></dd></dl> <p>Superficially, this definition is formally analogous to that of the derivative of a real function. However, complex derivatives and differentiable functions behave in significantly different ways compared to their real counterparts. In particular, for this limit to exist, the value of the difference quotient must approach the same complex number, regardless of the manner in which we approach <span class="mwe-math-element"><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_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e72d1d86e86355892b39b8eb32b964834e113bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{0}}"></span> in the complex plane. Consequently, complex differentiability has much stronger implications than real differentiability. For instance, holomorphic functions are <a href="/wiki/Infinitely_differentiable" class="mw-redirect" title="Infinitely differentiable">infinitely differentiable</a>, whereas the existence of the <i>n</i>th derivative need not imply the existence of the (<i>n</i> + 1)th derivative for real functions. Furthermore, all holomorphic functions satisfy the stronger condition of <a href="/wiki/Analytic_function" title="Analytic function">analyticity</a>, meaning that the function is, at every point in its domain, locally given by a convergent power series. In essence, this means that functions holomorphic on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">&#x03A9;<!-- Ω --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span> can be approximated arbitrarily well by polynomials in some neighborhood of every point in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">&#x03A9;<!-- Ω --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span>. This stands in sharp contrast to differentiable real functions; there are infinitely differentiable real functions that are <i>nowhere</i> analytic; see <a href="/wiki/Non-analytic_smooth_function#A_smooth_function_which_is_nowhere_real_analytic" title="Non-analytic smooth function">Non-analytic smooth function §&#160;A smooth function which is nowhere real analytic</a>. </p><p>Most elementary functions, including the <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a>, the <a href="/wiki/Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric functions</a>, and all <a href="/wiki/Polynomial" title="Polynomial">polynomial functions</a>, extended appropriately to complex arguments as functions <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} \to \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">&#x2192;<!-- → --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} \to \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e75067bdd107a9ed06c72de34a4bca18e427ca22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.97ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} \to \mathbb {C} }"></span>,</span> are holomorphic over the entire complex plane, making them <i>entire</i> <i>functions</i>, while rational functions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p/q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p/q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fa5bd4cf049744deac0ac4a04c07998bd6befa9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:3.491ex; height:2.843ex;" alt="{\displaystyle p/q}"></span>, where <i>p</i> and <i>q</i> are polynomials, are holomorphic on domains that exclude points where <i>q</i> is zero. Such functions that are holomorphic everywhere except a set of isolated points are known as <i>meromorphic functions</i>. On the other hand, the functions <span class="nowrap"><span class="mwe-math-element"><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\mapsto \Re (z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo stretchy="false">&#x21A6;<!-- ↦ --></mo> <mi mathvariant="normal">&#x211C;<!-- ℜ --></mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\mapsto \Re (z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67f8895d4033efe380374a430630600b6c097b23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.524ex; height:2.843ex;" alt="{\displaystyle z\mapsto \Re (z)}"></span>,</span> <span class="nowrap"><span class="mwe-math-element"><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\mapsto |z|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo stretchy="false">&#x21A6;<!-- ↦ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\mapsto |z|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcf1ef78ab357dc17fcb3c68b61f4787530e08ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.084ex; height:2.843ex;" alt="{\displaystyle z\mapsto |z|}"></span>,</span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\mapsto {\bar {z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo stretchy="false">&#x21A6;<!