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<div id="siteSub" class="noprint">Материал из Википедии — свободной энциклопедии</div> <div id="contentSub"><div id="mw-content-subtitle"><div id="mw-fr-revision-messages"><div class="cdx-message mw-fr-message-box cdx-message--block cdx-message--notice mw-fr-basic mw-fr-draft-not-synced plainlinks noprint"><span class="cdx-message__icon"></span><div class="cdx-message__content">Текущая версия страницы пока <a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%9F%D1%80%D0%BE%D0%B2%D0%B5%D1%80%D0%BA%D0%B0_%D1%81%D1%82%D0%B0%D1%82%D0%B5%D0%B9/%D0%9F%D0%BE%D1%8F%D1%81%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D0%B4%D0%BB%D1%8F_%D1%87%D0%B8%D1%82%D0%B0%D1%82%D0%B5%D0%BB%D0%B5%D0%B9" title="Википедия:Проверка статей/Пояснение для читателей">не проверялась</a> опытными участниками и может значительно отличаться от <a class="external text" href="https://ru.wikipedia.org/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&stable=1">версии, проверенной 20 июня 2022 года</a>; проверки требуют <a class="external text" href="https://ru.wikipedia.org/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&oldid=123432044&diff=cur&diffonly=0">6 правок</a>.</div></div></div></div></div> <div id="contentSub2"></div> <div id="jump-to-nav"></div> <a class="mw-jump-link" href="#mw-head">Перейти к навигации</a> <a class="mw-jump-link" href="#searchInput">Перейти к поиску</a> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ru" dir="ltr"><div class="hatnote navigation-not-searchable dabhide">Запрос «Однозначная комплексная аналитическая функция» перенаправляется сюда; см. также <a href="/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D0%B8%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Аналитическая функция">другие значения</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/%D0%A4%D0%B0%D0%B9%D0%BB:Conformal_map.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Conformal_map.svg/220px-Conformal_map.svg.png" decoding="async" width="220" height="385" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Conformal_map.svg/330px-Conformal_map.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Conformal_map.svg/440px-Conformal_map.svg.png 2x" data-file-width="535" data-file-height="937" /></a><figcaption>Голоморфная функция осуществляет <a href="/wiki/%D0%9A%D0%BE%D0%BD%D1%84%D0%BE%D1%80%D0%BC%D0%BD%D0%BE%D0%B5_%D0%BE%D1%82%D0%BE%D0%B1%D1%80%D0%B0%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5" title="Конформное отображение">конформное отображение</a>, преобразуя <i>ортогональную</i> сетку в такую же <i>ортогональную</i> (там, где комплексная производная не обращается в нуль).</figcaption></figure> <p><b>Голоморфная функция</b> или <b>однозначная комплексная аналитическая функция</b> (от греч. ὅλος — «весь, целый» и μορφή — «форма»), иногда называемая <b>регулярной функцией</b> — <a href="/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_%D0%BA%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D0%BE%D0%B3%D0%BE_%D0%BF%D0%B5%D1%80%D0%B5%D0%BC%D0%B5%D0%BD%D0%BD%D0%BE%D0%B3%D0%BE" class="mw-redirect" title="Функция комплексного переменного">функция комплексного переменного</a>, определённая на <a href="/wiki/%D0%9E%D1%82%D0%BA%D1%80%D1%8B%D1%82%D0%BE%D0%B5_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="Открытое множество">открытом</a> подмножестве <a href="/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D0%B0%D1%8F_%D0%BF%D0%BB%D0%BE%D1%81%D0%BA%D0%BE%D1%81%D1%82%D1%8C" title="Комплексная плоскость">комплексной плоскости</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 \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span> и <a href="/wiki/%D0%9C%D0%BE%D0%BD%D0%BE%D0%B3%D0%B5%D0%BD%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Моногенная функция">комплексно дифференцируемая</a> в каждой точке. </p><p>В отличие от вещественного случая, это условие означает, что функция бесконечно дифференцируема и может быть представлена сходящимся к ней <a href="/wiki/%D0%A0%D1%8F%D0%B4_%D0%A2%D0%B5%D0%B9%D0%BB%D0%BE%D1%80%D0%B0" title="Ряд Тейлора">рядом Тейлора</a>. </p><p>Голоморфные функции также называют иногда <i>аналитическими</i>, хотя второе понятие гораздо более широкое, так как аналитическая функция <a href="/wiki/%D0%9C%D0%BD%D0%BE%D0%B3%D0%BE%D0%B7%D0%BD%D0%B0%D1%87%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Многозначная аналитическая функция">может быть многозначной</a>, а также <a href="/wiki/%D0%92%D0%B5%D1%89%D0%B5%D1%81%D1%82%D0%B2%D0%B5%D0%BD%D0%BD%D0%BE-%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" class="mw-redirect" title="Вещественно-аналитическая функция">может рассматриваться и для вещественных чисел</a>. </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="ru" dir="ltr"><h2 id="mw-toc-heading">Содержание</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Определение"><span class="tocnumber">1</span> <span class="toctext">Определение</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Связанные_определения"><span class="tocnumber">2</span> <span class="toctext">Связанные определения</span></a></li> <li class="toclevel-1 tocsection-3"><a href="#Свойства"><span class="tocnumber">3</span> <span class="toctext">Свойства</span></a></li> <li class="toclevel-1 tocsection-4"><a href="#Примеры"><span class="tocnumber">4</span> <span class="toctext">Примеры</span></a></li> <li class="toclevel-1 tocsection-5"><a href="#История"><span class="tocnumber">5</span> <span class="toctext">История</span></a></li> <li class="toclevel-1 tocsection-6"><a href="#Вариации_и_обобщения"><span class="tocnumber">6</span> <span class="toctext">Вариации и обобщения</span></a> <ul> <li class="toclevel-2 tocsection-7"><a href="#Многомерный_случай"><span class="tocnumber">6.1</span> <span class="toctext">Многомерный случай</span></a> <ul> <li class="toclevel-3 tocsection-8"><a href="#С-линейность"><span class="tocnumber">6.1.1</span> <span class="toctext">С-линейность</span></a></li> <li class="toclevel-3 tocsection-9"><a href="#С-дифференцируемость"><span class="tocnumber">6.1.2</span> <span class="toctext">С-дифференцируемость</span></a></li> <li class="toclevel-3 tocsection-10"><a href="#Голоморфность"><span class="tocnumber">6.1.3</span> <span class="toctext">Голоморфность</span></a></li> <li class="toclevel-3 tocsection-11"><a href="#Квазианалитичность"><span class="tocnumber">6.1.4</span> <span class="toctext">Квазианалитичность</span></a></li> </ul> </li> </ul> </li> <li class="toclevel-1 tocsection-12"><a href="#Примечания"><span class="tocnumber">7</span> <span class="toctext">Примечания</span></a></li> <li class="toclevel-1 tocsection-13"><a href="#Литература"><span class="tocnumber">8</span> <span class="toctext">Литература</span></a></li> <li class="toclevel-1 tocsection-14"><a href="#Ссылки"><span class="tocnumber">9</span> <span class="toctext">Ссылки</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Определение"><span id=".D0.9E.D0.BF.D1.80.D0.B5.D0.B4.D0.B5.D0.BB.D0.B5.D0.BD.D0.B8.D0.B5"></span>Определение</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=1" title="Редактировать раздел «Определение»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=1" title="Редактировать код раздела «Определение»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Пусть <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> — <a href="/wiki/%D0%9E%D1%82%D0%BA%D1%80%D1%8B%D1%82%D0%BE%D0%B5_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="Открытое множество">открытое</a> подмножество в <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span> и <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:U\to \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>U</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:U\to \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0a9baac865b214fba8fce6842fb374f7e740361" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.29ex; height:2.509ex;" alt="{\displaystyle f:U\to \mathbb {C} }"></span> — комплекснозначная функция на <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span>. Функцию называют <b>голоморфной</b> на множестве <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span>, если выполняется одно из следующих равносильных условий: </p> <ol><li>У функции существует комплексная производная в каждой точке множества <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span>, то есть предел <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(z)=\lim _{h\to 0}{\frac {f(z+h)-f(z)}{h}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'(z)=\lim _{h\to 0}{\frac {f(z+h)-f(z)}{h}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba0de657dac2b3a29ff6c981ae36466ee732ed4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.655ex; height:5.843ex;" alt="{\displaystyle f'(z)=\lim _{h\to 0}{\frac {f(z+h)-f(z)}{h}}.