Finite difference - Wikipedia

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<span class="vector-toc-numb">2</span> <span>Relation with derivatives</span> </div> </a> <ul id="toc-Relation_with_derivatives-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Higher-order_differences" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Higher-order_differences"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Higher-order differences</span> </div> </a> <ul id="toc-Higher-order_differences-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polynomials" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Polynomials"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Polynomials</span> </div> </a> <button aria-controls="toc-Polynomials-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon 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id="toc-Inductive_step-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Application" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Application"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Application</span> </div> </a> <ul id="toc-Application-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Arbitrarily_sized_kernels" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Arbitrarily_sized_kernels"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Arbitrarily sized kernels</span> </div> </a> <button aria-controls="toc-Arbitrarily_sized_kernels-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Arbitrarily sized kernels subsection</span> </button> <ul id="toc-Arbitrarily_sized_kernels-sublist" class="vector-toc-list"> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-In_differential_equations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_differential_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>In differential equations</span> </div> </a> <ul id="toc-In_differential_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Newton's_series" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Newton's_series"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Newton's series</span> </div> </a> <ul id="toc-Newton's_series-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Calculus_of_finite_differences" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Calculus_of_finite_differences"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Calculus of finite differences</span> </div> </a> <button aria-controls="toc-Calculus_of_finite_differences-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Calculus of finite differences subsection</span> </button> <ul id="toc-Calculus_of_finite_differences-sublist" class="vector-toc-list"> <li id="toc-Rules_for_calculus_of_finite_difference_operators" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rules_for_calculus_of_finite_difference_operators"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Rules for calculus of finite difference operators</span> </div> </a> <ul id="toc-Rules_for_calculus_of_finite_difference_operators-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Generalizations</span> </div> </a> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multivariate_finite_differences" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Multivariate_finite_differences"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Multivariate finite differences</span> </div> </a> <ul id="toc-Multivariate_finite_differences-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" 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href="https://az.wikipedia.org/wiki/F%C9%99rq_operatoru" title="Fərq operatoru – Azerbaijani" lang="az" hreflang="az" data-title="Fərq operatoru" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D1%80%D0%B0%D0%B9%D0%BD%D0%B0_%D1%80%D0%B0%D0%B7%D0%BB%D0%B8%D0%BA%D0%B0" title="Крайна разлика – Bulgarian" lang="bg" hreflang="bg" data-title="Крайна разлика" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Difer%C3%A8ncia_finita" title="Diferència finita – Catalan" lang="ca" hreflang="ca" data-title="Diferència finita" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CE%B5%CF%80%CE%B5%CF%81%CE%B1%CF%83%CE%BC%CE%AD%CE%BD%CE%B7_%CE%B4%CE%B9%CE%B1%CF%86%CE%BF%CF%81%CE%AC" title="Πεπερασμένη διαφορά – Greek" lang="el" hreflang="el" data-title="Πεπερασμένη διαφορά" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Diferencia_finita" title="Diferencia finita – Spanish" lang="es" hreflang="es" data-title="Diferencia finita" data-language-autonym="Español" data-language-local-name="Spanish" 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%D9%81%D8%A7%D8%B6%D9%84_%D9%85%D8%AD%D8%AF%D9%88%D8%AF" title="تفاضل محدود – Persian" lang="fa" hreflang="fa" data-title="تفاضل محدود" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Diff%C3%A9rence_finie" title="Différence finie – French" lang="fr" hreflang="fr" data-title="Différence finie" data-language-autonym="Français" data-language-local-name="French" 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/%EC%9C%A0%ED%95%9C%EC%B0%A8%EB%B6%84" title="유한차분 – Korean" lang="ko" hreflang="ko" data-title="유한차분" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8E%D5%A5%D6%80%D5%BB%D5%A1%D5%BE%D5%B8%D6%80_%D5%BF%D5%A1%D6%80%D5%A2%D5%A5%D6%80%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6%D5%B6%D5%A5%D6%80%D5%AB_%D5%B0%D5%A1%D5%B7%D5%AB%D5%BE" title="Վերջավոր տարբերությունների հաշիվ – Armenian" lang="hy" hreflang="hy" data-title="Վերջավոր տարբերությունների հաշիվ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A4%B0%E0%A4%BF%E0%A4%AE%E0%A4%BF%E0%A4%A4_%E0%A4%85%E0%A4%82%E0%A4%A4%E0%A4%B0" title="परिमित अंतर – Hindi" lang="hi" hreflang="hi" data-title="परिमित अंतर" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Differenza_finita" title="Differenza finita – Italian" lang="it" hreflang="it" data-title="Differenza finita" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A2%D3%A9%D0%B3%D1%81%D0%B3%D3%A9%D0%BB%D3%A9%D0%B3_%D1%8F%D0%BB%D0%B3%D0%B0%D0%B2%D0%B0%D1%80" title="Төгсгөлөг ялгавар – Mongolian" lang="mn" hreflang="mn" data-title="Төгсгөлөг ялгавар" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Eindige_differentie" title="Eindige differentie – Dutch" lang="nl" hreflang="nl" data-title="Eindige differentie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%9C%89%E9%99%90%E5%B7%AE%E5%88%86" title="有限差分 – Japanese" lang="ja" hreflang="ja" data-title="有限差分" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rachunek_r%C3%B3%C5%BCnicowy" title="Rachunek różnicowy – Polish" lang="pl" hreflang="pl" data-title="Rachunek różnicowy" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Operador_de_diferen%C3%A7a" title="Operador de diferença – Portuguese" lang="pt" hreflang="pt" data-title="Operador de diferença" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BD%D0%B5%D1%87%D0%BD%D1%8B%D0%B5_%D1%80%D0%B0%D0%B7%D0%BD%D0%BE%D1%81%D1%82%D0%B8" title="Конечные разности – Russian" lang="ru" hreflang="ru" data-title="Конечные разности" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BD%D0%B0%D1%87%D0%BD%D0%B0_%D1%80%D0%B0%D0%B7%D0%BB%D0%B8%D0%BA%D0%B0" title="Коначна разлика – Serbian" lang="sr" hreflang="sr" data-title="Коначна разлика" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Differensoperator" title="Differensoperator – Swedish" lang="sv" hreflang="sv" data-title="Differensoperator" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Sonlu_fark" title="Sonlu fark – Turkish" lang="tr" hreflang="tr" data-title="Sonlu fark" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%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 – Ukrainian" lang="uk" hreflang="uk" data-title="Скінченні різниці" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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<div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p class="mw-empty-elt"> </p> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Discrete analog of a derivative</div> <p> A <b>finite difference</b> is a mathematical expression of the form <span class="texhtml"><i>f</i>(<i>x</i> + <i>b</i>) − <i>f</i>(<i>x</i> + <i>a</i>)</span>. Finite differences (or the associated <a href="/wiki/Difference_quotient" title="Difference quotient">difference quotients</a>) are often used as approximations of derivatives, such as in <a href="/wiki/Numerical_differentiation" title="Numerical differentiation">numerical differentiation</a>. </p><p>The <a href="/wiki/Difference_operator" class="mw-redirect" title="Difference operator">difference operator</a>, commonly denoted <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 \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" /></span>, is the <a href="/wiki/Operator_(mathematics)" title="Operator (mathematics)">operator</a> that maps a function <span class="texhtml mvar" style="font-style:italic;">f</span> to the function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta [f]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta [f]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/978a8399050eeaf584e6c0f7a57196f57d9bdb71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.508ex; height:2.843ex;" alt="{\displaystyle \Delta [f]}" /></span> defined by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta [f](x)=f(x+1)-f(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta [f](x)=f(x+1)-f(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15da4d892ccab64158a6e556b809b3f365da7057" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.071ex; height:2.843ex;" alt="{\displaystyle \Delta [f](x)=f(x+1)-f(x).}" /></span> A <a href="/wiki/Difference_equation" class="mw-redirect" title="Difference equation">difference equation</a> is a <a href="/wiki/Functional_equation" title="Functional equation">functional equation</a> that involves the finite difference operator in the same way as a <a href="/wiki/Differential_equation" title="Differential equation">differential equation</a> involves <a href="/wiki/Derivative" title="Derivative">derivatives</a>. There are many similarities between difference equations and differential equations. Certain <a href="/wiki/Recurrence_relation#Relationship_to_difference_equations_narrowly_defined" title="Recurrence relation">recurrence relations</a> can be written as difference equations by replacing iteration notation with finite differences. </p><p>In <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerical analysis</a>, finite differences are widely used for <a href="#Relation_with_derivatives">approximating derivatives</a>, and the term "finite difference" is often used as an abbreviation of "finite difference approximation of derivatives".<sup id="cite_ref-WilmottHowison1995_1-0" class="reference"><a href="#cite_note-WilmottHowison1995-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Olver2013_2-0" class="reference"><a href="#cite_note-Olver2013-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Chaudhry2007_3-0" class="reference"><a href="#cite_note-Chaudhry2007-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Finite differences were introduced by <a href="/wiki/Brook_Taylor" title="Brook Taylor">Brook Taylor</a> in 1715 and have also been studied as abstract self-standing mathematical objects in works by <a href="/wiki/George_Boole" title="George Boole">George Boole</a> (1860), <a href="/wiki/L._M._Milne-Thomson" title="L. M. Milne-Thomson">L. M. Milne-Thomson</a> (1933), and <a href="/w/index.php?title=K%C3%A1roly_Jordan&action=edit&redlink=1" class="new" title="Károly Jordan (page does not exist)">Károly Jordan</a><span class="noprint" style="font-size:85%; font-style: normal;"> [<a href="https://de.wikipedia.org/wiki/K%C3%A1roly_Jordan" class="extiw" title="de:Károly Jordan">de</a>]</span> (1939). Finite differences trace their origins back to one of <a href="/wiki/Jost_B%C3%BCrgi" title="Jost Bürgi">Jost Bürgi</a>'s algorithms (<abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 1592</span>) and work by others including <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a>. The formal calculus of finite differences can be viewed as an alternative to the <a href="/wiki/Calculus" title="Calculus">calculus</a> of <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimals</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Basic_types">Basic types</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_difference&action=edit&section=1" title="Edit section: Basic types"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Finite_difference_method.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Finite_difference_method.svg/330px-Finite_difference_method.svg.png" decoding="async" width="307" height="207" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Finite_difference_method.svg/461px-Finite_difference_method.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/90/Finite_difference_method.svg/614px-Finite_difference_method.svg.png 2x" data-file-width="496" data-file-height="334" /></a><figcaption>The three types of the finite differences. The central difference about <i>x</i> gives the best approximation of the derivative of the function at <i>x</i>.</figcaption></figure> <p>Three basic types are commonly considered: <i>forward</i>, <i>backward</i>, and <i>central</i> finite differences.