-- ↦ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">&#x00AF;<!-- ¯ --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\mapsto {\bar {z}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4049edc8b476632bf03063651c9b076e379b913" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.998ex; height:2.009ex;" alt="{\displaystyle z\mapsto {\bar {z}}}"></span> are not holomorphic anywhere on the complex plane, as can be shown by their failure to satisfy the Cauchy–Riemann conditions (see below). </p><p>An important property of holomorphic functions is the relationship between the partial derivatives of their real and imaginary components, known as the <a href="/wiki/Cauchy%E2%80%93Riemann_conditions" class="mw-redirect" title="Cauchy–Riemann conditions">Cauchy–Riemann conditions</a>. If <span class="mwe-math-element"><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 {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">&#x2192;<!-- → --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {C} \to \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61c4e3a3b342f10ee3f8556c374a36eafa559d0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {C} \to \mathbb {C} }"></span>, defined by <span class="nowrap"><span class="mwe-math-element"><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)=f(x+iy)=u(x,y)+iv(x,y)}"> <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> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>v</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z)=f(x+iy)=u(x,y)+iv(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07fa0c5ef6ae36c4b48104ead58d2949891ce19f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.346ex; height:2.843ex;" alt="{\displaystyle f(z)=f(x+iy)=u(x,y)+iv(x,y)}"></span>,</span> where <span class="nowrap"><span class="mwe-math-element"><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,y,u(x,y),v(x,y)\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>&#x2208;<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,u(x,y),v(x,y)\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5e1e2620066fd827b876ae8606eac987f7a6ec6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.22ex; height:2.843ex;" alt="{\displaystyle x,y,u(x,y),v(x,y)\in \mathbb {R} }"></span>,</span> is holomorphic on a <a href="/wiki/Region_(mathematics)" class="mw-redirect" title="Region (mathematics)">region</a> <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">&#x03A9;<!-- Ω --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span>,</span> then for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{0}\in \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>&#x2208;<!-- ∈ --></mo> <mi mathvariant="normal">&#x03A9;<!-- Ω --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{0}\in \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d623682aed0b658d48900e7e25ba8814b8998ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.654ex; height:2.509ex;" alt="{\displaystyle z_{0}\in \Omega }"></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 {\frac {\partial f}{\partial {\bar {z}}}}(z_{0})=0,\ {\text{where }}{\frac {\partial }{\partial {\bar {z}}}}\mathrel {:=} {\frac {1}{2}}\left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">&#x00AF;<!-- ¯ --></mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mtext>&#xA0;</mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>where&#xA0;</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">&#x00AF;<!-- ¯ --></mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-REL"> <mo>:=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial f}{\partial {\bar {z}}}}(z_{0})=0,\ {\text{where }}{\frac {\partial }{\partial {\bar {z}}}}\mathrel {:=} {\frac {1}{2}}\left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2b8d55fec75ca73d4a82f3969086594b127a3d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.271ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial f}{\partial {\bar {z}}}}(z_{0})=0,\ {\text{where }}{\frac {\partial }{\partial {\bar {z}}}}\mathrel {:=} {\frac {1}{2}}\left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right).}"></span></dd></dl> <p>In terms of the real and imaginary parts of the function, <i>u</i> and <i>v</i>, this is equivalent to the pair of equations <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{x}=v_{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{x}=v_{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/383bad63dfc8b966525d0bbf8b96cdf948fdc35f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.778ex; height:2.343ex;" alt="{\displaystyle u_{x}=v_{y}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{y}=-v_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mo>&#x2212;<!