CINXE.COM

<!DOCTYPE html> <html class="client-nojs" lang="uk" dir="ltr"> <head> <meta charset="UTF-8"> <title>Функція експоненційного типу — Вікіпедія</title> <script>(function(){var className="client-js";var cookie=document.cookie.match(/(?:^|; )ukwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":[",\t."," \t,"],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy","wgMonthNames":["","січень","лютий","березень","квітень","травень","червень","липень","серпень","вересень","жовтень","листопад","грудень"],"wgRequestId":"2963e1d0-31d3-4b4a-9b84-1b36e2127f64","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Функція_експоненційного_типу","wgTitle":"Функція експоненційного типу", "wgCurRevisionId":30214443,"wgRevisionId":0,"wgArticleId":4062487,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Сторінки з використанням розширення JsonConfig","Комплексний аналіз"],"wgPageViewLanguage":"uk","wgPageContentLanguage":"uk","wgPageContentModel":"wikitext","wgRelevantPageName":"Функція_експоненційного_типу","wgRelevantArticleId":4062487,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":true,"wgFlaggedRevsParams":{"tags":{"accuracy":{"levels":3}}},"wgStableRevisionId":0,"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"uk","pageLanguageDir":"ltr","pageVariantFallbacks":"uk"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true, "tagline":true,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":9000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":true,"wgVector2022LanguageInHeader":false,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q5421531","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.math.styles":"ready","ext.cite.styles":"ready","skins.vector.styles.legacy":"ready","ext.flaggedRevs.basic":"ready","mediawiki.codex.messagebox.styles": "ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","codex-search-styles":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","site","mediawiki.page.ready","mediawiki.toc","skins.vector.legacy.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.flaggedRevs.advanced","ext.gadget.CurIDLink","ext.gadget.collapserefs","ext.gadget.showContributorContent","ext.gadget.switcher","ext.gadget.edittop","ext.gadget.new-section","ext.gadget.newTopicOnTop","ext.gadget.MonobookToolbarStandard","ext.gadget.ProtectionIndicator","ext.gadget.Statistics","ext.gadget.interwiki-langlist","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.compactlinks", "ext.uls.interface","ext.cx.eventlogging.campaigns","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=uk&modules=codex-search-styles%7Cext.cite.styles%7Cext.flaggedRevs.basic%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cmediawiki.codex.messagebox.styles%7Cskins.vector.styles.legacy%7Cwikibase.client.init&only=styles&skin=vector"> <script async="" src="/w/load.php?lang=uk&modules=startup&only=scripts&raw=1&skin=vector"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=uk&modules=site.styles&only=styles&skin=vector"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/b/b2/ExtremeGaussian.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="742"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/b/b2/ExtremeGaussian.png/800px-ExtremeGaussian.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="495"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/b/b2/ExtremeGaussian.png/640px-ExtremeGaussian.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="396"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Функція експоненційного типу — Вікіпедія"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//uk.m.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83"> <link rel="alternate" type="application/x-wiki" title="Редагувати" href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Вікіпедія (uk)"> <link rel="EditURI" type="application/rsd+xml" href="//uk.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://uk.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.uk"> <link rel="alternate" type="application/atom+xml" title="Вікіпедія — Atom-стрічка" href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9D%D0%BE%D0%B2%D1%96_%D1%80%D0%B5%D0%B4%D0%B0%D0%B3%D1%83%D0%B2%D0%B0%D0%BD%D0%BD%D1%8F&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin-vector-legacy mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Функція_експоненційного_типу rootpage-Функція_експоненційного_типу skin-vector action-view"><div id="mw-page-base" class="noprint"></div> <div id="mw-head-base" class="noprint"></div> <div id="content" class="mw-body" role="main"> <a id="top"></a> <div id="siteNotice"></div> <div class="mw-indicators"> <div id="mw-indicator-indicator-fr-review-status" class="mw-indicator"><indicator name="fr-review-status" class="mw-fr-review-status-indicator" id="mw-fr-revision-toggle"><span class="cdx-fr-css-icon-review--status--unchecked"></span><b><a href="/wiki/%D0%92%D1%96%D0%BA%D1%96%D0%BF%D0%B5%D0%B4%D1%96%D1%8F:%D0%9F%D0%B0%D1%82%D1%80%D1%83%D0%BB%D1%8E%D0%B2%D0%B0%D0%BD%D0%BD%D1%8F" title="Вікіпедія:Патрулювання">Неперевірена версія</a></b> <a href="/wiki/%D0%92%D1%96%D0%BA%D1%96%D0%BF%D0%B5%D0%B4%D1%96%D1%8F:%D0%9F%D0%B0%D1%82%D1%80%D1%83%D0%BB%D1%8E%D0%B2%D0%B0%D0%BD%D0%BD%D1%8F/%D0%A9%D0%BE_%D1%80%D0%BE%D0%B1%D0%B8%D1%82%D0%B8%3F" title="Вікіпедія:Патрулювання/Що робити?">(що робити?)</a></indicator></div> </div> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Функція експоненційного типу</span></h1> <div id="bodyContent" class="vector-body"> <div id="siteSub" class="noprint">Матеріал з Вікіпедії — вільної енциклопедії.</div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="contentSub2"></div> <div id="jump-to-nav"></div> <a class="mw-jump-link" href="#mw-head">Перейти до навігації</a> <a class="mw-jump-link" href="#searchInput">Перейти до пошуку</a> <div id="mw-content-text" class="mw-body-content"><div id="mw-fr-revision-details" class="mw-fr-revision-details-dialog" style="display:none;"><div tabindex="0"></div><div class="cdx-dialog cdx-dialog--horizontal-actions"><header class="cdx-dialog__header cdx-dialog__header--default"><div class="cdx-dialog__header__title-group"><h2 class="cdx-dialog__header__title">Статус версії сторінки</h2><p class="cdx-dialog__header__subtitle">Сторінка не перевірена</p></div><button class="cdx-button cdx-button--action-default cdx-button--weight-quiet
							cdx-button--size-medium cdx-button--icon-only cdx-dialog__header__close-button" aria-label="Закрити" onclick="document.getElementById("mw-fr-revision-details").style.display = "none";" type="submit"><span class="cdx-icon cdx-icon--medium
