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class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ancient"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Ancient</span> </div> </a> <ul id="toc-Ancient-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Medieval" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Medieval"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Medieval</span> </div> </a> <ul id="toc-Medieval-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modern" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Modern"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Modern</span> </div> </a> <ul id="toc-Modern-sublist" class="vector-toc-list"> <li id="toc-Foundations" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Foundations"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3.1</span> <span>Foundations</span> </div> </a> <ul id="toc-Foundations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modernization" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Modernization"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3.2</span> <span>Modernization</span> </div> </a> <ul id="toc-Modernization-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Important_concepts" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Important_concepts"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Important concepts</span> </div> </a> <button aria-controls="toc-Important_concepts-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Important concepts subsection</span> </button> <ul id="toc-Important_concepts-sublist" class="vector-toc-list"> <li id="toc-Metric_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Metric_spaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Metric spaces</span> </div> </a> <ul id="toc-Metric_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sequences_and_limits" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sequences_and_limits"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Sequences and limits</span> </div> </a> <ul id="toc-Sequences_and_limits-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Main_branches" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Main_branches"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Main branches</span> </div> </a> <button aria-controls="toc-Main_branches-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Main branches subsection</span> </button> <ul id="toc-Main_branches-sublist" class="vector-toc-list"> <li id="toc-Calculus" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Calculus"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Calculus</span> </div> </a> <ul id="toc-Calculus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Real_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Real_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Real analysis</span> </div> </a> <ul id="toc-Real_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complex_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complex_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Complex analysis</span> </div> </a> <ul id="toc-Complex_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Functional_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Functional_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Functional analysis</span> </div> </a> <ul id="toc-Functional_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Harmonic_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Harmonic_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Harmonic analysis</span> </div> </a> <ul id="toc-Harmonic_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Differential_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Differential_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Differential equations</span> </div> </a> <ul id="toc-Differential_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Measure_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Measure_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.7</span> <span>Measure theory</span> </div> </a> <ul id="toc-Measure_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Numerical_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Numerical_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.8</span> <span>Numerical analysis</span> </div> </a> <ul id="toc-Numerical_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vector_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vector_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.9</span> <span>Vector analysis</span> </div> </a> <ul id="toc-Vector_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Scalar_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Scalar_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.10</span> <span>Scalar analysis</span> </div> </a> <ul id="toc-Scalar_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tensor_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tensor_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.11</span> <span>Tensor analysis</span> </div> </a> <ul id="toc-Tensor_analysis-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_topics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Other_topics"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Other topics</span> </div> </a> <ul id="toc-Other_topics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Applications</span> </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Applications subsection</span> </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Physical_sciences" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Physical_sciences"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Physical sciences</span> </div> </a> <ul id="toc-Physical_sciences-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Signal_processing" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Signal_processing"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Signal processing</span> </div> </a> <ul id="toc-Signal_processing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_areas_of_mathematics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_areas_of_mathematics"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Other areas of mathematics</span> </div> </a> <ul id="toc-Other_areas_of_mathematics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Famous_Textbooks" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Famous_Textbooks"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Famous Textbooks</span> </div> </a> <ul id="toc-Famous_Textbooks-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</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"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</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">Mathematical 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 114 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-114" 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">114 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Wiskundige_analise" title="Wiskundige analise – Afrikaans" lang="af" hreflang="af" data-title="Wiskundige analise" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Analysis" title="Analysis – Alemannic" lang="gsw" hreflang="gsw" data-title="Analysis" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-anp mw-list-item"><a href="https://anp.wikipedia.org/wiki/%E0%A4%95%E0%A4%B2%E0%A4%A8_%E0%A4%B6%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B0" title="कलन शास्त्र – Angika" lang="anp" hreflang="anp" data-title="कलन शास्त्र" data-language-autonym="अंगिका" data-language-local-name="Angika" class="interlanguage-link-target"><span>अंगिका</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%AD%D9%84%D9%8A%D9%84_%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A" title="تحليل رياضي – Arabic" lang="ar" hreflang="ar" data-title="تحليل رياضي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Anal%C3%ADs_matematica" title="Analís matematica – Aragonese" lang="an" hreflang="an" data-title="Analís matematica" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%97%E0%A6%BE%E0%A6%A3%E0%A6%BF%E0%A6%A4%E0%A6%BF%E0%A6%95_%E0%A6%AC%E0%A6%BF%E0%A6%B6%E0%A7%8D%E0%A6%B2%E0%A7%87%E0%A6%B7%E0%A6%A3" title="গাণিতিক বিশ্লেষণ – Assamese" lang="as" hreflang="as" data-title="গাণিতিক বিশ্লেষণ" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Anal%C3%ADs_matem%C3%A1ticu" title="Analís matemáticu – Asturian" lang="ast" hreflang="ast" data-title="Analís matemáticu" 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/Riyazi_analiz" title="Riyazi analiz – Azerbaijani" lang="az" hreflang="az" data-title="Riyazi analiz" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A6%BE%E0%A6%A3%E0%A6%BF%E0%A6%A4%E0%A6%BF%E0%A6%95_%E0%A6%AC%E0%A6%BF%E0%A6%B6%E0%A7%8D%E0%A6%B2%E0%A7%87%E0%A6%B7%E0%A6%A3" title="গাণিতিক বিশ্লেষণ – Bangla" lang="bn" hreflang="bn" data-title="গাণিতিক বিশ্লেষণ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0_%D0%B1%D2%AF%D0%BB%D0%B5%D0%B3%D0%B5)" title="Анализ (математика бүлеге) – Bashkir" lang="ba" hreflang="ba" data-title="Анализ (математика бүлеге)" 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%9C%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D1%87%D0%BD%D1%8B_%D0%B0%D0%BD%D0%B0%D0%BB%D1%96%D0%B7" title="Матэматычны аналіз – Belarusian" lang="be" hreflang="be" data-title="Матэматычны аналіз" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D1%87%D0%BD%D1%8B_%D0%B0%D0%BD%D0%B0%D0%BB%D1%96%D0%B7" title="Матэматычны аналіз – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Матэматычны аналіз" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%AC%E0%A4%BF%E0%A4%B8%E0%A5%8D%E0%A4%B2%E0%A5%87%E0%A4%B7%E0%A4%A3" title="गणितीय बिस्लेषण – Bhojpuri" lang="bh" hreflang="bh" data-title="गणितीय बिस्लेषण" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Математически анализ – Bulgarian" lang="bg" hreflang="bg" data-title="Математически анализ" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Matemati%C4%8Dka_analiza" title="Matematička analiza – Bosnian" lang="bs" hreflang="bs" data-title="Matematička analiza" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/An%C3%A0lisi_matem%C3%A0tica" title="Anàlisi matemàtica – Catalan" lang="ca" hreflang="ca" data-title="Anàlisi matemàtica" 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%90%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0_%D0%BF%D0%B0%D0%B9%C4%95)" title="Анализ (математика пайĕ) – Chuvash" lang="cv" hreflang="cv" data-title="Анализ (математика пайĕ)" 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/Matematick%C3%A1_anal%C3%BDza" title="Matematická analýza – Czech" lang="cs" hreflang="cs" data-title="Matematická analýza" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-co mw-list-item"><a href="https://co.wikipedia.org/wiki/Analisa" title="Analisa – Corsican" lang="co" hreflang="co" data-title="Analisa" data-language-autonym="Corsu" data-language-local-name="Corsican" class="interlanguage-link-target"><span>Corsu</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Dadansoddiad_mathemategol" title="Dadansoddiad mathemategol – Welsh" lang="cy" hreflang="cy" data-title="Dadansoddiad mathemategol" 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/Matematisk_analyse" title="Matematisk analyse – Danish" lang="da" hreflang="da" data-title="Matematisk 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/Analysis" title="Analysis – German" lang="de" hreflang="de" data-title="Analysis" 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/Matemaatiline_anal%C3%BC%C3%BCs" title="Matemaatiline analüüs – Estonian" lang="et" hreflang="et" data-title="Matemaatiline analüüs" 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%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AE_%CE%B1%CE%BD%CE%AC%CE%BB%CF%85%CF%83%CE%B7" title="Μαθηματική ανάλυση – Greek" lang="el" hreflang="el" data-title="Μαθηματική 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_matem%C3%A1tico" title="Análisis matemático – Spanish" lang="es" hreflang="es" data-title="Análisis matemático" 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/Analitiko" title="Analitiko – Esperanto" lang="eo" hreflang="eo" data-title="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_matematiko" title="Analisi matematiko – Basque" lang="eu" hreflang="eu" data-title="Analisi matematiko" 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_%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C" title="آنالیز ریاضی – Persian" lang="fa" hreflang="fa" data-title="آنالیز ریاضی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Mathematical_analysis" title="Mathematical analysis – Fiji Hindi" lang="hif" hreflang="hif" data-title="Mathematical analysis" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Analyse_(math%C3%A9matiques)" title="Analyse (mathématiques) – French" lang="fr" hreflang="fr" data-title="Analyse (mathématiques)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Anail%C3%ADs_mhatamaitici%C3%BAil" title="Anailís mhatamaiticiúil – Irish" lang="ga" hreflang="ga" data-title="Anailís mhatamaiticiúil" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Anailis_mhatamataigeach" title="Anailis mhatamataigeach – Scottish Gaelic" lang="gd" hreflang="gd" data-title="Anailis mhatamataigeach" data-language-autonym="Gàidhlig" data-language-local-name="Scottish Gaelic" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/An%C3%A1lise_matem%C3%A1tica" title="Análise matemática – Galician" lang="gl" hreflang="gl" data-title="Análise matemática" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E6%95%B8%E5%AD%B8%E5%88%86%E6%9E%90" title="數學分析 – Gan" lang="gan" hreflang="gan" data-title="數學分析" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%95%B4%EC%84%9D%ED%95%99_(%EC%88%98%ED%95%99)" title="해석학 (수학) – Korean" lang="ko" hreflang="ko" data-title="해석학 (수학)" 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/%D5%84%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1%D5%AF%D5%A1%D5%B6_%D5%A1%D5%B6%D5%A1%D5%AC%D5%AB%D5%A6" title="Մաթեմատիկական անալիզ – Armenian" lang="hy" hreflang="hy" data-title="Մաթեմատիկական 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%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A5%80%E0%A4%AF_%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="गणितीय विश्लेषण – Hindi" lang="hi" hreflang="hi" data-title="गणितीय विश्लेषण" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Matemati%C4%8Dka_analiza" title="Matematička analiza – Croatian" lang="hr" hreflang="hr" data-title="Matematička analiza" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Analitiko" title="Analitiko – Ido" lang="io" hreflang="io" data-title="Analitiko" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Analisis_matematis" title="Analisis matematis – Indonesian" lang="id" hreflang="id" data-title="Analisis matematis" 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-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Analyse_(mathematica)" title="Analyse (mathematica) – Interlingua" lang="ia" hreflang="ia" data-title="Analyse (mathematica)" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/St%C3%A6r%C3%B0fr%C3%A6%C3%B0igreining" title="Stærðfræðigreining – Icelandic" lang="is" hreflang="is" data-title="Stærðfræðigreining" 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_matematica" title="Analisi matematica – Italian" lang="it" hreflang="it" data-title="Analisi matematica" 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%AA%D7%9E%D7%98%D7%99%D7%AA" title="אנליזה מתמטית – Hebrew" lang="he" hreflang="he" data-title="אנליזה מתמטית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B5%E0%B2%BF%E0%B2%B6%E0%B3%8D%E0%B2%B2%E0%B3%87%E0%B2%B7%E0%B2%A3_%E0%B2%97%E0%B2%A3%E0%B2%BF%E0%B2%A4" title="ವಿಶ್ಲೇಷಣ ಗಣಿತ – Kannada" lang="kn" hreflang="kn" data-title="ವಿಶ್ಲೇಷಣ ಗಣಿತ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%90%E1%83%97%E1%83%94%E1%83%9B%E1%83%90%E1%83%A2%E1%83%98%E1%83%99%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="მათემატიკური ანალიზი – Georgian" lang="ka" hreflang="ka" data-title="მათემატიკური ანალიზი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D2%9B_%D1%82%D0%B0%D0%BB%D0%B4%D0%B0%D1%83" title="Математикалық талдау – Kazakh" lang="kk" hreflang="kk" data-title="Математикалық талдау" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Uchambuzi_wa_kihisabati" title="Uchambuzi wa kihisabati – Swahili" lang="sw" hreflang="sw" data-title="Uchambuzi wa kihisabati" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Analiz_(mat%C3%A9matik)" title="Analiz (matématik) – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Analiz (matématik)" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D0%BA_%D1%82%D0%B0%D0%BB%D0%B4%D0%BE%D0%BE" title="Математикалык талдоо – Kyrgyz" lang="ky" hreflang="ky" data-title="Математикалык талдоо" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%81%E0%BA%B2%E0%BA%99%E0%BA%A7%E0%BA%B4%E0%BB%80%E0%BA%84%E0%BA%B2%E0%BA%B0%E0%BA%97%E0%BA%B2%E0%BA%87%E0%BA%84%E0%BA%B0%E0%BA%99%E0%BA%B4%E0%BA%94%E0%BA%AA%E0%BA%B2%E0%BA%94" title="ການວິເຄາະທາງຄະນິດສາດ – Lao" lang="lo" hreflang="lo" data-title="ການວິເຄາະທາງຄະນິດສາດ" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Analysis_mathematica" title="Analysis mathematica – Latin" lang="la" hreflang="la" data-title="Analysis mathematica" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Matem%C4%81tisk%C4%81_anal%C4%ABze" title="Matemātiskā analīze – Latvian" lang="lv" hreflang="lv" data-title="Matemātiskā analīze" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Analys_(Mathematik)" title="Analys (Mathematik) – Luxembourgish" lang="lb" hreflang="lb" data-title="Analys (Mathematik)" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Matematin%C4%97_analiz%C4%97" title="Matematinė analizė – Lithuanian" lang="lt" hreflang="lt" data-title="Matematinė analizė" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lij mw-list-item"><a href="https://lij.wikipedia.org/wiki/Analixi_matematica" title="Analixi matematica – Ligurian" lang="lij" hreflang="lij" data-title="Analixi