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Analüütiline funktsioon – Vikipeedia

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Saadaval 37 keeles" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-37" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">37 keelt</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AF%D8%A7%D9%84%D8%A9_%D8%AA%D8%AD%D9%84%D9%8A%D9%84%D9%8A%D8%A9" title="دالة تحليلية – araabia" lang="ar" hreflang="ar" data-title="دالة تحليلية" data-language-autonym="العربية" data-language-local-name="araabia" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D1%96%D1%82%D1%8B%D1%87%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%8B%D1%8F" title="Аналітычная функцыя – valgevene" lang="be" hreflang="be" data-title="Аналітычная функцыя" data-language-autonym="Беларуская" data-language-local-name="valgevene" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D0%B8%D1%82%D0%B8%D1%87%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Аналитична функция – bulgaaria" lang="bg" hreflang="bg" data-title="Аналитична функция" data-language-autonym="Български" data-language-local-name="bulgaaria" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Funci%C3%B3_anal%C3%ADtica" title="Funció analítica – katalaani" lang="ca" hreflang="ca" data-title="Funció analítica" data-language-autonym="Català" data-language-local-name="katalaani" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Analytick%C3%A1_funkce" title="Analytická funkce – tšehhi" lang="cs" hreflang="cs" data-title="Analytická funkce" data-language-autonym="Čeština" data-language-local-name="tšehhi" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Analytische_Funktion" title="Analytische Funktion – saksa" lang="de" hreflang="de" data-title="Analytische Funktion" data-language-autonym="Deutsch" data-language-local-name="saksa" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%BD%CE%B1%CE%BB%CF%85%CF%84%CE%B9%CE%BA%CE%AE_%CF%83%CF%85%CE%BD%CE%AC%CF%81%CF%84%CE%B7%CF%83%CE%B7" title="Αναλυτική συνάρτηση – kreeka" lang="el" hreflang="el" data-title="Αναλυτική συνάρτηση" data-language-autonym="Ελληνικά" data-language-local-name="kreeka" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Analytic_function" title="Analytic function – inglise" lang="en" hreflang="en" data-title="Analytic function" data-language-autonym="English" data-language-local-name="inglise" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Funci%C3%B3n_anal%C3%ADtica" title="Función analítica – hispaania" lang="es" hreflang="es" data-title="Función analítica" data-language-autonym="Español" data-language-local-name="hispaania" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%A7%D8%A8%D8%B9_%D8%AA%D8%AD%D9%84%DB%8C%D9%84%DB%8C" title="تابع تحلیلی – pärsia" lang="fa" hreflang="fa" data-title="تابع تحلیلی" data-language-autonym="فارسی" data-language-local-name="pärsia" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Fonction_analytique" title="Fonction analytique – prantsuse" lang="fr" hreflang="fr" data-title="Fonction analytique" data-language-autonym="Français" data-language-local-name="prantsuse" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%95%B4%EC%84%9D_%ED%95%A8%EC%88%98" title="해석 함수 – korea" lang="ko" hreflang="ko" data-title="해석 함수" data-language-autonym="한국어" data-language-local-name="korea" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B1%D5%B6%D5%A1%D5%AC%D5%AB%D5%BF%D5%AB%D5%AF_%D6%86%D5%B8%D6%82%D5%B6%D5%AF%D6%81%D5%AB%D5%A1" title="Անալիտիկ ֆունկցիա – armeenia" lang="hy" hreflang="hy" data-title="Անալիտիկ ֆունկցիա" data-language-autonym="Հայերեն" data-language-local-name="armeenia" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A5%88%E0%A4%B6%E0%A5%8D%E2%80%8D%E0%A4%B2%E0%A5%87%E0%A4%B7%E0%A4%BF%E0%A4%95_%E0%A4%AB%E0%A4%B2%E0%A4%A8" title="वैश्‍लेषिक फलन – hindi" lang="hi" hreflang="hi" data-title="वैश्‍लेषिक फलन" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/F%C3%A1ga%C3%B0_fall" title="Fágað fall – islandi" lang="is" hreflang="is" data-title="Fágað fall" data-language-autonym="Íslenska" data-language-local-name="islandi" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Funzione_analitica" title="Funzione analitica – itaalia" lang="it" hreflang="it" data-title="Funzione