							cdx-fr-css-icon--close"></span></button></header><div class="cdx-dialog__body">Немає <a href="/wiki/%D0%92%D1%96%D0%BA%D1%96%D0%BF%D0%B5%D0%B4%D1%96%D1%8F:%D0%9F%D0%B0%D1%82%D1%80%D1%83%D0%BB%D1%8E%D0%B2%D0%B0%D0%BD%D0%BD%D1%8F" title="Вікіпедія:Патрулювання">перевірених версій</a> цієї сторінки; ймовірно, її ще <b>не</b> перевіряли на відповідність правилам проекту.</div></div><div tabindex="0"></div></div><div class="mw-content-ltr mw-parser-output" lang="uk" dir="ltr"><figure typeof="mw:File/Thumb"><a href="/wiki/%D0%A4%D0%B0%D0%B9%D0%BB:ExtremeGaussian.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/ExtremeGaussian.png/550px-ExtremeGaussian.png" decoding="async" width="550" height="340" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/ExtremeGaussian.png/825px-ExtremeGaussian.png 1.5x, //upload.wikimedia.org/wikipedia/commons/b/b2/ExtremeGaussian.png 2x" data-file-width="1017" data-file-height="629" /></a><figcaption>Графік сірої функції <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-\pi z^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-\pi z^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee7c6a5178dd2b5ef29ce92b23fe96061e476691" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.139ex; height:3.009ex;" alt="{\displaystyle e^{-\pi z^{2}}}"></span>, обмежений функцією щільности нормального розподілу у просторі дійсних чисел. Щільність нормального розподілу не є експоненційного типу, але червона і синя функції є її односторонніми наближеннями експоненційного типу <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 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }"></span>.</figcaption></figure> <p>В <a href="/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D0%B8%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D1%96%D0%B7" title="Комплексний аналіз">комплексному аналізі</a>, галузі <a href="/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика">математики</a>, <a href="/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">голоморфну функцію</a> називають <b>функцією</b> <b>експоненціального типу C</b>, якщо її зростання обмежене <a href="/wiki/%D0%9F%D0%BE%D0%BA%D0%B0%D0%B7%D0%BD%D0%B8%D0%BA%D0%BE%D0%B2%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F" title="Показникова функція">експоненційною функцією</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{C|z|}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{C|z|}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d826a644c6ce96ec13c3c22fb5356986710093c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.249ex; height:2.843ex;" alt="{\displaystyle e^{C|z|}}"></span>, для деякої <a href="/wiki/%D0%94%D1%96%D0%B9%D1%81%D0%BD%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Дійсне число">дійсної</a> константи <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> при <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|\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/878409f5f9775844a8d31df9dd60e41eadd1f889" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.32ex; height:2.843ex;" alt="{\displaystyle |z|\to \infty }"></span>. Якщо функція має таку властивість, то її можна виразити як певного роду збіжних сум рядів комплексних функцій, а також ясність щодо того, коли можна застосовувати такі прийоми, як <a href="/wiki/%D0%97%D0%B1%D1%96%D0%B6%D0%BD%D1%96%D1%81%D1%82%D1%8C_%D0%B7%D0%B0_%D0%91%D0%BE%D1%80%D0%B5%D0%BB%D0%B5%D0%BC" title="Збіжність за Борелем">сумування за Борелем</a>, чи наприклад, застосувати <a href="/wiki/%D0%9F%D0%B5%D1%80%D0%B5%D1%82%D0%B2%D0%BE%D1%80%D0%B5%D0%BD%D0%BD%D1%8F_%D0%9C%D0%B5%D0%BB%D0%BB%D1%96%D0%BD%D0%B0" title="Перетворення Мелліна">перетворення Мелліна</a>, або подати фукцію у вигляді розкладу <a href="/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0_%D0%95%D0%B9%D0%BB%D0%B5%D1%80%D0%B0_%E2%80%94_%D0%9C%D0%B0%D0%BA%D0%BB%D0%BE%D1%80%D0%B5%D0%BD%D0%B0" title="Формула Ейлера — Маклорена">Ейлера–Маклорена</a>. У загальному випадку пояснюється теоремою Начбіна, яка визначає аналогічне поняття <b>Ψ-типу</b> в просторі всіх функцій <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 \Psi (z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi (z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c6b5f6fa77dbd3d99daa3164fc10cdff322a62e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.706ex; height:2.843ex;" alt="{\displaystyle \Psi (z)}"></span> порівняно з <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4772def31b56e642df3e4d1160cadff3d80ba45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.085ex; height:2.343ex;" alt="{\displaystyle e^{z}}"></span>. </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="uk" dir="ltr"><h2 id="mw-toc-heading">Зміст</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Основна_ідея"><span class="tocnumber">1</span> <span class="toctext">Основна ідея</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Формальне_означення"><span class="tocnumber">2</span> <span class="toctext">Формальне означення</span></a></li> <li class="toclevel-1 tocsection-3"><a href="#Експоненційний_тип_відносно_симетричного_опуклого_тіла"><span class="tocnumber">3</span> <span class="toctext">Експоненційний тип відносно симетричного опуклого тіла</span></a></li> <li class="toclevel-1 tocsection-4"><a href="#Простір_Фреше"><span class="tocnumber">4</span> <span class="toctext">Простір Фреше</span></a></li> <li class="toclevel-1 tocsection-5"><a href="#Див._також"><span class="tocnumber">5</span> <span class="toctext">Див. також</span></a></li> <li class="toclevel-1 tocsection-6"><a href="#Примітки"><span class="tocnumber">6</span> <span class="toctext">Примітки</span></a></li> <li class="toclevel-1 tocsection-7"><a href="#Література"><span class="tocnumber">7</span> <span class="toctext">Література</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Основна_ідея"><span id=".D0.9E.D1.81.D0.BD.D0.BE.D0.B2.D0.BD.D0.B0_.D1.96.D0.B4.D0.B5.D1.8F"></span>Основна ідея</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83&veaction=edit&section=1" title="Редагувати розділ: Основна ідея" class="mw-editsection-visualeditor"><span>ред.</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83&action=edit&section=1" title="Редагувати вихідний код розділу: Основна ідея"><span>ред. код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Кажуть, що функція <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>, визначена на <a href="/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D0%B0_%D0%BF%D0%BB%D0%BE%D1%89%D0%B8%D0%BD%D0%B0" title="Комплексна площина">комплексній площині</a>, є експоненційним типом, якщо існують дійсні константи <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> і <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 \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> такі, що </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(re^{i\theta })|\leq Me^{\tau r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>r</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>M</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> <mi>r</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f(re^{i\theta })|\leq Me^{\tau r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/346d9863d7bf7d277a3b0dae3ece0f460ffc538c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.532ex; height:3.176ex;" alt="{\displaystyle |f(re^{i\theta })|\leq Me^{\tau r}}"></span></dd></dl> <p>при <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 