}"></span></dd></dl></li> <li>Функция является комплексно-дифференцируемой в каждой точке <span class="mwe-math-element"><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\in U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>∈</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\in U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60ddfa0dda66703a57db65136203350c952e1feb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.711ex; height:2.176ex;" alt="{\displaystyle z\in U}"></span>, то есть существует число <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L\in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L\in \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61fdf721ef1f9cdba745e9e7c07a2b78dbb73c30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.102ex; height:2.176ex;" alt="{\displaystyle L\in \mathbb {C} }"></span> такое, что в <a href="/wiki/%D0%9E%D0%BA%D1%80%D0%B5%D1%81%D1%82%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Окрестность">окрестности</a> точки <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z+h)=f(z)+Lh+o(h)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>L</mi> <mi>h</mi> <mo>+</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z+h)=f(z)+Lh+o(h)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28d17ce60b4315f63efdeee71cee9780e2e09e97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.508ex; height:2.843ex;" alt="{\displaystyle f(z+h)=f(z)+Lh+o(h)}"></span></dd></dl></li> <li>Функция является вещественно-дифференцируемой и в каждой точке <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=x+iy\in U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> <mo>∈</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=x+iy\in U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7f18c128415d5c03fc858deea2b56a7c1d64817" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.938ex; height:2.509ex;" alt="{\displaystyle z=x+iy\in U}"></span> выполняются <a href="/wiki/%D0%A3%D1%81%D0%BB%D0%BE%D0%B2%D0%B8%D1%8F_%D0%9A%D0%BE%D1%88%D0%B8_%E2%80%94_%D0%A0%D0%B8%D0%BC%D0%B0%D0%BD%D0%B0" title="Условия Коши — Римана">условия Коши — Римана</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 {\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13846b6da38e7b6446fe509e1784a93f4b6472b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.892ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}}}"></span> и <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/222c51bed74d29713303156cf47a029e65e5060d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.521ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}}.}"></span> Здесь <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10bbd250e62d7bc323b96b0719e29be7f227beb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.227ex; height:2.843ex;" alt="{\displaystyle u(z)}"></span> и <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7079b24d1daefdf8f5e1b35292032a3bf64968c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.025ex; height:2.843ex;" alt="{\displaystyle v(z)}"></span> — вещественная и мнимая части рассматриваемой функции.</li> <li>Функция является вещественно-дифференцируемой и в каждой точке <span class="mwe-math-element"><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\in U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>∈</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\in U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60ddfa0dda66703a57db65136203350c952e1feb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.711ex; height:2.176ex;" alt="{\displaystyle z\in U}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial f}{\partial {\bar {z}}}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial f}{\partial {\bar {z}}}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4afe340d0d86658b6564e5d1e7a441f31dadc612" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.711ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial f}{\partial {\bar {z}}}}=0}"></span>, где <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial }{\partial {\bar {z}}}}={1 \over 2}\left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial {\bar {z}}}}={1 \over 2}\left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5471013f2ea4c3d270ae2829c7cfb4eb4eede501" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.826ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial }{\partial {\bar {z}}}}={1 \over 2}\left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right),}"></span>.</li> <li><a href="/wiki/%D0%A0%D1%8F%D0%B4_%D0%A2%D0%B5%D0%B9%D0%BB%D0%BE%D1%80%D0%B0" title="Ряд Тейлора">Ряд Тейлора</a> функции в каждой точке <span class="mwe-math-element"><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\in U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>∈</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\in U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60ddfa0dda66703a57db65136203350c952e1feb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.711ex; height:2.176ex;" alt="{\displaystyle z\in U}"></span> имеет ненулевой радиус сходимости, и его сумма равна <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8dd568d570b390c337c0a911f0a1c5c214e8240" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.176ex; height:2.843ex;" alt="{\displaystyle f(z)}"></span> в некоторой окрестности <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>.</li> <li>Функция непрерывна и <a href="/wiki/%D0%98%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB" title="Интеграл">интеграл</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 \int \limits _{\Gamma }\,f(z)\,dz=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </munder> <mspace width="thinmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \limits _{\Gamma }\,f(z)\,dz=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6dea3dcfab655a38b77a7a07bea72db67d0d69b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:14.096ex; height:7.343ex;" alt="{\displaystyle \int \limits _{\Gamma }\,f(z)\,dz=0}"></span> для любой замкнутой кривой <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \subset U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>⊂</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma \subset U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7864ab0d130751bfc90869945b62a15d89068ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.334ex; height:2.176ex;" alt="{\displaystyle \Gamma \subset U}"></span>.</li></ol> <p>Тот факт, что все эти определения эквивалентны является нетривиальным и весьма замечательным результатом комплексного анализа. </p><p>Функцию <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> называют <b>голоморфной в точке</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{0}\in U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{0}\in U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fda199ea9aa9cfb9f0e3d11e65206620e7041a15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.759ex; height:2.509ex;" alt="{\displaystyle z_{0}\in U}"></span>, если она голоморфна в некоторой окрестности <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e72d1d86e86355892b39b8eb32b964834e113bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{0}}"></span>. </p><p>Функцию <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> называют <b>голоморфной</b>, если она комплексно дифференцируема в области определения. </p> <div class="mw-heading mw-heading2"><h2 id="Связанные_определения"><span id=".D0.A1.D0.B2.D1.8F.D0.B7.D0.B0.D0.BD.D0.BD.D1.8B.D0.B5_.D0.BE.D0.BF.D1.80.D0.B5.D0.B4.D0.B5.D0.BB.D0.B5.D0.BD.D0.B8.D1.8F"></span>Связанные определения</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=2" title="Редактировать раздел «Связанные определения»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=2" title="Редактировать код раздела «Связанные определения»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><i><a href="/wiki/%D0%A6%D0%B5%D0%BB%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Целая функция">Целая функция</a></i> — функция, голоморфная на всей комплексной плоскости.</li> <li><i><a href="/wiki/%D0%9C%D0%B5%D1%80%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Мероморфная функция">Мероморфная функция</a></i> — функция, голоморфная в области <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega \setminus \{{z_{i}},\;i\in I\subset {\mathbb {N}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mi>i</mi> <mo>∈</mo> <mi>I</mi> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega \setminus \{{z_{i}},\;i\in I\subset {\mathbb {N}}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e38de5de5f2b1a64a92173d5f7faecdfb1ee0aee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.349ex; height:2.843ex;" alt="{\displaystyle \Omega \setminus \{{z_{i}},\;i\in I\subset {\mathbb {N}}\}}"></span> и имеющая во всех своих особых точках <span class="mwe-math-element"><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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c6e920bac39ad09fff4efef16254595091a1025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.881ex; height:2.009ex;" alt="{\displaystyle z_{i}}"></span> <a href="/wiki/%D0%9F%D0%BE%D0%BB%D1%8E%D1%81_(%D0%BA%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D1%8B%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7)" title="Полюс (комплексный анализ)">полюс</a>.