<sup id="cite_ref-WilmottHowison1995_1-1" class="reference"><a href="#cite_note-WilmottHowison1995-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Olver2013_2-1" class="reference"><a href="#cite_note-Olver2013-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Chaudhry2007_3-1" class="reference"><a href="#cite_note-Chaudhry2007-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>A <b><style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="forward_difference"></span><span class="vanchor-text">forward difference</span></span></b>, denoted <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 \Delta _{h}[f],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{h}[f],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df1b2280f8d8410699154ebec9166fc24f713e55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.334ex; height:2.843ex;" alt="{\displaystyle \Delta _{h}[f],}" /></span> of a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> <span class="texhtml mvar" style="font-style:italic;">f</span> is a function defined as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{h}[f](x)=f(x+h)-f(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{h}[f](x)=f(x+h)-f(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b4b1b463ac6a532d0089dad4ef7f7ec1a91acb9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.426ex; height:2.843ex;" alt="{\displaystyle \Delta _{h}[f](x)=f(x+h)-f(x).}" /></span> </p><p>Depending on the application, the spacing <span class="texhtml mvar" style="font-style:italic;">h</span> may be variable or constant. When omitted, <span class="texhtml mvar" style="font-style:italic;">h</span> is taken to be 1; that is, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta [f](x)=\Delta _{1}[f](x)=f(x+1)-f(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta [f](x)=\Delta _{1}[f](x)=f(x+1)-f(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c34fb96bcb57be4378a378d93aa636c18f9bda94" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.871ex; height:2.843ex;" alt="{\displaystyle \Delta [f](x)=\Delta _{1}[f](x)=f(x+1)-f(x).}" /></span> </p><p>A <b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509" /><span class="vanchor"><span id="backward_difference"></span><span class="vanchor-text">backward difference</span></span></b> uses the function values at <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml"><i>x</i> − <i>h</i></span>, instead of the values at <span class="texhtml"><i>x</i> + <i>h</i></span> and <span class="texhtml mvar" style="font-style:italic;">x</span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla _{h}[f](x)=f(x)-f(x-h)=\Delta _{h}[f](x-h).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{h}[f](x)=f(x)-f(x-h)=\Delta _{h}[f](x-h).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a100757e5e1977e4c05a91e35bacf18bc054954c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.53ex; height:2.843ex;" alt="{\displaystyle \nabla _{h}[f](x)=f(x)-f(x-h)=\Delta _{h}[f](x-h).}" /></span> </p><p>Finally, the <b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509" /><span class="vanchor"><span id="central_difference"></span><span class="vanchor-text">central difference</span></span></b> is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{h}[f](x)=f(x+{\tfrac {h}{2}})-f(x-{\tfrac {h}{2}})=\Delta _{h/2}[f](x)+\nabla _{h/2}[f](x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>h</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>h</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msub> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msub> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{h}[f](x)=f(x+{\tfrac {h}{2}})-f(x-{\tfrac {h}{2}})=\Delta _{h/2}[f](x)+\nabla _{h/2}[f](x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2499836d521433992592446e489e3f7f5ff77de" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:59.469ex; height:3.676ex;" alt="{\displaystyle \delta _{h}[f](x)=f(x+{\tfrac {h}{2}})-f(x-{\tfrac {h}{2}})=\Delta _{h/2}[f](x)+\nabla _{h/2}[f](x).}" /></span> </p> <div class="mw-heading mw-heading2"><h2 id="Relation_with_derivatives">Relation with derivatives</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_difference&action=edit&section=2" title="Edit section: Relation with derivatives"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Difference_quotient" title="Difference quotient">difference quotient</a></div> <p><span class="anchor" id="finite_difference_approximation"></span> The approximation of <a href="/wiki/Derivative" title="Derivative">derivatives</a> by finite differences plays a central role in <a href="/wiki/Finite_difference_method" title="Finite difference method">finite difference methods</a> for the <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerical</a> solution of <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a>, especially <a href="/wiki/Boundary_value_problem" title="Boundary value problem">boundary value problems</a>. </p><p>The <a href="/wiki/Derivative" title="Derivative">derivative</a> of a function <span class="texhtml mvar" style="font-style:italic;">f</span> at a point <span class="texhtml mvar" style="font-style:italic;">x</span> is defined by the <a href="/wiki/Limit_of_a_function" title="Limit of a function">limit</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(x)=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'(x)=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f07ddc190e96e17d5d7e1ab262e8e5baf865949" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:29.38ex; height:5.843ex;" alt="{\displaystyle f'(x)=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}.}" /></span> </p><p>If <span class="texhtml mvar" style="font-style:italic;">h</span> has a fixed (non-zero) value instead of approaching zero, then the right-hand side of the above equation would be written <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {f(x+h)-f(x)}{h}}={\frac {\Delta _{h}[f](x)}{h}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {f(x+h)-f(x)}{h}}={\frac {\Delta _{h}[f](x)}{h}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/123e4ca301fc04c05f9129394da0feb303394251" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:30.099ex; height:5.843ex;" alt="{\displaystyle {\frac {f(x+h)-f(x)}{h}}={\frac {\Delta _{h}[f](x)}{h}}.}" /></span> </p><p>Hence, the forward difference divided by <span class="texhtml mvar" style="font-style:italic;">h</span> approximates the derivative when <span class="texhtml mvar" style="font-style:italic;">h</span> is small. The error in this approximation can be derived from <a href="/wiki/Taylor%27s_theorem" title="Taylor's theorem">Taylor's theorem</a>. Assuming that <span class="texhtml mvar" style="font-style:italic;">f</span> is twice differentiable, we have <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\Delta _{h}[f](x)}{h}}-f'(x)=o(h)\to 0\quad {\text{as }}h\to 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>−</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mn>0</mn> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>as </mtext> </mrow> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\Delta _{h}[f](x)}{h}}-f'(x)=o(h)\to 0\quad {\text{as }}h\to 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5f5875b9160e0ba4a348d43bf66e303847c76d8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:41.542ex; height:5.843ex;" alt="{\displaystyle {\frac {\Delta _{h}[f](x)}{h}}-f'(x)=o(h)\to 0\quad {\text{as }}h\to 0.}" /></span> </p><p>The same formula holds for the backward difference: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\nabla _{h}[f](x)}{h}}-f'(x)=o(h)\to 0\quad {\text{as }}h\to 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>−</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mn>0</mn> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>as </mtext> </mrow> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\nabla _{h}[f](x)}{h}}-f'(x)=o(h)\to 0\quad {\text{as }}h\to 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce60536b1d454af6d06ae1c823f4c304ad3f8605" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:41.542ex; height:5.843ex;" alt="{\displaystyle {\frac {\nabla _{h}[f](x)}{h}}-f'(x)=o(h)\to 0\quad {\text{as }}h\to 0.}" /></span> </p><p>However, the central (also called centered) difference yields a more accurate approximation. If <span class="texhtml mvar" style="font-style:italic;">f</span> is three times differentiable, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\delta _{h}[f](x)}{h}}-f'(x)=o\left(h^{2}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>−</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>o</mi> <mrow> <mo>(</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\delta _{h}[f](x)}{h}}-f'(x)=o\left(h^{2}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96e009427707a4ee9bb06e51bf683b1a505dd929" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.914ex; height:5.843ex;" alt="{\displaystyle {\frac {\delta _{h}[f](x)}{h}}-f'(x)=o\left(h^{2}\right).}" /></span> </p><p>The main problem<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (December 2017)">citation needed</span></a></i>]</sup> with the central difference method, however, is that oscillating functions can yield zero derivative. If <span class="texhtml"><i>f</i>(<i>nh</i>) = 1</span> for <span class="texhtml mvar" style="font-style:italic;">n</span> odd, and <span class="texhtml"><i>f</i>(<i>nh</i>) = 2</span> for <span class="texhtml mvar" style="font-style:italic;">n</span> even, then <span class="texhtml"><i>f</i>′(<i>nh</i>) = 0</span> if it is calculated with the <a href="/wiki/Central_difference_scheme" class="mw-redirect" title="Central difference scheme">central difference scheme</a>. This is particularly troublesome if the domain of <span class="texhtml mvar" style="font-style:italic;">f</span> is discrete. See also <a href="/wiki/Symmetric_derivative" title="Symmetric derivative">Symmetric derivative</a>. </p><p>Authors for whom finite differences mean finite difference approximations define the forward/backward/central differences as the quotients given in this section (instead of employing the definitions given in the previous section).<sup id="cite_ref-WilmottHowison1995_1-2" class="reference"><a href="#cite_note-WilmottHowison1995-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Olver2013_2-2" class="reference"><a href="#cite_note-Olver2013-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Chaudhry2007_3-2" class="reference"><a href="#cite_note-Chaudhry2007-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Higher-order_differences">Higher-order differences</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_difference&action=edit&section=3" title="Edit section: Higher-order differences"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Finite_difference" title="Special:EditPage/Finite difference">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a> in this section. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Finite+difference%22">"Finite difference"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Finite+difference%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Finite+difference%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Finite+difference%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Finite+difference%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Finite+difference%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">July 2018</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In an analogous way, one can obtain finite difference approximations to higher order derivatives and differential operators. For example, by using the above central difference formula for <span class="texhtml"><i>f</i>′(<i>x</i> + <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"><i>h</i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>)</span> and <span class="texhtml"><i>f</i>′(<i>x</i> − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num"><i>h</i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>)</span> and applying a central difference formula for the derivative of <span class="texhtml"><i>f</i>′</span> at <span class="texhtml mvar" style="font-style:italic;">x</span>, we obtain the central difference approximation of the second derivative of <span class="texhtml mvar" style="font-style:italic;">f</span>: </p> <dl><dt>Second-order central</dt> <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''(x)\approx {\frac {\delta _{h}^{2}[f](x)}{h^{2}}}={\frac {{\frac {f(x+h)-f(x)}{h}}-{\frac {f(x)-f(x-h)}{h}}}{h}}={\frac {f(x+h)-2f(x)+f(x-h)}{h^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f''(x)\approx {\frac {\delta _{h}^{2}[f](x)}{h^{2}}}={\frac {{\frac {f(x+h)-f(x)}{h}}-{\frac {f(x)-f(x-h)}{h}}}{h}}={\frac {f(x+h)-2f(x)+f(x-h)}{h^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e2aa8bf4fbaf676294b73c0f8028b86e675f0e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:78.44ex; height:7.343ex;" alt="{\displaystyle f''(x)\approx {\frac {\delta _{h}^{2}[f](x)}{h^{2}}}={\frac {{\frac {f(x+h)-f(x)}{h}}-{\frac {f(x)-f(x-h)}{h}}}{h}}={\frac {f(x+h)-2f(x)+f(x-h)}{h^{2}}}.}" /></span></dd></dl> <p>Similarly we can apply other differencing formulas in a recursive manner. </p> <dl><dt>Second order forward</dt> <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''(x)\approx {\frac {\Delta _{h}^{2}[f](x)}{h^{2}}}={\frac {{\frac {f(x+2h)-f(x+h)}{h}}-{\frac {f(x+h)-f(x)}{h}}}{h}}={\frac {f(x+2h)-2f(x+h)+f(x)}{h^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f''(x)\approx {\frac {\Delta _{h}^{2}[f](x)}{h^{2}}}={\frac {{\frac {f(x+2h)-f(x+h)}{h}}-{\frac {f(x+h)-f(x)}{h}}}{h}}={\frac {f(x+2h)-2f(x+h)+f(x)}{h^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76c1f08384bb48e0365b40faeac78f83b7b2e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:83.553ex; height:7.343ex;" alt="{\displaystyle f''(x)\approx {\frac {\Delta _{h}^{2}[f](x)}{h^{2}}}={\frac {{\frac {f(x+2h)-f(x+h)}{h}}-{\frac {f(x+h)-f(x)}{h}}}{h}}={\frac {f(x+2h)-2f(x+h)+f(x)}{h^{2}}}.