analitica" data-language-autonym="Italiano" data-language-local-name="itaalia" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%90%D7%A0%D7%9C%D7%99%D7%98%D7%99%D7%AA" title="פונקציה אנליטית – heebrea" lang="he" hreflang="he" data-title="פונקציה אנליטית" data-language-autonym="עברית" data-language-local-name="heebrea" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D0%B8%D1%82%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D2%9B_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Аналитикалық функция – kasahhi" lang="kk" hreflang="kk" data-title="Аналитикалық функция" data-language-autonym="Қазақша" data-language-local-name="kasahhi" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D0%B8%D1%82%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D0%BA_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Аналитикалык функция – kirgiisi" lang="ky" hreflang="ky" data-title="Аналитикалык функция" data-language-autonym="Кыргызча" data-language-local-name="kirgiisi" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Analizin%C4%97_funkcija" title="Analizinė funkcija – leedu" lang="lt" hreflang="lt" data-title="Analizinė funkcija" data-language-autonym="Lietuvių" data-language-local-name="leedu" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Val%C3%B3s_analitikus_f%C3%BCggv%C3%A9ny" title="Valós analitikus függvény – ungari" lang="hu" hreflang="hu" data-title="Valós analitikus függvény" data-language-autonym="Magyar" data-language-local-name="ungari" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B5%E0%B4%BF%E0%B4%B6%E0%B5%8D%E0%B4%B2%E0%B5%87%E0%B4%B7%E0%B4%95%E0%B4%AB%E0%B4%B2%E0%B4%A8%E0%B4%82" title="വിശ്ലേഷകഫലനം – malajalami" lang="ml" hreflang="ml" data-title="വിശ്ലേഷകഫലനം" data-language-autonym="മലയാളം" data-language-local-name="malajalami" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Analytische_functie" title="Analytische functie – hollandi" lang="nl" hreflang="nl" data-title="Analytische functie" data-language-autonym="Nederlands" data-language-local-name="hollandi" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%A7%A3%E6%9E%90%E9%96%A2%E6%95%B0" title="解析関数 – jaapani" lang="ja" hreflang="ja" data-title="解析関数" data-language-autonym="日本語" data-language-local-name="jaapani" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Analytisk_funksjon" title="Analytisk funksjon – norra bokmål" lang="nb" hreflang="nb" data-title="Analytisk funksjon" data-language-autonym="Norsk bokmål" data-language-local-name="norra bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Analytisk_funksjon" title="Analytisk funksjon – uusnorra" lang="nn" hreflang="nn" data-title="Analytisk funksjon" data-language-autonym="Norsk nynorsk" data-language-local-name="uusnorra" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Analitik_funksiya" title="Analitik funksiya – usbeki" lang="uz" hreflang="uz" data-title="Analitik funksiya" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="usbeki" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wz%C3%B3r_Taylora#Szereg_Taylora" title="Wzór Taylora – poola" lang="pl" hreflang="pl" data-title="Wzór Taylora" data-language-autonym="Polski" data-language-local-name="poola" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Fun%C3%A7%C3%A3o_anal%C3%ADtica" title="Função analítica – portugali" lang="pt" hreflang="pt" data-title="Função analítica" data-language-autonym="Português" data-language-local-name="portugali" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Func%C8%9Bie_analitic%C4%83" title="Funcție analitică – rumeenia" lang="ro" hreflang="ro" data-title="Funcție analitică" data-language-autonym="Română" data-language-local-name="rumeenia" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D0%B8%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Аналитическая функция – vene" lang="ru" hreflang="ru" data-title="Аналитическая функция" data-language-autonym="Русский" data-language-local-name="vene" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Analyyttinen_funktio" title="Analyyttinen funktio – soome" lang="fi" hreflang="fi" data-title="Analyyttinen