-- − --></mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{y}=-v_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92f2c2e2240bd29bbc0b2731501af69e7d2f7195" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.586ex; height:2.676ex;" alt="{\displaystyle u_{y}=-v_{x}}"></span>, where the subscripts indicate partial differentiation. However, the Cauchy–Riemann conditions do not characterize holomorphic functions, without additional continuity conditions (see <a href="/wiki/Looman%E2%80%93Menchoff_theorem" title="Looman–Menchoff theorem">Looman–Menchoff theorem</a>). </p><p>Holomorphic functions exhibit some remarkable features. For instance, <a href="/wiki/Picard_theorem" title="Picard theorem">Picard's theorem</a> asserts that the range of an entire function can take only three possible forms: <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>,</span> <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} \setminus \{z_{0}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo class="MJX-variant">&#x2216;<!-- ∖ --></mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} \setminus \{z_{0}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3ec15546a090988b5414e1e54a82343428dbaa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.333ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} \setminus \{z_{0}\}}"></span>,</span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{z_{0}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{z_{0}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a232a53cce19e680d87c87f0d07b3877e6e4f583" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.46ex; height:2.843ex;" alt="{\displaystyle \{z_{0}\}}"></span> for some <span class="nowrap"><span class="mwe-math-element"><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_{0}\in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>&#x2208;<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{0}\in \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f9230ede854d9253de13e7548acc5aaa66ac1ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.654ex; height:2.509ex;" alt="{\displaystyle z_{0}\in \mathbb {C} }"></span>.</span> In other words, if two distinct complex numbers <span class="mwe-math-element"><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> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}"></span> are not in the range of an entire function <span class="nowrap"><span class="mwe-math-element"><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>,</span> then <span class="mwe-math-element"><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> is a constant function. Moreover, a holomorphic function on a connected open set is determined by its restriction to any nonempty open subset. </p> <div class="mw-heading mw-heading2"><h2 id="Conformal_map">Conformal map</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complex_analysis&amp;action=edit&amp;section=4" title="Edit section: Conformal map"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><style data-mw-deduplicate="TemplateStyles:r1066933788">.mw-parser-output .excerpt-hat .mw-editsection-like{font-style:normal}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Conformal_map" title="Conformal map">Conformal map</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Conformal_map&amp;action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Conformal_map.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Conformal_map.svg/220px-Conformal_map.svg.png" decoding="async" width="220" height="385" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Conformal_map.svg/330px-Conformal_map.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Conformal_map.svg/440px-Conformal_map.svg.png 2x" data-file-width="535" data-file-height="937" /></a><figcaption>A rectangular grid (top) and its image under a conformal map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> (bottom). It is seen that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> maps pairs of lines intersecting at 90° to pairs of curves still intersecting at 90°.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <a href="/wiki/Conformal_map" title="Conformal map">conformal map</a> is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> that locally preserves <a href="/wiki/Angle" title="Angle">angles</a>, but not necessarily lengths. </p><p>More formally, let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> be open subsets of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>. A function <span class="mwe-math-element"><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:U\to V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>U</mi> <mo stretchy="false">&#x2192;<!