}" /></span></dd> <dt>Second order backward</dt> <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''(x)\approx {\frac {\nabla _{h}^{2}[f](x)}{h^{2}}}={\frac {{\frac {f(x)-f(x-h)}{h}}-{\frac {f(x-h)-f(x-2h)}{h}}}{h}}={\frac {f(x)-2f(x-h)+f(x-2h)}{h^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f''(x)\approx {\frac {\nabla _{h}^{2}[f](x)}{h^{2}}}={\frac {{\frac {f(x)-f(x-h)}{h}}-{\frac {f(x-h)-f(x-2h)}{h}}}{h}}={\frac {f(x)-2f(x-h)+f(x-2h)}{h^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84b0f013627aae29797cf6a175b1cb810b2e84ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:83.553ex; height:7.343ex;" alt="{\displaystyle f''(x)\approx {\frac {\nabla _{h}^{2}[f](x)}{h^{2}}}={\frac {{\frac {f(x)-f(x-h)}{h}}-{\frac {f(x-h)-f(x-2h)}{h}}}{h}}={\frac {f(x)-2f(x-h)+f(x-2h)}{h^{2}}}.}" /></span></dd></dl> <p>More generally, the <b><span class="texhtml mvar" style="font-style:italic;">n</span>-th order forward, backward, and central</b> differences are given by, respectively, </p> <dl><dt>Forward</dt> <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 \Delta _{h}^{n}[f](x)=\sum _{i=0}^{n}(-1)^{n-i}{\binom {n}{i}}f{\bigl (}x+ih{\bigr )},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>i</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>i</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{h}^{n}[f](x)=\sum _{i=0}^{n}(-1)^{n-i}{\binom {n}{i}}f{\bigl (}x+ih{\bigr )},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b169fd39e2dc70b58dcb348f61a0ea08d04f82a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.346ex; height:6.843ex;" alt="{\displaystyle \Delta _{h}^{n}[f](x)=\sum _{i=0}^{n}(-1)^{n-i}{\binom {n}{i}}f{\bigl (}x+ih{\bigr )},}" /></span></dd> <dt>Backward</dt> <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 \nabla _{h}^{n}[f](x)=\sum _{i=0}^{n}(-1)^{i}{\binom {n}{i}}f(x-ih),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>i</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>i</mi> <mi>h</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{h}^{n}[f](x)=\sum _{i=0}^{n}(-1)^{i}{\binom {n}{i}}f(x-ih),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2d02b454bc5bec308d6b40c4b1d6fd0e5177a6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.761ex; height:6.843ex;" alt="{\displaystyle \nabla _{h}^{n}[f](x)=\sum _{i=0}^{n}(-1)^{i}{\binom {n}{i}}f(x-ih),}" /></span></dd> <dt>Central</dt> <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 \delta _{h}^{n}[f](x)=\sum _{i=0}^{n}(-1)^{i}{\binom {n}{i}}f\left(x+\left({\frac {n}{2}}-i\right)h\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>i</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>i</mi> </mrow> <mo>)</mo> </mrow> <mi>h</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{h}^{n}[f](x)=\sum _{i=0}^{n}(-1)^{i}{\binom {n}{i}}f\left(x+\left({\frac {n}{2}}-i\right)h\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfd7b5b17a0cd910ace9cca88d7093843ccd6799" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.853ex; height:6.843ex;" alt="{\displaystyle \delta _{h}^{n}[f](x)=\sum _{i=0}^{n}(-1)^{i}{\binom {n}{i}}f\left(x+\left({\frac {n}{2}}-i\right)h\right).}" /></span></dd></dl> <p>These equations use <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficients</a> after the summation sign shown as <span class="texhtml"><big><big>(</big></big><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:center"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>n</i></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>i</i></sub></span></span><big><big>)</big></big></span>. Each row of <a href="/wiki/Pascal%27s_triangle" title="Pascal's triangle">Pascal's triangle</a> provides the coefficient for each value of <span class="texhtml mvar" style="font-style:italic;">i</span>. </p><p>Note that the central difference will, for odd <span class="texhtml mvar" style="font-style:italic;">n</span>, have <span class="texhtml mvar" style="font-style:italic;">h</span> multiplied by non-integers. This is often a problem because it amounts to changing the interval of discretization. The problem may be remedied substituting the average of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \delta ^{n}[f](\ x-{\tfrac {\ h\ }{2}}\ )\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mtext> </mtext> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mtext> </mtext> <mi>h</mi> <mtext> </mtext> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mtext> </mtext> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \delta ^{n}[f](\ x-{\tfrac {\ h\ }{2}}\ )\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c11968b5bda24990a46f892d66b61befa9ffd5f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.091ex; height:3.676ex;" alt="{\displaystyle \ \delta ^{n}[f](\ x-{\tfrac {\ h\ }{2}}\ )\ }" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \delta ^{n}[f](\ x+{\tfrac {\ h\ }{2}}\ )~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mtext> </mtext> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mtext> </mtext> <mi>h</mi> <mtext> </mtext> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mtext> </mtext> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \delta ^{n}[f](\ x+{\tfrac {\ h\ }{2}}\ )~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c81cb79a336a879f9bbc4b1bf443b4a69625fac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.737ex; height:3.676ex;" alt="{\displaystyle \ \delta ^{n}[f](\ x+{\tfrac {\ h\ }{2}}\ )~.}" /></span> </p><p>Forward differences applied to a <a href="/wiki/Sequence" title="Sequence">sequence</a> are sometimes called the <a href="/wiki/Binomial_transform" title="Binomial transform">binomial transform</a> of the sequence, and have a number of interesting combinatorial properties. Forward differences may be evaluated using the <a href="/wiki/N%C3%B6rlund%E2%80%93Rice_integral" class="mw-redirect" title="Nörlund–Rice integral">Nörlund–Rice integral</a>. The integral representation for these types of series is interesting, because the integral can often be evaluated using <a href="/wiki/Asymptotic_expansion" title="Asymptotic expansion">asymptotic expansion</a> or <a href="/wiki/Saddle-point" class="mw-redirect" title="Saddle-point">saddle-point</a> techniques; by contrast, the forward difference series can be extremely hard to evaluate numerically, because the binomial coefficients grow rapidly for large <span class="texhtml mvar" style="font-style:italic;">n</span>. </p><p>The relationship of these higher-order differences with the respective derivatives is straightforward, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d^{n}f}{dx^{n}}}(x)={\frac {\Delta _{h}^{n}[f](x)}{h^{n}}}+o(h)={\frac {\nabla _{h}^{n}[f](x)}{h^{n}}}+o(h)={\frac {\delta _{h}^{n}[f](x)}{h^{n}}}+o\left(h^{2}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>f</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mi>o</mi> <mrow> <mo>(</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{n}f}{dx^{n}}}(x)={\frac {\Delta _{h}^{n}[f](x)}{h^{n}}}+o(h)={\frac {\nabla _{h}^{n}[f](x)}{h^{n}}}+o(h)={\frac {\delta _{h}^{n}[f](x)}{h^{n}}}+o\left(h^{2}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55ee74cdb94f0b1cd8dd904c720643510b06652a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:69.402ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{n}f}{dx^{n}}}(x)={\frac {\Delta _{h}^{n}[f](x)}{h^{n}}}+o(h)={\frac {\nabla _{h}^{n}[f](x)}{h^{n}}}+o(h)={\frac {\delta _{h}^{n}[f](x)}{h^{n}}}+o\left(h^{2}\right).}" /></span> </p><p>Higher-order differences can also be used to construct better approximations. As mentioned above, the first-order difference approximates the first-order derivative up to a term of order <span class="texhtml mvar" style="font-style:italic;">h</span>. However, the combination <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\Delta _{h}[f](x)-{\frac {1}{2}}\Delta _{h}^{2}[f](x)}{h}}=-{\frac {f(x+2h)-4f(x+h)+3f(x)}{2h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>4</mn> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>3</mn> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>h</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\Delta _{h}[f](x)-{\frac {1}{2}}\Delta _{h}^{2}[f](x)}{h}}=-{\frac {f(x+2h)-4f(x+h)+3f(x)}{2h}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d9cddb870f225e7259dcbe96bacce30960cdb45" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:59.509ex; height:6.509ex;" alt="{\displaystyle {\frac {\Delta _{h}[f](x)-{\frac {1}{2}}\Delta _{h}^{2}[f](x)}{h}}=-{\frac {f(x+2h)-4f(x+h)+3f(x)}{2h}}}" /></span> approximates <span class="texhtml"><i>f</i>′(<i>x</i>)</span> up to a term of order <span class="texhtml"><i>h</i><sup>2</sup></span>. This can be proven by expanding the above expression in <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a>, or by using the calculus of finite differences, explained below. </p><p>If necessary, the finite difference can be centered about any point by mixing forward, backward, and central differences. </p> <div class="mw-heading mw-heading2"><h2 id="Polynomials">Polynomials</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_difference&action=edit&section=4" title="Edit section: Polynomials"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a given <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> of degree <span class="texhtml"><i>n</i> ≥ 1</span>, expressed in the function <span class="texhtml"><i>P</i>(<i>x</i>)</span>, with real numbers <span class="texhtml"><i>a</i> ≠ 0</span> and <span class="texhtml"><i>b</i></span> and <i>lower order terms</i> (if any) marked as <span class="texhtml"><i>l.o.t.</i></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)=ax^{n}+bx^{n-1}+l.o.t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mi>l</mi> <mo>.</mo> <mi>o</mi> <mo>.</mo> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=ax^{n}+bx^{n-1}+l.o.t.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4722f0f4abe87bc3896ca3a496a4ec3bbdb83774" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.463ex; height:3.176ex;" alt="{\displaystyle P(x)=ax^{n}+bx^{n-1}+l.o.t.}" /></span> </p><p>After <span class="texhtml"><i>n</i></span> pairwise differences, the following result can be achieved, where <span class="texhtml"><i>h</i> ≠ 0</span> is a real number marking the arithmetic difference:<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{h}^{n}[P](x)=ah^{n}n!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo stretchy="false">[</mo> <mi>P</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>n</mi> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{h}^{n}[P](x)=ah^{n}n!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aa329fa60dd1adaa02e3f417991a89ef5ae1c65" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.26ex; height:3.009ex;" alt="{\displaystyle \Delta _{h}^{n}[P](x)=ah^{n}n!}" /></span> </p><p>Only the coefficient of the highest-order term remains. As this result is constant with respect to <span class="texhtml"><i>x</i></span>, any further pairwise differences will have the value <span class="texhtml">0</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Inductive_proof">Inductive proof</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_difference&action=edit&section=5" title="Edit section: Inductive proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Base_case">Base case</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_difference&action=edit&section=6" title="Edit section: Base case"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="texhtml"><i>Q</i>(<i>x</i>)</span> be a polynomial of degree <span class="texhtml">1</span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{h}[Q](x)=Q(x+h)-Q(x)=[a(x+h)+b]-[ax+b]=ah=ah^{1}1!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>Q</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>−</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>a</mi> <mi>h</mi> <mo>=</mo> <mi>a</mi> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mn>1</mn> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{h}[Q](x)=Q(x+h)-Q(x)=[a(x+h)+b]-[ax+b]=ah=ah^{1}1!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12f0ffaff4491b644d3cde2b6381222368037319" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:70.967ex; height:3.176ex;" alt="{\displaystyle \Delta _{h}[Q](x)=Q(x+h)-Q(x)=[a(x+h)+b]-[ax+b]=ah=ah^{1}1!}" /></span> </p><p>This proves it for the base case. </p> <div class="mw-heading mw-heading4"><h4 id="Inductive_step">Inductive step</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_difference&action=edit&section=7" title="Edit section: Inductive step"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="texhtml"><i>R</i>(<i>x</i>)</span> be a polynomial of degree <span class="texhtml"><i>m</i> − 1</span> where <span class="texhtml"><i>m</i> ≥ 2</span> and the coefficient of the highest-order term be <span class="texhtml"><i>a</i> ≠ 0</span>. Assuming the following holds true for all polynomials of degree <span class="texhtml"><i>m</i> − 1</span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{h}^{m-1}[R](x)=ah^{m-1}(m-1)!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">[</mo> <mi>R</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>m</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{h}^{m-1}[R](x)=ah^{m-1}(m-1)!