matematica" data-language-autonym="Ligure" data-language-local-name="Ligurian" class="interlanguage-link-target"><span>Ligure</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Analise_matematical" title="Analise matematical – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Analise matematical" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Analisi_matematega" title="Analisi matematega – Lombard" lang="lmo" hreflang="lmo" data-title="Analisi matematega" 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/Matematikai_anal%C3%ADzis" title="Matematikai analízis – Hungarian" lang="hu" hreflang="hu" data-title="Matematikai 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%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%BA%D0%B0_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7%D0%B0" title="Математичка анализа – Macedonian" lang="mk" hreflang="mk" data-title="Математичка анализа" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%85%E0%B4%A8%E0%B4%BE%E0%B4%B2%E0%B4%BF%E0%B4%B8%E0%B4%BF%E0%B4%B8%E0%B5%8D_(%E0%B4%97%E0%B4%A3%E0%B4%BF%E0%B4%A4%E0%B4%82)" title="അനാലിസിസ് (ഗണിതം) – Malayalam" lang="ml" hreflang="ml" data-title="അനാലിസിസ് (ഗണിതം)" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Analisi_matematika" title="Analisi matematika – Maltese" lang="mt" hreflang="mt" data-title="Analisi matematika" data-language-autonym="Malti" data-language-local-name="Maltese" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Analisis_matematik" title="Analisis matematik – Malay" lang="ms" hreflang="ms" data-title="Analisis matematik" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mwl mw-list-item"><a href="https://mwl.wikipedia.org/wiki/An%C3%A1leze_matem%C3%A1tica" title="Análeze matemática – Mirandese" lang="mwl" hreflang="mwl" data-title="Análeze matemática" data-language-autonym="Mirandés" data-language-local-name="Mirandese" class="interlanguage-link-target"><span>Mirandés</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%81%E1%80%BD%E1%80%B2%E1%80%81%E1%80%BC%E1%80%99%E1%80%BA%E1%80%B8%E1%80%85%E1%80%AD%E1%80%90%E1%80%BA%E1%80%96%E1%80%BC%E1%80%AC%E1%80%9E%E1%80%84%E1%80%BA%E1%80%B9%E1%80%81%E1%80%BB%E1%80%AC" title="ခွဲခြမ်းစိတ်ဖြာသင်္ချာ – Burmese" lang="my" hreflang="my" data-title="ခွဲခြမ်းစိတ်ဖြာသင်္ချာ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Analyse_(wiskunde)" title="Analyse (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="Analyse (wiskunde)" 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%A7%A3%E6%9E%90%E5%AD%A6" title="解析学 – Japanese" lang="ja" hreflang="ja" data-title="解析学" 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-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Analysis" title="Analysis – Northern Frisian" lang="frr" hreflang="frr" data-title="Analysis" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Matematisk_analyse" title="Matematisk analyse – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Matematisk 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-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Matematisk_analyse" title="Matematisk analyse – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Matematisk analyse" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Analisi_matematica" title="Analisi matematica – Occitan" lang="oc" hreflang="oc" data-title="Analisi matematica" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Matematik_analiz" title="Matematik analiz – Uzbek" lang="uz" hreflang="uz" data-title="Matematik analiz" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%97%E0%A8%A3%E0%A8%BF%E0%A8%A4_%E0%A8%B5%E0%A8%BF%E0%A8%B8%E0%A8%BC%E0%A8%B2%E0%A9%87%E0%A8%B8%E0%A8%BC%E0%A8%A3" title="ਗਣਿਤ ਵਿਸ਼ਲੇਸ਼ਣ – Punjabi" lang="pa" hreflang="pa" data-title="ਗਣਿਤ ਵਿਸ਼ਲੇਸ਼ਣ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%85%DB%8C%D8%AA%DA%BE%D9%85%DB%8C%D9%B9%DB%8C%DA%A9%D9%84_%D8%A7%D9%86%DB%8C%D9%84%DB%8C%D8%B3%D8%B2" title="میتھمیٹیکل انیلیسز – Western Punjabi" lang="pnb" hreflang="pnb" data-title="میتھمیٹیکل انیلیسز" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Matimatikal_analisis" title="Matimatikal analisis – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Matimatikal analisis" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/An%C3%A0lisi_matem%C3%A0tica" title="Anàlisi matemàtica – Piedmontese" lang="pms" hreflang="pms" data-title="Anàlisi matemàtica" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Analysis" title="Analysis – Low German" lang="nds" hreflang="nds" data-title="Analysis" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Analiza_matematyczna" title="Analiza matematyczna – Polish" lang="pl" hreflang="pl" data-title="Analiza matematyczna" 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_matem%C3%A1tica" title="Análise matemática – Portuguese" lang="pt" hreflang="pt" data-title="Análise matemática" 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/Analiza_matematic%C4%83" title="Analiza matematică – Romanian" lang="ro" hreflang="ro" data-title="Analiza matematică" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D1%96%D1%87%D0%BD%D0%B0_%D0%B0%D0%BD%D0%B0%D0%BB%D1%96%D0%B7%D0%B0" title="Математічна аналіза – Rusyn" lang="rue" hreflang="rue" data-title="Математічна аналіза" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7_(%D1%80%D0%B0%D0%B7%D0%B4%D0%B5%D0%BB_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B8)" title="Анализ (раздел математики) – Russian" lang="ru" hreflang="ru" data-title="Анализ (раздел математики)" 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/Mathematical_analysis" title="Mathematical analysis – Scots" lang="sco" hreflang="sco" data-title="Mathematical 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_matematikore" title="Analiza matematikore – Albanian" lang="sq" hreflang="sq" data-title="Analiza matematikore" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/An%C3%A0lisi_(matim%C3%A0tica)" title="Anàlisi (matimàtica) – Sicilian" lang="scn" hreflang="scn" data-title="Anàlisi (matimàtica)" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%9C%E0%B6%AB%E0%B7%92%E0%B6%AD%E0%B6%B8%E0%B6%BA_%E0%B7%80%E0%B7%92%E0%B7%81%E0%B7%8A%E0%B6%BD%E0%B7%9A%E0%B7%82%E0%B6%AB%E0%B6%BA" title="ගණිතමය විශ්ලේෂණය – Sinhala" lang="si" hreflang="si" data-title="ගණිතමය විශ්ලේෂණය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Mathematical_analysis" title="Mathematical analysis – Simple English" lang="en-simple" hreflang="en-simple" data-title="Mathematical analysis" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Matematick%C3%A1_anal%C3%BDza" title="Matematická analýza – Slovak" lang="sk" hreflang="sk" data-title="Matematická analýza" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Matemati%C4%8Dna_analiza" title="Matematična analiza – Slovenian" lang="sl" hreflang="sl" data-title="Matematična analiza" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%B4%DB%8C%DA%A9%D8%A7%D8%B1%DB%8C%DB%8C_%D9%85%D8%A7%D8%AA%D9%85%D8%A7%D8%AA%DB%8C%DA%A9%DB%8C" title="شیکاریی ماتماتیکی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="شیکاریی ماتماتیکی" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%BA%D0%B0_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7%D0%B0" title="Математичка анализа – Serbian" lang="sr" hreflang="sr" data-title="Математичка анализа" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Matemati%C4%8Dka_analiza" title="Matematička analiza – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Matematička analiza" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Analyysi_(matematiikka)" title="Analyysi (matematiikka) – Finnish" lang="fi" hreflang="fi" data-title="Analyysi (matematiikka)" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Matematisk_analys" title="Matematisk analys – Swedish" lang="sv" hreflang="sv" data-title="Matematisk analys" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Pagsusuring_matematikal" title="Pagsusuring matematikal – Tagalog" lang="tl" hreflang="tl" data-title="Pagsusuring matematikal" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AE%95%E0%AF%81%E0%AE%B5%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="பகுவியல் (கணிதம்) – Tamil" lang="ta" hreflang="ta" data-title="பகுவியல் (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Математик анализ – Tatar" lang="tt" hreflang="tt" data-title="Математик анализ" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%A7%E0%B8%B4%E0%B9%80%E0%B8%84%E0%B8%A3%E0%B8%B2%E0%B8%B0%E0%B8%AB%E0%B9%8C" title="คณิตวิเคราะห์ – Thai" lang="th" hreflang="th" data-title="คณิตวิเคราะห์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Analiz_(matematik)" title="Analiz (matematik) – Turkish" lang="tr" hreflang="tr" data-title="Analiz (matematik)" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tk mw-list-item"><a href="https://tk.wikipedia.org/wiki/Analiz" title="Analiz – Turkmen" lang="tk" hreflang="tk" data-title="Analiz" data-language-autonym="Türkmençe" data-language-local-name="Turkmen" class="interlanguage-link-target"><span>Türkmençe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%BD%D0%B8%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D1%96%D0%B7" title="Математичний аналіз – Ukrainian" lang="uk" hreflang="uk" data-title="Математичний аналіз" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA%DB%8C_%D8%AA%D8%AD%D9%84%DB%8C%D9%84" title="ریاضیاتی تحلیل – Urdu" lang="ur" hreflang="ur" data-title="ریاضیاتی تحلیل" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/An%C3%A0%C5%82ixi_matem%C3%A0tica" title="Anàłixi matemàtica – Venetian" lang="vec" hreflang="vec" data-title="Anàłixi matemàtica" data-language-autonym="Vèneto" data-language-local-name="Venetian" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vep mw-list-item"><a href="https://vep.wikipedia.org/wiki/Analiz_(matematikan_jaguz)" title="Analiz (matematikan jaguz) – Veps" lang="vep" hreflang="vep" data-title="Analiz (matematikan jaguz)" data-language-autonym="Vepsän kel’" data-language-local-name="Veps" class="interlanguage-link-target"><span>Vepsän kel’</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Gi%E1%BA%A3i_t%C3%ADch_to%C3%A1n_h%E1%BB%8Dc" title="Giải tích toán học – Vietnamese" lang="vi" hreflang="vi" data-title="Giải tích toán học" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%88%86%E6%9E%90%E5%AD%B8" title="分析學 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="分析學" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Analisis_matematikal" title="Analisis matematikal – Waray" lang="war" hreflang="war" data-title="Analisis matematikal" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90" title="数学分析 – Wu" lang="wuu" hreflang="wuu" data-title="数学分析" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-ts mw-list-item"><a href="https://ts.wikipedia.org/wiki/Vukambisisi_bya_Dyondzo-Tinhlayo" title="Vukambisisi bya Dyondzo-Tinhlayo – Tsonga" lang="ts" hreflang="ts" data-title="Vukambisisi bya Dyondzo-Tinhlayo" data-language-autonym="Xitsonga" data-language-local-name="Tsonga" class="interlanguage-link-target"><span>Xitsonga</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a 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href="/wiki/Template_talk:Math_topics_sidebar" title="Template talk:Math topics sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Math_topics_sidebar" title="Special:EditPage/Template:Math topics sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Attracteur_%C3%A9trange_de_Lorenz.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Attracteur_%C3%A9trange_de_Lorenz.png/260px-Attracteur_%C3%A9trange_de_Lorenz.png" decoding="async" width="260" height="211" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Attracteur_%C3%A9trange_de_Lorenz.png/390px-Attracteur_%C3%A9trange_de_Lorenz.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/95/Attracteur_%C3%A9trange_de_Lorenz.png/520px-Attracteur_%C3%A9trange_de_Lorenz.png 2x" data-file-width="965" data-file-height="784" /></a><figcaption>A <a href="/wiki/Strange_attractor" class="mw-redirect" title="Strange attractor">strange attractor</a> arising from a <a href="/wiki/Differential_equation" title="Differential equation">differential equation</a>. Differential equations are an important area of mathematical analysis with many applications in science and engineering.</figcaption></figure> <p><b>Analysis</b> is the branch of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> dealing with <a href="/wiki/Continuous_function" title="Continuous function">continuous functions</a>, <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limits</a>, and related theories, such as <a href="/wiki/Derivative" title="Derivative">differentiation</a>, <a href="/wiki/Integral" title="Integral">integration</a>, <a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">measure</a>, <a href="/wiki/Infinite_sequence" class="mw-redirect" title="Infinite sequence">infinite sequences</a>, <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">series</a>, and <a href="/wiki/Analytic_function" title="Analytic function">analytic functions</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Stillwell_Analysis_2-0" class="reference"><a href="#cite_note-Stillwell_Analysis-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>These theories are usually studied in the context of <a href="/wiki/Real_number" title="Real number">real</a> and <a href="/wiki/Complex_number" title="Complex number">complex</a> numbers and <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a>. Analysis evolved from <a href="/wiki/Calculus" title="Calculus">calculus</a>, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from <a href="/wiki/Geometry" title="Geometry">geometry</a>; however, it can be applied to any <a href="/wiki/Space_(mathematics)" title="Space (mathematics)">space</a> of <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical objects</a> that has a definition of nearness (a <a href="/wiki/Topological_space" title="Topological space">topological space</a>) or specific distances between objects (a <a href="/wiki/Metric_space" title="Metric space">metric space</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=Mathematical_analysis&action=edit&section=1" title="Edit section: History"><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:Archimedes_pi.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Archimedes_pi.svg/300px-Archimedes_pi.svg.png" decoding="async" width="300" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Archimedes_pi.svg/450px-Archimedes_pi.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Archimedes_pi.svg/600px-Archimedes_pi.svg.png 2x" data-file-width="750" data-file-height="250" /></a><figcaption><a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> used the <a href="/wiki/Method_of_exhaustion" title="Method of exhaustion">method of exhaustion</a> to compute the <a href="/wiki/Area" title="Area">area</a> inside a circle by finding the area of <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygons</a> with more and more sides. This was an early but informal example of a <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limit</a>, one of the most basic concepts in mathematical analysis.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Ancient">Ancient</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_analysis&action=edit&section=2" title="Edit section: Ancient"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mathematical analysis formally developed in the 17th century during the <a href="/wiki/Scientific_Revolution" title="Scientific Revolution">Scientific Revolution</a>,<sup id="cite_ref-analysis_3-0" class="reference"><a href="#cite_note-analysis-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of <a href="/wiki/Greek_mathematics" title="Greek mathematics">ancient Greek mathematics</a>. For instance, an <a href="/wiki/Geometric_series" title="Geometric series">infinite geometric sum</a> is implicit in <a href="/wiki/Zeno_of_Elea" title="Zeno of Elea">Zeno's</a> <a href="/wiki/Zeno%27s_paradoxes#Dichotomy_paradox" title="Zeno's paradoxes">paradox of the dichotomy</a>.<sup id="cite_ref-Stillwell_2004_4-0" class="reference"><a href="#cite_note-Stillwell_2004-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> (Strictly speaking, the point of the paradox is to deny that the infinite sum exists.) Later, <a href="/wiki/Greek_mathematics" title="Greek mathematics">Greek mathematicians</a> such as <a href="/wiki/Eudoxus_of_Cnidus" title="Eudoxus of Cnidus">Eudoxus</a> and <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> made more explicit, but informal, use of the concepts of limits and convergence when they used the <a href="/wiki/Method_of_exhaustion" title="Method of exhaustion">method of exhaustion</a> to compute the area and volume of regions and solids.<sup id="cite_ref-Smith_1958_5-0" class="reference"><a href="#cite_note-Smith_1958-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The explicit use of <a href="/wiki/Infinitesimals" class="mw-redirect" title="Infinitesimals">infinitesimals</a> appears in Archimedes' <i><a href="/wiki/The_Method_of_Mechanical_Theorems" title="The Method of Mechanical Theorems">The Method of Mechanical Theorems</a></i>, a work rediscovered in the 20th century.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> In Asia, the <a href="/wiki/Chinese_mathematics" title="Chinese mathematics">Chinese mathematician</a> <a href="/wiki/Liu_Hui" title="Liu Hui">Liu Hui</a> used the method of exhaustion in the 3rd century CE to find the area of a circle.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> From Jain literature, it appears that Hindus were in possession of the formulae for the sum of the <a href="/wiki/Arithmetic_series" class="mw-redirect" title="Arithmetic series">arithmetic</a> and <a href="/wiki/Geometric_series" title="Geometric series">geometric</a> series as early as the 4th century BCE.