funktio" data-language-autonym="Suomi" data-language-local-name="soome" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AE%95%E0%AF%81%E0%AE%AE%E0%AF%81%E0%AE%B1%E0%AF%88%E0%AE%9A%E0%AF%8D_%E0%AE%9A%E0%AE%BE%E0%AE%B0%E0%AF%8D%E0%AE%AA%E0%AF%81" title="பகுமுறைச் சார்பு – tamili" lang="ta" hreflang="ta" data-title="பகுமுறைச் சார்பு" data-language-autonym="தமிழ்" data-language-local-name="tamili" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%C3%A0m_gi%E1%BA%A3i_t%C3%ADch" title="Hàm giải tích – vietnami" lang="vi" hreflang="vi" data-title="Hàm giải tích" data-language-autonym="Tiếng Việt" data-language-local-name="vietnami" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Analitik_fonksiyon" title="Analitik fonksiyon – türgi" lang="tr" hreflang="tr" data-title="Analitik fonksiyon" data-language-autonym="Türkçe" data-language-local-name="türgi" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a 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href="/w/index.php?title=Koonduv_rida&amp;action=edit&amp;redlink=1" class="new" title="Koonduv rida (pole veel kirjutatud)">koonduva</a> <a href="/wiki/Astmerida" title="Astmerida">astmereana</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">&#91;</span>1<span class="cite-bracket">&#93;</span></a></sup> </p><p>Eristatakse <b>reaalmuutuja analüütilisi funktsioone</b> ja <b>kompleksmuutuja analüütilisi funktsioone</b>. Analüütilisus on <a href="/wiki/Kompleksmuutuja_funktsioon" title="Kompleksmuutuja funktsioon">kompleksmuutuja funktsioonide</a> mõnevõrra tugevam omadus kui <a href="/wiki/Reaalmuutuja_funktsioon" title="Reaalmuutuja funktsioon">reaalmuutuja funktsioonide</a> jaoks. Kompleksmuutuja funktsioon on analüütiline <a href="/wiki/Parajasti_siis,_kui" title="Parajasti siis, kui">parajasti siis, kui</a> see on <a href="/wiki/Regulaarne_funktsioon" title="Regulaarne funktsioon">regulaarne</a>. </p><p>Analüütilised funktsioonid on <a href="/wiki/L%C3%B5pmatult_diferentseeruv_funktsioon" title="Lõpmatult diferentseeruv funktsioon">lõpmatult diferentseeruvad</a>. Funktsioon on analüütiline <a href="/wiki/Parajasti_siis,_kui" title="Parajasti siis, kui">parajasti siis, kui</a> see võrdub oma <a href="/wiki/Taylori_rida" title="Taylori rida">Taylori reaga</a> iga punkti mõnes ümbruses. </p> <div class="mw-heading mw-heading2"><h2 id="Definitsioon">Definitsioon</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Anal%C3%BC%C3%BCtiline_funktsioon&amp;veaction=edit&amp;section=1" title="Muuda alaosa &quot;Definitsioon&quot;" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Anal%C3%BC%C3%BCtiline_funktsioon&amp;action=edit&amp;section=1" title="Muuda alaosa &quot;Definitsioon&quot; lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Reaalmuutuja funktsioon ƒ on <i>analüütiline</i> <a href="/wiki/Reaalsirge" title="Reaalsirge">reaalsirgel</a> asuval <a href="/wiki/Lahtine_hulk" title="Lahtine hulk">lahtisel hulgal</a> <i>D</i>, kui iga hulka <i>D</i> kuuluva punkti <i>x</i><sub>0</sub> jaoks kehtib </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 {\begin{aligned}f(x)&amp;=\sum _{n=0}^{\infty }a_{n}\left(x-x_{0}\right)^{n}\\&amp;=a_{0}+a_{1}(x-x_{0})+a_{2}(x-x_{0})^{2}+a_{3}(x-x_{0})^{3}+\cdots \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">&#x221E;<!-- ∞ --></mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>&#x2212;<!-- − --></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>&#x2212;<!-- − --></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>&#x2212;<!-- − --></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>&#x2212;<!-- − --></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mo>&#x22EF;<!-- ⋯ --></mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f(x)&amp;=\sum _{n=0}^{\infty }a_{n}\left(x-x_{0}\right)^{n}\\&amp;=a_{0}+a_{1}(x-x_{0})+a_{2}(x-x_{0})^{2}+a_{3}(x-x_{0})^{3}+\cdots \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1e43a68478ea6dc9159a3ad4a38c50a949131c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:58.687ex; height:10.176ex;" alt="{\displaystyle {\begin{aligned}f(x)&amp;=\sum _{n=0}^{\infty }a_{n}\left(x-x_{0}\right)^{n}\\&amp;=a_{0}+a_{1}(x-x_{0})+a_{2}(x-x_{0})^{2}+a_{3}(x-x_{0})^{3}+\cdots \end{aligned}}}"></span></dd></dl> <p>kus kordajad <i>a</i><sub>0</sub>, <i>a</i><sub>1</sub>, ... on <a href="/wiki/Reaalarvud" class="mw-redirect" title="Reaalarvud">reaalarvud</a> ja