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5c0ef619cccc3509b7b3211b977b0676721f8c4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.85ex; height:3.343ex;" alt="{\displaystyle \Delta _{h}^{m-1}[R](x)=ah^{m-1}(m-1)!}" /></span> </p><p>Let <span class="texhtml"><i>S</i>(<i>x</i>)</span> be a polynomial of degree <span class="texhtml"><i>m</i></span>. With one pairwise difference: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{h}[S](x)=[a(x+h)^{m}+b(x+h)^{m-1}+{\text{l.o.t.}}]-[ax^{m}+bx^{m-1}+{\text{l.o.t.}}]=ahmx^{m-1}+{\text{l.o.t.}}=T(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>S</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>l.o.t.</mtext> </mrow> <mo stretchy="false">]</mo> <mo>−</mo> <mo stretchy="false">[</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>l.o.t.</mtext> </mrow> <mo stretchy="false">]</mo> <mo>=</mo> <mi>a</mi> <mi>h</mi> <mi>m</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>l.o.t.</mtext> </mrow> <mo>=</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{h}[S](x)=[a(x+h)^{m}+b(x+h)^{m-1}+{\text{l.o.t.}}]-[ax^{m}+bx^{m-1}+{\text{l.o.t.}}]=ahmx^{m-1}+{\text{l.o.t.}}=T(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c73f767533e0d2112cb6cc4c58c49039095a7773" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:99.077ex; height:3.176ex;" alt="{\displaystyle \Delta _{h}[S](x)=[a(x+h)^{m}+b(x+h)^{m-1}+{\text{l.o.t.}}]-[ax^{m}+bx^{m-1}+{\text{l.o.t.}}]=ahmx^{m-1}+{\text{l.o.t.}}=T(x)}" /></span> </p><p>As <span class="texhtml"><i>ahm</i> ≠ 0</span>, this results in a polynomial <span class="texhtml"><i>T</i>(<i>x</i>)</span> of degree <span class="texhtml"><i>m</i> − 1</span>, with <span class="texhtml"><i>ahm</i></span> as the coefficient of the highest-order term. Given the assumption above and <span class="texhtml"><i>m</i> − 1</span> pairwise differences (resulting in a total of <span class="texhtml"><i>m</i></span> pairwise differences for <span class="texhtml"><i>S</i>(<i>x</i>)</span>), it can be found that: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{h}^{m-1}[T](x)=ahm\cdot h^{m-1}(m-1)!=ah^{m}m!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">[</mo> <mi>T</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mi>h</mi> <mi>m</mi> <mo>⋅</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>m</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>=</mo> <mi>a</mi> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mi>m</mi> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{h}^{m-1}[T](x)=ahm\cdot h^{m-1}(m-1)!=ah^{m}m!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d56a6ec947597e0186d4673e02255d19e8efa919" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:44.811ex; height:3.343ex;" alt="{\displaystyle \Delta _{h}^{m-1}[T](x)=ahm\cdot h^{m-1}(m-1)!=ah^{m}m!}" /></span> </p><p>This completes the proof. </p> <div class="mw-heading mw-heading3"><h3 id="Application">Application</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_difference&action=edit&section=8" title="Edit section: Application"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This identity can be used to find the lowest-degree polynomial that intercepts a number of points <span class="texhtml">(<i>x</i>, <i>y</i>)</span> where the difference on the <i>x</i>-axis from one point to the next is a constant <span class="texhtml"><i>h</i> ≠ 0</span>. For example, given the following points: </p> <table class="wikitable"> <tbody><tr> <th><i>x</i></th> <th><i>y</i> </th></tr> <tr> <td>1</td> <td>4 </td></tr> <tr> <td>4</td> <td>109 </td></tr> <tr> <td>7</td> <td>772 </td></tr> <tr> <td>10</td> <td>2641 </td></tr> <tr> <td>13</td> <td>6364 </td></tr></tbody></table> <p>We can use a differences table, where for all cells to the right of the first <span class="texhtml"><i>y</i></span>, the following relation to the cells in the column immediately to the left exists for a cell <span class="texhtml">(<i>a</i> + 1, <i>b</i> + 1)</span>, with the top-leftmost cell being at coordinate <span class="texhtml">(0, 0)</span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+1,b+1)=(a,b+1)-(a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>b</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+1,b+1)=(a,b+1)-(a,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704317e69e3f8d9a29dde1310c958bb1af0ebbf6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.159ex; height:2.843ex;" alt="{\displaystyle (a+1,b+1)=(a,b+1)-(a,b)}" /></span> </p><p>To find the first term, the following table can be used: </p> <table class="wikitable"> <tbody><tr> <th><span class="texhtml"><i>x</i></span></th> <th><span class="texhtml"><i>y</i></span></th> <th><span class="texhtml">Δ<i>y</i></span></th> <th><span class="texhtml">Δ<sup>2</sup><i>y</i></span></th> <th><span class="texhtml">Δ<sup>3</sup><i>y</i></span> </th></tr> <tr> <th>1 </th> <td>4 </td></tr> <tr> <th>4 </th> <td>109</td> <td>105 </td></tr> <tr> <th>7 </th> <td>772</td> <td>663</td> <td>558 </td></tr> <tr> <th>10 </th> <td>2641</td> <td>1869</td> <td>1206</td> <td>648 </td></tr> <tr> <th>13 </th> <td>6364</td> <td>3723</td> <td>1854</td> <td>648 </td></tr></tbody></table> <p>This arrives at a constant <span class="texhtml">648</span>. The arithmetic difference is <span class="texhtml"><i>h</i> = 3</span>, as established above. Given the number of pairwise differences needed to reach the constant, it can be surmised this is a polynomial of degree <span class="texhtml">3</span>. Thus, using the identity above: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 648=a\cdot 3^{3}\cdot 3!=a\cdot 27\cdot 6=a\cdot 162}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>648</mn> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⋅</mo> <mn>3</mn> <mo>!</mo> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mn>27</mn> <mo>⋅</mo> <mn>6</mn> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mn>162</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 648=a\cdot 3^{3}\cdot 3!=a\cdot 27\cdot 6=a\cdot 162}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b84483b2a9db515d5b6200294b262dbb22a8b752" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:35.868ex; height:2.676ex;" alt="{\displaystyle 648=a\cdot 3^{3}\cdot 3!=a\cdot 27\cdot 6=a\cdot 162}" /></span> </p><p>Solving for <span class="texhtml"><i>a</i></span>, it can be found to have the value <span class="texhtml">4</span>. Thus, the first term of the polynomial is <span class="texhtml">4<i>x</i><sup>3</sup></span>. </p><p>Then, subtracting out the first term, which lowers the polynomial's degree, and finding the finite difference again: </p> <table class="wikitable"> <tbody><tr> <th><span class="texhtml mvar" style="font-style:italic;">x</span></th> <th><span class="texhtml mvar" style="font-style:italic;">y</span></th> <th><span class="texhtml">Δ<i>y</i></span></th> <th><span class="texhtml">Δ<sup>2</sup><i>y</i></span> </th></tr> <tr> <th>1 </th> <td><span class="texhtml">4 − 4(1)<sup>3</sup> = 4 − 4 = 0</span> </td></tr> <tr> <th>4 </th> <td><span class="texhtml">109 − 4(4)<sup>3</sup> = 109 − 256 = −147</span></td> <td>−147 </td></tr> <tr> <th>7 </th> <td><span class="texhtml">772 − 4(7)<sup>3</sup> = 772 − 1372 = −600</span></td> <td>−453</td> <td>−306 </td></tr> <tr> <th>10 </th> <td><span class="texhtml">2641 − 4(10)<sup>3</sup> = 2641 − 4000 = −1359</span></td> <td>−759</td> <td>−306 </td></tr> <tr> <th>13 </th> <td><span class="texhtml">6364 − 4(13)<sup>3</sup> = 6364 − 8788 = −2424</span></td> <td>−1065</td> <td>−306 </td></tr></tbody></table> <p>Here, the constant is achieved after only two pairwise differences, thus the following result: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -306=a\cdot 3^{2}\cdot 2!=a\cdot 18}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>306</mn> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mn>2</mn> <mo>!</mo> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mn>18</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -306=a\cdot 3^{2}\cdot 2!=a\cdot 18}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a10585dd427fcc25d5a9bb4190fc3f57256eabf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:25.34ex; height:2.843ex;" alt="{\displaystyle -306=a\cdot 3^{2}\cdot 2!=a\cdot 18}" /></span> </p><p>Solving for <span class="texhtml"><i>a</i></span>, which is <span class="texhtml">−17</span>, the polynomial's second term is <span class="texhtml">−17<i>x</i><sup>2</sup></span>. </p><p>Moving on to the next term, by subtracting out the second term: </p> <table class="wikitable"> <tbody><tr> <th><span class="texhtml"><i>x</i></span></th> <th><span class="texhtml"><i>y</i></span></th> <th><span class="texhtml">Δ<i>y</i></span> </th></tr> <tr> <th>1 </th> <td><span class="texhtml">0 − (−17(1)<sup>2</sup>) = 0 + 17 = 17</span> </td></tr> <tr> <th>4 </th> <td><span class="texhtml">−147 − (−17(4)<sup>2</sup>) = −147 + 272 = 125</span></td> <td>108 </td></tr> <tr> <th>7 </th> <td><span class="texhtml">−600 − (−17(7)<sup>2</sup>) = −600 + 833 = 233 </span></td> <td>108 </td></tr> <tr> <th>10 </th> <td><span class="texhtml">−1359 − (−17(10)<sup>2</sup>) = −1359 + 1700 = 341 </span></td> <td>108 </td></tr> <tr> <th>13 </th> <td><span class="texhtml">−2424 − (−17(13)<sup>2</sup>) = −2424 + 2873 = 449 </span></td> <td>108 </td></tr></tbody></table> <p>Thus the constant is achieved after only one pairwise difference: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 108=a\cdot 3^{1}\cdot 1!=a\cdot 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>108</mn> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <mn>1</mn> <mo>!</mo> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 108=a\cdot 3^{1}\cdot 1!=a\cdot 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9e305169474d01217b5c0b1bbae4e66f655cbf2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:22.37ex; height:2.676ex;" alt="{\displaystyle 108=a\cdot 3^{1}\cdot 1!=a\cdot 3}" /></span> </p><p>It can be found that <span class="texhtml"><i>a</i> = 36</span> and thus the third term of the polynomial is <span class="texhtml"><b>36<i>x</i></b></span>. Subtracting out the third term: </p> <table class="wikitable"> <tbody><tr> <th><span class="texhtml"><i>x</i></span></th> <th><span class="texhtml"><i>y</i></span> </th></tr> <tr> <th>1 </th> <td><span class="texhtml">17 − 36(1) = 17 − 36 = −19</span> </td></tr> <tr> <th>4 </th> <td><span class="texhtml">125 − 36(4) = 125 − 144 = −19</span> </td></tr> <tr> <th>7 </th> <td><span class="texhtml">233 − 36(7) = 233 − 252 = −19</span> </td></tr> <tr> <th>10 </th> <td><span class="texhtml">341 − 36(10) = 341 − 360 = −19</span> </td></tr> <tr> <th>13 </th> <td><span class="texhtml">449 − 36(13) = 449 − 468 = −19</span> </td></tr></tbody></table> <p>Without any pairwise differences, it is found that the 4th and final term of the polynomial is the constant <span class="texhtml">−19</span>. Thus, the lowest-degree polynomial intercepting all the points in the first table is found: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4x^{3}-17x^{2}+36x-19}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>17</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>36</mn> <mi>x</mi> <mo>−</mo> <mn>19</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4x^{3}-17x^{2}+36x-19}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a984fddb9e358de4f917d2c424aca3d2d0430eed" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:22.756ex; height:2.843ex;" alt="{\displaystyle 4x^{3}-17x^{2}+36x-19}" /></span> </p> <div class="mw-heading mw-heading2"><h2 id="Arbitrarily_sized_kernels">Arbitrarily sized kernels</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_difference&action=edit&section=9" title="Edit section: Arbitrarily sized kernels"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Finite_difference_coefficient" title="Finite difference coefficient">Finite difference coefficient</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Five-point_stencil" title="Five-point stencil">Five-point stencil</a></div> <p>Using linear algebra one can construct finite difference approximations which utilize an arbitrary number of points to the left and a (possibly different) number of points to the right of the evaluation point, for any order derivative. This involves solving a linear system such that the <a href="/wiki/Taylor_expansion" class="mw-redirect" title="Taylor expansion">Taylor expansion</a> of the sum of those points around the evaluation point best approximates the Taylor expansion of the desired derivative. Such formulas can be represented graphically on a hexagonal or diamond-shaped grid.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> This is useful for differentiating a function on a grid, where, as one approaches the edge of the grid, one must sample fewer and fewer points on one side.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Finite difference approximations for non-standard (and even non-integer) stencils given an arbitrary stencil and a desired derivative order may be constructed.