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Bhadrabahu" class="mw-redirect" title="Bhadrabahu">Ācārya Bhadrabāhu</a> uses the sum of a geometric series in his Kalpasūtra in 433 <span title="Before Common Era">BCE</span>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Medieval">Medieval</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_analysis&action=edit&section=3" title="Edit section: Medieval"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Zu_Chongzhi" title="Zu Chongzhi">Zu Chongzhi</a> established a method that would later be called <a href="/wiki/Cavalieri%27s_principle" title="Cavalieri's principle">Cavalieri's principle</a> to find the volume of a <a href="/wiki/Sphere" title="Sphere">sphere</a> in the 5th century.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> In the 12th century, the <a href="/wiki/Indian_mathematics" title="Indian mathematics">Indian mathematician</a> <a href="/wiki/Bh%C4%81skara_II" title="Bhāskara II">Bhāskara II</a> used infinitesimal and used what is now known as <a href="/wiki/Rolle%27s_theorem" title="Rolle's theorem">Rolle's theorem</a>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>In the 14th century, <a href="/wiki/Madhava_of_Sangamagrama" title="Madhava of Sangamagrama">Madhava of Sangamagrama</a> developed <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">infinite series</a> expansions, now called <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a>, of functions such as <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">sine</a>, <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">cosine</a>, <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">tangent</a> and <a href="/wiki/Inverse_trigonometric_functions" title="Inverse trigonometric functions">arctangent</a>.<sup id="cite_ref-rajag78_12-0" class="reference"><a href="#cite_note-rajag78-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Alongside his development of Taylor series of <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a>, he also estimated the magnitude of the error terms resulting of truncating these series, and gave a rational approximation of some infinite series. His followers at the <a href="/wiki/Kerala_School_of_Astronomy_and_Mathematics" class="mw-redirect" title="Kerala School of Astronomy and Mathematics">Kerala School of Astronomy and Mathematics</a> further expanded his works, up to the 16th century. </p> <div class="mw-heading mw-heading3"><h3 id="Modern">Modern</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_analysis&action=edit&section=4" title="Edit section: Modern"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Foundations">Foundations</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_analysis&action=edit&section=5" title="Edit section: Foundations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The modern foundations of mathematical analysis were established in 17th century Europe.<sup id="cite_ref-analysis_3-1" class="reference"><a href="#cite_note-analysis-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> This began when <a href="/wiki/Fermat" class="mw-redirect" title="Fermat">Fermat</a> and <a href="/wiki/Descartes" class="mw-redirect" title="Descartes">Descartes</a> developed <a href="/wiki/Analytic_geometry" title="Analytic geometry">analytic geometry</a>, which is the precursor to modern calculus. Fermat's method of <a href="/wiki/Adequality" title="Adequality">adequality</a> allowed him to determine the maxima and minima of functions and the tangents of curves.<sup id="cite_ref-Pellegrino_13-0" class="reference"><a href="#cite_note-Pellegrino-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Descartes's publication of <i><a href="/wiki/La_G%C3%A9om%C3%A9trie" title="La Géométrie">La Géométrie</a></i> in 1637, which introduced the <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a>, is considered to be the establishment of mathematical analysis. It would be a few decades later that <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a> and <a href="/wiki/Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Leibniz</a> independently developed <a href="/wiki/Infinitesimal_calculus" class="mw-redirect" title="Infinitesimal calculus">infinitesimal calculus</a>, which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as the <a href="/wiki/Calculus_of_variations" title="Calculus of variations">calculus of variations</a>, <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary</a> and <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a>, <a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a>, and <a href="/wiki/Generating_function" title="Generating function">generating functions</a>. During this period, calculus techniques were applied to approximate <a href="/wiki/Discrete_mathematics" title="Discrete mathematics">discrete problems</a> by continuous ones. </p> <div class="mw-heading mw-heading4"><h4 id="Modernization">Modernization</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_analysis&action=edit&section=6" title="Edit section: Modernization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the 18th century, <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a> introduced the notion of a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">mathematical function</a>.<sup id="cite_ref-function_14-0" class="reference"><a href="#cite_note-function-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Real analysis began to emerge as an independent subject when <a href="/wiki/Bernard_Bolzano" title="Bernard Bolzano">Bernard Bolzano</a> introduced the modern definition of continuity in 1816,<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> but Bolzano's work did not become widely known until the 1870s. In 1821, <a href="/wiki/Augustin_Louis_Cauchy" class="mw-redirect" title="Augustin Louis Cauchy">Cauchy</a> began to put calculus on a firm logical foundation by rejecting the principle of the <a href="/wiki/Generality_of_algebra" title="Generality of algebra">generality of algebra</a> widely used in earlier work, particularly by Euler. Instead, Cauchy formulated calculus in terms of geometric ideas and <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimals</a>. Thus, his definition of continuity required an infinitesimal change in <i>x</i> to correspond to an infinitesimal change in <i>y</i>. He also introduced the concept of the <a href="/wiki/Cauchy_sequence" title="Cauchy sequence">Cauchy sequence</a>, and started the formal theory of <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a>. <a href="/wiki/Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Poisson</a>, <a href="/wiki/Joseph_Liouville" title="Joseph Liouville">Liouville</a>, <a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Fourier</a> and others studied partial differential equations and <a href="/wiki/Harmonic_analysis" title="Harmonic analysis">harmonic analysis</a>. The contributions of these mathematicians and others, such as <a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Weierstrass</a>, developed the <a href="/wiki/(%CE%B5,_%CE%B4)-definition_of_limit" class="mw-redirect" title="(ε, δ)-definition of limit">(ε, δ)-definition of limit</a> approach, thus founding the modern field of mathematical analysis. Around the same time, <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Riemann</a> introduced his theory of <a href="/wiki/Integral" title="Integral">integration</a>, and made significant advances in complex analysis. </p><p>Towards the end of the 19th century, mathematicians started worrying that they were assuming the existence of a <a href="/wiki/Continuum_(set_theory)" title="Continuum (set theory)">continuum</a> of <a href="/wiki/Real_number" title="Real number">real numbers</a> without proof. <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Dedekind</a> then constructed the real numbers by <a href="/wiki/Dedekind_cut" title="Dedekind cut">Dedekind cuts</a>, in which irrational numbers are formally defined, which serve to fill the "gaps" between rational numbers, thereby creating a <a href="/wiki/Complete_metric_space" title="Complete metric space">complete</a> set: the continuum of real numbers, which had already been developed by <a href="/wiki/Simon_Stevin" title="Simon Stevin">Simon Stevin</a> in terms of <a href="/wiki/Decimal_expansion" class="mw-redirect" title="Decimal expansion">decimal expansions</a>. Around that time, the attempts to refine the <a href="/wiki/Theorem" title="Theorem">theorems</a> of <a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integration</a> led to the study of the "size" of the set of <a href="/wiki/Classification_of_discontinuities" title="Classification of discontinuities">discontinuities</a> of real functions. </p><p>Also, various <a href="/wiki/Pathological_(mathematics)" title="Pathological (mathematics)">pathological objects</a>, (such as <a href="/wiki/Nowhere_continuous_function" title="Nowhere continuous function">nowhere continuous functions</a>, continuous but <a href="/wiki/Weierstrass_function" title="Weierstrass function">nowhere differentiable functions</a>, and <a href="/wiki/Space-filling_curve" title="Space-filling curve">space-filling curves</a>), commonly known as "monsters", began to be investigated. In this context, <a href="/wiki/Camille_Jordan" title="Camille Jordan">Jordan</a> developed his theory of <a href="/wiki/Jordan_measure" class="mw-redirect" title="Jordan measure">measure</a>, <a href="/wiki/Georg_Cantor" title="Georg Cantor">Cantor</a> developed what is now called <a href="/wiki/Naive_set_theory" title="Naive set theory">naive set theory</a>, and <a href="/wiki/Ren%C3%A9-Louis_Baire" title="René-Louis Baire">Baire</a> proved the <a href="/wiki/Baire_category_theorem" title="Baire category theorem">Baire category theorem</a>. In the early 20th century, calculus was formalized using an axiomatic <a href="/wiki/Set_theory" title="Set theory">set theory</a>. <a href="/wiki/Henri_Lebesgue" title="Henri Lebesgue">Lebesgue</a> greatly improved measure theory, and introduced his own theory of integration, now known as <a href="/wiki/Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integration</a>, which proved to be a big improvement over Riemann's. <a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a> introduced <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert spaces</a> to solve <a href="/wiki/Integral_equation" title="Integral equation">integral equations</a>. The idea of <a href="/wiki/Normed_vector_space" title="Normed vector space">normed vector space</a> was in the air, and in the 1920s <a href="/wiki/Stefan_Banach" title="Stefan Banach">Banach</a> created <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Important_concepts">Important concepts</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_analysis&action=edit&section=7" title="Edit section: Important concepts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Metric_spaces">Metric spaces</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_analysis&action=edit&section=8" title="Edit section: Metric spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style 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">Main article: <a href="/wiki/Metric_space" title="Metric space">Metric space</a></div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a metric space is a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> where a notion of <a href="/wiki/Distance" title="Distance">distance</a> (called a <a href="/wiki/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a>) between elements of the set is defined. </p><p>Much of analysis happens in some metric space; the most commonly used are the <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a>, the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>, <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, other <a href="/wiki/Vector_space" title="Vector space">vector spaces</a>, and the <a href="/wiki/Integer" title="Integer">integers</a>. Examples of analysis without a metric include <a href="/wiki/Measure_theory" class="mw-redirect" title="Measure theory">measure theory</a> (which describes size rather than distance) and <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a> (which studies <a href="/wiki/Topological_vector_space" title="Topological vector space">topological vector spaces</a> that need not have any sense of distance). </p><p>Formally, a metric space is an <a href="/wiki/Ordered_pair" title="Ordered pair">ordered pair</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 (M,d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M,d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d78e6f2ddf5baee227ee2a9f164726ba0c23c263" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.501ex; height:2.843ex;" alt="{\displaystyle (M,d)}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> is a set 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 d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> is a <a href="/wiki/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a> 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 M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span>, i.e., a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> </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 d\colon M\times M\rightarrow \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>:</mo> <mi>M</mi> <mo>×</mo> <mi>M</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\colon M\times M\rightarrow \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcf5a1da6629cc651072dab329ed2e4874d19e34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.267ex; height:2.176ex;" alt="{\displaystyle d\colon M\times M\rightarrow \mathbb {R} }"></span></dd></dl> <p>such that for any <span class="mwe-math-element"><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,z\in M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>∈</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z\in M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba283a127121ad64c98d3f69ced0ac4a86ec6414" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.924ex; height:2.509ex;" alt="{\displaystyle x,y,z\in M}"></span>, the following holds: </p> <ol><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1309cec878e6d9effccd8a2d2ed065a8e82bfa82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.805ex; height:2.843ex;" alt="{\displaystyle d(x,y)\geq 0}"></span>, with equality <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/409a91214d63eabe46ec10ff3cbba689ab687366" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle x=y}"></span>    (<i><a href="/wiki/Identity_of_indiscernibles" title="Identity of indiscernibles">identity of indiscernibles</a></i>),</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)=d(y,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)=d(y,x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7fea33d0e60116abd16287351eb6bf142a61fdd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.187ex; height:2.843ex;" alt="{\displaystyle d(x,y)=d(y,x)}"></span>    (<i>symmetry</i>), and</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,z)\leq d(x,y)+d(y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,z)\leq d(x,y)+d(y,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43ae751284c2944886e1effbfe4e0c1293f98419" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.263ex; height:2.843ex;" alt="{\displaystyle d(x,z)\leq d(x,y)+d(y,z)}"></span>    (<i><a href="/wiki/Triangle_inequality" title="Triangle inequality">triangle inequality</a></i>).</li></ol> <p>By taking the third property and letting <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a61c98d99e4f3fea95d3ace9687c25dcab83ac44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.516ex; height:1.676ex;" alt="{\displaystyle z=x}"></span>, it can be shown 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 d(x,y)\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1309cec878e6d9effccd8a2d2ed065a8e82bfa82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.805ex; height:2.843ex;" alt="{\displaystyle d(x,y)\geq 0}"></span>     (<i>non-negative</i>). </p> <div class="mw-heading mw-heading3"><h3 id="Sequences_and_limits">Sequences and limits</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_analysis&action=edit&section=9" title="Edit section: Sequences and limits"><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/Sequence" title="Sequence">Sequence</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">Limit of a sequence</a></div> <p>A sequence is an ordered list. Like a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a>, it contains <a href="/wiki/Element_(mathematics)" title="Element (mathematics)">members</a> (also called <i>elements</i>, or <i>terms</i>). Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Most precisely, a sequence can be defined as a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> whose domain is a <a href="/wiki/Countable" class="mw-redirect" title="Countable">countable</a> <a href="/wiki/Totally_ordered" class="mw-redirect" title="Totally ordered">totally ordered</a> set, such as the <a href="/wiki/Natural_numbers" class="mw-redirect" title="Natural numbers">natural numbers</a>. </p><p>One of the most important properties of a sequence is <i>convergence</i>. Informally, a sequence converges if it has a <i>limit</i>. Continuing informally, a (<a href="#Finite_and_infinite">singly-infinite</a>) sequence has a limit if it approaches some point <i>x</i>, called the limit, as <i>n</i> becomes very large. That is, for an abstract sequence (<i>a</i><sub><i>n</i></sub>) (with <i>n</i> running from 1 to infinity understood) the distance between <i>a</i><sub><i>n</i></sub> and <i>x</i> approaches 0 as <i>n</i> → ∞, denoted </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 \lim _{n\to \infty }a_{n}=x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }a_{n}=x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39ceac2fba6ab5c1246eb71dac752723cef1635d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.183ex; height:3.676ex;" alt="{\displaystyle \lim _{n\to \infty }a_{n}=x.