antud <a href="/wiki/Astmerida" title="Astmerida">astmerida</a> <a href="/w/index.php?title=Koonduv_rida&amp;action=edit&amp;redlink=1" class="new" title="Koonduv rida (pole veel kirjutatud)">koondub</a> funktsiooniks ƒ(<i>x</i>) iga punkti <i>x</i> korral, mis kuulub punkti <i>x</i><sub>0</sub> mõnda ümbrusesse. </p><p>Alternatiivselt võib analüütilise funktsiooni defineerida kui <a href="/wiki/L%C3%B5pmatult_diferentseeruv_funktsioon" title="Lõpmatult diferentseeruv funktsioon">lõpmatult diferentseeruva funktsiooni</a>, mille jaoks <a href="/wiki/Taylori_rida" title="Taylori rida">Taylori rida</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 T(x)=\sum _{n=0}^{\infty }{\frac {f^{(n)}(x_{0})}{n!}}(x-x_{0})^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">&#x221E;<!-- ∞ --></mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>&#x2212;<!-- − --></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(x)=\sum _{n=0}^{\infty }{\frac {f^{(n)}(x_{0})}{n!}}(x-x_{0})^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94ed8d65a0565a4eace4df21c0410a4adc657112" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.045ex; height:7.176ex;" alt="{\displaystyle T(x)=\sum _{n=0}^{\infty }{\frac {f^{(n)}(x_{0})}{n!}}(x-x_{0})^{n}}"></span></dd></dl> <p>igas selle funktsiooni <a href="/wiki/M%C3%A4%C3%A4ramispiirkond" title="Määramispiirkond">määramispiirkonda</a> kuuluvas punktis <i>x</i><sub>0</sub> koondub funktsiooniks ƒ(<i>x</i>) iga punkti <i>x</i><sub>0</sub> mõnda ümbrusesse kuuluva punkti <i>x</i> korral. </p><p>Öeldakse, et funktsiooni ƒ on analüütiline punktis <i>x</i>, kui leidub punkti <i>x</i> ümbrus, millel ƒ on analüütiline. </p><p><b>Kompleksmuutuja analüütilise funktsiooni</b> defineerimiseks tuleb ülalantud definitsioonis asendada reaalarvud <a href="/wiki/Kompleksarv" title="Kompleksarv">kompleksarvudega</a> ja reaalsirge <a href="/wiki/Komplekstasand" title="Komplekstasand">komplekstasandiga</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Vaata_ka">Vaata ka</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Anal%C3%BC%C3%BCtiline_funktsioon&amp;veaction=edit&amp;section=2" title="Muuda alaosa &quot;Vaata ka&quot;" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Anal%C3%BC%C3%BCtiline_funktsioon&amp;action=edit&amp;section=2" title="Muuda alaosa &quot;Vaata ka&quot; lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=Anal%C3%BC%C3%BCtiline_j%C3%A4tkamine&amp;action=edit&amp;redlink=1" class="new" title="Analüütiline jätkamine (pole veel kirjutatud)">Analüütiline jätkamine</a></li> <li><a href="/wiki/Regulaarne_funktsioon" title="Regulaarne funktsioon">Regulaarne funktsioon</a></li> <li><a href="/wiki/L%C3%B5pmatult_diferentseeruv_funktsioon" title="Lõpmatult diferentseeruv funktsioon">Lõpmatult diferentseeruv funktsioon</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Viited">Viited</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Anal%C3%BC%C3%BCtiline_funktsioon&amp;veaction=edit&amp;section=3" title="Muuda alaosa &quot;Viited&quot;" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Anal%C3%BC%C3%BCtiline_funktsioon&amp;action=edit&amp;section=3" title="Muuda alaosa &quot;Viited&quot; lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <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"> Kaasik, Ü. (2002). <i>Matemaatikaleksikon</i>. Tartu. </span> </li> </ol></div></div> <!-- NewPP limit report Parsed by mw‐web.eqiad.main‐6c869c4f4d‐gzpdl Cached time: 20241121082118 Cache expiry: 2592000 Reduced expiry: false Complications: [] CPU time usage: 0.015 seconds Real time usage: 0.076 seconds Preprocessor visited node count: 107/1000000 Post‐expand include size: 148/2097152 bytes Template argument size: 0/2097152 bytes Highest expansion depth: 5/100 Expensive parser function count: 0/500 Unstrip recursion depth: 0/20 Unstrip post‐expand size: 478/5000000 bytes Number of Wikibase entities loaded: 0/400 --> <!-- Transclusion expansion time report (%,ms,calls,template) 100.00% 3.305 1 Mall:Viited 100.00% 3.305 1 -total --> <!-- Saved in parser cache with key etwiki:pcache:294454:|#|:idhash:canonical and timestamp 20241121082118 and revision id 3798407. 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