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Properties">Properties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_difference&action=edit&section=10" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>For all positive <span class="texhtml mvar" style="font-style:italic;">k</span> and <span class="texhtml mvar" style="font-style:italic;">n</span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{kh}^{n}(f,x)=\sum \limits _{i_{1}=0}^{k-1}\sum \limits _{i_{2}=0}^{k-1}\cdots \sum \limits _{i_{n}=0}^{k-1}\Delta _{h}^{n}\left(f,x+i_{1}h+i_{2}h+\cdots +i_{n}h\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mo>⋯</mo> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <mi>f</mi> <mo>,</mo> <mi>x</mi> <mo>+</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>h</mi> <mo>+</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>h</mi> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>h</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{kh}^{n}(f,x)=\sum \limits _{i_{1}=0}^{k-1}\sum \limits _{i_{2}=0}^{k-1}\cdots \sum \limits _{i_{n}=0}^{k-1}\Delta _{h}^{n}\left(f,x+i_{1}h+i_{2}h+\cdots +i_{n}h\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ae1f986f22e70f17167e907edbc132faaf13de3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:61.288ex; height:7.676ex;" alt="{\displaystyle \Delta _{kh}^{n}(f,x)=\sum \limits _{i_{1}=0}^{k-1}\sum \limits _{i_{2}=0}^{k-1}\cdots \sum \limits _{i_{n}=0}^{k-1}\Delta _{h}^{n}\left(f,x+i_{1}h+i_{2}h+\cdots +i_{n}h\right).}" /></span></li> <li><a href="/wiki/Leibniz_rule_(generalized_product_rule)" class="mw-redirect" title="Leibniz rule (generalized product rule)">Leibniz rule</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{h}^{n}(fg,x)=\sum \limits _{k=0}^{n}{\binom {n}{k}}\Delta _{h}^{k}(f,x)\Delta _{h}^{n-k}(g,x+kh).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mi>g</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>g</mi> <mo>,</mo> <mi>x</mi> <mo>+</mo> <mi>k</mi> <mi>h</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{h}^{n}(fg,x)=\sum \limits _{k=0}^{n}{\binom {n}{k}}\Delta _{h}^{k}(f,x)\Delta _{h}^{n-k}(g,x+kh).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fadf6bac0c542733b0cc28246b373ce4f4077fa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.56ex; height:7.009ex;" alt="{\displaystyle \Delta _{h}^{n}(fg,x)=\sum \limits _{k=0}^{n}{\binom {n}{k}}\Delta _{h}^{k}(f,x)\Delta _{h}^{n-k}(g,x+kh).}" /></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="In_differential_equations">In differential equations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_difference&action=edit&section=11" title="Edit section: In differential equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Finite_difference_method" title="Finite difference method">Finite difference method</a></div> <p>An important application of finite differences is in <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerical analysis</a>, especially in <a href="/wiki/Numerical_partial_differential_equations" class="mw-redirect" title="Numerical partial differential equations">numerical differential equations</a>, which aim at the numerical solution of <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary</a> and <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a>. The idea is to replace the derivatives appearing in the differential equation by finite differences that approximate them. The resulting methods are called <a href="/wiki/Finite_difference_method" title="Finite difference method">finite difference methods</a>. </p><p>Common applications of the finite difference method are in computational science and engineering disciplines, such as <a href="/wiki/Thermal_engineering" title="Thermal engineering">thermal engineering</a>, <a href="/wiki/Fluid_mechanics" title="Fluid mechanics">fluid mechanics</a>, etc. </p> <div class="mw-heading mw-heading2"><h2 id="Newton's_series"><span id="Newton.27s_series"></span>Newton's series</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_difference&action=edit&section=12" title="Edit section: Newton's series"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b><a href="/wiki/Newton_polynomial" title="Newton polynomial">Newton series</a></b> consists of the terms of the <b>Newton forward difference equation</b>, named after <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a>; in essence, it is the <b>Gregory–Newton interpolation formula</b><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> (named after <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> and <a href="/wiki/James_Gregory_(mathematician)" title="James Gregory (mathematician)">James Gregory</a>), first published in his <i><a href="/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica" title="Philosophiæ Naturalis Principia Mathematica">Principia Mathematica</a></i> in 1687,<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> namely the discrete analog of the continuous Taylor expansion, </p> <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\sum _{k=0}^{\infty }{\frac {\Delta ^{k}[f](a)}{k!}}\,(x-a)_{k}=\sum _{k=0}^{\infty }{\binom {x-a}{k}}\,\Delta ^{k}[f](a),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>a</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>x</mi> <mo>−</mo> <mi>a</mi> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\sum _{k=0}^{\infty }{\frac {\Delta ^{k}[f](a)}{k!}}\,(x-a)_{k}=\sum _{k=0}^{\infty }{\binom {x-a}{k}}\,\Delta ^{k}[f](a),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dcfd8646a4b3c907637c9a16dfef5fabeefac80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:54.747ex; height:7.176ex;" alt="{\displaystyle f(x)=\sum _{k=0}^{\infty }{\frac {\Delta ^{k}[f](a)}{k!}}\,(x-a)_{k}=\sum _{k=0}^{\infty }{\binom {x-a}{k}}\,\Delta ^{k}[f](a),}" /></span> </p> </div> <p>which holds for any <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> function <span class="texhtml mvar" style="font-style:italic;">f</span> and for many (but not all) <a href="/wiki/Analytic_function" title="Analytic function">analytic functions</a>. (It does not hold when <span class="texhtml mvar" style="font-style:italic;">f</span> is <a href="/wiki/Exponential_type" title="Exponential type">exponential type</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 \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>. This is easily seen, as the sine function vanishes at integer multiples of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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>; the corresponding Newton series is identically zero, as all finite differences are zero in this case. Yet clearly, the sine function is not zero.) Here, the expression <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\binom {x}{k}}={\frac {(x)_{k}}{k!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>x</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\binom {x}{k}}={\frac {(x)_{k}}{k!}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cde80b462b069841875eb8e59d8a53e5d4b6c2f0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.913ex; height:6.343ex;" alt="{\displaystyle {\binom {x}{k}}={\frac {(x)_{k}}{k!}}}" /></span> is the <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficient</a>, and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x)_{k}=x(x-1)(x-2)\cdots (x-k+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x)_{k}=x(x-1)(x-2)\cdots (x-k+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0c0eab4568ea37b287be10f59f1853547aa5e35" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.63ex; height:2.843ex;" alt="{\displaystyle (x)_{k}=x(x-1)(x-2)\cdots (x-k+1)}" /></span> is the "<a href="/wiki/Falling_factorial" class="mw-redirect" title="Falling factorial">falling factorial</a>" or "lower factorial", while the <a href="/wiki/Empty_product" title="Empty product">empty product</a> <span class="texhtml">(<i>x</i>)<sub>0</sub></span> is defined to be 1. In this particular case, there is an assumption of unit steps for the changes in the values of <span class="texhtml"><i>x</i>, <i>h</i> = 1</span> of the generalization below. </p><p>Note the formal correspondence of this result to <a href="/wiki/Taylor%27s_theorem" title="Taylor's theorem">Taylor's theorem</a>. Historically, this, as well as the <a href="/wiki/Chu%E2%80%93Vandermonde_identity" class="mw-redirect" title="Chu–Vandermonde identity">Chu–Vandermonde identity</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x+y)_{n}=\sum _{k=0}^{n}{\binom {n}{k}}(x)_{n-k}\,(y)_{k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mo stretchy="false">(</mo> <mi>y</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+y)_{n}=\sum _{k=0}^{n}{\binom {n}{k}}(x)_{n-k}\,(y)_{k},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0577d089e860225343b7a8d340270a633d164e40" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.59ex; height:7.009ex;" alt="{\displaystyle (x+y)_{n}=\sum _{k=0}^{n}{\binom {n}{k}}(x)_{n-k}\,(y)_{k},}" /></span> (following from it, and corresponding to the <a href="/wiki/Binomial_theorem" title="Binomial theorem">binomial theorem</a>), are included in the observations that matured to the system of <a href="/wiki/Umbral_calculus" title="Umbral calculus">umbral calculus</a>. </p><p>Newton series expansions can be superior to Taylor series expansions when applied to discrete quantities like quantum spins (see <a href="/wiki/Holstein%E2%80%93Primakoff_transformation" title="Holstein–Primakoff transformation">Holstein–Primakoff transformation</a>), <a href="/wiki/Normal_order#Bosonic_operator_functions" title="Normal order">bosonic operator functions</a> or discrete counting statistics.<sup id="cite_ref-Hucht_12-0" class="reference"><a href="#cite_note-Hucht-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>To illustrate how one may use Newton's formula in actual practice, consider the first few terms of doubling the <a href="/wiki/Fibonacci_sequence" title="Fibonacci sequence">Fibonacci sequence</a> <span class="texhtml"><i>f</i> = 2, 2, 4, ...</span> One can find a <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> that reproduces these values, by first computing a difference table, and then substituting the differences that correspond to <span class="texhtml"><i>x</i><sub>0</sub></span> (underlined) into the formula as follows, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}{\begin{array}{|c||c|c|c|}\hline x&f=\Delta ^{0}&\Delta ^{1}&\Delta ^{2}\\\hline 1&{\underline {2}}&&\\&&{\underline {0}}&\\2&2&&{\underline {2}}\\&&2&\\3&4&&\\\hline \end{array}}&\quad {\begin{aligned}f(x)&=\Delta ^{0}\cdot 1+\Delta ^{1}\cdot {\dfrac {(x-x_{0})_{1}}{1!}}+\Delta ^{2}\cdot {\dfrac {(x-x_{0})_{2}}{2!}}\quad (x_{0}=1)\\\\&=2\cdot 1+0\cdot {\dfrac {x-1}{1}}+2\cdot {\dfrac {(x-1)(x-2)}{2}}\\\\&=2+(x-1)(x-2)\\\end{aligned}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center center" rowspacing="4pt" columnspacing="1em" rowlines="solid none none none none" columnlines="solid solid solid" frame="solid"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi>f</mi> <mo>=</mo> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <munder> <mn>2</mn> <mo>_</mo> </munder> </mrow> </mtd> <mtd></mtd> <mtd></mtd> </mtr> <mtr> <mtd></mtd> <mtd></mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <munder> <mn>0</mn> <mo>_</mo> </munder> </mrow> </mtd> <mtd></mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd></mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <munder> <mn>2</mn> <mo>_</mo> </munder> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd></mtd> <mtd> <mn>2</mn> </mtd> <mtd></mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd></mtd> <mtd></mtd> </mtr> </mtable> </mrow> </mtd> <mtd> <mspace width="1em"></mspace> <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> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>⋅</mo> <mn>1</mn> <mo>+</mo> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>!</mo> </mrow> </mfrac> </mstyle> </mrow> <mo>+</mo> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mstyle> </mrow> <mspace width="1em"></mspace> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd></mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mn>1</mn> <mo>+</mo> <mn>0</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>1</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\begin{array}{|c||c|c|c|}\hline x&f=\Delta ^{0}&\Delta ^{1}&\Delta ^{2}\\\hline 1&{\underline {2}}&&\\&&{\underline {0}}&\\2&2&&{\underline {2}}\\&&2&\\3&4&&\\\hline \end{array}}&\quad {\begin{aligned}f(x)&=\Delta ^{0}\cdot 1+\Delta ^{1}\cdot {\dfrac {(x-x_{0})_{1}}{1!}}+\Delta ^{2}\cdot {\dfrac {(x-x_{0})_{2}}{2!}}\quad (x_{0}=1)\\\\&=2\cdot 1+0\cdot {\dfrac {x-1}{1}}+2\cdot {\dfrac {(x-1)(x-2)}{2}}\\\\&=2+(x-1)(x-2)\\\end{aligned}}\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f90c1dfd4e990a0bb22b18040fa55038c2bbcb17" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.838ex; width:90.054ex; height:20.843ex;" alt="{\displaystyle {\begin{matrix}{\begin{array}{|c||c|c|c|}\hline x&f=\Delta ^{0}&\Delta ^{1}&\Delta ^{2}\\\hline 1&{\underline {2}}&&\\&&{\underline {0}}&\\2&2&&{\underline {2}}\\&&2&\\3&4&&\\\hline \end{array}}&\quad {\begin{aligned}f(x)&=\Delta ^{0}\cdot 1+\Delta ^{1}\cdot {\dfrac {(x-x_{0})_{1}}{1!}}+\Delta ^{2}\cdot {\dfrac {(x-x_{0})_{2}}{2!