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Main_branches">Main branches</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_analysis&action=edit&section=10" title="Edit section: Main branches"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Calculus">Calculus</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_analysis&action=edit&section=11" title="Edit section: Calculus"><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/Calculus" title="Calculus">Calculus</a></div> <div class="mw-heading mw-heading3"><h3 id="Real_analysis">Real analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_analysis&action=edit&section=12" title="Edit section: Real analysis"><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/Real_analysis" title="Real analysis">Real analysis</a></div> <p>Real analysis (traditionally, the "theory of functions of a real variable") is a branch of mathematical analysis dealing with the <a href="/wiki/Real_number" title="Real number">real numbers</a> and real-valued functions of a real variable.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> In particular, it deals with the analytic properties of real <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> and <a href="/wiki/Sequence" title="Sequence">sequences</a>, including <a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">convergence</a> and <a href="/wiki/Limit_of_a_function" title="Limit of a function">limits</a> of <a href="/wiki/Sequence" title="Sequence">sequences</a> of real numbers, the <a href="/wiki/Calculus" title="Calculus">calculus</a> of the real numbers, and <a href="/wiki/Continuous_function" title="Continuous function">continuity</a>, <a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">smoothness</a> and related properties of real-valued functions. </p> <div class="mw-heading mw-heading3"><h3 id="Complex_analysis">Complex analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_analysis&action=edit&section=13" title="Edit section: Complex analysis"><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/Complex_analysis" title="Complex analysis">Complex analysis</a></div> <p>Complex analysis (traditionally known as the "theory of functions of a complex variable") is the branch of mathematical analysis 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>.<sup id="cite_ref-Ahlfors_1979_18-0" class="reference"><a href="#cite_note-Ahlfors_1979-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> It is useful 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/Applied_mathematics" title="Applied mathematics">applied mathematics</a>; as well as in <a href="/wiki/Physics" title="Physics">physics</a>, including <a href="/wiki/Hydrodynamics" class="mw-redirect" title="Hydrodynamics">hydrodynamics</a>, <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a>, <a href="/wiki/Mechanical_engineering" title="Mechanical engineering">mechanical engineering</a>, <a href="/wiki/Electrical_engineering" title="Electrical engineering">electrical engineering</a>, and particularly, <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>. </p><p>Complex analysis is particularly concerned with the <a href="/wiki/Analytic_function" title="Analytic function">analytic functions</a> of complex variables (or, more generally, <a href="/wiki/Meromorphic_function" title="Meromorphic function">meromorphic functions</a>). Because the separate <a href="/wiki/Real_number" title="Real number">real</a> and <a href="/wiki/Imaginary_number" title="Imaginary number">imaginary</a> parts of any analytic function must satisfy <a href="/wiki/Laplace%27s_equation" title="Laplace's equation">Laplace's equation</a>, complex analysis is widely applicable to two-dimensional problems in <a href="/wiki/Physics" title="Physics">physics</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Functional_analysis">Functional analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_analysis&action=edit&section=14" title="Edit section: Functional analysis"><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/Functional_analysis" title="Functional analysis">Functional analysis</a></div> <p>Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of <a href="/wiki/Vector_space" title="Vector space">vector spaces</a> endowed with some kind of limit-related structure (e.g. <a href="/wiki/Inner_product_space#Definition" title="Inner product space">inner product</a>, <a href="/wiki/Norm_(mathematics)#Definition" title="Norm (mathematics)">norm</a>, <a href="/wiki/Topological_space#Definitions" title="Topological space">topology</a>, etc.) and the <a href="/wiki/Linear_transformation" class="mw-redirect" title="Linear transformation">linear operators</a> acting upon these spaces and respecting these structures in a suitable sense.<sup id="cite_ref-Rudin_1991_19-0" class="reference"><a href="#cite_note-Rudin_1991-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Conway_1994_20-0" class="reference"><a href="#cite_note-Conway_1994-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> The historical roots of functional analysis lie in the study of <a href="/wiki/Function_space" title="Function space">spaces of functions</a> and the formulation of properties of transformations of functions such as the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> as transformations defining <a href="/wiki/Continuous_function" title="Continuous function">continuous</a>, <a href="/wiki/Unitary_operator" title="Unitary operator">unitary</a> etc. operators between function spaces. This point of view turned out to be particularly useful for the study of <a href="/wiki/Differential_equations" class="mw-redirect" title="Differential equations">differential</a> and <a href="/wiki/Integral_equations" class="mw-redirect" title="Integral equations">integral equations</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Harmonic_analysis">Harmonic analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_analysis&action=edit&section=15" title="Edit section: Harmonic analysis"><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/Harmonic_analysis" title="Harmonic analysis">Harmonic analysis</a></div> <p>Harmonic analysis is a branch of mathematical analysis concerned with the representation of <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> and <a href="/wiki/Signal" title="Signal">signals</a> as the superposition of basic <a href="/wiki/Wave" title="Wave">waves</a>. This includes the study of the notions of <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a> and <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transforms</a> (<a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a>), and of their generalizations. Harmonic analysis has applications in areas as diverse as <a href="/wiki/Music_theory" title="Music theory">music theory</a>, <a href="/wiki/Number_theory" title="Number theory">number theory</a>, <a href="/wiki/Representation_theory" title="Representation theory">representation theory</a>, <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a>, <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, <a href="/wiki/Tidal_analysis" class="mw-redirect" title="Tidal analysis">tidal analysis</a>, and <a href="/wiki/Neuroscience" title="Neuroscience">neuroscience</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Differential_equations">Differential equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_analysis&action=edit&section=16" title="Edit section: Differential equations"><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/Differential_equation" title="Differential equation">Differential equation</a></div> <p>A differential equation is a <a href="/wiki/Mathematics" title="Mathematics">mathematical</a> <a href="/wiki/Equation" title="Equation">equation</a> for an unknown <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> of one or several <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a> that relates the values of the function itself and its <a href="/wiki/Derivative" title="Derivative">derivatives</a> of various <a href="/wiki/Derivative#Higher_derivatives" title="Derivative">orders</a>.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Evans_1998_23-0" class="reference"><a href="#cite_note-Evans_1998-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> Differential equations play a prominent role in <a href="/wiki/Engineering" title="Engineering">engineering</a>, <a href="/wiki/Physics" title="Physics">physics</a>, <a href="/wiki/Economics" title="Economics">economics</a>, <a href="/wiki/Biology" title="Biology">biology</a>, and other disciplines. </p><p>Differential equations arise in many areas of science and technology, specifically whenever a <a href="/wiki/Deterministic_system_(mathematics)" class="mw-redirect" title="Deterministic system (mathematics)">deterministic</a> relation involving some continuously varying quantities (modeled by functions) and their rates of change in space or time (expressed as derivatives) is known or postulated. This is illustrated in <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a>, where the motion of a body is described by its position and velocity as the time value varies. <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws</a> allow one (given the position, velocity, acceleration and various forces acting on the body) to express these variables dynamically as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called an <a href="/wiki/Equations_of_motion" title="Equations of motion">equation of motion</a>) may be solved explicitly. </p> <div class="mw-heading mw-heading3"><h3 id="Measure_theory">Measure theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_analysis&action=edit&section=17" title="Edit section: Measure theory"><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/Measure_(mathematics)" title="Measure (mathematics)">Measure (mathematics)</a></div> <p>A measure on a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> is a systematic way to assign a number to each suitable <a href="/wiki/Subset" title="Subset">subset</a> of that set, intuitively interpreted as its size.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a> on a <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, which assigns the conventional <a href="/wiki/Length" title="Length">length</a>, <a href="/wiki/Area" title="Area">area</a>, and <a href="/wiki/Volume" title="Volume">volume</a> of <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a> to suitable subsets of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-dimensional Euclidean space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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>. For instance, the Lebesgue measure of the <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</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 \left[0,1\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mn>0</mn> <mo>,</mo> <mn>1</mn> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[0,1\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c57121c2b6c63c0b2f38eb96b1f7a543b5d1c522" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle \left[0,1\right]}"></span> in the <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real numbers</a> is its length in the everyday sense of the word – specifically, 1. </p><p>Technically, a measure is a function that assigns a non-negative real number or <a href="/wiki/Extended_real_number_line" title="Extended real number line">+∞</a> to (certain) subsets of a set <span class="mwe-math-element"><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/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>. It must assign 0 to the <a href="/wiki/Empty_set" title="Empty set">empty set</a> and be (<a href="/wiki/Countably" class="mw-redirect" title="Countably">countably</a>) additive: the measure of a 'large' subset that can be decomposed into a finite (or countable) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate a <i>consistent</i> size to <i>each</i> subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the <a href="/wiki/Counting_measure" title="Counting measure">counting measure</a>. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called <i>measurable</i> subsets, which are required to form a <a href="/wiki/Sigma-algebra" class="mw-redirect" title="Sigma-algebra"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" 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 \sigma }"></span>-algebra</a>. This means that the empty set, countable <a href="/wiki/Union_(set_theory)" title="Union (set theory)">unions</a>, countable <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersections</a> and <a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complements</a> of measurable subsets are measurable. <a href="/wiki/Non-measurable_set" title="Non-measurable set">Non-measurable sets</a> in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. Indeed, their existence is a non-trivial consequence of the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Numerical_analysis">Numerical analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_analysis&action=edit&section=18" title="Edit section: Numerical analysis"><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/Numerical_analysis" title="Numerical analysis">Numerical analysis</a></div> <p>Numerical analysis is the study of <a href="/wiki/Algorithm" title="Algorithm">algorithms</a> that use numerical <a href="/wiki/Approximation" title="Approximation">approximation</a> (as opposed to general <a href="/wiki/Symbolic_computation" class="mw-redirect" title="Symbolic computation">symbolic manipulations</a>) for the problems of mathematical analysis (as distinguished from <a href="/wiki/Discrete_mathematics" title="Discrete mathematics">discrete mathematics</a>).<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p><p>Modern numerical analysis does not seek exact answers, because exact answers are often impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors. </p><p>Numerical analysis naturally finds applications in all fields of engineering and the physical sciences, but in the 21st century, the life sciences and even the arts have adopted elements of scientific computations. <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary differential equations</a> appear in <a href="/wiki/Celestial_mechanics" title="Celestial mechanics">celestial mechanics</a> (planets, stars and galaxies); <a href="/wiki/Numerical_linear_algebra" title="Numerical linear algebra">numerical linear algebra</a> is important for data analysis; <a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">stochastic differential equations</a> and <a href="/wiki/Markov_chain" title="Markov chain">Markov chains</a> are essential in simulating living cells for medicine and biology. </p> <div class="mw-heading mw-heading3"><h3 id="Vector_analysis">Vector analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_analysis&action=edit&section=19" title="Edit section: Vector analysis"><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/Vector_calculus" title="Vector calculus">Vector calculus</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/A_History_of_Vector_Analysis" title="A History of Vector Analysis">A History of Vector Analysis</a> and <a href="/wiki/Vector_Analysis" title="Vector Analysis">Vector Analysis</a></div> <p><i>Vector analysis</i>, also called <i>vector calculus</i>, is a branch of mathematical analysis dealing with <a href="/wiki/Vector-valued_function" title="Vector-valued function">vector-valued functions</a>.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Scalar_analysis">Scalar analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_analysis&action=edit&section=20" title="Edit section: Scalar analysis"><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/Scalar_(mathematics)" title="Scalar (mathematics)">Scalar (mathematics)</a></div> <p>Scalar analysis is a branch of mathematical analysis dealing with values related to scale as opposed to direction. Values such as temperature are scalar because they describe the magnitude of a value without regard to direction, force, or displacement that value may or may not have. </p> <div class="mw-heading mw-heading3"><h3 id="Tensor_analysis">Tensor analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_analysis&action=edit&section=21" title="Edit section: Tensor analysis"><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/Tensor_field" title="Tensor field">Tensor field</a></div> <div class="mw-heading mw-heading2"><h2 id="Other_topics">Other topics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_analysis&action=edit&section=22" title="Edit section: Other topics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Calculus_of_variations" title="Calculus of variations">Calculus of variations</a> deals with extremizing <a href="/wiki/Functional_(mathematics)" title="Functional (mathematics)">functionals</a>, as opposed to ordinary <a href="/wiki/Calculus" title="Calculus">calculus</a> which deals with <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a>.</li> <li><a href="/wiki/Harmonic_analysis" title="Harmonic analysis">Harmonic analysis</a> deals with the representation of <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> or signals as the <a href="/wiki/Superposition_principle" title="Superposition principle">superposition</a> of basic <a href="/wiki/Wave" title="Wave">waves</a>.</li> <li><a href="/wiki/Geometric_analysis" title="Geometric analysis">Geometric analysis</a> involves the use of geometrical methods in the study of <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a> and the application of the theory of partial differential equations to geometry.</li> <li><a href="/wiki/Clifford_analysis" title="Clifford analysis">Clifford analysis</a>, the study of Clifford valued functions that are annihilated by Dirac or Dirac-like operators, termed in general as monogenic or Clifford analytic functions.</li> <li><a href="/wiki/P-adic_analysis" title="P-adic analysis"><i>p</i>-adic analysis</a>, the study of analysis within the context of <a href="/wiki/P-adic_number" title="P-adic number"><i>p</i>-adic numbers</a>, which differs in some interesting and surprising ways from its real and complex counterparts.