r\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcd3a85ea2e3d6b4027434e502cace4177d7a3e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.986ex; height:1.843ex;" alt="{\displaystyle r\to \infty }"></span>. Тут <a href="/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D0%B8%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D1%96%D0%B7" title="Комплексний аналіз">комплексна змінна</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 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> записана у формі <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=re^{i\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>r</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=re^{i\theta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5b36ddd965193c2b7d6ea24a7c3678814d0dc8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.889ex; height:2.676ex;" alt="{\displaystyle z=re^{i\theta }}"></span> аби підкреслити, що границя має виконуватися в усіх напрямках <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 \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span>. Якщо <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 \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> інфімум усіх таких <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 \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span>, тоді кажуть, що функція <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> має експоненційний тип <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 \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span>. </p><p>Наприклад, нехай <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)=\sin(\pi 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>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z)=\sin(\pi z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b3b505d7edad17e5ff5278517f6309e2e6a2814" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.36ex; height:2.843ex;" alt="{\displaystyle f(z)=\sin(\pi z)}"></span>. Тоді кажуть що <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\pi z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\pi z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18452bfc35a2d0c4a640da105259a76299482986" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.085ex; height:2.843ex;" alt="{\displaystyle \sin(\pi z)}"></span> експоненційного типу <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span>, оскільки <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span> є найменшим числом, яке обмежує зростання <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\pi z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\pi z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18452bfc35a2d0c4a640da105259a76299482986" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.085ex; height:2.843ex;" alt="{\displaystyle \sin(\pi z)}"></span> вздовж уявної осі. Отже, у цьому прикладі не можна застосувати теорему Карлсона, оскільки для її застосування потрібно функції експоненційного типу менше <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span>. Подібним чином не можна застосовувати <a href="/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0_%D0%95%D0%B9%D0%BB%D0%B5%D1%80%D0%B0_%E2%80%94_%D0%9C%D0%B0%D0%BA%D0%BB%D0%BE%D1%80%D0%B5%D0%BD%D0%B0" title="Формула Ейлера — Маклорена">формулу Ейлера – Маклауріна</a>, оскільки вона також виражає твердження, що базується на <a href="/wiki/%D0%A1%D0%BA%D1%96%D0%BD%D1%87%D0%B5%D0%BD%D0%BD%D1%96_%D1%80%D1%96%D0%B7%D0%BD%D0%B8%D1%86%D1%96" title="Скінченні різниці">скінченних різниць</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Формальне_означення"><span id=".D0.A4.D0.BE.D1.80.D0.BC.D0.B0.D0.BB.D1.8C.D0.BD.D0.B5_.D0.BE.D0.B7.D0.BD.D0.B0.D1.87.D0.B5.D0.BD.D0.BD.D1.8F"></span>Формальне означення</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83&veaction=edit&section=2" title="Редагувати розділ: Формальне означення" class="mw-editsection-visualeditor"><span>ред.</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83&action=edit&section=2" title="Редагувати вихідний код розділу: Формальне означення"><span>ред. код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Про <a href="/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">голоморфну функцію</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 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/5cf298d777120df944559c5e985b88a824debb80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.638ex; height:2.843ex;" alt="{\displaystyle F(z)}"></span> кажуть, що вона <b>експоненційного типу</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 \sigma >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/762ecd0f0905dd0d4d7a07f80fa8bfb324b9b021" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle \sigma >0}"></span>, якщо для кожного <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 \varepsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e04ec3670b50384a3ce48aca42e7cc5131a06b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}"></span> існує дійсне 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 A_{\varepsilon }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\varepsilon }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12f25e3f7957fec98a4ba67f9f422a8f852db016" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.742ex; height:2.509ex;" alt="{\displaystyle A_{\varepsilon }}"></span>, що </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)|\leq A_{\varepsilon }e^{(\sigma +\varepsilon )|z|}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>σ</mi> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |F(z)|\leq A_{\varepsilon }e^{(\sigma +\varepsilon )|z|}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e11da7d57a60cbe3249682e26bf3eac58f1fc44c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.036ex; height:3.343ex;" alt="{\displaystyle |F(z)|\leq A_{\varepsilon }e^{(\sigma +\varepsilon )|z|}}"></span></dd></dl> <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 |z|\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/878409f5f9775844a8d31df9dd60e41eadd1f889" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.32ex; height:2.843ex;" alt="{\displaystyle |z|\to \infty }"></span> де <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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\in \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/169fae60c23a2027ece2aa7fd4b5047492887e91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:2.176ex;" alt="{\displaystyle z\in \mathbb {C} }"></span>. Кажуть <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/5cf298d777120df944559c5e985b88a824debb80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.638ex; height:2.843ex;" alt="{\displaystyle F(z)}"></span>, що вона експоненційного типу, якщо <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/5cf298d777120df944559c5e985b88a824debb80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.638ex; height:2.843ex;" alt="{\displaystyle F(z)}"></span> експоненційного типу <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"></span> для якогось <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/762ecd0f0905dd0d4d7a07f80fa8bfb324b9b021" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle \sigma >0}"></span>. Число </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 \tau (F)=\sigma =\displaystyle \limsup _{|z|\rightarrow \infty }|z|^{-1}\log |F(z)|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>σ</mi> <mo>=</mo> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau (F)=\sigma =\displaystyle \limsup _{|z|\rightarrow \infty }|z|^{-1}\log |F(z)|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3938c4c78a326d7942d0851b3ee1ab649fb656e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.177ex; height:5.509ex;" alt="{\displaystyle \tau (F)=\sigma =\displaystyle \limsup _{|z|\rightarrow \infty }|z|^{-1}\log |F(z)|}"></span></dd></dl> <p>є експоненційним типом <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/5cf298d777120df944559c5e985b88a824debb80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.638ex; height:2.843ex;" alt="{\displaystyle F(z)}"></span>. Ця <a href="/wiki/%D0%92%D0%B5%D1%80%D1%85%D0%BD%D1%8F_%D1%96_%D0%BD%D0%B8%D0%B6%D0%BD%D1%8F_%D0%B3%D1%80%D0%B0%D0%BD%D0%B8%D1%86%D1%96" title="Верхня і нижня границі">верхня границя</a> означає границю супремуму відносини за межами заданого радіуса, як радіус прагне до нескінченності. Це теж межа, відмінний від максимального коефіцієнта в заданому радіусі, а радіус прагне до нескінченності. Верхня межа може існувати, навіть якщо максимуму на радіусі <i>р</i> не має меж, як <i>Р</i> йде в нескінченність. Наприклад, для функції </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)=\sum _{n=1}^{\infty }{\frac {z^{10^{n!}}}{(10^{n!})!}}}"> <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> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>!</mo> </mrow> </msup> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>!</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(z)=\sum _{n=1}^{\infty }{\frac {z^{10^{n!}}}{(10^{n!})!}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a3442c9a990ed006ae52b3e6d1d6c655276f71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.772ex; height:7.176ex;" alt="{\displaystyle F(z)=\sum _{n=1}^{\infty }{\frac {z^{10^{n!}}}{(10^{n!})!}}}"></span></dd></dl> <p>значення </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 (\max _{|z|=r}\log |F(z)|)/r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>r</mi> </mrow> </munder> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\max _{|z|=r}\log |F(z)|)/r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b6e4e41c476324503f4b69c87067e97b9aefe14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.024ex; height:4.509ex;" alt="{\displaystyle (\max _{|z|=r}\log |F(z)|)/r}"></span></dd></dl> <p>при <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 r=10^{n!-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>!</mo> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=10^{n!-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b650de5b25f20c9c77518e82cf13909f0ecf5da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.248ex; height:2.676ex;" alt="{\displaystyle r=10^{n!-1}}"></span> асимптотичному 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 (\log 10^{(n-1)!(n-1)-1})/10^{(n-1)!(n-1)-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\log 10^{(n-1)!(n-1)-1})/10^{(n-1)!(n-1)-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c0192e7d908af8b07e0d7921ed13fa6efb7ce1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.025ex; height:3.343ex;" alt="{\displaystyle (\log 10^{(n-1)!(n-1)-1})/10^{(n-1)!(n-1)-1}}"></span>, а тому прямує до нуля, при <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> до нескінченності<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>, проте <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/5cf298d777120df944559c5e985b88a824debb80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.638ex; height:2.843ex;" alt="{\displaystyle F(z)}"></span> все ж є експоненційним типом 1, у чому можна переконатись розглянувши точку <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=10^{n!}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>!</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=10^{n!}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76e467ed10287bcaeaa7159f0256da1217193efc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.187ex; height:2.676ex;" alt="{\displaystyle z=10^{n!}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Експоненційний_тип_відносно_симетричного_опуклого_тіла"><span id=".D0.95.D0.BA.D1.81.D0.BF.D0.BE.D0.BD.D0.B5.D0.BD.D1.86.D1.96.D0.B9.D0.BD.D0.B8.D0.B9_.D1.82.D0.B8.D0.BF_.D0.B2.D1.96.D0.B4.D0.BD.D0.BE.D1.81.D0.BD.D0.BE_.D1.81.D0.B8.D0.BC.D0.B5.D1.82.D1.80.D0.B8.D1.87.D0.BD.D0.BE.D0.B3.D0.BE_.D0.BE.D0.BF.D1.83.D0.BA.D0.BB.D0.BE.D0.B3.D0.BE_.D1.82.D1.96.D0.BB.D0.B0"></span>Експоненційний тип відносно симетричного опуклого тіла</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83&veaction=edit&section=3" title="Редагувати розділ: Експоненційний тип відносно симетричного опуклого тіла" class="mw-editsection-visualeditor"><span>ред.</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83&action=edit&section=3" title="Редагувати вихідний код розділу: Експоненційний тип відносно симетричного опуклого тіла"><span>ред. код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="#CITEREFStein1957">Stein, (1957)</a> узагальнив поняття експоненційний тип для <a href="/wiki/%D0%A6%D1%96%D0%BB%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F" title="Ціла функція">цілої функції</a> <a href="/w/index.php?title=%D0%9A%D1%96%D0%BB%D1%8C%D0%BA%D0%B0_%D1%81%D0%BA%D0%BB%D0%B0%D0%B4%D0%BD%D0%B8%D1%85_%D0%B7%D0%BC%D1%96%D0%BD%D0%BD%D0%B8%D1%85&action=edit&redlink=1" class="new" title="Кілька складних змінних (ще не написана)">кількох комплексних змінних</a>. Нехай <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> <a href="/wiki/%D0%9E%D0%BF%D1%83%D0%BA%D0%BB%D0%B0_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B8%D0%BD%D0%B0" title="Опукла множина">опукла</a>, <a href="/w/index.php?title=%D0%9A%D0%BE%D0%BC%D0%BF%D0%B0%D0%BA%D1%82%D0%BD%D0%B8%D0%B9_%D0%B5%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82&action=edit&redlink=1" class="new" title="Компактний елемент (ще не написана)">компактна</a> і <a href="/wiki/%D0%A1%D0%B8%D0%BC%D0%B5%D1%82%D1%80%D1%96%D1%8F" title="Симетрія">симетрична</a> підмножина <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>. Відомо, що для кожної такої <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> існує відповідна <a href="/wiki/%D0%9D%D0%BE%D1%80%D0%BC%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Норма (математика)">норма</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 \|\cdot \|_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{K}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d415781ab15cfce5b0d7e59fb959f19f49050f28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.697ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|_{K}}"></span> з властивістю </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 K=\{x\in \mathbb {R} ^{n}:\|x\|_{K}\leq 1\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>:</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo>≤</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=\{x\in \mathbb {R} ^{n}:\|x\|_{K}\leq 1\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07e913a74eec08e03264be330571a6a089f5901f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.749ex; height:2.843ex;" alt="{\displaystyle K=\{x\in \mathbb {R} ^{n}:\|x\|_{K}\leq 1\}.