}}\quad (x_{0}=1)\\\\&=2\cdot 1+0\cdot {\dfrac {x-1}{1}}+2\cdot {\dfrac {(x-1)(x-2)}{2}}\\\\&=2+(x-1)(x-2)\\\end{aligned}}\end{matrix}}}" /></span> </p><p>For the case of nonuniform steps in the values of <span class="texhtml mvar" style="font-style:italic;">x</span>, Newton computes the <a href="/wiki/Divided_differences" title="Divided differences">divided differences</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{j,0}=y_{j},\qquad \Delta _{j,k}={\frac {\Delta _{j+1,k-1}-\Delta _{j,k-1}}{x_{j+k}-x_{j}}}\quad \ni \quad \left\{k>0,\;j\leq \max \left(j\right)-k\right\},\qquad \Delta 0_{k}=\Delta _{0,k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mspace width="2em"></mspace> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mspace width="1em"></mspace> <mo>∋</mo> <mspace width="1em"></mspace> <mrow> <mo>{</mo> <mrow> <mi>k</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mspace width="thickmathspace"></mspace> <mi>j</mi> <mo>≤</mo> <mo movablelimits="true" form="prefix">max</mo> <mrow> <mo>(</mo> <mi>j</mi> <mo>)</mo> </mrow> <mo>−</mo> <mi>k</mi> </mrow> <mo>}</mo> </mrow> <mo>,</mo> <mspace width="2em"></mspace> <mi mathvariant="normal">Δ</mi> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{j,0}=y_{j},\qquad \Delta _{j,k}={\frac {\Delta _{j+1,k-1}-\Delta _{j,k-1}}{x_{j+k}-x_{j}}}\quad \ni \quad \left\{k>0,\;j\leq \max \left(j\right)-k\right\},\qquad \Delta 0_{k}=\Delta _{0,k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bef477a7406a826cc4971124c9d6d392521f01d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:90.713ex; height:6.343ex;" alt="{\displaystyle \Delta _{j,0}=y_{j},\qquad \Delta _{j,k}={\frac {\Delta _{j+1,k-1}-\Delta _{j,k-1}}{x_{j+k}-x_{j}}}\quad \ni \quad \left\{k>0,\;j\leq \max \left(j\right)-k\right\},\qquad \Delta 0_{k}=\Delta _{0,k}}" /></span> the series of products, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {P_{0}}=1,\quad \quad P_{k+1}=P_{k}\cdot \left(\xi -x_{k}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="1em"></mspace> <mspace width="1em"></mspace> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mi>ξ</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {P_{0}}=1,\quad \quad P_{k+1}=P_{k}\cdot \left(\xi -x_{k}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb91e61dacad9ef7271bf201c408299d45e56405" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.659ex; height:2.843ex;" alt="{\displaystyle {P_{0}}=1,\quad \quad P_{k+1}=P_{k}\cdot \left(\xi -x_{k}\right),}" /></span> and the resulting polynomial is the <a href="/wiki/Scalar_product" class="mw-redirect" title="Scalar product">scalar product</a>,<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\xi )=\Delta 0\cdot P\left(\xi \right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mn>0</mn> <mo>⋅</mo> <mi>P</mi> <mrow> <mo>(</mo> <mi>ξ</mi> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\xi )=\Delta 0\cdot P\left(\xi \right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/116a2cc0e40cd05b56dd3a08dc397053dcc371b4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18ex; height:2.843ex;" alt="{\displaystyle f(\xi )=\Delta 0\cdot P\left(\xi \right).}" /></span> </p><p>In analysis with <a href="/wiki/P-adic_number" title="P-adic number"><span class="texhtml mvar" style="font-style:italic;">p</span>-adic numbers</a>, <a href="/wiki/Mahler%27s_theorem" title="Mahler's theorem">Mahler's theorem</a> states that the assumption that <span class="texhtml mvar" style="font-style:italic;">f</span> is a polynomial function can be weakened all the way to the assumption that <span class="texhtml mvar" style="font-style:italic;">f</span> is merely continuous. </p><p><a href="/wiki/Carlson%27s_theorem" title="Carlson's theorem">Carlson's theorem</a> provides necessary and sufficient conditions for a Newton series to be unique, if it exists. However, a Newton series does not, in general, exist. </p><p>The Newton series, together with the <a href="/wiki/Stirling_series" class="mw-redirect" title="Stirling series">Stirling series</a> and the <a href="/wiki/Selberg_class" title="Selberg class">Selberg series</a>, is a special case of the general <a href="/wiki/Difference_series" class="mw-redirect" title="Difference series">difference series</a>, all of which are defined in terms of suitably scaled forward differences. </p><p>In a compressed and slightly more general form and equidistant nodes the formula reads <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\sum _{k=0}{\binom {\frac {x-a}{h}}{k}}\sum _{j=0}^{k}(-1)^{k-j}{\binom {k}{j}}f(a+jh).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mi>a</mi> </mrow> <mi>h</mi> </mfrac> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mi>j</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>k</mi> <mi>j</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>j</mi> <mi>h</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\sum _{k=0}{\binom {\frac {x-a}{h}}{k}}\sum _{j=0}^{k}(-1)^{k-j}{\binom {k}{j}}f(a+jh).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19bf5be68e5c3253097a4e53fc9a76659ef779ef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:44.905ex; height:7.676ex;" alt="{\displaystyle f(x)=\sum _{k=0}{\binom {\frac {x-a}{h}}{k}}\sum _{j=0}^{k}(-1)^{k-j}{\binom {k}{j}}f(a+jh).}" /></span> </p> <div class="mw-heading mw-heading2"><h2 id="Calculus_of_finite_differences">Calculus of finite differences</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_difference&action=edit&section=13" title="Edit section: Calculus of finite differences"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The forward difference can be considered as an <a href="/wiki/Operator_(mathematics)" title="Operator (mathematics)">operator</a>, called the <a href="/wiki/Difference_operator" class="mw-redirect" title="Difference operator">difference operator</a>, which maps the function <span class="texhtml mvar" style="font-style:italic;">f</span> to <span class="texhtml">Δ<sub><i>h</i></sub>[<i>f</i>]</span>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> This operator amounts to <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{h}=\operatorname {T} _{h}-\operatorname {I} \ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo>=</mo> <msub> <mi mathvariant="normal">T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo>−</mo> <mi mathvariant="normal">I</mi> <mo>⁡</mo> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{h}=\operatorname {T} _{h}-\operatorname {I} \ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aa01fa4a0d9d1a0341ab42e3bf4b600727ad619" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.107ex; height:2.509ex;" alt="{\displaystyle \Delta _{h}=\operatorname {T} _{h}-\operatorname {I} \ ,}" /></span> where <span class="texhtml">T<sub><i>h</i></sub></span> is the <a href="/wiki/Shift_operator" title="Shift operator">shift operator</a> with step <span class="texhtml mvar" style="font-style:italic;">h</span>, defined by <span class="nowrap"><span class="texhtml">T<sub><i>h</i></sub>[<i>f</i>](<i>x</i>) = <i>f</i>(<i>x</i> + <i>h</i>)</span>,</span> and <span class="texhtml">I</span> is the <a href="/wiki/Identity_operator" class="mw-redirect" title="Identity operator">identity operator</a>. </p><p>The finite difference of higher orders can be defined in recursive manner as <span class="nowrap"><span class="texhtml">Δ<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>n</i></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>h</i></sub></span></span> ≡ Δ<sub><i>h</i></sub>(Δ<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>n</i> − 1</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>h</i></sub></span></span>)</span>.</span> Another equivalent definition is <span class="nowrap"><span class="texhtml">Δ<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>n</i></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>h</i></sub></span></span> ≡ [T<sub><i>h</i></sub> − I]<sup><i>n</i></sup></span>.</span> </p><p>The difference operator <span class="texhtml">Δ<sub><i>h</i></sub></span> is a <a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">linear operator</a>, as such it satisfies <span class="nowrap"><span class="texhtml">Δ<sub><i>h</i></sub>[<i>α f</i> + <i>β g</i>](<i>x</i>) = <i>α</i> Δ<sub><i>h</i></sub>[<i>f</i>](<i>x</i>) + <i>β</i> Δ<sub><i>h</i></sub>[<i>g</i>](<i>x</i>)</span>.</span> </p><p>It also satisfies a special <a href="/wiki/Leibniz_rule_(generalized_product_rule)" class="mw-redirect" title="Leibniz rule (generalized product rule)">Leibniz rule</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 \ \operatorname {\Delta } _{h}{\bigl (}\ f(x)\ g(x)\ {\bigr )}\ =\ {\bigl (}\ \operatorname {\Delta } _{h}f(x)\ {\bigr )}\ g(x+h)\ +\ f(x)\ {\bigl (}\ \operatorname {\Delta } _{h}g(x)\ {\bigr )}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">Δ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">Δ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo>⁡</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mtext> </mtext> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>+</mo> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">Δ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo>⁡</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \operatorname {\Delta } _{h}{\bigl (}\ f(x)\ g(x)\ {\bigr )}\ =\ {\bigl (}\ \operatorname {\Delta } _{h}f(x)\ {\bigr )}\ g(x+h)\ +\ f(x)\ {\bigl (}\ \operatorname {\Delta } _{h}g(x)\ {\bigr )}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e48d9fd054eb54ba0c86c2fc2a26412477507c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:63.549ex; height:3.176ex;" alt="{\displaystyle \ \operatorname {\Delta } _{h}{\bigl (}\ f(x)\ g(x)\ {\bigr )}\ =\ {\bigl (}\ \operatorname {\Delta } _{h}f(x)\ {\bigr )}\ g(x+h)\ +\ f(x)\ {\bigl (}\ \operatorname {\Delta } _{h}g(x)\ {\bigr )}~.}" /></span></dd></dl> <p>Similar Leibniz rules hold for the backward and central differences. </p><p>Formally applying the <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> with respect to <span class="texhtml mvar" style="font-style:italic;">h</span>, yields the operator equation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {\Delta } _{h}=h\operatorname {D} +{\frac {1}{2!}}h^{2}\operatorname {D} ^{2}+{\frac {1}{3!}}h^{3}\operatorname {D} ^{3}+\cdots =e^{h\operatorname {D} }-\operatorname {I} \ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">Δ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo>=</mo> <mi>h</mi> <mi mathvariant="normal">D</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi mathvariant="normal">D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi mathvariant="normal">D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mi mathvariant="normal">D</mi> </mrow> </msup> <mo>−</mo> <mi mathvariant="normal">I</mi> <mo>⁡</mo> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {\Delta } _{h}=h\operatorname {D} +{\frac {1}{2!}}h^{2}\operatorname {D} ^{2}+{\frac {1}{3!}}h^{3}\operatorname {D} ^{3}+\cdots =e^{h\operatorname {D} }-\operatorname {I} \ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f74342c37f1ddae8d006543e3f50789b826176f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:48.221ex; height:5.343ex;" alt="{\displaystyle \operatorname {\Delta } _{h}=h\operatorname {D} +{\frac {1}{2!}}h^{2}\operatorname {D} ^{2}+{\frac {1}{3!}}h^{3}\operatorname {D} ^{3}+\cdots =e^{h\operatorname {D} }-\operatorname {I} \ ,}" /></span> where <span class="texhtml">D</span> denotes the conventional, continuous derivative operator, mapping <span class="texhtml mvar" style="font-style:italic;">f</span> to its derivative <span class="nowrap"><span class="texhtml"><i>f</i>′</span>.</span> The expansion is valid when both sides act on <a href="/wiki/Analytic_function" title="Analytic function">analytic functions</a>, for sufficiently small <span class="texhtml mvar" style="font-style:italic;">h</span>; in the special case that the series of derivatives terminates (when the function operated on is a finite <a href="/wiki/Polynomial" title="Polynomial">polynomial</a>) the expression is exact, for <i>all</i> finite stepsizes, <span class="nowrap"> <span class="texhtml mvar" style="font-style:italic;">h</span> .</span> Thus <span class="nowrap"><span class="texhtml"> T<sub><i>h</i></sub> = <i>e</i><sup><i>h</i> D</sup></span>,</span> and formally inverting the exponential yields <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\operatorname {D} =\ln(1+\Delta _{h})=\Delta _{h}-{\tfrac {1}{2}}\,\Delta _{h}^{2}+{\tfrac {1}{3}}\,\Delta _{h}^{3}-\cdots ~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mi mathvariant="normal">D</mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace"></mspace> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace"></mspace> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>−</mo> <mo>⋯</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\operatorname {D} =\ln(1+\Delta _{h})=\Delta _{h}-{\tfrac {1}{2}}\,\Delta _{h}^{2}+{\tfrac {1}{3}}\,\Delta _{h}^{3}-\cdots ~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17eb81d31aa4f39b659d66999cdd7b04acc841ce" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:46.859ex; height:3.676ex;" alt="{\displaystyle h\operatorname {D} =\ln(1+\Delta _{h})=\Delta _{h}-{\tfrac {1}{2}}\,\Delta _{h}^{2}+{\tfrac {1}{3}}\,\Delta _{h}^{3}-\cdots ~.