</li> <li><a href="/wiki/Non-standard_analysis" class="mw-redirect" title="Non-standard analysis">Non-standard analysis</a>, which investigates the <a href="/wiki/Hyperreal_number" title="Hyperreal number">hyperreal numbers</a> and their functions and gives a <a href="/wiki/Rigour#Mathematical_rigour" title="Rigour">rigorous</a> treatment of <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimals</a> and infinitely large numbers.</li> <li><a href="/wiki/Computable_analysis" title="Computable analysis">Computable analysis</a>, the study of which parts of analysis can be carried out in a <a href="/wiki/Computability_theory" title="Computability theory">computable</a> manner.</li> <li><a href="/wiki/Stochastic_calculus" title="Stochastic calculus">Stochastic calculus</a> – analytical notions developed for <a href="/wiki/Stochastic_processes" class="mw-redirect" title="Stochastic processes">stochastic processes</a>.</li> <li><a href="/wiki/Set-valued_analysis" class="mw-redirect" title="Set-valued analysis">Set-valued analysis</a> – applies ideas from analysis and topology to set-valued functions.</li> <li><a href="/wiki/Convex_analysis" title="Convex analysis">Convex analysis</a>, the study of convex sets and functions.</li> <li><a href="/wiki/Idempotent_analysis" title="Idempotent analysis">Idempotent analysis</a> – analysis in the context of an <a href="/wiki/Idempotent_semiring" class="mw-redirect" title="Idempotent semiring">idempotent semiring</a>, where the lack of an additive inverse is compensated somewhat by the idempotent rule A + A = A. <ul><li><a href="/wiki/Tropical_analysis" title="Tropical analysis">Tropical analysis</a> – analysis of the idempotent semiring called the <a href="/wiki/Tropical_semiring" title="Tropical semiring">tropical semiring</a> (or <a href="/wiki/Max-plus_algebra" class="mw-redirect" title="Max-plus algebra">max-plus algebra</a>/<a href="/wiki/Min-plus_algebra" class="mw-redirect" title="Min-plus algebra">min-plus algebra</a>).</li></ul></li> <li><a href="/wiki/Constructive_analysis" title="Constructive analysis">Constructive analysis</a>, which is built upon a foundation of <a href="/wiki/Constructive_logic" class="mw-redirect" title="Constructive logic">constructive</a>, rather than classical, logic and set theory.</li> <li><a href="/wiki/Intuitionistic_analysis" class="mw-redirect" title="Intuitionistic analysis">Intuitionistic analysis</a>, which is developed from constructive logic like constructive analysis but also incorporates <a href="/wiki/Choice_sequence" title="Choice sequence">choice sequences</a>.</li> <li><a href="/wiki/Paraconsistent_analysis" class="mw-redirect" title="Paraconsistent analysis">Paraconsistent analysis</a>, which is built upon a foundation of <a href="/wiki/Paraconsistent_logic" title="Paraconsistent logic">paraconsistent</a>, rather than classical, logic and set theory.</li> <li><a href="/wiki/Smooth_infinitesimal_analysis" title="Smooth infinitesimal analysis">Smooth infinitesimal analysis</a>, which is developed in a smooth topos.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_analysis&action=edit&section=23" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Techniques from analysis are also found in other areas such as: </p> <div class="mw-heading mw-heading3"><h3 id="Physical_sciences">Physical sciences</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_analysis&action=edit&section=24" title="Edit section: Physical sciences"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The vast majority of <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a>, <a href="/wiki/Theory_of_relativity" title="Theory of relativity">relativity</a>, and <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> is based on applied analysis, and <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a> in particular. Examples of important differential equations include <a href="/wiki/Newton%27s_second_law" class="mw-redirect" title="Newton's second law">Newton's second law</a>, the <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a>, and the <a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a>. </p><p><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a> is also a major factor in <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Signal_processing">Signal processing</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_analysis&action=edit&section=25" title="Edit section: Signal processing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When processing signals, such as <a href="/wiki/Sound" title="Sound">audio</a>, <a href="/wiki/Radio_wave" title="Radio wave">radio waves</a>, light waves, <a href="/wiki/Seismic_waves" class="mw-redirect" title="Seismic waves">seismic waves</a>, and even images, Fourier analysis can isolate individual components of a compound waveform, concentrating them for easier detection or removal. A large family of signal processing techniques consist of Fourier-transforming a signal, manipulating the Fourier-transformed data in a simple way, and reversing the transformation.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Other_areas_of_mathematics">Other areas of mathematics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_analysis&action=edit&section=26" title="Edit section: Other areas of mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Techniques from analysis are used in many areas of mathematics, including: </p> <ul><li><a href="/wiki/Analytic_number_theory" title="Analytic number theory">Analytic number theory</a></li> <li><a href="/wiki/Analytic_combinatorics" title="Analytic combinatorics">Analytic combinatorics</a></li> <li><a href="/wiki/Continuous_probability" class="mw-redirect" title="Continuous probability">Continuous probability</a></li> <li><a href="/wiki/Differential_entropy" title="Differential entropy">Differential entropy</a> in information theory</li> <li><a href="/wiki/Differential_game" title="Differential game">Differential games</a></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential geometry</a>, the application of calculus to specific mathematical spaces known as <a href="/wiki/Manifold" title="Manifold">manifolds</a> that possess a complicated internal structure but behave in a simple manner locally.</li> <li><a href="/wiki/Differentiable_manifolds" class="mw-redirect" title="Differentiable manifolds">Differentiable manifolds</a></li> <li><a href="/wiki/Differential_topology" title="Differential topology">Differential topology</a></li> <li><a href="/wiki/Partial_differential_equations" class="mw-redirect" title="Partial differential equations">Partial differential equations</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Famous_Textbooks">Famous Textbooks</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_analysis&action=edit&section=27" title="Edit section: Famous Textbooks"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Foundation of Analysis: The Arithmetic of Whole Rational, Irrational and Complex Numbers, by Edmund Landau</li> <li>Introductory Real Analysis, by <a href="/wiki/Andrey_Kolmogorov" title="Andrey Kolmogorov">Andrey Kolmogorov</a>, <a href="/wiki/Sergei_Fomin" title="Sergei Fomin">Sergei Fomin</a><sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup></li> <li>Differential and Integral Calculus (3 volumes), by <a href="/wiki/Grigorii_Fichtenholz" class="mw-redirect" title="Grigorii Fichtenholz">Grigorii Fichtenholz</a><sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup></li> <li>The Fundamentals of Mathematical Analysis (2 volumes), by <a href="/wiki/Grigorii_Fichtenholz" class="mw-redirect" title="Grigorii Fichtenholz">Grigorii Fichtenholz</a><sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup></li> <li>A Course Of Mathematical Analysis (2 volumes), by <a href="/wiki/Sergey_Nikolsky" title="Sergey Nikolsky">Sergey Nikolsky</a><sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup></li> <li>Mathematical Analysis (2 volumes), by <a href="/wiki/Vladimir_A._Zorich" title="Vladimir A. Zorich">Vladimir Zorich</a><sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup></li> <li>A Course of Higher Mathematics (5 volumes, 6 parts), by <a href="/wiki/Vladimir_Smirnov_(mathematician)" title="Vladimir Smirnov (mathematician)">Vladimir Smirnov</a><sup id="cite_ref-archive.org_38-0" class="reference"><a href="#cite_note-archive.org-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup></li> <li>Differential And Integral Calculus, by <a href="/wiki/Nikolai_Piskunov" title="Nikolai Piskunov">Nikolai Piskunov</a><sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup></li> <li>A Course of Mathematical Analysis, by <a href="/wiki/Aleksandr_Khinchin" title="Aleksandr Khinchin">Aleksandr Khinchin</a><sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup></li> <li>Mathematical Analysis: A Special Course, by <a href="/wiki/Georgiy_Shilov" title="Georgiy Shilov">Georgiy Shilov</a><sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup></li> <li>Theory of Functions of a Real Variable (2 volumes), by <a href="/wiki/Isidor_Natanson" title="Isidor Natanson">Isidor Natanson</a><sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup></li> <li>Problems in Mathematical Analysis, by <a href="/wiki/Boris_Demidovich" title="Boris Demidovich">Boris Demidovich</a><sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Problems_and_Theorems_in_Analysis" title="Problems and Theorems in Analysis">Problems and Theorems in Analysis</a> (2 volumes), by <a href="/wiki/George_P%C3%B3lya" title="George Pólya">George Pólya</a>, <a href="/wiki/G%C3%A1bor_Szeg%C5%91" title="Gábor Szegő">Gábor Szegő</a><sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup></li> <li>Mathematical Analysis: A Modern Approach to Advanced Calculus, by <a href="/wiki/Tom_M._Apostol" title="Tom M. Apostol">Tom Apostol</a><sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup></li> <li>Principles of Mathematical Analysis, by <a href="/wiki/Walter_Rudin" title="Walter Rudin">Walter Rudin</a><sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup></li> <li>Real Analysis: Measure Theory, Integration, and Hilbert Spaces, by <a href="/wiki/Elias_M._Stein" title="Elias M. Stein">Elias Stein</a><sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup></li> <li>Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable, by <a href="/wiki/Lars_Ahlfors" title="Lars Ahlfors">Lars Ahlfors</a><sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup></li> <li>Complex Analysis, by <a href="/wiki/Elias_M._Stein" title="Elias M. Stein">Elias Stein</a><sup id="cite_ref-55" class="reference"><a href="#cite_note-55"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup></li> <li>Functional Analysis: Introduction to Further Topics in Analysis, by <a href="/wiki/Elias_M._Stein" title="Elias M. Stein">Elias Stein</a><sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup></li> <li>Analysis (2 volumes), by <a href="/wiki/Terence_Tao" title="Terence Tao">Terence Tao</a><sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup></li> <li>Analysis (3 volumes), by Herbert Amann, Joachim Escher<sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-61" class="reference"><a href="#cite_note-61"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup></li> <li>Real and Functional Analysis, by Vladimir Bogachev, Oleg Smolyanov<sup id="cite_ref-62" class="reference"><a href="#cite_note-62"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup></li> <li>Real and Functional Analysis, by <a href="/wiki/Serge_Lang" title="Serge Lang">Serge Lang</a><sup id="cite_ref-63" class="reference"><a href="#cite_note-63"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup></li></ul> <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=Mathematical_analysis&action=edit&section=28" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 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href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span><span class="portalbox-link"><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></span></li></ul> <ul><li><a href="/wiki/Constructive_analysis" title="Constructive analysis">Constructive analysis</a></li> <li><a href="/wiki/History_of_calculus" title="History of calculus">History of calculus</a></li> <li><a href="/wiki/Hypercomplex_analysis" title="Hypercomplex analysis">Hypercomplex analysis</a></li> <li><a href="/wiki/Multiple_rule-based_problems" title="Multiple rule-based problems">Multiple rule-based problems</a></li> <li><a href="/wiki/Multivariable_calculus" title="Multivariable calculus">Multivariable calculus</a></li> <li><a href="/wiki/Paraconsistent_logic" title="Paraconsistent logic">Paraconsistent logic</a></li> <li><a href="/wiki/Smooth_infinitesimal_analysis" title="Smooth infinitesimal analysis">Smooth infinitesimal analysis</a></li> <li><a href="/wiki/Timeline_of_calculus_and_mathematical_analysis" title="Timeline of calculus and mathematical analysis">Timeline of calculus and mathematical analysis</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=Mathematical_analysis&action=edit&section=29" 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 mw-references-columns"><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"><a href="/wiki/Edwin_Hewitt" title="Edwin Hewitt">Edwin Hewitt</a> and Karl Stromberg, "Real and Abstract Analysis", Springer-Verlag, 1965</span> </li> <li id="cite_note-Stillwell_Analysis-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Stillwell_Analysis_2-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFStillwell" class="citation encyclopaedia cs1"><a href="/wiki/John_Colin_Stillwell" class="mw-redirect" title="John Colin Stillwell">Stillwell, John Colin</a>. <a rel="nofollow" class="external text" href="https://www.britannica.com/topic/analysis-mathematics">"analysis | mathematics"</a>. <i>Encyclopædia Britannica</i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150726223522/https://www.britannica.com/topic/analysis-mathematics">Archived</a> from the original on 2015-07-26<span class="reference-accessdate">. Retrieved <span class="nowrap">2015-07-31</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=analysis+%7C+mathematics&rft.btitle=Encyclop%C3%A6dia+Britannica&rft.aulast=Stillwell&rft.aufirst=John+Colin&rft_id=https%3A%2F%2Fwww.britannica.com%2Ftopic%2Fanalysis-mathematics&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-analysis-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-analysis_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-analysis_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJahnke2003" class="citation book cs1">Jahnke, Hans Niels (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CVRZEXFVsZkC&pg=PR7"><i>A History of Analysis</i></a>. History of Mathematics. Vol. 24. <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. p. 7. <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%2Fhmath%2F024">10.1090/hmath/024</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0821826232" title="Special:BookSources/978-0821826232"><bdi>978-0821826232</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160517180439/https://books.google.com/books?id=CVRZEXFVsZkC&pg=PR7">Archived</a> from the original on 2016-05-17<span class="reference-accessdate">. Retrieved <span class="nowrap">2015-11-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Analysis&rft.series=History+of+Mathematics&rft.pages=7&rft.pub=American+Mathematical+Society&rft.date=2003&rft_id=info%3Adoi%2F10.1090%2Fhmath%2F024&rft.isbn=978-0821826232&rft.aulast=Jahnke&rft.aufirst=Hans+Niels&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DCVRZEXFVsZkC%26pg%3DPR7&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-Stillwell_2004-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Stillwell_2004_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStillwell2004" class="citation book cs1"><a href="/wiki/John_Colin_Stillwell" class="mw-redirect" title="John Colin Stillwell">Stillwell, John Colin</a> (2004). "Infinite Series". <i>Mathematics and its History</i> (2nd ed.). <a href="/wiki/Springer_Science%2BBusiness_Media_Inc." class="mw-redirect" title="Springer Science+Business Media Inc.">Springer Science+Business Media Inc.</a> p. 170. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0387953366" title="Special:BookSources/978-0387953366"><bdi>978-0387953366</bdi></a>. <q>Infinite series were present in Greek mathematics, [...] There is no question that Zeno's paradox of the dichotomy (Section 4.1), for example, concerns the decomposition of the number 1 into the infinite series <sup>1</sup>⁄<sub>2</sub> + <sup>1</sup>⁄<sub>2</sub><sup>2</sup> + <sup>1</sup>⁄<sub>2</sub><sup>3</sup> + <sup>1</sup>⁄<sub>2</sub><sup>4</sup> + ... and that Archimedes found the area of the parabolic segment (Section 4.4) essentially by summing the infinite series 1 + <sup>1</sup>⁄<sub>4</sub> + <sup>1</sup>⁄<sub>4</sub><sup>2</sup> + <sup>1</sup>⁄<sub>4</sub><sup>3</sup> + ... = <sup>4</sup>⁄<sub>3</sub>. Both these examples are special cases of the result we express as summation of a geometric series</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Infinite+Series&rft.btitle=Mathematics+and+its+History&rft.pages=170&rft.edition=2nd&rft.pub=Springer+Science%2BBusiness+Media+Inc.&rft.date=2004&rft.isbn=978-0387953366&rft.aulast=Stillwell&rft.aufirst=John+Colin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-Smith_1958-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Smith_1958_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmith1958" class="citation book cs1"><a href="/wiki/David_Eugene_Smith" title="David Eugene Smith">Smith, David Eugene</a> (1958). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema0002smit"><i>History of Mathematics</i></a></span>. <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0486204307" title="Special:BookSources/978-0486204307"><bdi>978-0486204307</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+of+Mathematics&rft.pub=Dover+Publications&rft.date=1958&rft.isbn=978-0486204307&rft.aulast=Smith&rft.aufirst=David+Eugene&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofmathema0002smit&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPinto2004" class="citation book cs1">Pinto, J. Sousa (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bLbfhYrhyJUC&pg=PA7"><i>Infinitesimal Methods of Mathematical Analysis</i></a>. Horwood Publishing. p. 8. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1898563990" title="Special:BookSources/978-1898563990"><bdi>978-1898563990</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160611045431/https://books.google.com/books?id=bLbfhYrhyJUC&pg=PA7">Archived</a> from the original on 2016-06-11<span class="reference-accessdate">. Retrieved <span class="nowrap">2015-11-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Infinitesimal+Methods+of+Mathematical+Analysis&rft.pages=8&rft.pub=Horwood+Publishing&rft.date=2004&rft.isbn=978-1898563990&rft.aulast=Pinto&rft.aufirst=J.+Sousa&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DbLbfhYrhyJUC%26pg%3DPA7&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDunFanCohen1966" class="citation book cs1">Dun, Liu; Fan, Dainian; Cohen, Robert Sonné (1966). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jaQH6_8Ju-MC"><i>A comparison of Archimedes' and Liu Hui's studies of circles</i></a>. Chinese studies in the history and philosophy of science and technology. Vol. 130. Springer. p. 279. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7923-3463-7" title="Special:BookSources/978-0-7923-3463-7"><bdi>978-0-7923-3463-7</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160617055211/https://books.google.com/books?id=jaQH6_8Ju-MC">Archived</a> from the original on 2016-06-17<span class="reference-accessdate">. Retrieved <span class="nowrap">2015-11-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+comparison+of+Archimedes%27+and+Liu+Hui%27s+studies+of+circles&rft.series=Chinese+studies+in+the+history+and+philosophy+of+science+and+technology&rft.pages=279&rft.pub=Springer&rft.date=1966&rft.isbn=978-0-7923-3463-7&rft.aulast=Dun&rft.aufirst=Liu&rft.au=Fan%2C+Dainian&rft.au=Cohen%2C+Robert+Sonn%C3%A9&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DjaQH6_8Ju-MC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jaQH6_8Ju-MC&pg=PA279">Chapter, p. 279</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160526221958/https://books.google.com/books?id=jaQH6_8Ju-MC&pg=PA279">Archived</a> 2016-05-26 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSingh,_A._N.1936" class="citation journal cs1">Singh, A. N. (1936). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/301627">"On the Use of Series in Hindu Mathematics"</a>. <i>Osiris</i>. <b>1</b>: 606–628. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F368443">10.1086/368443</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/301627">301627</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:144760421">144760421</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Osiris&rft.atitle=On+the+Use+of+Series+in+Hindu+Mathematics&rft.volume=1&rft.pages=606-628&rft.date=1936&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A144760421%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F301627%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1086%2F368443&rft.au=Singh%2C+A.+N.&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F301627&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFK._B._Basant,_Satyananda_Panda2013" class="citation journal cs1">K. 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Basant, Satyananda Panda (2013). <a rel="nofollow" class="external text" href="https://insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol48_2_7_KBBasant.pdf">"Summation of Convergent Geometric Series and the concept of approachable Sunya"</a> <span class="cs1-format">(PDF)</span>. <i>Indian Journal of History of Science</i>. <b>48</b>: 291–313.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Indian+Journal+of+History+of+Science&rft.atitle=Summation+of+Convergent+Geometric+Series+and+the+concept+of+approachable+Sunya&rft.volume=48&rft.pages=291-313&rft.date=2013&rft.au=K.+B.+Basant%2C+Satyananda+Panda&rft_id=https%3A%2F%2Finsa.nic.in%2Fwritereaddata%2FUpLoadedFiles%2FIJHS%2FVol48_2_7_KBBasant.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZillWrightWright2009" class="citation book cs1">Zill, Dennis G.; Wright, Scott; Wright, Warren S. (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=R3Hk4Uhb1Z0C&pg=PR27"><i>Calculus: Early Transcendentals</i></a> (3 ed.). Jones & Bartlett Learning. p. xxvii. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0763759957" title="Special:BookSources/978-0763759957"><bdi>978-0763759957</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190421114230/https://books.google.com/books?id=R3Hk4Uhb1Z0C&pg=PR27">Archived</a> from the original on 2019-04-21<span class="reference-accessdate">. Retrieved <span class="nowrap">2015-11-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus%3A+Early+Transcendentals&rft.pages=xxvii&rft.edition=3&rft.pub=Jones+%26+Bartlett+Learning&rft.date=2009&rft.isbn=978-0763759957&rft.aulast=Zill&rft.aufirst=Dennis+G.&rft.au=Wright%2C+Scott&rft.au=Wright%2C+Warren+S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DR3Hk4Uhb1Z0C%26pg%3DPR27&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSeal1915" class="citation cs2">Seal, Sir Brajendranath (1915), "The positive sciences of the ancient Hindus", <i>Nature</i>, <b>97</b> (2426): 177, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1916Natur..97..177.">1916Natur..97..177.</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F097177a0">10.1038/097177a0</a>, <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/2027%2Fmdp.39015004845684">2027/mdp.39015004845684</a></span>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:3958488">3958488</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=The+positive+sciences+of+the+ancient+Hindus&rft.volume=97&rft.issue=2426&rft.pages=177&rft.date=1915&rft_id=info%3Ahdl%2F2027%2Fmdp.39015004845684&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A3958488%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1038%2F097177a0&rft_id=info%3Abibcode%2F1916Natur..97..177.&rft.aulast=Seal&rft.aufirst=Sir+Brajendranath&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-rajag78-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-rajag78_12-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRajagopalRangachari1978" class="citation journal cs1">Rajagopal, C. T.; Rangachari, M. S. (June 1978). "On an untapped source of medieval Keralese Mathematics". <i>Archive for History of Exact Sciences</i>. <b>18</b> (2): 89–102. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00348142">10.1007/BF00348142</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:51861422">51861422</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archive+for+History+of+Exact+Sciences&rft.atitle=On+an+untapped+source+of+medieval+Keralese+Mathematics&rft.volume=18&rft.issue=2&rft.pages=89-102&rft.date=1978-06&rft_id=info%3Adoi%2F10.1007%2FBF00348142&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A51861422%23id-name%3DS2CID&rft.aulast=Rajagopal&rft.aufirst=C.+T.&rft.au=Rangachari%2C+M.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-Pellegrino-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-Pellegrino_13-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPellegrino" class="citation web cs1">Pellegrino, Dana. <a rel="nofollow" class="external text" href="http://www.math.rutgers.edu/~cherlin/History/Papers2000/pellegrino.html">"Pierre de Fermat"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20081012024028/http://www.math.rutgers.edu/~cherlin/History/Papers2000/pellegrino.html">Archived</a> from the original on 2008-10-12<span class="reference-accessdate">. Retrieved <span class="nowrap">2008-02-24</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Pierre+de+Fermat&rft.aulast=Pellegrino&rft.aufirst=Dana&rft_id=http%3A%2F%2Fwww.math.rutgers.edu%2F~cherlin%2FHistory%2FPapers2000%2Fpellegrino.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-function-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-function_14-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDunham1999" class="citation book cs1">Dunham, William (1999). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/eulermasterofusa0000dunh"><i>Euler: The Master of Us All</i></a></span>. The Mathematical Association of America. p. <a rel="nofollow" class="external text" href="https://archive.org/details/eulermasterofusa0000dunh/page/17">17</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Euler%3A+The+Master+of+Us+All&rft.pages=17&rft.pub=The+Mathematical+Association+of+America&rft.date=1999&rft.aulast=Dunham&rft.aufirst=William&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Feulermasterofusa0000dunh&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">*<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCooke1997" class="citation book cs1"><a href="/w/index.php?title=Roger_Cooke_(mathematician)&action=edit&redlink=1" class="new" title="Roger Cooke (mathematician) (page does not exist)">Cooke, Roger</a> (1997). <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema0000cook/page/379">"Beyond the Calculus"</a>. <i>The History of Mathematics: A Brief Course</i>. Wiley-Interscience. p. <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema0000cook/page/379">379</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0471180821" title="Special:BookSources/978-0471180821"><bdi>978-0471180821</bdi></a>. <q>Real analysis began its growth as an independent subject with the introduction of the modern definition of continuity in 1816 by the Czech mathematician Bernard Bolzano (1781–1848)</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Beyond+the+Calculus&rft.btitle=The+History+of+Mathematics%3A+A+Brief+Course&rft.pages=379&rft.pub=Wiley-Interscience&rft.date=1997&rft.isbn=978-0471180821&rft.aulast=Cooke&rft.aufirst=Roger&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofmathema0000cook%2Fpage%2F379&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRudin1976" class="citation book cs1"><a href="/wiki/Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1976). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/principlesofmath00rudi"><i>Principles of Mathematical Analysis</i></a></span>. Walter Rudin Student Series in Advanced Mathematics (3rd ed.). McGraw–Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0070542358" title="Special:BookSources/978-0070542358"><bdi>978-0070542358</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Mathematical+Analysis&rft.series=Walter+Rudin+Student+Series+in+Advanced+Mathematics&rft.edition=3rd&rft.pub=McGraw%E2%80%93Hill&rft.date=1976&rft.isbn=978-0070542358&rft.aulast=Rudin&rft.aufirst=Walter&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fprinciplesofmath00rudi&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAbbott2001" class="citation book cs1">Abbott, Stephen (2001). <i>Understanding Analysis</i>. Undergraduate Texts in Mathematics. New York: Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0387950600" title="Special:BookSources/978-0387950600"><bdi>978-0387950600</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Understanding+Analysis&rft.place=New+York&rft.series=Undergraduate+Texts+in+Mathematics&rft.pub=Springer-Verlag&rft.date=2001&rft.isbn=978-0387950600&rft.aulast=Abbott&rft.aufirst=Stephen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-Ahlfors_1979-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ahlfors_1979_18-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAhlfors1979" class="citation book cs1"><a href="/wiki/Lars_Valerian_Ahlfors" class="mw-redirect" title="Lars Valerian Ahlfors">Ahlfors, Lars Valerian</a> (1979). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2MRuus-5GGoC"><i>Complex Analysis</i></a> (3rd ed.). New York: <a href="/wiki/McGraw-Hill" class="mw-redirect" title="McGraw-Hill">McGraw-Hill</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0070006577" title="Special:BookSources/978-0070006577"><bdi>978-0070006577</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complex+Analysis&rft.place=New+York&rft.edition=3rd&rft.pub=McGraw-Hill&rft.date=1979&rft.isbn=978-0070006577&rft.aulast=Ahlfors&rft.aufirst=Lars+Valerian&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D2MRuus-5GGoC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-Rudin_1991-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-Rudin_1991_19-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRudin1991" class="citation book cs1"><a href="/wiki/Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1991). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/functionalanalys0000rudi"><i>Functional Analysis</i></a></span>. <a href="/wiki/McGraw-Hill_Science" class="mw-redirect" title="McGraw-Hill Science">McGraw-Hill Science</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0070542365" title="Special:BookSources/978-0070542365"><bdi>978-0070542365</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Functional+Analysis&rft.pub=McGraw-Hill+Science&rft.date=1991&rft.isbn=978-0070542365&rft.aulast=Rudin&rft.aufirst=Walter&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffunctionalanalys0000rudi&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-Conway_1994-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-Conway_1994_20-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFConway1994" class="citation book cs1"><a href="/wiki/John_Bligh_Conway" class="mw-redirect" title="John Bligh Conway">Conway, John Bligh</a> (1994). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ix4P1e6AkeIC"><i>A Course in Functional Analysis</i></a> (2nd ed.). <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0387972459" title="Special:BookSources/978-0387972459"><bdi>978-0387972459</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200909165657/https://books.google.com/books?id=ix4P1e6AkeIC">Archived</a> from the original on 2020-09-09<span class="reference-accessdate">. Retrieved <span class="nowrap">2016-02-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Course+in+Functional+Analysis&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1994&rft.isbn=978-0387972459&rft.aulast=Conway&rft.aufirst=John+Bligh&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dix4P1e6AkeIC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFInce1956" class="citation book cs1">Ince, Edward L. (1956). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mbyqAAAAQBAJ"><i>Ordinary Differential Equations</i></a>. Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0486603490" title="Special:BookSources/978-0486603490"><bdi>978-0486603490</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ordinary+Differential+Equations&rft.pub=Dover+Publications&rft.date=1956&rft.isbn=978-0486603490&rft.aulast=Ince&rft.aufirst=Edward+L.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DmbyqAAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><a href="/wiki/Witold_Hurewicz" title="Witold Hurewicz">Witold Hurewicz</a>, <i>Lectures on Ordinary Differential Equations</i>, Dover Publications, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0486495108" title="Special:BookSources/0486495108">0486495108</a></span> </li> <li id="cite_note-Evans_1998-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-Evans_1998_23-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEvans1998" class="citation book cs1"><a href="/wiki/Lawrence_Craig_Evans" class="mw-redirect" title="Lawrence Craig Evans">Evans, Lawrence Craig</a> (1998). <i>Partial Differential Equations</i>. Providence: <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0821807729" title="Special:BookSources/978-0821807729"><bdi>978-0821807729</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Partial+Differential+Equations&rft.place=Providence&rft.pub=American+Mathematical+Society&rft.date=1998&rft.isbn=978-0821807729&rft.aulast=Evans&rft.aufirst=Lawrence+Craig&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTao2011" class="citation book cs1"><a href="/wiki/Terence_Tao" title="Terence Tao">Tao, Terence</a> (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=HoGDAwAAQBAJ"><i>An Introduction to Measure Theory</i></a>. Graduate Studies in Mathematics. Vol. 126. 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%2Fgsm%2F126">10.1090/gsm/126</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0821869192" title="Special:BookSources/978-0821869192"><bdi>978-0821869192</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191227145317/https://books.google.com/books?id=HoGDAwAAQBAJ">Archived</a> from the original on 2019-12-27<span class="reference-accessdate">. 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Том II"</a>. 1960.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D1%8B+%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B3%D0%BE+%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7%D0%B0.+%D0%A2%D0%BE%D0%BC+II&rft.date=1960&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2FB-001-014-359&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://archive.org/details/B-001-014-346">"Курс дифференциального и интегрального исчисления. Том III"</a>. 1960.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=%D0%9A%D1%83%D1%80%D1%81+%D0%B4%D0%B8%D1%84%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D0%B3%D0%BE+%D0%B8+%D0%B8%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D0%B3%D0%BE+%D0%B8%D1%81%D1%87%D0%B8%D1%81%D0%BB%D0%B5%D0%BD%D0%B8%D1%8F.