}"></span></dd></dl> <p>Тобто, <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> одинична куля 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} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> за мірою <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 \|\cdot \|_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{K}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d415781ab15cfce5b0d7e59fb959f19f49050f28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.697ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|_{K}}"></span>. Множину </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 K^{*}=\{y\in \mathbb {R} ^{n}:x\cdot y\leq 1{\text{ for all }}x\in {K}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>y</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>:</mo> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> <mo>≤</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K^{*}=\{y\in \mathbb {R} ^{n}:x\cdot y\leq 1{\text{ for all }}x\in {K}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96b1babd0bf8bdb98eeaaf27e0883e60b0e59d24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.047ex; height:2.843ex;" alt="{\displaystyle K^{*}=\{y\in \mathbb {R} ^{n}:x\cdot y\leq 1{\text{ for all }}x\in {K}\}}"></span></dd></dl> <p>називають полярною множиною, яка також є <a href="/wiki/%D0%9E%D0%BF%D1%83%D0%BA%D0%BB%D0%B0_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B8%D0%BD%D0%B0" title="Опукла множина">опуклою</a>, компактною та <a href="/wiki/%D0%A1%D0%B8%D0%BC%D0%B5%D1%82%D1%80%D1%96%D1%8F" title="Симетрія">симетричною</a> підмножиною <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>. До того ж можемо записати </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 \|x\|_{K}=\displaystyle \sup _{y\in K^{*}}|x\cdot y|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo>=</mo> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>∈</mo> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|_{K}=\displaystyle \sup _{y\in K^{*}}|x\cdot y|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f679139d2e95b83fd1fc53bd683645e96b16479" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.164ex; height:4.843ex;" alt="{\displaystyle \|x\|_{K}=\displaystyle \sup _{y\in K^{*}}|x\cdot y|.}"></span></dd></dl> <p>Довизначимо <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 \|\cdot \|_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{K}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d415781ab15cfce5b0d7e59fb959f19f49050f28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.697ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|_{K}}"></span> з <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> до <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} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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 \mathbb {C} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}"></span> як </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|z\|_{K}=\displaystyle \sup _{y\in K^{*}}|z\cdot y|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>z</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo>=</mo> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>∈</mo> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mo>⋅</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|z\|_{K}=\displaystyle \sup _{y\in K^{*}}|z\cdot y|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40f8572a14bc836db0eb0ba511eb7b07455178bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:18.68ex; height:4.843ex;" alt="{\displaystyle \|z\|_{K}=\displaystyle \sup _{y\in K^{*}}|z\cdot y|.}"></span></dd></dl> <p>Про цілу функцію <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/5cf298d777120df944559c5e985b88a824debb80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.638ex; height:2.843ex;" alt="{\displaystyle F(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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> -комплексних змінних кажуть що вона експоненційного типу відносно <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> якщо для кожного <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 \varepsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e04ec3670b50384a3ce48aca42e7cc5131a06b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}"></span> існує дійснозначна константа <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_{\varepsilon }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\varepsilon }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12f25e3f7957fec98a4ba67f9f422a8f852db016" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.742ex; height:2.509ex;" alt="{\displaystyle A_{\varepsilon }}"></span> така, що </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)|<A_{\varepsilon }e^{2\pi (1+\varepsilon )\|z\|_{K}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖</mo> <mi>z</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |F(z)|<A_{\varepsilon }e^{2\pi (1+\varepsilon )\|z\|_{K}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/925cc0d460db4f00510aafb31381b738682735a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.761ex; height:3.343ex;" alt="{\displaystyle |F(z)|<A_{\varepsilon }e^{2\pi (1+\varepsilon )\|z\|_{K}}}"></span></dd></dl> <p>для всіх <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>. </p> <div class="mw-heading mw-heading2"><h2 id="Простір_Фреше"><span id=".D0.9F.D1.80.D0.BE.D1.81.D1.82.D1.96.D1.80_.D0.A4.D1.80.D0.B5.D1.88.D0.B5"></span>Простір Фреше</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83&veaction=edit&section=4" title="Редагувати розділ: Простір Фреше" class="mw-editsection-visualeditor"><span>ред.