}" /></span> This formula holds in the sense that both operators give the same result when applied to a polynomial. </p><p>Even for analytic functions, the series on the right is not guaranteed to converge; it may be an <a href="/wiki/Asymptotic_series" class="mw-redirect" title="Asymptotic series">asymptotic series</a>. However, it can be used to obtain more accurate approximations for the derivative. For instance, retaining the first two terms of the series yields the second-order approximation to <span class="texhtml"><i>f</i> ′(<i>x</i>)</span> mentioned at the end of the section <i><a href="#Higher-order_differences">§ Higher-order differences</a></i>. </p><p>The analogous formulas for the backward and central difference operators are <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\operatorname {D} =-\ln(1-\nabla _{h})\quad {\text{ and }}\quad h\operatorname {D} =2\operatorname {arsinh} \left({\tfrac {1}{2}}\,\delta _{h}\right)~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mi mathvariant="normal">D</mi> <mo>=</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mspace width="1em"></mspace> <mi>h</mi> <mi mathvariant="normal">D</mi> <mo>=</mo> <mn>2</mn> <mi>arsinh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace"></mspace> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\operatorname {D} =-\ln(1-\nabla _{h})\quad {\text{ and }}\quad h\operatorname {D} =2\operatorname {arsinh} \left({\tfrac {1}{2}}\,\delta _{h}\right)~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f36f9f489799e7735270bf69fdfae62f7c83970" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:51.202ex; height:3.509ex;" alt="{\displaystyle h\operatorname {D} =-\ln(1-\nabla _{h})\quad {\text{ and }}\quad h\operatorname {D} =2\operatorname {arsinh} \left({\tfrac {1}{2}}\,\delta _{h}\right)~.}" /></span> </p><p>The calculus of finite differences is related to the <a href="/wiki/Umbral_calculus" title="Umbral calculus">umbral calculus</a> of combinatorics. This remarkably systematic correspondence is due to the identity of the <a href="/wiki/Commutators" class="mw-redirect" title="Commutators">commutators</a> of the umbral quantities to their continuum analogs (<span class="texhtml"><i>h</i> → 0</span> limits), </p> <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\frac {\Delta _{h}}{h}},x\,\operatorname {T} _{h}^{-1}\right]=[\operatorname {D} ,x]=I.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mi>h</mi> </mfrac> </mrow> <mo>,</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> <msubsup> <mi mathvariant="normal">T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mi mathvariant="normal">D</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>I</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\frac {\Delta _{h}}{h}},x\,\operatorname {T} _{h}^{-1}\right]=[\operatorname {D} ,x]=I.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dae337f5d41a42cb62f6f6345662b9ef59285d88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.003ex; height:6.176ex;" alt="{\displaystyle \left[{\frac {\Delta _{h}}{h}},x\,\operatorname {T} _{h}^{-1}\right]=[\operatorname {D} ,x]=I.}" /></span> </p> </div> <p>A large number of formal differential relations of standard calculus involving functions <span class="texhtml"><i>f</i>(<i>x</i>)</span> thus <i>systematically map to umbral finite-difference analogs</i> involving <span class="nowrap"><span class="texhtml"><i>f</i>( <i>x</i> T<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">−1</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>h</i></sub></span></span> )</span>.</span> </p><p>For instance, the umbral analog of a monomial <span class="texhtml mvar" style="font-style:italic;">x<sup>n</sup></span> is a generalization of the above falling factorial (<a href="/wiki/Pochhammer_k-symbol" title="Pochhammer k-symbol">Pochhammer k-symbol</a>), <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ (x)_{n}\equiv \left(\ x\ \operatorname {T} _{h}^{-1}\right)^{n}=x\left(x-h\right)\left(x-2h\right)\cdots {\bigl (}x-\left(n-1\right)\ h{\bigr )}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≡</mo> <msup> <mrow> <mo>(</mo> <mrow> <mtext> </mtext> <mi>x</mi> <mtext> </mtext> <msubsup> <mi mathvariant="normal">T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>h</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mi>h</mi> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>x</mi> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (x)_{n}\equiv \left(\ x\ \operatorname {T} _{h}^{-1}\right)^{n}=x\left(x-h\right)\left(x-2h\right)\cdots {\bigl (}x-\left(n-1\right)\ h{\bigr )}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ca3e7729c08dc621a13edbfe884b8a8db28cb96" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:59.814ex; height:3.343ex;" alt="{\displaystyle \ (x)_{n}\equiv \left(\ x\ \operatorname {T} _{h}^{-1}\right)^{n}=x\left(x-h\right)\left(x-2h\right)\cdots {\bigl (}x-\left(n-1\right)\ h{\bigr )}\ ,}" /></span> so that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\frac {\Delta _{h}}{h}}(x)_{n}=n\ (x)_{n-1}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mi>h</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>n</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\frac {\Delta _{h}}{h}}(x)_{n}=n\ (x)_{n-1}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fe3e8addc0a24f22e749f1104037d3e7a262b96" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.648ex; height:5.509ex;" alt="{\displaystyle \ {\frac {\Delta _{h}}{h}}(x)_{n}=n\ (x)_{n-1}\ ,}" /></span> hence the above Newton interpolation formula (by matching coefficients in the expansion of an arbitrary function <span class="texhtml"><i>f</i>(<i>x</i>)</span> in such symbols), and so on. </p><p>For example, the umbral sine is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \sin \left(x\ \operatorname {T} _{h}^{-1}\right)=x-{\frac {(x)_{3}}{3!}}+{\frac {(x)_{5}}{5!}}-{\frac {(x)_{7}}{7!}}+\cdots \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mtext> </mtext> <msubsup> <mi mathvariant="normal">T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mrow> <mrow> <mn>5</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> </mrow> <mrow> <mn>7</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \sin \left(x\ \operatorname {T} _{h}^{-1}\right)=x-{\frac {(x)_{3}}{3!}}+{\frac {(x)_{5}}{5!}}-{\frac {(x)_{7}}{7!}}+\cdots \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b37b06825b2219480cdfd7f69234a5d08bd6972" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:46.83ex; height:5.843ex;" alt="{\displaystyle \ \sin \left(x\ \operatorname {T} _{h}^{-1}\right)=x-{\frac {(x)_{3}}{3!}}+{\frac {(x)_{5}}{5!}}-{\frac {(x)_{7}}{7!}}+\cdots \ }" /></span> </p><p>As in the <a href="/wiki/Continuum_limit" title="Continuum limit">continuum limit</a>, the eigenfunction of  <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num">Δ<sub><i>h</i></sub></span><span class="sr-only">/</span><span class="den"><i>h</i></span></span>⁠</span></span>  also happens to be an exponential, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\frac {\Delta _{h}}{h}}(1+\lambda h)^{\frac {x}{h}}={\frac {\Delta _{h}}{h}}e^{\ln(1+\lambda h){\frac {x}{h}}}=\lambda e^{\ln(1+\lambda h){\frac {x}{h}}}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mi>h</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>λ</mi> <mi>h</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>h</mi> </mfrac> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mi>h</mi> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>λ</mi> <mi>h</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>h</mi> </mfrac> </mrow> </mrow> </msup> <mo>=</mo> <mi>λ</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>λ</mi> <mi>h</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>h</mi> </mfrac> </mrow> </mrow> </msup> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\frac {\Delta _{h}}{h}}(1+\lambda h)^{\frac {x}{h}}={\frac {\Delta _{h}}{h}}e^{\ln(1+\lambda h){\frac {x}{h}}}=\lambda e^{\ln(1+\lambda h){\frac {x}{h}}}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75b5690bab9a5da940581bdbed28380ec01d20c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:46.76ex; height:5.509ex;" alt="{\displaystyle \ {\frac {\Delta _{h}}{h}}(1+\lambda h)^{\frac {x}{h}}={\frac {\Delta _{h}}{h}}e^{\ln(1+\lambda h){\frac {x}{h}}}=\lambda e^{\ln(1+\lambda h){\frac {x}{h}}}\ ,}" /></span></dd></dl> <p>and hence <i>Fourier sums of continuum functions are readily, faithfully mapped to umbral Fourier sums</i>, i.e., involving the same Fourier coefficients multiplying these umbral basis exponentials.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> This umbral exponential thus amounts to the exponential <a href="/wiki/Generating_function" title="Generating function">generating function</a> of the <a href="/wiki/Pochhammer_symbol" class="mw-redirect" title="Pochhammer symbol">Pochhammer symbols</a>. </p><p>Thus, for instance, the <a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a> maps to its umbral correspondent, the <a href="/wiki/Sinc_function" title="Sinc function">cardinal sine function</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \delta (x)\mapsto {\frac {\sin \left[{\frac {\pi }{2}}\left(1+{\frac {x}{h}}\right)\right]}{\pi (x+h)}}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>h</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mrow> <mrow> <mi>π</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \delta (x)\mapsto {\frac {\sin \left[{\frac {\pi }{2}}\left(1+{\frac {x}{h}}\right)\right]}{\pi (x+h)}}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132d19a452781239067dc13d50672d2d0a37d929" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.223ex; height:8.343ex;" alt="{\displaystyle \ \delta (x)\mapsto {\frac {\sin \left[{\frac {\pi }{2}}\left(1+{\frac {x}{h}}\right)\right]}{\pi (x+h)}}\ ,}" /></span> and so forth.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Difference_equation" class="mw-redirect" title="Difference equation">Difference equations</a> can often be solved with techniques very similar to those for solving <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a>. </p><p>The inverse operator of the forward difference operator, so then the umbral integral, is the <a href="/wiki/Indefinite_sum" title="Indefinite sum">indefinite sum</a> or antidifference operator. </p> <div class="mw-heading mw-heading3"><h3 id="Rules_for_calculus_of_finite_difference_operators">Rules for calculus of finite difference operators</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_difference&action=edit&section=14" title="Edit section: Rules for calculus of finite difference operators"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Analogous to <a href="/wiki/Differentiation_rules" title="Differentiation rules">rules for finding the derivative</a>, we have: </p> <ul><li><b>Constant rule</b>: If <span class="texhtml mvar" style="font-style:italic;">c</span> is a <a href="/wiki/Constant_(mathematics)" title="Constant (mathematics)">constant</a>, then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \Delta c=0\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi mathvariant="normal">Δ</mi> <mi>c</mi> <mo>=</mo> <mn>0</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \Delta c=0\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad4c9a9f5a9778e48e3f0081d741a54584a398fc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.365ex; height:2.176ex;" alt="{\displaystyle \ \Delta c=0\ }" /></span></li> <li><b><a href="/wiki/Linearity_of_differentiation" title="Linearity of differentiation">Linearity</a></b>: If <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> are <a href="/wiki/Constant_(mathematics)" title="Constant (mathematics)">constants</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \Delta (a\ f+b\ g)=a\ \Delta f+b\ \Delta g\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mtext> </mtext> <mi>f</mi> <mo>+</mo> <mi>b</mi> <mtext> </mtext> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mtext> </mtext> <mi mathvariant="normal">Δ</mi> <mi>f</mi> <mo>+</mo> <mi>b</mi> <mtext> </mtext> <mi mathvariant="normal">Δ</mi> <mi>g</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \Delta (a\ f+b\ g)=a\ \Delta f+b\ \Delta g\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e593ad269f369395495a605beacf08e6889e5c4f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.124ex; height:2.843ex;" alt="{\displaystyle \ \Delta (a\ f+b\ g)=a\ \Delta f+b\ \Delta g\ }" /></span></li></ul> <p>All of the above rules apply equally well to any difference operator as to <span class="texhtml">Δ</span>, including <span class="texhtml">δ</span> and <span class="nowrap"> <span class="texhtml">∇</span> .</span> </p> <ul><li><b><a href="/wiki/Product_rule" title="Product rule">Product rule</a></b>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\ \Delta (fg)&=f\,\Delta g+g\,\Delta f+\Delta f\,\Delta g\\[4pt]\nabla (fg)&=f\,\nabla g+g\,\nabla f-\nabla f\,\nabla g\ \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="0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mtext> </mtext> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mi>g</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>f</mi> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">Δ</mi> <mi>g</mi> <mo>+</mo> <mi>g</mi> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">Δ</mi> <mi>f</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>f</mi> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">Δ</mi> <mi>g</mi> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∇</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mi>g</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>f</mi> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">∇</mi> <mi>g</mi> <mo>+</mo> <mi>g</mi> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">∇</mi> <mi>f</mi> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mi>f</mi> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">∇</mi> <mi>g</mi> <mtext> </mtext> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ \Delta (fg)&=f\,\Delta g+g\,\Delta f+\Delta f\,\Delta g\\[4pt]\nabla (fg)&=f\,\nabla g+g\,\nabla f-\nabla f\,\nabla g\ \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa7ca1ad1827a69f52c511fa0d967d50bdbff396" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.92ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}\ \Delta (fg)&=f\,\Delta g+g\,\Delta f+\Delta f\,\Delta g\\[4pt]\nabla (fg)&=f\,\nabla g+g\,\nabla f-\nabla f\,\nabla g\ \end{aligned}}}" /></span></li> <li><b><a href="/wiki/Quotient_rule" title="Quotient rule">Quotient rule</a></b>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \nabla \left({\frac {f}{g}}\right)=\left.