-- → --></mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:U\to V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99d431c81a43bf61447672a1c6e62d56fc73026a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.4ex; height:2.509ex;" alt="{\displaystyle f:U\to V}"></span> is called conformal (or angle-preserving) at a point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{0}\in U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>&#x2208;<!-- ∈ --></mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{0}\in U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/133e383f80a11def53dfb9316d0d4f71d32b20da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.007ex; height:2.509ex;" alt="{\displaystyle u_{0}\in U}"></span> if it preserves angles between directed <a href="/wiki/Curve" title="Curve">curves</a> through <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c7425f9c7ab645587060423c0af62f8a61fbc65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle u_{0}}"></span>, as well as preserving orientation. Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or <a href="/wiki/Curvature" title="Curvature">curvature</a>. </p><p>The conformal property may be described in terms of the <a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a> derivative matrix of a <a href="/wiki/Coordinate_transformation" class="mw-redirect" title="Coordinate transformation">coordinate transformation</a>. The transformation is conformal whenever the Jacobian at each point is a positive scalar times a <a href="/wiki/Rotation_matrix" title="Rotation matrix">rotation matrix</a> (<a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal</a> with determinant one). Some authors define conformality to include orientation-reversing mappings whose Jacobians can be written as any scalar times any orthogonal matrix.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">&#91;</span>3<span class="cite-bracket">&#93;</span></a></sup> </p><p>For mappings in two dimensions, the (orientation-preserving) conformal mappings are precisely the locally invertible <a href="/wiki/Holomorphic_function" title="Holomorphic function">complex analytic</a> functions. In three and higher dimensions, <a href="/wiki/Liouville%27s_theorem_(conformal_mappings)" title="Liouville&#39;s theorem (conformal mappings)">Liouville's theorem</a> sharply limits the conformal mappings to a few types. </p> The notion of conformality generalizes in a natural way to maps between <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian</a> or <a href="/wiki/Semi-Riemannian_manifold" class="mw-redirect" title="Semi-Riemannian manifold">semi-Riemannian manifolds</a>.</div></div> <div class="mw-heading mw-heading2"><h2 id="Major_results">Major results</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complex_analysis&amp;action=edit&amp;section=5" title="Edit section: Major results"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Complex-plot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Complex-plot.png/262px-Complex-plot.png" decoding="async" width="262" height="201" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Complex-plot.png/393px-Complex-plot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Complex-plot.png/524px-Complex-plot.png 2x" data-file-width="579" data-file-height="445" /></a><figcaption><a href="/wiki/Domain_coloring" title="Domain coloring">Color wheel graph</a> of the function <span class="texhtml"><i>f</i>(<i>x</i>) = <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">&#8288;<span class="tion"><span class="num">(<i>x</i><sup>2</sup> − 1)(<i>x</i> − 2 − <i>i</i>)<sup>2</sup></span><span class="sr-only">/</span><span class="den"><i>x</i><sup>2</sup> + 2 + 2<i>i</i></span></span>&#8288;</span></span>.<br /> <a href="/wiki/Hue" title="Hue">Hue</a> represents the <a href="/wiki/Argument_(complex_analysis)" title="Argument (complex analysis)">argument</a>, <a href="/wiki/Brightness" title="Brightness">brightness</a> the <a href="/wiki/Absolute_value#Complex_numbers" title="Absolute value">magnitude.