</li> <li>Функция <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> называется голоморфной на <a href="/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%B0%D0%BA%D1%82%D0%BD%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Компактное пространство">компакте</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 K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span>, если существует <a href="/wiki/%D0%9E%D1%82%D0%BA%D1%80%D1%8B%D1%82%D0%BE%D0%B5_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="Открытое множество">открытое множество</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 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/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span>, содержащее <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span>, такое что <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> голоморфна в <span class="mwe-math-element"><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/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Свойства"><span id=".D0.A1.D0.B2.D0.BE.D0.B9.D1.81.D1.82.D0.B2.D0.B0"></span>Свойства</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=3" title="Редактировать раздел «Свойства»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=3" title="Редактировать код раздела «Свойства»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Комплексная функция <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u+iv=f(x+iy)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>+</mo> <mi>i</mi> <mi>v</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u+iv=f(x+iy)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/054a1693b087571547cf4d9c834581c21acf6840" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.415ex; height:2.843ex;" alt="{\displaystyle u+iv=f(x+iy)}"></span> является голоморфной тогда и только тогда, когда выполняются <a href="/wiki/%D0%A3%D1%81%D0%BB%D0%BE%D0%B2%D0%B8%D1%8F_%D0%9A%D0%BE%D1%88%D0%B8_%E2%80%94_%D0%A0%D0%B8%D0%BC%D0%B0%D0%BD%D0%B0" title="Условия Коши — Римана">условия Коши — Римана</a> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}};\quad {\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>;</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}};\quad {\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82ff1a7cdd946503fab949a1d0ab80b1595819e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.123ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}};\quad {\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}}}"></span></dd></dl></li></ul> <dl><dd>и частные производные <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial u}{\partial x}},\;{\frac {\partial u}{\partial y}},\;{\frac {\partial v}{\partial x}},\;{\frac {\partial v}{\partial y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial u}{\partial x}},\;{\frac {\partial u}{\partial y}},\;{\frac {\partial v}{\partial x}},\;{\frac {\partial v}{\partial y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08b6bacb819d0b5f304bf7c0005354bd77ce011c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.799ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial u}{\partial x}},\;{\frac {\partial u}{\partial y}},\;{\frac {\partial v}{\partial x}},\;{\frac {\partial v}{\partial y}}}"></span> непрерывны.</dd></dl> <ul><li>Сумма и произведение голоморфных функций — голоморфная функция, что следует из линейности дифференцирования и выполнения правила Лейбница. Частное голоморфных функций также голоморфно во всех точках, где знаменатель не обращается в 0.</li> <li><a href="/wiki/%D0%9F%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%BE%D0%B4%D0%BD%D0%B0%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Производная (математика)">Производная</a> голоморфной функции опять является голоморфной, поэтому голоморфные функции являются бесконечно дифференцируемыми в своей области определения.</li> <li>Голоморфные функции могут быть представлены в виде сходящегося в некоторой окрестности каждой точки <a href="/wiki/%D0%A0%D1%8F%D0%B4_%D0%A2%D0%B5%D0%B9%D0%BB%D0%BE%D1%80%D0%B0" title="Ряд Тейлора">ряда Тейлора</a>.</li> <li>Из любой голоморфной функции можно выделить её вещественную и мнимую часть, каждая из которых будет решением уравнения Лапласа в <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span>. То есть если <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x+iy)=u(x,\;y)+iv(x,\;y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>v</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x+iy)=u(x,\;y)+iv(x,\;y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39b7da6fb308b969abe5c6d4432eb491ed0850ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.362ex; height:2.843ex;" alt="{\displaystyle f(x+iy)=u(x,\;y)+iv(x,\;y)}"></span> — голоморфная функция, то <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"></span> и <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> — <a href="/wiki/%D0%93%D0%B0%D1%80%D0%BC%D0%BE%D0%BD%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Гармоническая функция">гармонические</a> функции.</li> <li>Если <a href="/wiki/%D0%90%D0%B1%D1%81%D0%BE%D0%BB%D1%8E%D1%82%D0%BD%D0%B0%D1%8F_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D0%B0" title="Абсолютная величина">абсолютная величина</a> голоморфной функции достигает локального максимума во внутренней точке своей области определения, то функция постоянна (предполагается, что область определения связна). Отсюда следует, что максимум (и минимум, если он не равен нулю) абсолютной величины голоморфной функции могут достигаться лишь на границе области.</li> <li>В области, где первая производная голоморфной функции не обращается в 0, а функция <a href="/wiki/%D0%9E%D0%B4%D0%BD%D0%BE%D0%BB%D0%B8%D1%81%D1%82%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Однолистная функция">однолистна</a>, она осуществляет <a href="/wiki/%D0%9A%D0%BE%D0%BD%D1%84%D0%BE%D1%80%D0%BC%D0%BD%D0%BE%D0%B5_%D0%BE%D1%82%D0%BE%D0%B1%D1%80%D0%B0%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5" title="Конформное отображение">конформное отображение</a>.</li> <li><a href="/wiki/%D0%98%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D1%84%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0_%D0%9A%D0%BE%D1%88%D0%B8" title="Интегральная формула Коши">Интегральная формула Коши</a> связывает значение функции во внутренней точке области с её значениями на границе этой области.</li> <li>С алгебраической точки зрения, множество голоморфных на открытом множестве функций — это коммутативное <a href="/wiki/%D0%9A%D0%BE%D0%BB%D1%8C%D1%86%D0%BE_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" class="mw-redirect" title="Кольцо (алгебра)">кольцо</a> и комплексное <a href="/wiki/%D0%9B%D0%B8%D0%BD%D0%B5%D0%B9%D0%BD%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" class="mw-redirect" title="Линейное пространство">линейное пространство</a>. Это локально выпуклое <a href="/wiki/%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%B2%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%BD%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Топологическое векторное пространство">топологическое векторное пространство</a> с <a href="/wiki/%D0%9F%D0%BE%D0%BB%D1%83%D0%BD%D0%BE%D1%80%D0%BC%D0%B0" title="Полунорма">полунормой</a>, равной <a href="/wiki/%D0%A1%D1%83%D0%BF%D1%80%D0%B5%D0%BC%D1%83%D0%BC" class="mw-redirect" title="Супремум">супремуму</a> на компактных подмножествах.</li> <li>Согласно теореме <a href="/wiki/%D0%92%D0%B5%D0%B9%D0%B5%D1%80%D1%88%D1%82%D1%80%D0%B0%D1%81%D1%81" class="mw-redirect" title="Вейерштрасс">Вейерштрасса</a>, если ряд голоморфных функций в области <span class="mwe-math-element"><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/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> равномерно сходится на любом <a href="/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%B0%D0%BA%D1%82" class="mw-redirect" title="Компакт">компакте</a> в <span class="mwe-math-element"><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> <mo>,</mo> </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/ab9cb9bd7fcb5f2a543b128c1b876019b158c0fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.571ex; height:2.509ex;" alt="{\displaystyle D,}"></span> то его сумма также голоморфна, причём её производная является пределом производных частичных сумм ряда<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>.</li> <li>Если <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6649b7c907c78aeaff76ce9e5be30e982e34d0f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.903ex; height:3.009ex;" alt="{\displaystyle f'(z)}"></span> в области <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> не обращается в ноль, то <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8053d85b2cab33b923c697dfcdb95bb2f48894f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.551ex; height:3.176ex;" alt="{\displaystyle f^{-1}(z)}"></span> голоморфна в <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>.</li></ul> <p>Некоторые свойства голоморфных функций близки к свойствам <a href="/wiki/%D0%9C%D0%BD%D0%BE%D0%B3%D0%BE%D1%87%D0%BB%D0%B5%D0%BD" title="Многочлен">многочленов</a>, что, впрочем, и неудивительно — разложимость голоморфных функций в ряды Тейлора свидетельствует о том, что функции — в некотором роде предельные варианты многочленов. Допустим, согласно <a href="/wiki/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D0%B0%D1%8F_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D1%8B" title="Основная теорема алгебры">основной теореме алгебры</a> любой многочлен может иметь нулей числом не более его степени. Для голомофных функций справедливо аналогичное утверждение, вытекающее из <a href="/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D0%B8%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D0%B4%D0%BE%D0%BB%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5#Единственность" title="Аналитическое продолжение">теоремы единственности</a> в альтернативной форме: </p> <ul><li>Если множество нулей голоморфной в односвязной области функции имеет в этой области <a href="/wiki/%D0%9F%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D1%82%D0%BE%D1%87%D0%BA%D0%B0" title="Предельная точка">предельную точку</a>, то функция тождественно равна нулю.