</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83&action=edit&section=4" title="Редагувати вихідний код розділу: Простір Фреше"><span>ред. код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Набір функцій експоненційного типу <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 \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> можуть утворювати <a href="/wiki/%D0%9F%D0%BE%D0%B2%D0%BD%D0%B8%D0%B9_%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%B8%D0%B9_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%96%D1%80" title="Повний метричний простір">повний</a> <a href="/wiki/%D0%A0%D1%96%D0%B2%D0%BD%D0%BE%D0%BC%D1%96%D1%80%D0%BD%D0%B8%D0%B9_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%96%D1%80" title="Рівномірний простір">рівномірний простір</a>, а саме <a href="/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82%D1%96%D1%80_%D0%A4%D1%80%D0%B5%D1%88%D0%B5" title="Простір Фреше">простір Фреше</a>, до <a href="/wiki/%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D1%96%D1%87%D0%BD%D0%B8%D0%B9_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%96%D1%80" title="Топологічний простір">топологічно</a> індукований зліченним сімейством <a href="/wiki/%D0%9D%D0%BE%D1%80%D0%BC%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Норма (математика)">норм</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{n}=\sup _{z\in \mathbb {C} }\exp \left[-\left(\tau +{\frac {1}{n}}\right)|z|\right]|f(z)|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mrow> </munder> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mi>τ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{n}=\sup _{z\in \mathbb {C} }\exp \left[-\left(\tau +{\frac {1}{n}}\right)|z|\right]|f(z)|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0afab228894be9dfa1e50ff655815efb5392c58a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.591ex; height:6.176ex;" alt="{\displaystyle \|f\|_{n}=\sup _{z\in \mathbb {C} }\exp \left[-\left(\tau +{\frac {1}{n}}\right)|z|\right]|f(z)|.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Див._також"><span id=".D0.94.D0.B8.D0.B2._.D1.82.D0.B0.D0.BA.D0.BE.D0.B6"></span>Див. також</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83&veaction=edit&section=5" title="Редагувати розділ: Див. також" class="mw-editsection-visualeditor"><span>ред.</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83&action=edit&section=5" title="Редагувати вихідний код розділу: Див. також"><span>ред. код</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Теорема Пелі–Вінера</li> <li>Пелі–Вінера простору</li></ul> <div class="mw-heading mw-heading2"><h2 id="Примітки"><span id=".D0.9F.D1.80.D0.B8.D0.BC.D1.96.D1.82.D0.BA.D0.B8"></span>Примітки</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83&veaction=edit&section=6" title="Редагувати розділ: Примітки" class="mw-editsection-visualeditor"><span>ред.</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83&action=edit&section=6" title="Редагувати вихідний код розділу: Примітки"><span>ред. код</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r43816068">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist" 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">Насправді, навіть <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 (\max _{|z|=r}\log \log |F(z)|)/(\log r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>r</mi> </mrow> </munder> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\max _{|z|=r}\log \log |F(z)|)/(\log r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29e2d7e930c3a76b8b133b8347296379feaee28f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.551ex; height:4.509ex;" alt="{\displaystyle (\max _{|z|=r}\log \log |F(z)|)/(\log r)}"></span> прямує до нуля в точці <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 r=10^{n!-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>!</mo> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=10^{n!-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b650de5b25f20c9c77518e82cf13909f0ecf5da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.248ex; height:2.676ex;" alt="{\displaystyle r=10^{n!-1}}"></span> при <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d55d9b32f6fa8fab6a84ea444a6b5a24bb45e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle n\to \infty }"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Література"><span id=".D0.9B.D1.96.D1.82.D0.B5.D1.80.D0.B0.D1.82.D1.83.D1.80.D0.B0"></span>Література</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83&veaction=edit&section=7" title="Редагувати розділ: Література" class="mw-editsection-visualeditor"><span>ред.</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83&action=edit&section=7" title="Редагувати вихідний код розділу: Література"><span>ред. код</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r43245077">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ref-lang{font-size:85%;cursor:help;margin-left:0.2em;color:var(--color-subtle,#54595d)}.mw-parser-output .cs1-ref-lg{font-style:normal;cursor:help}.mw-parser-output .cs1-ref-lg-text{color:#252525;text-decoration:inherit;text-decoration-color:#252525}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}html.skin-theme-clientpref-night .mw-parser-output .cs1-ref-lg-text{color:#dadad6;text-decoration-color:#dadad6}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}html.skin-theme-clientpref-night .mw-parser-output .cs1-ref-lg-text{color:#dadad6;text-decoration-color:#dadad6}}</style><cite class="citation cs2"><a href="/w/index.php?title=Elias_M._Stein&action=edit&redlink=1" class="new" title="Elias M. Stein (ще не написана)">Stein, E.M.</a> (1957), Functions of exponential type, <i>Ann. of Math.</i>, 2, <b>65</b>: 582—592, <a href="/wiki/%D0%A6%D0%B8%D1%84%D1%80%D0%BE%D0%B2%D0%B8%D0%B9_%D1%96%D0%B4%D0%B5%D0%BD%D1%82%D0%B8%D1%84%D1%96%D0%BA%D0%B0%D1%82%D0%BE%D1%80_%D0%BE%D0%B1%27%D1%94%D0%BA%D1%82%D0%B0" title="Цифровий ідентифікатор об'єкта">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1970066">10.2307/1970066</a>, <a href="/wiki/JSTOR" title="JSTOR">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1970066">1970066</a>, <a href="/wiki/Mathematical_Reviews" title="Mathematical Reviews">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0085342">0085342</a></cite></li></ul></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Отримано з <a dir="ltr" href="https://uk.wikipedia.org/wiki/Функція_експоненційного_типу">https://uk.wikipedia.org/wiki/Функція_експоненційного_типу</a></div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D1%96%D1%97" title="Спеціальна:Категорії">Категорія</a>: <ul><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D1%96%D1%8F:%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D0%B8%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D1%96%D0%B7" title="Категорія:Комплексний аналіз">Комплексний аналіз</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Прихована категорія: <ul><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D1%96%D1%8F:%D0%A1%D1%82%D0%BE%D1%80%D1%96%D0%BD%D0%BA%D0%B8_%D0%B7_%D0%B2%D0%B8%D0%BA%D0%BE%D1%80%D0%B8%D1%81%D1%82%D0%B0%D0%BD%D0%BD%D1%8F%D0%BC_%D1%80%D0%BE%D0%B7%D1%88%D0%B8%D1%80%D0%B5%D0%BD%D0%BD%D1%8F_JsonConfig" title="Категорія:Сторінки з використанням розширення JsonConfig">Сторінки з використанням розширення JsonConfig</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Навігаційне меню</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">Особисті інструменти</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item"><span title="Сторінка користувача для вашої IP-адреси">Ви не увійшли до системи</span></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9C%D0%BE%D1%94_%D0%BE%D0%B1%D0%B3%D0%BE%D0%B2%D0%BE%D1%80%D0%B5%D0%BD%D0%BD%D1%8F" title="Обговорення