</a></figcaption></figure> <p>One of the central tools in complex analysis is the <a href="/wiki/Line_integral" title="Line integral">line integral</a>. The line integral around a closed path of a function that is holomorphic everywhere inside the area bounded by the closed path is always zero, as is stated by the <a href="/wiki/Cauchy_integral_theorem" class="mw-redirect" title="Cauchy integral theorem">Cauchy integral theorem</a>. The values of such a holomorphic function inside a disk can be computed by a path integral on the disk's boundary (as shown in <a href="/wiki/Cauchy%27s_integral_formula" title="Cauchy&#39;s integral formula">Cauchy's integral formula</a>). Path integrals in the complex plane are often used to determine complicated real integrals, and here the theory of <a href="/wiki/Residue_(complex_analysis)" title="Residue (complex analysis)">residues</a> among others is applicable (see <a href="/wiki/Methods_of_contour_integration" class="mw-redirect" title="Methods of contour integration">methods of contour integration</a>). A "pole" (or <a href="/wiki/Isolated_singularity" title="Isolated singularity">isolated singularity</a>) of a function is a point where the function's value becomes unbounded, or "blows up". If a function has such a pole, then one can compute the function's residue there, which can be used to compute path integrals involving the function; this is the content of the powerful <a href="/wiki/Residue_theorem" title="Residue theorem">residue theorem</a>. The remarkable behavior of holomorphic functions near essential singularities is described by <a href="/wiki/Picard_theorem#Big_Picard" title="Picard theorem">Picard's theorem</a>. Functions that have only poles but no <a href="/wiki/Essential_singularity" title="Essential singularity">essential singularities</a> are called <a href="/wiki/Meromorphic" class="mw-redirect" title="Meromorphic">meromorphic</a>. <a href="/wiki/Laurent_series" title="Laurent series">Laurent series</a> are the complex-valued equivalent to <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a>, but can be used to study the behavior of functions near singularities through infinite sums of more well understood functions, such as polynomials. </p><p>A <a href="/wiki/Bounded_function" title="Bounded function">bounded function</a> that is holomorphic in the entire complex plane must be constant; this is <a href="/wiki/Liouville%27s_theorem_(complex_analysis)" title="Liouville&#39;s theorem (complex analysis)">Liouville's theorem</a>. It can be used to provide a natural and short proof for the <a href="/wiki/Fundamental_Theorem_of_Algebra" class="mw-redirect" title="Fundamental Theorem of Algebra">fundamental theorem of algebra</a> which states that the <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> of complex numbers is <a href="/wiki/Algebraically_closed_field" title="Algebraically closed field">algebraically closed</a>. </p><p>If a function is holomorphic throughout a <a href="/wiki/Connected_space" title="Connected space">connected</a> domain then its values are fully determined by its values on any smaller subdomain. The function on the larger domain is said to be <a href="/wiki/Analytic_continuation" title="Analytic continuation">analytically continued</a> from its values on the smaller domain. This allows the extension of the definition of functions, such as the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>, which are initially defined in terms of infinite sums that converge only on limited domains to almost the entire complex plane. Sometimes, as in the case of the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a>, it is impossible to analytically continue a holomorphic function to a non-simply connected domain in the complex plane but it is possible to extend it to a holomorphic function on a closely related surface known as a <a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surface</a>. </p><p>All this refers to complex analysis in one variable. There is also a very rich theory of <a href="/wiki/Function_of_several_complex_variables" title="Function of several complex variables">complex analysis in more than one complex dimension</a> in which the analytic properties such as <a href="/wiki/Power_series" title="Power series">power series</a> expansion carry over whereas most of the geometric properties of holomorphic functions in one complex dimension (such as <a href="/wiki/Conformality" class="mw-redirect" title="Conformality">conformality</a>) do not carry over. The <a href="/wiki/Riemann_mapping_theorem" title="Riemann mapping theorem">Riemann mapping theorem</a> about the conformal relationship of certain domains in the complex plane, which may be the most important result in the one-dimensional theory, fails dramatically in higher dimensions. </p><p>A major application of certain <a href="/wiki/Complex_Hilbert_space" class="mw-redirect" title="Complex Hilbert space">complex spaces</a> is in <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> as <a href="/wiki/Wave_function" title="Wave function">wave