</li> <li>Для функции от нескольких действительных переменных дифференцируемости по каждой из переменных недостаточно для дифференцируемости функции. Для функции от нескольких комплексных переменных голоморфности по каждой из переменных достаточно для голоморфности функции (<a href="/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%A5%D0%B0%D1%80%D1%82%D0%BE%D0%B3%D1%81%D0%B0" title="Теорема Хартогса">Теорема Хартогса</a>).</li></ul> <div class="mw-heading mw-heading2"><h2 id="Примеры"><span id=".D0.9F.D1.80.D0.B8.D0.BC.D0.B5.D1.80.D1.8B"></span>Примеры</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=4" title="Редактировать раздел «Примеры»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=4" title="Редактировать код раздела «Примеры»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Все многочлены от z являются голоморфными функциями на всей плоскости <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>. </p><p>Далее, голоморфными, хотя и <i>не на всей</i> комплексной плоскости, являются <a href="/wiki/%D0%A0%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B%D0%B5_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%B8" class="mw-redirect" title="Рациональные функции">рациональные функции</a>, <a href="/wiki/%D0%9F%D0%BE%D0%BA%D0%B0%D0%B7%D0%B0%D1%82%D0%B5%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Показательная функция">показательная функция</a>, <a href="/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Логарифм">логарифм</a>, <a href="/wiki/%D0%A2%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B5_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%B8" title="Тригонометрические функции">тригонометрические функции</a>, <a href="/wiki/%D0%9E%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D1%8B%D0%B5_%D1%82%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B5_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%B8" title="Обратные тригонометрические функции">обратные тригонометрические функции</a> и многие другие классы функций, а также суммы, разности, произведения, частные голоморфными функций. </p><p>Функции <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)=|z|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z)=|z|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7024d62077bbfdf683793896ba3703f476225839" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.656ex; height:2.843ex;" alt="{\displaystyle f(z)=|z|}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)={\overline {z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z)={\overline {z}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75311667f3ed9db08d4f87510c37e372a2c87d3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.482ex; height:2.843ex;" alt="{\displaystyle f(z)={\overline {z}}}"></span> неголоморфны на <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>, поскольку они не имеют комплексной производной ни в одной точке. При этом сужение <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)={\overline {z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z)={\overline {z}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75311667f3ed9db08d4f87510c37e372a2c87d3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.482ex; height:2.843ex;" alt="{\displaystyle f(z)={\overline {z}}}"></span> на вещественную ось является аналитической функцией <i>вещественного</i> переменного (так как оно полностью совпадает с сужением функции <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)=z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z)=z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a47d2cd0ea040d1b1af7532f5321ed581026e61f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.363ex; height:2.843ex;" alt="{\displaystyle f(z)=z}"></span>). </p> <div class="mw-heading mw-heading2"><h2 id="История"><span id=".D0.98.D1.81.D1.82.D0.BE.D1.80.D0.B8.D1.8F"></span>История</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=5" title="Редактировать раздел «История»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=5" title="Редактировать код раздела «История»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Термин «голоморфная функция» был введён двумя учениками <a href="/wiki/%D0%9A%D0%BE%D1%88%D0%B8,_%D0%9E%D0%B3%D1%8E%D1%81%D1%82%D0%B5%D0%BD_%D0%9B%D1%83%D0%B8" title="Коши, Огюстен Луи">Коши</a>, Брио (<a href="/wiki/1817" class="mw-redirect" title="1817">1817</a>—<a href="/wiki/1882" class="mw-redirect" title="1882">1882</a>) и Буке (<a href="/wiki/1819" class="mw-redirect" title="1819">1819</a>—<a href="/wiki/1895" class="mw-redirect" title="1895">1895</a>). Термин «аналитическая функция» употребляют обычно для более общего случая, когда функции многозначны и их удобно рассматривать как функции, заданные на подходящей <a href="/wiki/%D0%A0%D0%B8%D0%BC%D0%B0%D0%BD%D0%BE%D0%B2%D0%B0_%D0%BF%D0%BE%D0%B2%D0%B5%D1%80%D1%85%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Риманова поверхность">римановой поверхности</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Вариации_и_обобщения"><span id=".D0.92.D0.B0.D1.80.D0.B8.D0.B0.D1.86.D0.B8.D0.B8_.D0.B8_.D0.BE.D0.B1.D0.BE.D0.B1.D1.89.D0.B5.D0.BD.D0.B8.D1.8F"></span>Вариации и обобщения</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=6" title="Редактировать раздел «Вариации и обобщения»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=6" title="Редактировать код раздела «Вариации и обобщения»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Многомерный_случай"><span id=".D0.9C.D0.BD.D0.BE.D0.B3.D0.BE.D0.BC.D0.B5.D1.80.D0.BD.D1.8B.D0.B9_.D1.81.D0.BB.D1.83.D1.87.D0.B0.D0.B9"></span>Многомерный случай</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=7" title="Редактировать раздел «Многомерный случай»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=7" title="Редактировать код раздела «Многомерный случай»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Существует также определение голоморфности функций многих комплексных переменных </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \mathbb {C} ^{n}\to \mathbb {C} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon \mathbb {C} ^{n}\to \mathbb {C} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12ab1792d613cd2931c9ea25faa1e33ed2c8822f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.148ex; height:2.676ex;" alt="{\displaystyle f\colon \mathbb {C} ^{n}\to \mathbb {C} .}"></span></dd></dl> <p>Для определения используются понятия <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>-дифференцируемости и <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>-линейности таких функций </p> <div class="mw-heading mw-heading4"><h4 id="С-линейность"><span id=".D0.A1-.D0.BB.D0.B8.D0.BD.D0.B5.D0.B9.D0.BD.D0.BE.D1.81.D1.82.D1.8C"></span>С-линейность</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=8" title="Редактировать раздел «С-линейность»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=8" title="Редактировать код раздела «С-линейность»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Функция <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> называется <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>-линейной если удовлетворяются условия: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z'+z'')=f(z')+f(z''),\quad z',\;z''\in \mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mi>z</mi> <mo>′</mo> </msup> <mo>+</mo> <msup> <mi>z</mi> <mo>″</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mi>z</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mi>z</mi> <mo>″</mo> </msup> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <msup> <mi>z</mi> <mo>′</mo> </msup> <mo>,</mo> <mspace width="thickmathspace" /> <msup> <mi>z</mi> <mo>″</mo> </msup> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z'+z'')=f(z')+f(z''),\quad z',\;z''\in \mathbb {C} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b89a7a2f80ea58b36e0085c34b6edec46679c338" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.823ex; height:3.009ex;" alt="{\displaystyle f(z'+z'')=f(z')+f(z''),\quad z',\;z''\in \mathbb {C} ^{n}}"></span>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\lambda z)=\lambda f(z),\quad z\in \mathbb {C} ^{n},\quad \lambda \in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>λ</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>z</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mi>λ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\lambda z)=\lambda f(z),\quad z\in \mathbb {C} ^{n},\quad \lambda \in \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f844b2155a178df5aef7566941726cbb15f99361" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.573ex; height:2.843ex;" alt="{\displaystyle f(\lambda z)=\lambda f(z),\quad z\in \mathbb {C} ^{n},\quad \lambda \in \mathbb {C} }"></span></li></ul> <p>(для <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>-линейных функций <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda \in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b87bc1622689bc998795834cd65eecdb4955a785" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.874ex; height:2.176ex;" alt="{\displaystyle \lambda \in \mathbb {R} }"></span>). </p> <ul><li>Для любой <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>-линейной функции <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> существуют <a href="/wiki/%D0%9F%D0%BE%D1%81%D0%BB%D0%B5%D0%B4%D0%BE%D0%B2%D0%B0%D1%82%D0%B5%D0%BB%D1%8C%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Последовательность">последовательности</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 \{a_{n}\},\;\{b_{n}\}\subset \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mspace width="thickmathspace" /> <mo fence="false" stretchy="false">{</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{a_{n}\},\;\{b_{n}\}\subset \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c023ea129b4d2abe15b55f0587e2b99106f3fbe5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.77ex; height:2.843ex;" alt="{\displaystyle \{a_{n}\},\;\{b_{n}\}\subset \mathbb {C} }"></span>, такие, что <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=\sum _{i=1}^{n}(a_{i}z_{i}+b_{i}{\bar {z}}_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=\sum _{i=1}^{n}(a_{i}z_{i}+b_{i}{\bar {z}}_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7476b6cdb08db78c3a12bec92068d4e491941b84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.185ex; height:6.843ex;" alt="{\displaystyle f=\sum _{i=1}^{n}(a_{i}z_{i}+b_{i}{\bar {z}}_{i})}"></span>.