редагувань з цієї IP-адреси [n]" accesskey="n"><span>Обговорення</span></a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9C%D1%96%D0%B9_%D0%B2%D0%BD%D0%B5%D1%81%D0%BE%D0%BA" title="Список редагувань, зроблених з цієї IP-адреси [y]" accesskey="y"><span>Внесок</span></a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%A1%D1%82%D0%B2%D0%BE%D1%80%D0%B8%D1%82%D0%B8_%D0%BE%D0%B1%D0%BB%D1%96%D0%BA%D0%BE%D0%B2%D0%B8%D0%B9_%D0%B7%D0%B0%D0%BF%D0%B8%D1%81&returnto=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F+%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE+%D1%82%D0%B8%D0%BF%D1%83" title="Пропонуємо створити обліковий запис і увійти в систему; однак, це не обов'язково"><span>Створити обліковий запис</span></a></li><li id="pt-login" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%92%D1%85%D1%96%D0%B4&returnto=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F+%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE+%D1%82%D0%B8%D0%BF%D1%83" title="Заохочуємо Вас увійти в систему, але це необов'язково. [o]" accesskey="o"><span>Увійти</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">Простори назв</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%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83" title="Вміст статті [c]" accesskey="c"><span>Стаття</span></a></li><li id="ca-talk" class="new mw-list-item"><a href="/w/index.php?title=%D0%9E%D0%B1%D0%B3%D0%BE%D0%B2%D0%BE%D1%80%D0%B5%D0%BD%D0%BD%D1%8F:%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83&action=edit&redlink=1" rel="discussion" class="new" title="Обговорення сторінки (ще не написана) [t]" accesskey="t"><span>Обговорення</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">українська</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">Перегляди</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected mw-list-item"><a href="/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83"><span>Читати</span></a></li><li id="ca-ve-edit" class="mw-list-item"><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83&veaction=edit" title="Редагувати цю сторінку [v]" accesskey="v"><span>Редагувати</span></a></li><li id="ca-edit" class="collapsible mw-list-item"><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83&action=edit" title="Редагувати вихідний код сторінки [e]" accesskey="e"><span>Редагувати код</span></a></li><li id="ca-history" class="mw-list-item"><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83&action=history" title="Журнал змін сторінки [h]" accesskey="h"><span>Переглянути історію</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="Більше опцій" > <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">Більше</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 >Пошук</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="Пошук у Вікіпедії" aria-label="Пошук у Вікіпедії" autocapitalize="sentences" title="Шукати у Вікіпедії [f]" accesskey="f" id="searchInput" > <input type="hidden" name="title" value="Спеціальна:Пошук"> <input id="mw-searchButton" class="searchButton mw-fallbackSearchButton" type="submit" name="fulltext" title="Знайти сторінки, що містять зазначений текст" value="Знайти"> <input id="searchButton" class="searchButton" type="submit" name="go" title="Перейти до сторінки, що має точно таку назву (якщо вона існує)" 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%93%D0%BE%D0%BB%D0%BE%D0%B2%D0%BD%D0%B0_%D1%81%D1%82%D0%BE%D1%80%D1%96%D0%BD%D0%BA%D0%B0" title="Перейти на головну сторінку"></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">Навігація</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%93%D0%BE%D0%BB%D0%BE%D0%B2%D0%BD%D0%B0_%D1%81%D1%82%D0%BE%D1%80%D1%96%D0%BD%D0%BA%D0%B0" title="Перейти на головну сторінку [z]" accesskey="z"><span>Головна сторінка</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%9F%D0%BE%D1%82%D0%BE%D1%87%D0%BD%D1%96_%D0%BF%D0%BE%D0%B4%D1%96%D1%97" title="Список поточних подій"><span>Поточні події</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9D%D0%BE%D0%B2%D1%96_%D1%80%D0%B5%D0%B4%D0%B0%D0%B3%D1%83%D0%B2%D0%B0%D0%BD%D0%BD%D1%8F" title="Список останніх змін у цій вікі [r]" accesskey="r"><span>Нові редагування</span></a></li><li id="n-newpages" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9D%D0%BE%D0%B2%D1%96_%D1%81%D1%82%D0%BE%D1%80%D1%96%D0%BD%D0%BA%D0%B8"><span>Нові сторінки</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%92%D0%B8%D0%BF%D0%B0%D0%B4%D0%BA%D0%BE%D0%B2%D0%B0_%D1%81%D1%82%D0%BE%D1%80%D1%96%D0%BD%D0%BA%D0%B0" title="Переглянути випадкову сторінку [x]" accesskey="x"><span>Випадкова стаття</span></a></li> </ul> </div> </nav> <nav id="p-Участь" class="mw-portlet mw-portlet-Участь vector-menu-portal portal vector-menu" aria-labelledby="p-Участь-label" > <h3 id="p-Участь-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Участь</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-portal" class="mw-list-item"><a href="/wiki/%D0%92%D1%96%D0%BA%D1%96%D0%BF%D0%B5%D0%B4%D1%96%D1%8F:%D0%9F%D0%BE%D1%80%D1%82%D0%B0%D0%BB_%D1%81%D0%BF%D1%96%D0%BB%D1%8C%D0%BD%D0%BE%D1%82%D0%B8" title="Про проєкт, про те, що Ви можете зробити, і що де шукати"><span>Портал спільноти</span></a></li><li id="n-tavern" class="mw-list-item"><a href="/wiki/%D0%92%D1%96%D0%BA%D1%96%D0%BF%D0%B5%D0%B4%D1%96%D1%8F:%D0%9A%D0%BD%D0%B0%D0%B9%D0%BF%D0%B0" title="Місце для обговорення більшості питань"><span>Кнайпа</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/%D0%92%D1%96%D0%BA%D1%96%D0%BF%D0%B5%D0%B4%D1%96%D1%8F:%D0%94%D0%BE%D0%B2%D1%96%D0%B4%D0%BA%D0%B0" title="Довідка з проєкту"><span>Довідка</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_uk.wikipedia.org&uselang=uk" title="Підтримайте проєкт"><span>Пожертвувати</span></a></li><li id="n-Сторінка-для-медіа" class="mw-list-item"><a href="/wiki/%D0%92%D1%96%D0%BA%D1%96%D0%BF%D0%B5%D0%B4%D1%96%D1%8F:%D0%A1%D1%82%D0%BE%D1%80%D1%96%D0%BD%D0%BA%D0%B0_%D0%B4%D0%BB%D1%8F_%D0%BC%D0%B5%D0%B4%D1%96%D0%B0"><span>Сторінка для медіа</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">Інструменти</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%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9F%D0%BE%D1%81%D0%B8%D0%BB%D0%B0%D0%BD%D0%BD%D1%8F_%D1%81%D1%8E%D0%B4%D0%B8/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83" title="Перелік усіх сторінок, які посилаються на цю сторінку [j]" accesskey="j"><span>Посилання сюди</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9F%D0%BE%D0%B2%27%D1%8F%D0%B7%D0%B0%D0%BD%D1%96_%D1%80%D0%B5%D0%B4%D0%B0%D0%B3%D1%83%D0%B2%D0%B0%D0%BD%D0%BD%D1%8F/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83" rel="nofollow" title="Останні зміни на сторінках, на які посилається ця сторінка [k]" accesskey="k"><span>Пов'язані редагування</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%81%D1%82%D0%BE%D1%80%D1%96%D0%BD%D0%BA%D0%B8" title="Перелік спеціальних сторінок [q]" accesskey="q"><span>Спеціальні сторінки</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83&action=info" title="Додаткові відомості про цю сторінку"><span>Інформація