functions</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complex_analysis&amp;action=edit&amp;section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Complex_geometry" title="Complex geometry">Complex geometry</a></li> <li><a href="/wiki/Hypercomplex_analysis" title="Hypercomplex analysis">Hypercomplex analysis</a></li> <li><a href="/wiki/Vector_calculus" title="Vector calculus">Vector calculus</a></li> <li><a href="/wiki/List_of_complex_analysis_topics" title="List of complex analysis topics">List of complex analysis topics</a></li> <li><a href="/wiki/Monodromy_theorem" title="Monodromy theorem">Monodromy theorem</a></li> <li><a href="/wiki/Real_analysis" title="Real analysis">Real analysis</a></li> <li><a href="/wiki/Riemann%E2%80%93Roch_theorem" title="Riemann–Roch theorem">Riemann–Roch theorem</a></li> <li><a href="/wiki/Runge%27s_theorem" title="Runge&#39;s theorem">Runge's theorem</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complex_analysis&amp;action=edit&amp;section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://gateway.newton.ac.uk/event/ofbw51">"Industrial Applications of Complex Analysis"</a>. <i>Newton Gateway to Mathematics</i>. October 30, 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">November 20,</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=unknown&amp;rft.jtitle=Newton+Gateway+to+Mathematics&amp;rft.atitle=Industrial+Applications+of+Complex+Analysis&amp;rft.date=2019-10-30&amp;rft_id=https%3A%2F%2Fgateway.newton.ac.uk%2Fevent%2Fofbw51&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+analysis" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRudin1987" class="citation book cs1">Rudin, Walter (1987). <a rel="nofollow" class="external text" href="https://59clc.files.wordpress.com/2011/01/real-and-complex-analysis.pdf#page=212"><i>Real and Complex Analysis</i></a> <span class="cs1-format">(PDF)</span>. McGraw-Hill Education. p.&#160;197. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-07-054234-1" title="Special:BookSources/978-0-07-054234-1"><bdi>978-0-07-054234-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Real+and+Complex+Analysis&amp;rft.pages=197&amp;rft.pub=McGraw-Hill+Education&amp;rft.date=1987&amp;rft.isbn=978-0-07-054234-1&amp;rft.aulast=Rudin&amp;rft.aufirst=Walter&amp;rft_id=https%3A%2F%2F59clc.files.wordpress.com%2F2011%2F01%2Freal-and-complex-analysis.pdf%23page%3D212&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+analysis" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlair2000" class="citation book cs1">Blair, David (2000-08-17). <i>Inversion Theory and Conformal Mapping</i>. The Student Mathematical Library. Vol.&#160;9. Providence, Rhode Island: American Mathematical Society. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fstml%2F009">10.1090/stml/009</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-8218-2636-2" title="Special:BookSources/978-0-8218-2636-2"><bdi>978-0-8218-2636-2</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a>&#160;<a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118752074">118752074</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Inversion+Theory+and+Conformal+Mapping&amp;rft.place=Providence%2C+Rhode+Island&amp;rft.series=The+Student+Mathematical+Library&amp;rft.pub=American+Mathematical+Society&amp;rft.date=2000-08-17&amp;rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118752074%23id-name%3DS2CID&amp;rft_id=info%3Adoi%2F10.1090%2Fstml%2F009&amp;rft.isbn=978-0-8218-2636-2&amp;rft.aulast=Blair&amp;rft.aufirst=David&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplex+analysis" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complex_analysis&amp;action=edit&amp;section=8" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Mark_J._Ablowitz" title="Mark J. Ablowitz">Ablowitz, M. J.</a> &amp; <a href="/wiki/Athanassios_S._Fokas" class="mw-redirect" title="Athanassios S. Fokas">A. S. Fokas</a>, <i>Complex Variables: Introduction and Applications</i> (Cambridge, 2003).</li> <li><a href="/wiki/Lars_Ahlfors" title="Lars Ahlfors">Ahlfors, L.</a>, <i>Complex Analysis</i> (McGraw-Hill, 1953).</li> <li><a href="/wiki/Henri_Cartan" title="Henri Cartan">Cartan, H.</a>, <i>Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes.</i> (Hermann, 1961). English translation, <i>Elementary Theory of Analytic Functions of One or Several Complex Variables.</i> (Addison-Wesley, 1963).</li> <li><a href="/wiki/Constantin_Carath%C3%A9odory" title="Constantin Carathéodory">Carathéodory, C.</a>, <i>Funktionentheorie.</i> (Birkhäuser, 1950). English translation, <i>Theory of Functions of a Complex Variable</i> (Chelsea, 1954). [2 volumes.]</li> <li><a href="/wiki/George_F._Carrier" title="George F. Carrier">Carrier, G. F.