</li> <li>Для любой <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>-линейной функции <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> существуют <a href="/wiki/%D0%9F%D0%BE%D1%81%D0%BB%D0%B5%D0%B4%D0%BE%D0%B2%D0%B0%D1%82%D0%B5%D0%BB%D1%8C%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Последовательность">последовательность</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 \{a_{n}\}\subset \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{a_{n}\}\subset \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df98a2414da08fddb5cca9c7506806cc92b0ca25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.55ex; height:2.843ex;" alt="{\displaystyle \{a_{n}\}\subset \mathbb {C} }"></span>, такая, что <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=\sum _{i=1}^{n}a_{i}z_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=\sum _{i=1}^{n}a_{i}z_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/879bd927d455d5552159377029b36f0a961f8418" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.029ex; height:6.843ex;" alt="{\displaystyle f=\sum _{i=1}^{n}a_{i}z_{i}}"></span>.</li></ul> <div class="mw-heading mw-heading4"><h4 id="С-дифференцируемость"><span id=".D0.A1-.D0.B4.D0.B8.D1.84.D1.84.D0.B5.D1.80.D0.B5.D0.BD.D1.86.D0.B8.D1.80.D1.83.D0.B5.D0.BC.D0.BE.D1.81.D1.82.D1.8C"></span>С-дифференцируемость</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=9" title="Редактировать раздел «С-дифференцируемость»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=9" title="Редактировать код раздела «С-дифференцируемость»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Функция <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> называется <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>-дифференцируемой в точке <span class="mwe-math-element"><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\in \mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\in \mathbb {C} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/977ed3d7cb322555a0d9266b40f2cf50ae26d7f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.825ex; height:2.343ex;" alt="{\displaystyle z\in \mathbb {C} ^{n}}"></span> если существуют функции <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span> и <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c1031f61947aa3d1cf3a70ec3e4904df2c3675d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle o}"></span> такие, что в <a href="/wiki/%D0%9E%D0%BA%D1%80%D0%B5%D1%81%D1%82%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Окрестность">окрестности</a> точки <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z+h)=f(z)+l(h)+o(h),\quad \lim _{h\to 0}{\frac {o(h)}{h}}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>l</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>o</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z+h)=f(z)+l(h)+o(h),\quad \lim _{h\to 0}{\frac {o(h)}{h}}=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd3112de438bed8798159f94bc810495b9a1ae4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:46.603ex; height:5.843ex;" alt="{\displaystyle f(z+h)=f(z)+l(h)+o(h),\quad \lim _{h\to 0}{\frac {o(h)}{h}}=0,}"></span></dd></dl> <p>где <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span> — <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>-линейная (для <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>-дифференцируемости — <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>-линейная) функция. </p> <div class="mw-heading mw-heading4"><h4 id="Голоморфность"><span id=".D0.93.D0.BE.D0.BB.D0.BE.D0.BC.D0.BE.D1.80.D1.84.D0.BD.D0.BE.D1.81.D1.82.D1.8C"></span>Голоморфность</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=10" title="Редактировать раздел «Голоморфность»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=10" title="Редактировать код раздела «Голоморфность»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Функция <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> называется голоморфной в области <span class="mwe-math-element"><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> <mo>,</mo> </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/ab9cb9bd7fcb5f2a543b128c1b876019b158c0fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.571ex; height:2.509ex;" alt="{\displaystyle D,}"></span> если она <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>-дифференцируема в <a href="/wiki/%D0%9E%D0%BA%D1%80%D0%B5%D1%81%D1%82%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Окрестность">окрестности</a> каждой точки этой области. </p> <div class="mw-heading mw-heading4"><h4 id="Квазианалитичность"><span id=".D0.9A.D0.B2.D0.B0.D0.B7.D0.B8.D0.B0.D0.BD.D0.B0.D0.BB.D0.B8.D1.82.D0.B8.D1.87.D0.BD.D0.BE.D1.81.D1.82.D1.8C"></span>Квазианалитичность</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=11" title="Редактировать раздел «Квазианалитичность»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=11" title="Редактировать код раздела «Квазианалитичность»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hatnote navigation-not-searchable">Основная статья: <b><a href="/wiki/%D0%9A%D0%B2%D0%B0%D0%B7%D0%B8%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Квазианалитическая функция">Квазианалитическая функция</a></b></div> <div class="mw-heading mw-heading2"><h2 id="Примечания"><span id=".D0.9F.D1.80.D0.B8.D0.BC.D0.B5.D1.87.D0.B0.D0.BD.D0.B8.D1.8F"></span>Примечания</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=12" title="Редактировать раздел «Примечания»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=12" title="Редактировать код раздела «Примечания»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist columns" style="list-style-type: decimal;"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r141305934">.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 a::after,.mw-parser-output .id-lock-limited a::after,.mw-parser-output .id-lock-registration a::after,.mw-parser-output .id-lock-subscription a::after,.mw-parser-output .cs1-ws-icon a::after{content:"";width:1.1em;height:1.1em;display:inline-block;vertical-align:middle;background-position:center;background-repeat:no-repeat;background-size:contain}.mw-parser-output .id-lock-free.id-lock-free a::after{background-image:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")}.mw-parser-output .id-lock-limited.id-lock-limited a::after,.mw-parser-output .id-lock-registration.id-lock-registration a::after{background-image:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")}.mw-parser-output .id-lock-subscription.id-lock-subscription a::after{background-image:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")}.mw-parser-output .cs1-ws-icon a::after{background-image:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")}.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}html.skin-theme-clientpref-night .mw-parser-output .id-lock-free a::after,html.skin-theme-clientpref-night .mw-parser-output .id-lock-limited a::after,html.skin-theme-clientpref-night .mw-parser-output .id-lock-registration a::after,html.skin-theme-clientpref-night .mw-parser-output .id-lock-subscription a::after{filter:invert(1)hue-rotate(180deg)}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}html.skin-theme-clientpref-os .mw-parser-output .id-lock-free a::after,html.skin-theme-clientpref-os .mw-parser-output .id-lock-limited a::after,html.skin-theme-clientpref-os .mw-parser-output .id-lock-registration a::after,html.skin-theme-clientpref-os .mw-parser-output .id-lock-subscription a::after{filter:invert(1)hue-rotate(180deg)}}</style><span class="citation no-wikidata" data-wikidata-property-id="P1343"><i>А. В. Домрин, А. Г. Сергеев.</i> Лекции по комплексному анализу. Первое полугодие. — <abbr title="Москва">М.</abbr>: МИАН, 2004. — С. 79. — <a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%98%D1%81%D1%82%D0%BE%D1%87%D0%BD%D0%B8%D0%BA%D0%B8_%D0%BA%D0%BD%D0%B8%D0%B3/5984190079" class="internal mw-magiclink-isbn">ISBN 5-98419-007-9</a>.