про сторінку</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:UrlShortener&url=https%3A%2F%2Fuk.wikipedia.org%2Fwiki%2F%25D0%25A4%25D1%2583%25D0%25BD%25D0%25BA%25D1%2586%25D1%2596%25D1%258F_%25D0%25B5%25D0%25BA%25D1%2581%25D0%25BF%25D0%25BE%25D0%25BD%25D0%25B5%25D0%25BD%25D1%2586%25D1%2596%25D0%25B9%25D0%25BD%25D0%25BE%25D0%25B3%25D0%25BE_%25D1%2582%25D0%25B8%25D0%25BF%25D1%2583"><span>Отримати вкорочену URL-адресу</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:QrCode&url=https%3A%2F%2Fuk.wikipedia.org%2Fwiki%2F%25D0%25A4%25D1%2583%25D0%25BD%25D0%25BA%25D1%2586%25D1%2596%25D1%258F_%25D0%25B5%25D0%25BA%25D1%2581%25D0%25BF%25D0%25BE%25D0%25BD%25D0%25B5%25D0%25BD%25D1%2586%25D1%2596%25D0%25B9%25D0%25BD%25D0%25BE%25D0%25B3%25D0%25BE_%25D1%2582%25D0%25B8%25D0%25BF%25D1%2583"><span>Завантажити QR-код</span></a></li> </ul> </div> </nav> <nav id="p-coll-print_export" class="mw-portlet mw-portlet-coll-print_export vector-menu-portal portal vector-menu" aria-labelledby="p-coll-print_export-label" > <h3 id="p-coll-print_export-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Друк/експорт</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9A%D0%BD%D0%B8%D0%B3%D0%B0&bookcmd=book_creator&referer=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F+%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE+%D1%82%D0%B8%D0%BF%D1%83"><span>Створити книгу</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:DownloadAsPdf&page=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83&action=show-download-screen"><span>Завантажити як PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83&printable=yes" title="Версія цієї сторінки для друку [p]" accesskey="p"><span>Версія до друку</span></a></li> </ul> </div> </nav> <nav id="p-wikibase-otherprojects" class="mw-portlet mw-portlet-wikibase-otherprojects vector-menu-portal portal vector-menu" aria-labelledby="p-wikibase-otherprojects-label" > <h3 id="p-wikibase-otherprojects-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">В інших проєктах</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q5421531" title="Посилання на пов’язаний елемент сховища даних [g]" accesskey="g"><span>Елемент Вікіданих</span></a></li> </ul> </div> </nav> <nav id="p-lang" class="mw-portlet mw-portlet-lang vector-menu-portal portal vector-menu" aria-labelledby="p-lang-label" > <h3 id="p-lang-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Іншими мовами</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Exponential_type" title="Exponential type — англійська" lang="en" hreflang="en" data-title="Exponential type" data-language-autonym="English" data-language-local-name="англійська" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Type_exponentiel" title="Type exponentiel — французька" lang="fr" hreflang="fr" data-title="Type exponentiel" data-language-autonym="Français" data-language-local-name="французька" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%8C%87%E6%95%B0%E5%9E%8B" title="指数型 — китайська" lang="zh" hreflang="zh" data-title="指数型" data-language-autonym="中文" data-language-local-name="китайська" class="interlanguage-link-target"><span>中文</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/Q5421531#sitelinks-wikipedia" title="Редагувати міжмовні посилання" class="wbc-editpage">Редагувати посилання</a></span></div> </div> </nav> </div> </div> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-copyright">Текст доступний на умовах ліцензії <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.uk">Creative Commons Attribution-ShareAlike</a>; також можуть діяти додаткові умови. Детальніше див. <a class="external text" href="https://foundation.wikimedia.org/wiki/Policy:Terms_of_Use/uk">Умови використання</a>.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Політика конфіденційності</a></li> <li id="footer-places-about"><a href="/wiki/%D0%92%D1%96%D0%BA%D1%96%D0%BF%D0%B5%D0%B4%D1%96%D1%8F:%D0%9F%D1%80%D0%BE">Про Вікіпедію</a></li> <li id="footer-places-disclaimers"><a href="/wiki/%D0%92%D1%96%D0%BA%D1%96%D0%BF%D0%B5%D0%B4%D1%96%D1%8F:%D0%92%D1%96%D0%B4%D0%BC%D0%BE%D0%B2%D0%B0_%D0%B2%D1%96%D0%B4_%D0%B2%D1%96%D0%B4%D0%BF%D0%BE%D0%B2%D1%96%D0%B4%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D1%81%D1%82%D1%96">Відмова від відповідальності</a></li> <li id="footer-places-contact"><a href="//uk.wikipedia.org/wiki/Вікіпедія:Зворотний_зв%27язок">Зворотний зв'язок</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Кодекс поведінки</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Розробники</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/uk.wikipedia.org">Статистика</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Куки</a></li> <li id="footer-places-mobileview"><a href="//uk.m.wikipedia.org/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Мобільний вигляд</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> <script>(RLQ=window.RLQ||[]).push(function(){mw.log.warn("This page is using the deprecated ResourceLoader module \"codex-search-styles\".\n[1.43] Use a CodexModule with codexComponents to set your specific components used: https://www.mediawiki.org/wiki/Codex#Using_a_limited_subset_of_components");mw.config.set({"wgHostname":"mw-web.codfw.main-f69cdc8f6-9m94g","wgBackendResponseTime":201,"wgPageParseReport":{"limitreport":{"cputime":"0.193","walltime":"0.303","ppvisitednodes":{"value":547,"limit":1000000},"postexpandincludesize":{"value":1728,"limit":2097152},"templateargumentsize":{"value":0,"limit":2097152},"expansiondepth":{"value":5,"limit":100},"expensivefunctioncount":{"value":1,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":6725,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 136.432 1 -total"," 71.82% 97.982 1 Шаблон:Citation"," 16.26% 22.183 1 Шаблон:Harvtxt"," 11.11% 15.155 1 Шаблон:Reflist"]},"scribunto":{"limitreport-timeusage":{"value":"0.072","limit":"10.000"},"limitreport-memusage":{"value":2555355,"limit":52428800}},"cachereport":{"origin":"mw-api-int.codfw.main-6c94b8d879-bvvqs","timestamp":"20241114014413","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"\u0424\u0443\u043d\u043a\u0446\u0456\u044f \u0435\u043a\u0441\u043f\u043e\u043d\u0435\u043d\u0446\u0456\u0439\u043d\u043e\u0433\u043e \u0442\u0438\u043f\u0443","url":"https:\/\/uk.wikipedia.org\/wiki\/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D1%96%D0%B9%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B8%D0%BF%D1%83","sameAs":"http:\/\/www.wikidata.org\/entity\/Q5421531","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q5421531","author":{"@type":"Organization","name":"\u0423\u0447\u0430\u0441\u043d\u0438\u043a\u0438 \u043f\u0440\u043e\u0435\u043a\u0442\u0456\u0432 \u0412\u0456\u043a\u0456\u043c\u0435\u0434\u0456\u0430"},"publisher":{"@type":"Organization","name":"\u0424\u043e\u043d\u0434 \u0412\u0456\u043a\u0456\u043c\u0435\u0434\u0456\u0430","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2020-12-04T13:27:20Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/b\/b2\/ExtremeGaussian.png"}</script> </body> </html>