+%D0%A2%D0%BE%D0%BC+III&rft.date=1960&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2FB-001-014-346&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><i>The Fundamentals of Mathematical Analysis: International Series in Pure and Applied Mathematics, Volume 1</i>. <a 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title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=A+Course+of+Mathematical+Analysis+Vol+2&rft.date=1987&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fnikolsky-a-course-of-mathematical-analysis-vol-2-mir&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><i>Mathematical Analysis I</i>. <a href="/wiki/Amazon_Standard_Identification_Number" title="Amazon Standard Identification Number">ASIN</a> <a rel="nofollow" class="external text" href="https://www.amazon.ca/dp/3662569558">3662569558</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Analysis+I&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><i>Mathematical Analysis II</i>. <a href="/wiki/Amazon_Standard_Identification_Number" title="Amazon Standard Identification Number">ASIN</a> <a rel="nofollow" class="external text" href="https://www.amazon.ca/dp/3662569663">3662569663</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Analysis+II&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-archive.org-38"><span 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href="https://archive.org/details/smirnov-a-course-of-higher-mathematics-vol-3-2-complex-variables-special-functions">"A Course of Higher Mathematics Vol 3-2 Complex Variables Special Functions"</a>. 1964.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=A+Course+of+Higher+Mathematics+Vol+3-2+Complex+Variables+Special+Functions&rft.date=1964&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsmirnov-a-course-of-higher-mathematics-vol-3-2-complex-variables-special-functions&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://archive.org/details/smirnov-a-course-of-higher-mathematics-vol-4-integral-and-partial-differential-equations">"A Course of Higher Mathematics Vol 4 Integral and Partial Differential Equations"</a>. 1964.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=A+Course+of+Higher+Mathematics+Vol+4+Integral+and+Partial+Differential+Equations&rft.date=1964&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsmirnov-a-course-of-higher-mathematics-vol-4-integral-and-partial-differential-equations&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" 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title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Analysis%3A+A+Special+Course&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-46">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://archive.org/details/theoryoffunction00nata">"Theory of functions of a real variable (Teoria functsiy veshchestvennoy peremennoy, chapters I to IX)"</a>. 1955.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Theory+of+functions+of+a+real+variable+%28Teoria+functsiy+veshchestvennoy+peremennoy%2C+chapters+I+to+IX%29&rft.date=1955&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftheoryoffunction00nata&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-47">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://archive.org/details/theoryoffunction0002nata">"Theory of functions of a real variable =Teoria functsiy veshchestvennoy peremennoy"</a>. 1955.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Theory+of+functions+of+a+real+variable+%3DTeoria+functsiy+veshchestvennoy+peremennoy&rft.date=1955&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftheoryoffunction0002nata&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-48">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://archive.org/details/DemidovichEtAlProblemsInMathematicalAnalysisMir1970">"Problems in Mathematical Analysis"</a>. 1970.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Problems+in+Mathematical+Analysis&rft.date=1970&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2FDemidovichEtAlProblemsInMathematicalAnalysisMir1970&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-49">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><i>Problems and Theorems in Analysis I: Series. Integral Calculus. Theory of Functions</i>. <a href="/wiki/Amazon_Standard_Identification_Number" title="Amazon Standard Identification Number">ASIN</a> <a rel="nofollow" class="external text" href="https://www.amazon.ca/dp/3540636404">3540636404</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Problems+and+Theorems+in+Analysis+I%3A+Series.+Integral+Calculus.+Theory+of+Functions&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-50">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><i>Problems and Theorems in Analysis II: Theory of Functions. Zeros. Polynomials. Determinants. Number Theory. Geometry</i>. <a href="/wiki/Amazon_Standard_Identification_Number" title="Amazon Standard Identification Number">ASIN</a> <a rel="nofollow" class="external text" href="https://www.amazon.ca/dp/3540636862">3540636862</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Problems+and+Theorems+in+Analysis+II%3A+Theory+of+Functions.+Zeros.+Polynomials.+Determinants.+Number+Theory.+Geometry&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-51">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><i>Mathematical Analysis: A Modern Approach to Advanced Calculus, 2nd Edition</i>. <a href="/wiki/Amazon_Standard_Identification_Number" title="Amazon Standard Identification Number">ASIN</a> <a rel="nofollow" class="external text" href="https://www.amazon.ca/dp/0201002884">0201002884</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Analysis%3A+A+Modern+Approach+to+Advanced+Calculus%2C+2nd+Edition&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-52"><span class="mw-cite-backlink"><b><a href="#cite_ref-52">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><i>Principles of Mathematical Analysis</i>. <a href="/wiki/Amazon_Standard_Identification_Number" title="Amazon Standard Identification Number">ASIN</a> <a rel="nofollow" class="external text" href="https://www.amazon.ca/dp/0070856133">0070856133</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Mathematical+Analysis&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-53">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><i>Real Analysis: Measure Theory, Integration, and Hilbert Spaces</i>. <a href="/wiki/Amazon_Standard_Identification_Number" title="Amazon Standard Identification Number">ASIN</a> <a rel="nofollow" class="external text" href="https://www.amazon.ca/dp/0691113866">0691113866</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Real+Analysis%3A+Measure+Theory%2C+Integration%2C+and+Hilbert+Spaces&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-54"><span class="mw-cite-backlink"><b><a href="#cite_ref-54">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><i>Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable</i>. 1979-01-01. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0070006577" title="Special:BookSources/978-0070006577"><bdi>978-0070006577</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complex+Analysis%3A+An+Introduction+to+the+Theory+of+Analytic+Functions+of+One+Complex+Variable&rft.date=1979-01-01&rft.isbn=978-0070006577&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-55"><span class="mw-cite-backlink"><b><a href="#cite_ref-55">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><i>Complex Analysis</i>. <a href="/wiki/Amazon_Standard_Identification_Number" title="Amazon Standard Identification Number">ASIN</a> <a rel="nofollow" class="external text" href="https://www.amazon.ca/dp/0691113858">0691113858</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complex+Analysis&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-56"><span class="mw-cite-backlink"><b><a href="#cite_ref-56">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><i>Functional Analysis: Introduction to Further Topics in Analysis</i>. <a href="/wiki/Amazon_Standard_Identification_Number" title="Amazon Standard Identification Number">ASIN</a> <a rel="nofollow" class="external text" href="https://www.amazon.ca/dp/0691113874">0691113874</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Functional+Analysis%3A+Introduction+to+Further+Topics+in+Analysis&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-57"><span class="mw-cite-backlink"><b><a href="#cite_ref-57">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><i>Analysis I: Third Edition</i>. <a href="/wiki/Amazon_Standard_Identification_Number" title="Amazon Standard Identification Number">ASIN</a> <a rel="nofollow" class="external text" href="https://www.amazon.ca/dp/9380250649">9380250649</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analysis+I%3A+Third+Edition&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-58"><span class="mw-cite-backlink"><b><a href="#cite_ref-58">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><i>Analysis II: Third Edition</i>. <a href="/wiki/Amazon_Standard_Identification_Number" title="Amazon Standard Identification Number">ASIN</a> <a rel="nofollow" class="external text" href="https://www.amazon.ca/dp/9380250657">9380250657</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analysis+II%3A+Third+Edition&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-59"><span class="mw-cite-backlink"><b><a href="#cite_ref-59">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAmannEscher2004" class="citation book cs1">Amann, Herbert; Escher, Joachim (2004). <i>Analysis I</i>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3764371531" title="Special:BookSources/978-3764371531"><bdi>978-3764371531</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analysis+I&rft.date=2004&rft.isbn=978-3764371531&rft.aulast=Amann&rft.aufirst=Herbert&rft.au=Escher%2C+Joachim&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-60"><span class="mw-cite-backlink"><b><a href="#cite_ref-60">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAmannEscher2008" class="citation book cs1">Amann, Herbert; Escher, Joachim (2008-05-16). <i>Analysis II</i>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3764374723" title="Special:BookSources/978-3764374723"><bdi>978-3764374723</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analysis+II&rft.date=2008-05-16&rft.isbn=978-3764374723&rft.aulast=Amann&rft.aufirst=Herbert&rft.au=Escher%2C+Joachim&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-61"><span class="mw-cite-backlink"><b><a href="#cite_ref-61">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAmannEscher2009" class="citation book cs1">Amann, Herbert; Escher, Joachim (2009). <i>Analysis III</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3764374792" title="Special:BookSources/978-3764374792"><bdi>978-3764374792</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analysis+III&rft.pub=Springer&rft.date=2009&rft.isbn=978-3764374792&rft.aulast=Amann&rft.aufirst=Herbert&rft.au=Escher%2C+Joachim&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-62"><span class="mw-cite-backlink"><b><a href="#cite_ref-62">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBogachevSmolyanov2021" class="citation book cs1">Bogachev, Vladimir I.; Smolyanov, Oleg G. (2021). <i>Real and Functional Analysis</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3030382216" title="Special:BookSources/978-3030382216"><bdi>978-3030382216</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Real+and+Functional+Analysis&rft.pub=Springer&rft.date=2021&rft.isbn=978-3030382216&rft.aulast=Bogachev&rft.aufirst=Vladimir+I.&rft.au=Smolyanov%2C+Oleg+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> <li id="cite_note-63"><span class="mw-cite-backlink"><b><a href="#cite_ref-63">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLang2012" class="citation book cs1">Lang, Serge (2012). <i>Real and Functional Analysis</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1461269380" title="Special:BookSources/978-1461269380"><bdi>978-1461269380</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Real+and+Functional+Analysis&rft.pub=Springer&rft.date=2012&rft.isbn=978-1461269380&rft.aulast=Lang&rft.aufirst=Serge&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_analysis&action=edit&section=30" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="anchor" id="Mathematics:_Its_Content,_Methods,_and_Meaning"></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAleksandrovKolmogorovLavrent'ev1969" class="citation book cs1"><a href="/wiki/Aleksandr_Danilovich_Aleksandrov" class="mw-redirect" title="Aleksandr Danilovich Aleksandrov">Aleksandrov, A. D.</a>; <a href="/wiki/Andrey_Nikolaevich_Kolmogorov" class="mw-redirect" title="Andrey Nikolaevich Kolmogorov">Kolmogorov, A. N.</a>; <a href="/wiki/Mikhail_Alekseevich_Lavrentyev" class="mw-redirect" title="Mikhail Alekseevich Lavrentyev">Lavrent'ev, M. A.</a>, eds. (March 1969). <i>Mathematics: Its Content, Methods, and Meaning</i>. Vol. 1–3. Translated by Gould, S. H. (2nd ed.). Cambridge, Massachusetts: <a href="/wiki/The_M.I.T._Press" class="mw-redirect" title="The M.I.T. Press">The M.I.T. Press</a> / <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics%3A+Its+Content%2C+Methods%2C+and+Meaning&rft.place=Cambridge%2C+Massachusetts&rft.edition=2nd&rft.pub=The+M.I.T.+Press+%2F+American+Mathematical+Society&rft.date=1969-03&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFApostol1974" class="citation book cs1"><a href="/wiki/Tom_M._Apostol" title="Tom M. Apostol">Apostol, Tom M.</a> (1974). <i>Mathematical Analysis</i> (2nd ed.). <a href="/wiki/Addison%E2%80%93Wesley" class="mw-redirect" title="Addison–Wesley">Addison–Wesley</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0201002881" title="Special:BookSources/978-0201002881"><bdi>978-0201002881</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Analysis&rft.edition=2nd&rft.pub=Addison%E2%80%93Wesley&rft.date=1974&rft.isbn=978-0201002881&rft.aulast=Apostol&rft.aufirst=Tom+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBinmore1981" class="citation book cs1"><a href="/wiki/Kenneth_George_Binmore" class="mw-redirect" title="Kenneth George Binmore">Binmore, Kenneth George</a> (1981) [1981]. <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/foundationsofana0000binm"><i>The foundations of analysis: a straightforward introduction</i></a></span>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+foundations+of+analysis%3A+a+straightforward+introduction&rft.pub=Cambridge+University+Press&rft.date=1981&rft.aulast=Binmore&rft.aufirst=Kenneth+George&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffoundationsofana0000binm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohnsonbaughPfaffenberger1981" class="citation book cs1"><a href="/wiki/Richard_Johnsonbaugh" title="Richard Johnsonbaugh">Johnsonbaugh, Richard</a>; Pfaffenberger, William Elmer (1981). <i>Foundations of mathematical analysis</i>. New York: <a href="/wiki/M._Dekker" class="mw-redirect" title="M. Dekker">M. Dekker</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+mathematical+analysis&rft.place=New+York&rft.pub=M.+Dekker&rft.date=1981&rft.aulast=Johnsonbaugh&rft.aufirst=Richard&rft.au=Pfaffenberger%2C+William+Elmer&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNikol'skiĭ_[Нико́льский]2002" class="citation encyclopaedia cs1"><a href="/wiki/Sergey_Mikhailovich_Nikolsky" class="mw-redirect" title="Sergey Mikhailovich Nikolsky">Nikol'skiĭ [Нико́льский], Sergey Mikhailovich [Серге́й Миха́йлович]</a> (2002). <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Mathematical_analysis">"Mathematical analysis"</a>. In <a href="/wiki/Michiel_Hazewinkel" title="Michiel Hazewinkel">Hazewinkel, Michiel</a> (ed.). <i><a href="/wiki/Encyclopaedia_of_Mathematics" class="mw-redirect" title="Encyclopaedia of Mathematics">Encyclopaedia of Mathematics</a></i>. <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1402006098" title="Special:BookSources/978-1402006098"><bdi>978-1402006098</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Mathematical+analysis&rft.btitle=Encyclopaedia+of+Mathematics&rft.pub=Springer-Verlag&rft.date=2002&rft.isbn=978-1402006098&rft.aulast=Nikol%27ski%C4%AD+%5B%D0%9D%D0%B8%D0%BA%D0%BE%CC%81%D0%BB%D1%8C%D1%81%D0%BA%D0%B8%D0%B9%5D&rft.aufirst=Sergey+Mikhailovich+%5B%D0%A1%D0%B5%D1%80%D0%B3%D0%B5%CC%81%D0%B9+%D0%9C%D0%B8%D1%85%D0%B0%CC%81%D0%B9%D0%BB%D0%BE%D0%B2%D0%B8%D1%87%5D&rft_id=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FMathematical_analysis&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFuscoMarcelliniSbordone1996" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Nicola_Fusco" title="Nicola Fusco">Fusco, Nicola</a>; <a href="/wiki/Paolo_Marcellini" title="Paolo Marcellini">Marcellini, Paolo</a>; Sbordone, Carlo (1996). <i>Analisi Matematica Due</i> (in Italian). <a href="/w/index.php?title=Liguori_Editore&action=edit&redlink=1" class="new" title="Liguori Editore (page does not exist)">Liguori Editore</a><span class="noprint" style="font-size:85%; font-style: normal;"> [<a href="https://it.wikipedia.org/wiki/Liguori_Editore" class="extiw" title="it:Liguori Editore">it</a>]</span>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-8820726751" title="Special:BookSources/978-8820726751"><bdi>978-8820726751</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analisi+Matematica+Due&rft.pub=Liguori+Editore%3Cspan+class%3D%22noprint%22+style%3D%22font-size%3A85%25%3B+font-style%3A+normal%3B+%22%3E+%26%2391%3Bit%26%2393%3B%3C%2Fspan%3E&rft.date=1996&rft.isbn=978-8820726751&rft.aulast=Fusco&rft.aufirst=Nicola&rft.au=Marcellini%2C+Paolo&rft.au=Sbordone%2C+Carlo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRombaldi2004" class="citation book cs1 cs1-prop-foreign-lang-source">Rombaldi, Jean-Étienne (2004). <i>Éléments d'analyse réelle : CAPES et agrégation interne de mathématiques</i> (in French). <a href="/wiki/EDP_Sciences" title="EDP Sciences">EDP Sciences</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-2868836816" title="Special:BookSources/978-2868836816"><bdi>978-2868836816</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%C3%89l%C3%A9ments+d%27analyse+r%C3%A9elle+%3A+CAPES+et+agr%C3%A9gation+interne+de+math%C3%A9matiques&rft.pub=EDP+Sciences&rft.date=2004&rft.isbn=978-2868836816&rft.aulast=Rombaldi&rft.aufirst=Jean-%C3%89tienne&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRudin1976" class="citation book cs1"><a href="/wiki/Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1976). <i>Principles of Mathematical Analysis</i> (3rd ed.). New York: <a href="/wiki/McGraw-Hill" class="mw-redirect" title="McGraw-Hill">McGraw-Hill</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0070542358" title="Special:BookSources/978-0070542358"><bdi>978-0070542358</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Mathematical+Analysis&rft.place=New+York&rft.edition=3rd&rft.pub=McGraw-Hill&rft.date=1976&rft.isbn=978-0070542358&rft.aulast=Rudin&rft.aufirst=Walter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRudin1987" class="citation book cs1"><a href="/wiki/Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1987). <i>Real and Complex Analysis</i> (3rd ed.). New York: <a href="/wiki/McGraw-Hill" class="mw-redirect" title="McGraw-Hill">McGraw-Hill</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0070542341" title="Special:BookSources/978-0070542341"><bdi>978-0070542341</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Real+and+Complex+Analysis&rft.place=New+York&rft.edition=3rd&rft.pub=McGraw-Hill&rft.date=1987&rft.isbn=978-0070542341&rft.aulast=Rudin&rft.aufirst=Walter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhittakerWatson1927" class="citation book cs1"><a href="/wiki/Edmund_Taylor_Whittaker" class="mw-redirect" title="Edmund Taylor Whittaker">Whittaker, Edmund Taylor</a>; <a href="/wiki/George_Neville_Watson" class="mw-redirect" title="George Neville Watson">Watson, George Neville</a> (1927-01-02). <a href="/wiki/Whittaker_and_Watson" class="mw-redirect" title="Whittaker and Watson"><i>A Course Of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions</i></a> (4th ed.). Cambridge: <a href="/wiki/At_the_University_Press" class="mw-redirect" title="At the University Press">at the University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0521067944" title="Special:BookSources/0521067944"><bdi>0521067944</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Course+Of+Modern+Analysis%3A+An+Introduction+to+the+General+Theory+of+Infinite+Processes+and+of+Analytic+Functions%3B+with+an+Account+of+the+Principal+Transcendental+Functions&rft.place=Cambridge&rft.edition=4th&rft.pub=at+the+University+Press&rft.date=1927-01-02&rft.isbn=0521067944&rft.aulast=Whittaker&rft.aufirst=Edmund+Taylor&rft.au=Watson%2C+George+Neville&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span> (vi+608 pages) (reprinted: 1935, 1940, 1946, 1950, 1952, 1958, 1962, 1963, 1992)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.math.harvard.edu/~ctm/home/text/class/harvard/114/07/html/home/course/course.pdf">"Real Analysis – Course Notes"</a> <span class="cs1-format">(PDF)</span>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070419024458/http://www.math.harvard.edu/~ctm/home/text/class/harvard/114/07/html/home/course/course.