</span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Литература"><span id=".D0.9B.D0.B8.D1.82.D0.B5.D1.80.D0.B0.D1.82.D1.83.D1.80.D0.B0"></span>Литература</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=13" title="Редактировать раздел «Литература»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=13" title="Редактировать код раздела «Литература»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r141305934"><span class="citation no-wikidata" data-wikidata-property-id="P1343"><span data-wikidata-qualifier-id="P248"><a href="https://ru.wikisource.org/wiki/%D0%AD%D0%A1%D0%91%D0%95/%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" class="extiw" title="s:ЭСБЕ/Голоморфная функция">Голоморфная функция</a></span> // <a href="/wiki/%D0%AD%D0%BD%D1%86%D0%B8%D0%BA%D0%BB%D0%BE%D0%BF%D0%B5%D0%B4%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B9_%D1%81%D0%BB%D0%BE%D0%B2%D0%B0%D1%80%D1%8C_%D0%91%D1%80%D0%BE%D0%BA%D0%B3%D0%B0%D1%83%D0%B7%D0%B0_%D0%B8_%D0%95%D1%84%D1%80%D0%BE%D0%BD%D0%B0" title="Энциклопедический словарь Брокгауза и Ефрона">Энциклопедический словарь Брокгауза и Ефрона</a> : в 86 т. (82 т. и 4 доп.). — <abbr title="Санкт-Петербург">СПб.</abbr>, 1890—1907.</span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r141305934"><span class="citation no-wikidata" data-wikidata-property-id="P1343"><i><a href="/wiki/%D0%A8%D0%B0%D0%B1%D0%B0%D1%82,_%D0%91%D0%BE%D1%80%D0%B8%D1%81_%D0%92%D0%BB%D0%B0%D0%B4%D0%B8%D0%BC%D0%B8%D1%80%D0%BE%D0%B2%D0%B8%D1%87" title="Шабат, Борис Владимирович">Шабат Б. В.</a></i> Введение в комплексный анализ. — <abbr title="Москва">М.</abbr>: <a href="/wiki/%D0%9D%D0%B0%D1%83%D0%BA%D0%B0_(%D0%B8%D0%B7%D0%B4%D0%B0%D1%82%D0%B5%D0%BB%D1%8C%D1%81%D1%82%D0%B2%D0%BE)" title="Наука (издательство)">Наука</a>, <a href="/wiki/1969" class="mw-redirect" title="1969">1969</a>. — 577 с.</span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r141305934"><span class="citation no-wikidata" data-wikidata-property-id="P1343"><i>Титчмарш Е.</i> Теория функций: Пер. с англ. — 2-е изд., перераб. — <abbr title="Москва">М.</abbr>: <a href="/wiki/%D0%9D%D0%B0%D1%83%D0%BA%D0%B0_(%D0%B8%D0%B7%D0%B4%D0%B0%D1%82%D0%B5%D0%BB%D1%8C%D1%81%D1%82%D0%B2%D0%BE)" title="Наука (издательство)">Наука</a>, <a href="/wiki/1980" class="mw-redirect" title="1980">1980</a>. — 464 с.</span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r141305934"><span class="citation no-wikidata" data-wikidata-property-id="P1343"><i><a href="/wiki/%D0%9F%D1%80%D0%B8%D0%B2%D0%B0%D0%BB%D0%BE%D0%B2,_%D0%98%D0%B2%D0%B0%D0%BD_%D0%98%D0%B2%D0%B0%D0%BD%D0%BE%D0%B2%D0%B8%D1%87" title="Привалов, Иван Иванович">Привалов И. И.</a></i> Введение в теорию функций комплексного переменного: Пособие для высшей школы. — <abbr title="Москва">М.</abbr>—<abbr title="Ленинград">Л.</abbr>: Государственное издательство, <a href="/wiki/1927" class="mw-redirect" title="1927">1927</a>. — 316 с.</span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r141305934"><span class="citation no-wikidata" data-wikidata-property-id="P1343"><i><a href="/wiki/%D0%95%D0%B2%D0%B3%D1%80%D0%B0%D1%84%D0%BE%D0%B2,_%D0%9C%D0%B0%D1%80%D0%B0%D1%82_%D0%90%D0%BD%D0%B4%D1%80%D0%B5%D0%B5%D0%B2%D0%B8%D1%87" title="Евграфов, Марат Андреевич">Евграфов М. А.</a></i> Аналитические функции. — 2-е изд., перераб. и дополн. — <abbr title="Москва">М.</abbr>: <a href="/wiki/%D0%9D%D0%B0%D1%83%D0%BA%D0%B0_(%D0%B8%D0%B7%D0%B4%D0%B0%D1%82%D0%B5%D0%BB%D1%8C%D1%81%D1%82%D0%B2%D0%BE)" title="Наука (издательство)">Наука</a>, <a href="/wiki/1968" class="mw-redirect" title="1968">1968</a>. — 472 с.</span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r141305934"><span class="citation no-wikidata" data-wikidata-property-id="P1343" id="CITEREFBlakey1958"><i>Blakey, Joseph.</i> University Mathematics <small class="ref-info" style="cursor:help;" title="на неопределённом языке">(неопр.)</small>. — 2nd. — London: Blackie and Sons, 1958.</span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Ссылки"><span id=".D0.A1.D1.81.D1.8B.D0.BB.D0.BA.D0.B8"></span>Ссылки</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=14" title="Редактировать раздел «Ссылки»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=14" title="Редактировать код раздела «Ссылки»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r141305934"><cite id="CITEREFHazewinkel2001" class="citation cs2">Hazewinkel, Michiel, ed. (2001), <a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/index.php?title=p/a012240">"Analytic function"</a>, <i><a href="/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%B0%D1%8F_%D1%8D%D0%BD%D1%86%D0%B8%D0%BA%D0%BB%D0%BE%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F" title="Математическая энциклопедия">Encyclopedia of Mathematics</a></i>, <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%98%D1%81%D1%82%D0%BE%D1%87%D0%BD%D0%B8%D0%BA%D0%B8_%D0%BA%D0%BD%D0%B8%D0%B3/978-1-55608-010-4" title="Служебная:Источники книг/978-1-55608-010-4"><bdi>978-1-55608-010-4</bdi></a></cite><span 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href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F:%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%A1%D1%82%D0%B0%D1%82%D1%8C%D0%B8_%D1%81_%D0%BD%D0%B5%D0%BA%D0%BE%D1%80%D1%80%D0%B5%D0%BA%D1%82%D0%BD%D1%8B%D0%BC_%D0%B8%D1%81%D0%BF%D0%BE%D0%BB%D1%8C%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D0%B5%D0%BC_%D1%88%D0%B0%D0%B1%D0%BB%D0%BE%D0%BD%D0%BE%D0%B2:%D0%9A%D0%BD%D0%B8%D0%B3%D0%B0_(%D1%83%D0%BA%D0%B0%D0%B7%D0%B0%D0%BD_%D0%BD%D0%B5%D0%B2%D0%B5%D1%80%D0%BD%D1%8B%D0%B9_%D0%BA%D0%BE%D0%B4_%D1%8F%D0%B7%D1%8B%D0%BA%D0%B0)" title="Категория:Википедия:Статьи с некорректным использованием шаблонов:Книга (указан неверный код языка)">Википедия:Статьи с некорректным использованием шаблонов:Книга (указан неверный код языка)</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F:%D0%A1%D1%82%D1%80%D0%B0%D0%BD%D0%B8%D1%86%D1%8B,_%D0%B8%D1%81%D0%BF%D0%BE%D0%BB%D1%8C%D0%B7%D1%83%D1%8E%D1%89%D0%B8%D0%B5_%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%B8%D0%B5_%D1%82%D0%B5%D0%B3%D0%B8_%D0%B2_%D1%83%D1%81%D1%82%D0%B0%D1%80%D0%B5%D0%B2%D1%88%D0%B5%D0%BC_%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%82%D0%B5" title="Категория:Страницы, использующие математические теги в устаревшем формате">Страницы, использующие математические теги в устаревшем формате</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Навигация</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Персональные инструменты</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item"><span title="Страница участника для моего IP">Вы не представились системе</span></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%9C%D0%BE%D1%91_%D0%BE%D0%B1%D1%81%D1%83%D0%B6%D0%B4%D0%B5%D0%BD%D0%B8%D0%B5" title="Страница обсуждений для моего IP [n]" accesskey="n"><span>Обсуждение</span></a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%9C%D0%BE%D0%B9_%D0%B2%D0%BA%D0%BB%D0%B0%D0%B4" title="Список правок, сделанных с этого IP-адреса [y]" accesskey="y"><span>Вклад</span></a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%A1%D0%BE%D0%B7%D0%B4%D0%B0%D1%82%D1%8C_%D1%83%D1%87%D1%91%D1%82%D0%BD%D1%83%D1%8E_%D0%B7%D0%B0%D0%BF%D0%B8%D1%81%D1%8C&returnto=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F+%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Мы предлагаем вам создать учётную запись и войти в систему, хотя это и не обязательно."><span>Создать учётную запись</span></a></li><li id="pt-login" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%92%D1%85%D0%BE%D0%B4&returnto=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F+%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Здесь можно зарегистрироваться в системе, но это необязательно. [o]" accesskey="o"><span>Войти</span></a></li> </ul> </div> </nav> <div id="left-navigation"> <nav id="p-namespaces" class="mw-portlet mw-portlet-namespaces vector-menu-tabs vector-menu-tabs-legacy vector-menu" aria-labelledby="p-namespaces-label" > <h3 id="p-namespaces-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Пространства имён</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected mw-list-item"><a href="/wiki/%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Просмотреть контентную страницу [c]" accesskey="c"><span>Статья</span></a></li><li id="ca-talk" class="mw-list-item"><a href="/wiki/%D0%9E%D0%B1%D1%81%D1%83%D0%B6%D0%B4%D0%B5%D0%BD%D0%B8%D0%B5:%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" rel="discussion" title="Обсуждение основной страницы [t]" accesskey="t"><span>Обсуждение</span></a></li> </ul> </div> </nav> <nav id="p-variants" class="mw-portlet mw-portlet-variants emptyPortlet vector-menu-dropdown vector-menu" aria-labelledby="p-variants-label" > <input type="checkbox" id="p-variants-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-variants" class="vector-menu-checkbox" aria-labelledby="p-variants-label" > <label id="p-variants-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">русский</span> </label> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </nav> </div> <div id="right-navigation"> <nav id="p-views" class="mw-portlet mw-portlet-views vector-menu-tabs vector-menu-tabs-legacy vector-menu" aria-labelledby="p-views-label" > <h3 id="p-views-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Просмотры</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="mw-list-item"><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&stable=1"><span>Читать</span></a></li><li id="ca-current" class="collapsible selected mw-list-item"><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&stable=0&redirect=no" title="Показать текущую версию этой страницы [v]" accesskey="v"><span>Текущая версия</span></a></li><li id="ca-ve-edit" class="mw-list-item"><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit" title="Редактировать данную страницу [v]" accesskey="v"><span>Править</span></a></li><li id="ca-edit" class="collapsible mw-list-item"><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit" title="Править исходный текст этой страницы [e]" accesskey="e"><span>Править код</span></a></li><li id="ca-history" class="mw-list-item"><a href="/w/index.php?title=%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=history" title="Журнал изменений страницы [h]" accesskey="h"><span>История</span></a></li> </ul> </div> </nav> <nav id="p-cactions" class="mw-portlet mw-portlet-cactions emptyPortlet vector-menu-dropdown vector-menu" aria-labelledby="p-cactions-label" title="Больше возможностей" > <input type="checkbox" id="p-cactions-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-cactions" class="vector-menu-checkbox" aria-labelledby="p-cactions-label" > <label id="p-cactions-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Ещё</span> </label> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </nav> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <h3 >Поиск</h3> <form action="/w/index.php" id="searchform" class="vector-search-box-form"> <div id="simpleSearch" class="vector-search-box-inner" data-search-loc="header-navigation"> <input class="vector-search-box-input" type="search" name="search" placeholder="Искать в Википедии" aria-label="Искать в Википедии" autocapitalize="sentences" title="Искать в Википедии [f]" accesskey="f" id="searchInput" > <input type="hidden" name="title" value="Служебная:Поиск"> <input id="mw-searchButton" class="searchButton mw-fallbackSearchButton" type="submit" name="fulltext" title="Найти страницы, содержащие указанный текст" value="Найти"> <input id="searchButton" class="searchButton" type="submit" name="go" title="Перейти к странице, имеющей в точности такое название" value="Перейти"> </div> </form> </div> </div> </div> <div id="mw-panel" class="vector-legacy-sidebar"> <div id="p-logo" role="banner"> <a class="mw-wiki-logo" href="/wiki/%D0%97%D0%B0%D0%B3%D0%BB%D0%B0%D0%B2%D0%BD%D0%B0%D1%8F_%D1%81%D1%82%D1%80%D0%B0%D0%BD%D0%B8%D1%86%D0%B0" title="Перейти на заглавную страницу"></a> </div> <nav id="p-navigation" class="mw-portlet mw-portlet-navigation vector-menu-portal portal vector-menu" aria-labelledby="p-navigation-label" > <h3 id="p-navigation-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Навигация</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/%D0%97%D0%B0%D0%B3%D0%BB%D0%B0%D0%B2%D0%BD%D0%B0%D1%8F_%D1%81%D1%82%D1%80%D0%B0%D0%BD%D0%B8%D1%86%D0%B0" title="Перейти на заглавную страницу [z]" accesskey="z"><span>Заглавная страница</span></a></li><li id="n-content" class="mw-list-item"><a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%A1%D0%BE%D0%B4%D0%B5%D1%80%D0%B6%D0%B0%D0%BD%D0%B8%D0%B5"><span>Содержание</span></a></li><li id="n-featured" class="mw-list-item"><a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%98%D0%B7%D0%B1%D1%80%D0%B0%D0%BD%D0%BD%D1%8B%D0%B5_%D1%81%D1%82%D0%B0%D1%82%D1%8C%D0%B8" title="Статьи, считающиеся лучшими статьями проекта"><span>Избранные статьи</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%A1%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%B0%D1%8F_%D1%81%D1%82%D1%80%D0%B0%D0%BD%D0%B8%D1%86%D0%B0" title="Посмотреть случайно выбранную страницу [x]" accesskey="x"><span>Случайная статья</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/%D0%9F%D0%BE%D1%80%D1%82%D0%B0%D0%BB:%D0%A2%D0%B5%D0%BA%D1%83%D1%89%D0%B8%D0%B5_%D1%81%D0%BE%D0%B1%D1%8B%D1%82%D0%B8%D1%8F" title="Статьи о текущих событиях в мире"><span>Текущие события</span></a></li><li id="n-sitesupport" class="mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_ru.wikipedia.org&uselang=ru" title="Поддержите нас"><span>Пожертвовать</span></a></li> </ul> </div> </nav> <nav id="p-participation" class="mw-portlet mw-portlet-participation vector-menu-portal portal vector-menu" aria-labelledby="p-participation-label" > <h3 id="p-participation-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Участие</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-bug_in_article" class="mw-list-item"><a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%A1%D0%BE%D0%BE%D0%B1%D1%89%D0%B5%D0%BD%D0%B8%D1%8F_%D0%BE%D0%B1_%D0%BE%D1%88%D0%B8%D0%B1%D0%BA%D0%B0%D1%85" title="Сообщить об ошибке в этой статье"><span>Сообщить об ошибке</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D1%80%D0%B0%D0%B2%D0%BA%D0%B0:%D0%92%D0%B2%D0%B5%D0%B4%D0%B5%D0%BD%D0%B8%D0%B5"><span>Как править статьи</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%A1%D0%BE%D0%BE%D0%B1%D1%89%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="О проекте, о том, чем здесь можно заниматься, а также — где что находится"><span>Сообщество</span></a></li><li id="n-forum" class="mw-list-item"><a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%A4%D0%BE%D1%80%D1%83%D0%BC" title="Форум участников Википедии"><span>Форум</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%A1%D0%B2%D0%B5%D0%B6%D0%B8%D0%B5_%D0%BF%D1%80%D0%B0%D0%B2%D0%BA%D0%B8" title="Список последних изменений [r]" accesskey="r"><span>Свежие правки</span></a></li><li id="n-newpages" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%9D%D0%BE%D0%B2%D1%8B%D0%B5_%D1%81%D1%82%D1%80%D0%B0%D0%BD%D0%B8%D1%86%D1%8B" title="Список недавно созданных страниц"><span>Новые страницы</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%A1%D0%BF%D1%80%D0%B0%D0%B2%D0%BA%D0%B0" title="Место расположения Справки"><span>Справка</span></a></li> </ul> </div> </nav> <nav id="p-tb" class="mw-portlet mw-portlet-tb vector-menu-portal portal vector-menu" aria-labelledby="p-tb-label" > <h3 id="p-tb-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Инструменты</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%A1%D1%81%D1%8B%D0%BB%D0%BA%D0%B8_%D1%81%D1%8E%D0%B4%D0%B0/%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Список всех страниц, ссылающихся на данную [j]" accesskey="j"><span>Ссылки сюда</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%A1%D0%B2%D1%8F%D0%B7%D0%B0%D0%BD%D0%BD%D1%8B%D0%B5_%D0%BF%D1%80%D0%B0%D0%B2%D0%BA%D0%B8/%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" rel="nofollow" title="Последние изменения в страницах, на которые ссылается эта страница [k]" accesskey="k"><span>Связанные правки</span></a></li><li id="t-specialpages" class="mw-list-item"><a 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class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Funci%C3%B3n_holomorfa" title="Función holomorfa — астурийский" lang="ast" hreflang="ast" data-title="Función holomorfa" data-language-autonym="Asturianu" data-language-local-name="астурийский" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BB%D1%8B_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Голоморфлы функция — башкирский" lang="ba" hreflang="ba" data-title="Голоморфлы функция" data-language-autonym="Башҡортса" data-language-local-name="башкирский" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A5%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Холоморфна функция — болгарский" lang="bg" hreflang="bg" data-title="Холоморфна функция" data-language-autonym="Български" data-language-local-name="болгарский" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Funci%C3%B3_holomorfa" title="Funció holomorfa — каталанский" lang="ca" hreflang="ca" data-title="Funció holomorfa" data-language-autonym="Català" data-language-local-name="каталанский" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cbk-zam mw-list-item"><a href="https://cbk-zam.wikipedia.org/wiki/Holomorfo_funcion" title="Holomorfo funcion — Chavacano" lang="cbk" hreflang="cbk" data-title="Holomorfo funcion" data-language-autonym="Chavacano de Zamboanga" data-language-local-name="Chavacano" class="interlanguage-link-target"><span>Chavacano de Zamboanga</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Holomorfn%C3%AD_funkce" title="Holomorfní funkce — чешский" lang="cs" hreflang="cs" data-title="Holomorfní funkce" data-language-autonym="Čeština" data-language-local-name="чешский" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de badge-Q17437796 badge-featuredarticle mw-list-item" title="избранная статья"><a href="https://de.wikipedia.org/wiki/Holomorphe_Funktion" title="Holomorphe Funktion — немецкий" lang="de" hreflang="de" data-title="Holomorphe Funktion" data-language-autonym="Deutsch" data-language-local-name="немецкий" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9F%CE%BB%CF%8C%CE%BC%CE%BF%CF%81%CF%86%CE%B7_%CF%83%CF%85%CE%BD%CE%AC%CF%81%CF%84%CE%B7%CF%83%CE%B7" title="Ολόμορφη συνάρτηση — греческий" lang="el" hreflang="el" data-title="Ολόμορφη συνάρτηση" data-language-autonym="Ελληνικά" data-language-local-name="греческий" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Holomorphic_function" title="Holomorphic function — английский" lang="en" hreflang="en" data-title="Holomorphic function" data-language-autonym="English" data-language-local-name="английский" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Holomorfa_funkcio" title="Holomorfa funkcio — эсперанто" lang="eo" hreflang="eo" data-title="Holomorfa funkcio" data-language-autonym="Esperanto" data-language-local-name="эсперанто" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Funci%C3%B3n_holomorfa" title="Función