</a>, <a href="/wiki/Max_Krook" title="Max Krook">M. Krook</a>, &amp; C. E. Pearson, <a rel="nofollow" class="external text" href="https://archive.org/details/functionsofcompl00carr/"><i>Functions of a Complex Variable: Theory and Technique.</i></a> (McGraw-Hill, 1966).</li> <li><a href="/wiki/John_B._Conway" title="John B. Conway">Conway, J. B.</a>, <i>Functions of One Complex Variable.</i> (Springer, 1973).</li> <li>Fisher, S., <i>Complex Variables.</i> (Wadsworth &amp; Brooks/Cole, 1990).</li> <li><a href="/wiki/Andrew_Forsyth" title="Andrew Forsyth">Forsyth, A.</a>, <a rel="nofollow" class="external text" href="https://archive.org/details/theoryoffunction00fors/"><i>Theory of Functions of a Complex Variable</i></a> (Cambridge, 1893).</li> <li><a href="/wiki/Eberhard_Freitag" title="Eberhard Freitag">Freitag, E.</a> &amp; R. Busam, <i>Funktionentheorie</i>. (Springer, 1995). English translation, <i>Complex Analysis</i>. (Springer, 2005).</li> <li><a href="/wiki/%C3%89douard_Goursat" title="Édouard Goursat">Goursat, E.</a>, <a rel="nofollow" class="external text" href="https://archive.org/details/courseinmathemat02gouruoft/"><i>Cours d'analyse mathématique, tome 2</i></a>. (Gauthier-Villars, 1905). English translation, <a rel="nofollow" class="external text" href="https://archive.org/details/coursemathema0102gourrich/"><i>A course of mathematical analysis, vol. 2, part 1: Functions of a complex variable</i></a>. (Ginn, 1916).</li> <li><a href="/wiki/Peter_Henrici_(mathematician)" title="Peter Henrici (mathematician)">Henrici, P.</a>, <i>Applied and Computational Complex Analysis</i> (Wiley). [Three volumes: 1974, 1977, 1986.]</li> <li><a href="/wiki/Erwin_Kreyszig" title="Erwin Kreyszig">Kreyszig, E.</a>, <i>Advanced Engineering Mathematics.</i> (Wiley, 1962).</li> <li><a href="/wiki/Mikhail_Lavrentyev" title="Mikhail Lavrentyev">Lavrentyev, M.</a> &amp; B. Shabat, <i>Методы теории функций комплексного переменного.</i> (<i>Methods of the Theory of Functions of a Complex Variable</i>). (1951, in Russian).</li> <li><a href="/wiki/Aleksei_Ivanovich_Markushevich" class="mw-redirect" title="Aleksei Ivanovich Markushevich">Markushevich, A. I.</a>, <i>Theory of Functions of a Complex Variable</i>, (Prentice-Hall, 1965). [Three volumes.]</li> <li><a href="/wiki/Jerrold_E._Marsden" title="Jerrold E. Marsden">Marsden</a> &amp; Hoffman, <i>Basic Complex Analysis.</i> (Freeman, 1973).</li> <li><a href="/wiki/Tristan_Needham" title="Tristan Needham">Needham, T.</a>, <i>Visual Complex Analysis.</i> (Oxford, 1997). <a rel="nofollow" class="external free" href="http://usf.usfca.edu/vca/">http://usf.usfca.edu/vca/</a></li> <li><a href="/wiki/Reinhold_Remmert" title="Reinhold Remmert">Remmert, R.</a>, <i>Theory of Complex Functions</i>. (Springer, 1990).</li> <li><a href="/wiki/Walter_Rudin" title="Walter Rudin">Rudin, W.</a>, <i>Real and Complex Analysis.</i> (McGraw-Hill, 1966).</li> <li>Shaw, W. T., <i>Complex Analysis with Mathematica</i> (Cambridge, 2006).</li> <li><a href="/wiki/Elias_M._Stein" title="Elias M. Stein">Stein, E.</a> &amp; R. Shakarchi, <i>Complex Analysis.</i> (Princeton, 2003).</li> <li><a href="/wiki/Aleksei_Sveshnikov" title="Aleksei Sveshnikov">Sveshnikov, A. G.</a> &amp; <a href="/wiki/Andrey_Nikolayevich_Tikhonov" class="mw-redirect" title="Andrey Nikolayevich Tikhonov">A. N. Tikhonov</a>, <i>Теория функций комплексной переменной.</i> (Nauka, 1967). English translation, <a rel="nofollow" class="external text" href="https://archive.org/details/SveshnikovTikhonovTheTheoryOfFunctionsOfAComplexVariable"><i>The Theory Of Functions Of A Complex Variable</i></a> (MIR, 1978).</li> <li><a href="/wiki/Edward_Charles_Titchmarsh" title="Edward Charles Titchmarsh">Titchmarsh, E. C.</a>, <a rel="nofollow" class="external text" href="https://archive.org/details/in.ernet.dli.2015.2588/"><i>The Theory of Functions.</i></a> (Oxford, 1932).</li> <li>Wegert, E., <i>Visual Complex Functions</i>. (Birkhäuser, 2012).</li> <li><a href="/wiki/Edmund_T._Whittaker" class="mw-redirect" title="Edmund T. Whittaker">Whittaker, E. T.</a> &amp; <a href="/wiki/George_N._Watson" class="mw-redirect" title="George N. Watson">G. N. Watson</a>, <i><a href="/wiki/A_Course_of_Modern_Analysis" title="A Course of Modern Analysis">A Course of Modern Analysis</a>.</i> (Cambridge, 1902). <a rel="nofollow" class="external text" href="https://archive.org/details/courseofmodernan00whit/">3rd ed. (1920)</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complex_analysis&amp;action=edit&amp;section=9" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output 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