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2007-04-19.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Real+Analysis+%E2%80%93+Course+Notes&rft_id=http%3A%2F%2Fwww.math.harvard.edu%2F~ctm%2Fhome%2Ftext%2Fclass%2Fharvard%2F114%2F07%2Fhtml%2Fhome%2Fcourse%2Fcourse.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+analysis" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 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rel="nofollow" class="external text" href="http://www.economics.soton.ac.uk/staff/aldrich/Calculus%20and%20Analysis%20Earliest%20Uses.htm">Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis</a></li> <li><a rel="nofollow" class="external text" href="http://www.jirka.org/ra/">Basic Analysis: Introduction to Real Analysis</a> by Jiri Lebl (<a href="/wiki/Creative_Commons" title="Creative Commons">Creative Commons BY-NC-SA</a>)</li> <li><a rel="nofollow" class="external text" href="https://www.britannica.com/topic/analysis-mathematics">Mathematical Analysis – Encyclopædia Britannica</a></li> <li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/topics/CalculusandAnalysis.html">Calculus and Analysis</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid 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.navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Major_topics_in_mathematical_analysis" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Analysis-footer" title="Template:Analysis-footer"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Analysis-footer" title="Template talk:Analysis-footer"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Analysis-footer" title="Special:EditPage/Template:Analysis-footer"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Major_topics_in_mathematical_analysis" style="font-size:114%;margin:0 4em">Major topics in <a class="mw-selflink selflink">mathematical analysis</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><b><a href="/wiki/Calculus" title="Calculus">Calculus</a></b>: <a href="/wiki/Integral" title="Integral">Integration</a></li> <li><a href="/wiki/Derivative" title="Derivative">Differentiation</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential equations</a> <ul><li><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary</a></li> <li><a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial</a></li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">stochastic</a></li></ul></li> <li><a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem of calculus</a></li> <li><a href="/wiki/Calculus_of_variations" title="Calculus of variations">Calculus of variations</a></li> <li><a href="/wiki/Vector_calculus" title="Vector calculus">Vector calculus</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li> <li><a href="/wiki/Matrix_calculus" title="Matrix calculus">Matrix calculus</a></li> <li><a href="/wiki/Lists_of_integrals" title="Lists of integrals">Lists of integrals</a></li> <li><a href="/wiki/Table_of_derivatives" class="mw-redirect" title="Table of derivatives">Table of derivatives</a></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Real_analysis" title="Real analysis">Real analysis</a></li> <li><a href="/wiki/Complex_analysis" title="Complex analysis">Complex analysis</a></li> <li><a href="/wiki/Hypercomplex_analysis" title="Hypercomplex analysis">Hypercomplex analysis</a> (<a href="/wiki/Quaternionic_analysis" title="Quaternionic analysis">quaternionic analysis</a>)</li> <li><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a></li> <li><a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li> <li><a href="/wiki/Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li> <li><a href="/wiki/Harmonic_analysis" title="Harmonic analysis">Harmonic analysis</a></li> <li><a href="/wiki/P-adic_analysis" title="P-adic analysis">P-adic analysis</a> (<a href="/wiki/P-adic_number" title="P-adic number">P-adic numbers</a>)</li> <li><a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">Measure theory</a></li> <li><a href="/wiki/Representation_theory" title="Representation theory">Representation theory</a></li></ul> <ul><li><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Functions</a></li> <li><a href="/wiki/Continuous_function" title="Continuous function">Continuous function</a></li> <li><a href="/wiki/Special_functions" title="Special functions">Special functions</a></li> <li><a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">Limit</a></li> <li><a href="/wiki/Series_(mathematics)" title="Series (mathematics)">Series</a></li> <li><a href="/wiki/Infinity" title="Infinity">Infinity</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><b><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></b></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Major_mathematics_areas" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Areas_of_mathematics" title="Template:Areas of mathematics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Areas_of_mathematics" title="Template talk:Areas of mathematics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Areas_of_mathematics" title="Special:EditPage/Template:Areas of mathematics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Major_mathematics_areas" style="font-size:114%;margin:0 4em">Major <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> areas</div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/History_of_mathematics" title="History of mathematics">History</a> <ul><li><a href="/wiki/Timeline_of_mathematics" title="Timeline of mathematics">Timeline</a></li> <li><a href="/wiki/Future_of_mathematics" title="Future of mathematics">Future</a></li></ul></li> <li><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Lists</a></li> <li><a href="/wiki/Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></li> <li><a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li> <li><a href="/wiki/Set_theory" title="Set theory">Set theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Algebra" title="Algebra">Algebra</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_algebra" title="Abstract algebra">Abstract</a></li> <li><a href="/wiki/Commutative_algebra" title="Commutative algebra">Commutative</a></li> <li><a href="/wiki/Elementary_algebra" title="Elementary algebra">Elementary</a></li> <li><a href="/wiki/Group_theory" title="Group theory">Group theory</a></li> <li><a href="/wiki/Linear_algebra" title="Linear algebra">Linear</a></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear</a></li> <li><a href="/wiki/Universal_algebra" title="Universal algebra">Universal</a></li> <li><a href="/wiki/Homological_algebra" title="Homological algebra">Homological</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a class="mw-selflink selflink">Analysis</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Calculus" title="Calculus">Calculus</a></li> <li><a href="/wiki/Real_analysis" title="Real analysis">Real analysis</a></li> <li><a href="/wiki/Complex_analysis" title="Complex analysis">Complex analysis</a></li> <li><a href="/wiki/Hypercomplex_analysis" title="Hypercomplex analysis">Hypercomplex analysis</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential equations</a></li> <li><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a></li> <li><a href="/wiki/Harmonic_analysis" title="Harmonic analysis">Harmonic analysis</a></li> <li><a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">Measure theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Discrete_mathematics" title="Discrete mathematics">Discrete</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Combinatorics" title="Combinatorics">Combinatorics</a></li> <li><a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a></li> <li><a href="/wiki/Order_theory" title="Order theory">Order theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Geometry" title="Geometry">Geometry</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebraic_geometry" title="Algebraic geometry">Algebraic</a></li> <li><a href="/wiki/Analytic_geometry" title="Analytic geometry">Analytic</a></li> <li><a href="/wiki/Arithmetic_geometry" title="Arithmetic geometry">Arithmetic</a></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential</a></li> <li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete</a></li> <li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a></li> <li><a href="/wiki/Finite_geometry" title="Finite geometry">Finite</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Number_theory" title="Number theory">Number theory</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetic" title="Arithmetic">Arithmetic</a></li> <li><a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></li> <li><a href="/wiki/Analytic_number_theory" title="Analytic number theory">Analytic number theory</a></li> <li><a href="/wiki/Diophantine_geometry" title="Diophantine geometry">Diophantine geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Topology" title="Topology">Topology</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/General_topology" title="General topology">General</a></li> <li><a href="/wiki/Algebraic_topology" title="Algebraic topology">Algebraic</a></li> <li><a href="/wiki/Differential_topology" title="Differential topology">Differential</a></li> <li><a href="/wiki/Geometric_topology" title="Geometric topology">Geometric</a></li> <li><a href="/wiki/Homotopy_theory" title="Homotopy theory">Homotopy theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Applied_mathematics" title="Applied mathematics">Applied</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Engineering_mathematics" title="Engineering mathematics">Engineering mathematics</a></li> <li><a href="/wiki/Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Mathematical biology</a></li> <li><a href="/wiki/Mathematical_chemistry" title="Mathematical chemistry">Mathematical chemistry</a></li> <li><a href="/wiki/Mathematical_economics" title="Mathematical economics">Mathematical economics</a></li> <li><a href="/wiki/Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li> <li><a href="/wiki/Mathematical_physics" title="Mathematical physics">Mathematical physics</a></li> <li><a href="/wiki/Mathematical_psychology" title="Mathematical psychology">Mathematical psychology</a></li> <li><a href="/wiki/Mathematical_sociology" title="Mathematical sociology">Mathematical sociology</a></li> <li><a href="/wiki/Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li> <li><a href="/wiki/Probability_theory" title="Probability theory">Probability</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Systems_science" title="Systems science">Systems science</a> <ul><li><a href="/wiki/Control_theory" title="Control theory">Control theory</a></li> <li><a href="/wiki/Game_theory" title="Game theory">Game theory</a></li> <li><a href="/wiki/Operations_research" title="Operations research">Operations research</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computational_mathematics" title="Computational mathematics">Computational</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Computer_science" title="Computer science">Computer science</a></li> <li><a href="/wiki/Theory_of_computation" title="Theory of computation">Theory of computation</a></li> <li><a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">Computational complexity theory</a></li> <li><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a></li> <li><a href="/wiki/Mathematical_optimization" title="Mathematical optimization">Optimization</a></li> <li><a href="/wiki/Computer_algebra" title="Computer algebra">Computer algebra</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Related topics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematicians" class="mw-redirect" title="Mathematicians">Mathematicians</a> <ul><li><a href="/wiki/List_of_mathematicians" class="mw-redirect" title="List of mathematicians">lists</a></li></ul></li> <li><a href="/wiki/Informal_mathematics" title="Informal mathematics">Informal mathematics</a></li> <li><a href="/wiki/List_of_films_about_mathematicians" title="List of films about mathematicians">Films about mathematicians</a></li> <li><a href="/wiki/Recreational_mathematics" title="Recreational mathematics">Recreational mathematics</a></li> <li><a href="/wiki/Mathematics_and_art" title="Mathematics and art">Mathematics and art</a></li> <li><a href="/wiki/Mathematics_education" title="Mathematics education">Mathematics education</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><b><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" 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Mathematics">WikiProject</a></b></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Industrial_and_applied_mathematics" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Industrial_and_applied_mathematics" title="Template:Industrial and applied mathematics"><abbr title="View this template">v</abbr></a></li><li 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class="mw-redirect" title="Algorithm design">design</a></li> <li><a href="/wiki/Analysis_of_algorithms" title="Analysis of algorithms">analysis</a></li></ul></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata theory</a></li> <li><a href="/wiki/Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></li> <li><a href="/wiki/Coding_theory" title="Coding theory">Coding theory</a></li> <li><a href="/wiki/Computational_geometry" title="Computational geometry">Computational geometry</a></li> <li><a href="/wiki/Constraint_satisfaction_problem" title="Constraint satisfaction problem">Constraint satisfaction</a> <ul><li><a href="/wiki/Constraint_programming" title="Constraint programming">Constraint programming</a></li></ul></li> <li><a href="/wiki/Logic_in_computer_science" title="Logic in computer science">Computational logic</a></li> <li><a href="/wiki/Cryptography" title="Cryptography">Cryptography</a></li> <li><a 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href="/wiki/List_of_interactive_geometry_software" title="List of interactive geometry software">Interactive geometry software</a></li> <li><a href="/wiki/List_of_optimization_software" title="List of optimization software">Optimization software</a></li> <li><a href="/wiki/List_of_statistical_software" title="List of statistical software">Statistical software</a></li> <li><a href="/wiki/List_of_numerical-analysis_software" title="List of numerical-analysis software">Numerical-analysis software</a></li> <li><a href="/wiki/List_of_numerical-analysis_software" title="List of numerical-analysis software">Numerical libraries</a></li> <li><a href="/wiki/Solver" title="Solver">Solvers</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Discrete_mathematics" title="Discrete mathematics">Discrete</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Computer_algebra" title="Computer algebra">Computer algebra</a></li> <li><a href="/wiki/Computational_number_theory" title="Computational number theory">Computational number theory</a></li> <li><a href="/wiki/Combinatorics" title="Combinatorics">Combinatorics</a></li> <li><a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a></li> <li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a class="mw-selflink selflink">Analysis</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Approximation_theory" title="Approximation theory">Approximation theory</a></li> <li><a href="/wiki/Clifford_analysis" title="Clifford analysis">Clifford analysis</a> <ul><li><a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford 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