holomorfa — испанский" lang="es" hreflang="es" data-title="Función holomorfa" data-language-autonym="Español" data-language-local-name="испанский" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Regulaarne_funktsioon" title="Regulaarne funktsioon — эстонский" lang="et" hreflang="et" data-title="Regulaarne funktsioon" data-language-autonym="Eesti" data-language-local-name="эстонский" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Funtzio_holomorfiko" title="Funtzio holomorfiko — баскский" lang="eu" hreflang="eu" data-title="Funtzio holomorfiko" data-language-autonym="Euskara" data-language-local-name="баскский" 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%AA%D8%A7%D8%A8%D8%B9_%D9%87%D9%88%D9%84%D9%88%D9%85%D9%88%D8%B1%D9%81%DB%8C%DA%A9" title="تابع هولومورفیک — персидский" lang="fa" hreflang="fa" data-title="تابع هولومورفیک" data-language-autonym="فارسی" data-language-local-name="персидский" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Holomorfinen_funktio" title="Holomorfinen funktio — финский" lang="fi" hreflang="fi" data-title="Holomorfinen funktio" data-language-autonym="Suomi" data-language-local-name="финский" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Fonction_holomorphe" title="Fonction holomorphe — французский" lang="fr" hreflang="fr" data-title="Fonction holomorphe" data-language-autonym="Français" data-language-local-name="французский" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Funci%C3%B3n_holomorfa" title="Función holomorfa — галисийский" lang="gl" hreflang="gl" data-title="Función holomorfa" data-language-autonym="Galego" data-language-local-name="галисийский" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%94%D7%95%D7%9C%D7%95%D7%9E%D7%95%D7%A8%D7%A4%D7%99%D7%AA" title="פונקציה הולומורפית — иврит" lang="he" hreflang="he" data-title="פונקציה הולומורפית" data-language-autonym="עברית" data-language-local-name="иврит" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B9%E0%A5%8B%E0%A4%B2%E0%A5%8B%E0%A4%AE%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%AB%E0%A4%BF%E0%A4%95_%E0%A4%AB%E0%A4%B2%E0%A4%A8" title="होलोमार्फिक फलन — хинди" lang="hi" hreflang="hi" data-title="होलोमार्फिक फलन" data-language-autonym="हिन्दी" data-language-local-name="хинди" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Holomorf_f%C3%BCggv%C3%A9nyek" title="Holomorf függvények — венгерский" lang="hu" hreflang="hu" data-title="Holomorf függvények" data-language-autonym="Magyar" data-language-local-name="венгерский" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Funzione_olomorfa" title="Funzione olomorfa — итальянский" lang="it" hreflang="it" data-title="Funzione olomorfa" data-language-autonym="Italiano" data-language-local-name="итальянский" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%AD%A3%E5%89%87%E9%96%A2%E6%95%B0" title="正則関数 — японский" lang="ja" hreflang="ja" data-title="正則関数" data-language-autonym="日本語" data-language-local-name="японский" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%B0%E1%83%9D%E1%83%9A%E1%83%9D%E1%83%9B%E1%83%9D%E1%83%A0%E1%83%A4%E1%83%A3%E1%83%9A%E1%83%98_%E1%83%A4%E1%83%A3%E1%83%9C%E1%83%A5%E1%83%AA%E1%83%98%E1%83%90" title="ჰოლომორფული ფუნქცია — грузинский" lang="ka" hreflang="ka" data-title="ჰოლომორფული ფუნქცია" data-language-autonym="ქართული" data-language-local-name="грузинский" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%95%EC%B9%99_%ED%95%A8%EC%88%98" title="정칙 함수 — корейский" lang="ko" hreflang="ko" data-title="정칙 함수" data-language-autonym="한국어" data-language-local-name="корейский" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lmo badge-Q17437796 badge-featuredarticle mw-list-item" title="избранная статья"><a href="https://lmo.wikipedia.org/wiki/Funzi%C3%BA_ulumorfa" title="Funziú ulumorfa — Lombard" lang="lmo" hreflang="lmo" data-title="Funziú ulumorfa" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Holomorfin%C4%97_funkcija" title="Holomorfinė funkcija — литовский" lang="lt" hreflang="lt" data-title="Holomorfinė funkcija" data-language-autonym="Lietuvių" data-language-local-name="литовский" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Holomorfe_functie" title="Holomorfe functie — нидерландский" lang="nl" hreflang="nl" data-title="Holomorfe functie" data-language-autonym="Nederlands" data-language-local-name="нидерландский" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Holomorf_funksjon" title="Holomorf funksjon — нюнорск" lang="nn" hreflang="nn" data-title="Holomorf funksjon" data-language-autonym="Norsk nynorsk" data-language-local-name="нюнорск" 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/Foncion_olom%C3%B2rfa" title="Foncion olomòrfa — окситанский" lang="oc" hreflang="oc" data-title="Foncion olomòrfa" data-language-autonym="Occitan" data-language-local-name="окситанский" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Funkcja_holomorficzna" title="Funkcja holomorficzna — польский" lang="pl" hreflang="pl" data-title="Funkcja holomorficzna" data-language-autonym="Polski" data-language-local-name="польский" 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/Fun%C3%A7%C3%A3o_holomorfa" title="Função holomorfa — португальский" lang="pt" hreflang="pt" data-title="Função holomorfa" data-language-autonym="Português" data-language-local-name="португальский" 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/Func%C8%9Bie_olomorf%C4%83" title="Funcție olomorfă — румынский" lang="ro" hreflang="ro" data-title="Funcție olomorfă" data-language-autonym="Română" data-language-local-name="румынский" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Funzioni_olomorfa" title="Funzioni olomorfa — сицилийский" lang="scn" hreflang="scn" data-title="Funzioni olomorfa" data-language-autonym="Sicilianu" data-language-local-name="сицилийский" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Holomorphic_function" title="Holomorphic function — Simple English" lang="en-simple" hreflang="en-simple" data-title="Holomorphic function" 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/Holomorfn%C3%A1_funkcia" title="Holomorfná funkcia — словацкий" lang="sk" hreflang="sk" data-title="Holomorfná funkcia" data-language-autonym="Slovenčina" data-language-local-name="словацкий" 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/Holomorfna_funkcija" title="Holomorfna funkcija — словенский" lang="sl" hreflang="sl" data-title="Holomorfna funkcija" data-language-autonym="Slovenščina" data-language-local-name="словенский" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Funksioni_holomorfik" title="Funksioni holomorfik — албанский" lang="sq" hreflang="sq" data-title="Funksioni holomorfik" data-language-autonym="Shqip" data-language-local-name="албанский" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Holomorfna_funkcija" title="Holomorfna funkcija — сербский" lang="sr" hreflang="sr" data-title="Holomorfna funkcija" data-language-autonym="Српски / srpski" data-language-local-name="сербский" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Analytisk_funktion" title="Analytisk funktion — шведский" lang="sv" hreflang="sv" data-title="Analytisk funktion" data-language-autonym="Svenska" data-language-local-name="шведский" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AF%81%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AF%81%E0%AE%B0%E0%AF%81%E0%AE%B5%E0%AE%9A%E0%AF%8D_%E0%AE%9A%E0%AE%BE%E0%AE%B0%E0%AF%8D%E0%AE%AA%E0%AE%BF%E0%AE%AF%E0%AE%AE%E0%AF%8D" title="முற்றுருவச் சார்பியம் — тамильский" lang="ta" hreflang="ta" data-title="முற்றுருவச் சார்பியம்" data-language-autonym="தமிழ்" data-language-local-name="тамильский" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Holomorf_fonksiyon" title="Holomorf fonksiyon — турецкий" lang="tr" hreflang="tr" data-title="Holomorf fonksiyon" data-language-autonym="Türkçe" data-language-local-name="турецкий" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D1%80%D1%84%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F" title="Голоморфна функція — украинский" lang="uk" hreflang="uk" data-title="Голоморфна функція" data-language-autonym="Українська" data-language-local-name="украинский" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%C3%A0m_ch%E1%BB%89nh_h%C3%ACnh" title="Hàm chỉnh hình — вьетнамский" lang="vi" hreflang="vi" data-title="Hàm chỉnh hình" data-language-autonym="Tiếng Việt" data-language-local-name="вьетнамский" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%85%A8%E7%BA%AF%E5%87%BD%E6%95%B0" title="全纯函数 — китайский" lang="zh" hreflang="zh" data-title="全纯函数" data-language-autonym="中文" data-language-local-name="китайский" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%85%A8%E7%B4%94%E5%87%BD%E6%95%B8" title="全純函數 — кантонский" lang="yue" hreflang="yue" data-title="全純函數" data-language-autonym="粵語" data-language-local-name="кантонский" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q207476#sitelinks-wikipedia" title="Править ссылки на другие языки" class="wbc-editpage">Править ссылки</a></span></div> </div> </nav> </div> </div> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Эта страница в последний раз была отредактирована 19 июля 2023 в 12:28.</li> <li id="footer-info-copyright">Текст доступен по <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/deed.ru">лицензии Creative Commons «С указанием авторства — С сохранением условий» (CC BY-SA)</a>; в отдельных 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