\left(\det {\begin{bmatrix}\nabla f&\nabla g\\f&g\end{bmatrix}}\right)\right/\left(g\cdot \det {\begin{bmatrix}g&\nabla g\\1&1\end{bmatrix}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi mathvariant="normal">∇</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>f</mi> <mi>g</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mo>(</mo> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi mathvariant="normal">∇</mi> <mi>f</mi> </mtd> <mtd> <mi mathvariant="normal">∇</mi> <mi>g</mi> </mtd> </mtr> <mtr> <mtd> <mi>f</mi> </mtd> <mtd> <mi>g</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo fence="true" stretchy="true" symmetric="true">/</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>g</mi> <mo>⋅</mo> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>g</mi> </mtd> <mtd> <mi mathvariant="normal">∇</mi> <mi>g</mi> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \nabla \left({\frac {f}{g}}\right)=\left.\left(\det {\begin{bmatrix}\nabla f&\nabla g\\f&g\end{bmatrix}}\right)\right/\left(g\cdot \det {\begin{bmatrix}g&\nabla g\\1&1\end{bmatrix}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b8eb23cbd1951361a98ccf44b91eb51dcbe2a07" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:52.761ex; height:6.176ex;" alt="{\displaystyle \ \nabla \left({\frac {f}{g}}\right)=\left.\left(\det {\begin{bmatrix}\nabla f&\nabla g\\f&g\end{bmatrix}}\right)\right/\left(g\cdot \det {\begin{bmatrix}g&\nabla g\\1&1\end{bmatrix}}\right)}" /></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \left({\frac {f}{g}}\right)={\frac {g\,\nabla f-f\,\nabla g}{g\cdot (g-\nabla g)}}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>f</mi> <mi>g</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>g</mi> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">∇</mi> <mi>f</mi> <mo>−</mo> <mi>f</mi> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">∇</mi> <mi>g</mi> </mrow> <mrow> <mi>g</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mi>g</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \left({\frac {f}{g}}\right)={\frac {g\,\nabla f-f\,\nabla g}{g\cdot (g-\nabla g)}}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/053f34119b48ca85b681ac695f6b86727aea40b7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.65ex; height:6.343ex;" alt="{\displaystyle \nabla \left({\frac {f}{g}}\right)={\frac {g\,\nabla f-f\,\nabla g}{g\cdot (g-\nabla g)}}\ }" /></span></li> <li><b><a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Summation rules</a></b>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\ \sum _{n=a}^{b}\Delta f(n)&=f(b+1)-f(a)\\\sum _{n=a}^{b}\nabla f(n)&=f(b)-f(a-1)\ \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> <mtext> </mtext> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </munderover> <mi mathvariant="normal">Δ</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </munderover> <mi mathvariant="normal">∇</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mtext> </mtext> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ \sum _{n=a}^{b}\Delta f(n)&=f(b+1)-f(a)\\\sum _{n=a}^{b}\nabla f(n)&=f(b)-f(a-1)\ \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b04ba91f4d8e278c46b09d4c5baa295004732060" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:30.805ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}\ \sum _{n=a}^{b}\Delta f(n)&=f(b+1)-f(a)\\\sum _{n=a}^{b}\nabla f(n)&=f(b)-f(a-1)\ \end{aligned}}}" /></span></li></ul> <p>See references.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_difference&action=edit&section=15" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>A <b>generalized finite difference</b> is usually defined as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{h}^{\mu }[f](x)=\sum _{k=0}^{N}\mu _{k}f(x+kh),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msubsup> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>k</mi> <mi>h</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{h}^{\mu }[f](x)=\sum _{k=0}^{N}\mu _{k}f(x+kh),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28abc7dcba38138a34d65f0e23df02ea91bbbcad" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.656ex; height:7.509ex;" alt="{\displaystyle \Delta _{h}^{\mu }[f](x)=\sum _{k=0}^{N}\mu _{k}f(x+kh),}" /></span> where <span class="texhtml"><i>μ</i> = (<i>μ</i><sub>0</sub>, …, <i>μ<sub>N</sub></i>)</span> is its coefficient vector. An <b>infinite difference</b> is a further generalization, where the finite sum above is replaced by an <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">infinite series</a>. Another way of generalization is making coefficients <span class="texhtml"><i>μ<sub>k</sub></i></span> depend on point <span class="texhtml mvar" style="font-style:italic;">x</span>: <span class="texhtml"><i>μ<sub>k</sub></i> = <i>μ<sub>k</sub></i>(<i>x</i>)</span>, thus considering <b>weighted finite difference</b>. Also one may make the step <span class="texhtml mvar" style="font-style:italic;">h</span> depend on point <span class="texhtml mvar" style="font-style:italic;">x</span>: <span class="texhtml"><i>h</i> = <i>h</i>(<i>x</i>)</span>. Such generalizations are useful for constructing different <a href="/wiki/Modulus_of_continuity" title="Modulus of continuity">modulus of continuity</a>.</li> <li>The generalized difference can be seen as the polynomial rings <span class="texhtml"><i>R</i>[<i>T<sub>h</sub></i>]</span>. It leads to difference algebras.</li> <li>Difference operator generalizes to <a href="/wiki/M%C3%B6bius_inversion" class="mw-redirect" title="Möbius inversion">Möbius inversion</a> over a <a href="/wiki/Partially_ordered_set" title="Partially ordered set">partially ordered set</a>.</li> <li>As a convolution operator: Via the formalism of <a href="/wiki/Incidence_algebra" title="Incidence algebra">incidence algebras</a>, difference operators and other Möbius inversion can be represented by <a href="/wiki/Convolution" title="Convolution">convolution</a> with a function on the poset, called the <a href="/wiki/M%C3%B6bius_function_(combinatorics)" class="mw-redirect" title="Möbius function (combinatorics)">Möbius function</a> <span class="texhtml mvar" style="font-style:italic;">μ</span>; for the difference operator, <span class="texhtml mvar" style="font-style:italic;">μ</span> is the sequence <span class="nowrap">(1, −1, 0, 0, 0, …)</span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Multivariate_finite_differences">Multivariate finite differences</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_difference&action=edit&section=16" title="Edit section: Multivariate finite differences"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Finite differences can be considered in more than one variable. They are analogous to <a href="/wiki/Partial_derivative" title="Partial derivative">partial derivatives</a> in several variables. </p><p>Some partial derivative approximations are: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}f_{x}(x,y)&\approx {\frac {f(x+h,y)-f(x-h,y)}{2h}}\\f_{y}(x,y)&\approx {\frac {f(x,y+k)-f(x,y-k)}{2k}}\\f_{xx}(x,y)&\approx {\frac {f(x+h,y)-2f(x,y)+f(x-h,y)}{h^{2}}}\\f_{yy}(x,y)&\approx {\frac {f(x,y+k)-2f(x,y)+f(x,y-k)}{k^{2}}}\\f_{xy}(x,y)&\approx {\frac {f(x+h,y+k)-f(x+h,y-k)-f(x-h,y+k)+f(x-h,y-k)}{4hk}}.\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> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>h</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>h</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>h</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo>,</mo> <mi>y</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo>,</mo> <mi>y</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>h</mi> <mo>,</mo> <mi>y</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>h</mi> <mo>,</mo> <mi>y</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>4</mn> <mi>h</mi> <mi>k</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f_{x}(x,y)&\approx {\frac {f(x+h,y)-f(x-h,y)}{2h}}\\f_{y}(x,y)&\approx {\frac {f(x,y+k)-f(x,y-k)}{2k}}\\f_{xx}(x,y)&\approx {\frac {f(x+h,y)-2f(x,y)+f(x-h,y)}{h^{2}}}\\f_{yy}(x,y)&\approx {\frac {f(x,y+k)-2f(x,y)+f(x,y-k)}{k^{2}}}\\f_{xy}(x,y)&\approx {\frac {f(x+h,y+k)-f(x+h,y-k)-f(x-h,y+k)+f(x-h,y-k)}{4hk}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc455c54536691669cf12b085b9e05568ee79e45" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.338ex; width:81.786ex; height:29.843ex;" alt="{\displaystyle {\begin{aligned}f_{x}(x,y)&\approx {\frac {f(x+h,y)-f(x-h,y)}{2h}}\\f_{y}(x,y)&\approx {\frac {f(x,y+k)-f(x,y-k)}{2k}}\\f_{xx}(x,y)&\approx {\frac {f(x+h,y)-2f(x,y)+f(x-h,y)}{h^{2}}}\\f_{yy}(x,y)&\approx {\frac {f(x,y+k)-2f(x,y)+f(x,y-k)}{k^{2}}}\\f_{xy}(x,y)&\approx {\frac {f(x+h,y+k)-f(x+h,y-k)-f(x-h,y+k)+f(x-h,y-k)}{4hk}}.\end{aligned}}}" /></span> </p><p>Alternatively, for applications in which the computation of <span class="texhtml mvar" style="font-style:italic;">f</span> is the most costly step, and both first and second derivatives must be computed, a more efficient formula for the last case is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{xy}(x,y)\approx {\frac {f(x+h,y+k)-f(x+h,y)-f(x,y+k)+2f(x,y)-f(x-h,y)-f(x,y-k)+f(x-h,y-k)}{2hk}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo>,</mo> <mi>y</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>h</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>h</mi> <mo>,</mo> <mi>y</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>h</mi> <mi>k</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{xy}(x,y)\approx {\frac {f(x+h,y+k)-f(x+h,y)-f(x,y+k)+2f(x,y)-f(x-h,y)-f(x,y-k)+f(x-h,y-k)}{2hk}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f04e749083f6e4643334ce44975fa523d31734f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:110.416ex; height:5.843ex;" alt="{\displaystyle f_{xy}(x,y)\approx {\frac {f(x+h,y+k)-f(x+h,y)-f(x,y+k)+2f(x,y)-f(x-h,y)-f(x,y-k)+f(x-h,y-k)}{2hk}},}" /></span> since the only values to compute that are not already needed for the previous four equations are <span class="texhtml"><i>f</i>(<i>x</i> + <i>h</i>, <i>y</i> + <i>k</i>)</span> and <span class="texhtml"><i>f</i>(<i>x</i> − <i>h</i>, <i>y</i> − <i>k</i>)</span>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_difference&action=edit&section=17" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 20em;"> <ul><li><a href="/wiki/Discrete_calculus" title="Discrete calculus">Discrete calculus</a></li> <li><a href="/wiki/Divided_differences" title="Divided differences">Divided differences</a></li> <li><a href="/wiki/Finite-difference_time-domain_method" title="Finite-difference time-domain method">Finite-difference time-domain method</a> (FDTD)</li> <li><a href="/wiki/Finite_volume_method" title="Finite volume method">Finite volume method</a></li> <li><a href="/wiki/FTCS_scheme" title="FTCS scheme">FTCS scheme</a></li> <li><a href="/wiki/Gilbreath%27s_conjecture" title="Gilbreath's conjecture">Gilbreath's conjecture</a></li> <li><a href="/wiki/Sheffer_sequence" title="Sheffer sequence">Sheffer sequence</a></li> <li><a href="/wiki/Summation_by_parts" title="Summation by parts">Summation by parts</a></li> <li><a href="/wiki/Time_scale_calculus" class="mw-redirect" title="Time scale calculus">Time scale calculus</a></li> <li><a href="/wiki/Upwind_differencing_scheme_for_convection" title="Upwind differencing scheme for convection">Upwind differencing scheme for convection</a></li></ul></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_difference&action=edit&section=18" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-WilmottHowison1995-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-WilmottHowison1995_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-WilmottHowison1995_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-WilmottHowison1995_1-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFPaul_WilmottSam_HowisonJeff_Dewynne1995" class="citation book cs1">Paul Wilmott; Sam Howison; Jeff Dewynne (1995). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/mathematicsoffin00wilm/page/137"><i>The Mathematics of Financial Derivatives: A Student Introduction</i></a></span>. Cambridge University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicsoffin00wilm/page/137">137</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-49789-3" title="Special:BookSources/978-0-521-49789-3"><bdi>978-0-521-49789-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Mathematics+of+Financial+Derivatives%3A+A+Student+Introduction&rft.pages=137&rft.pub=Cambridge+University+Press&rft.date=1995&rft.isbn=978-0-521-49789-3&rft.au=Paul+Wilmott&rft.au=Sam+Howison&rft.au=Jeff+Dewynne&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicsoffin00wilm%2Fpage%2F137&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinite+difference" class="Z3988"></span></span> </li> <li id="cite_note-Olver2013-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Olver2013_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Olver2013_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Olver2013_2-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPeter_Olver2013" class="citation book cs1"><a href="/wiki/Peter_J._Olver" title="Peter J. Olver">Peter Olver</a> (2013). <i>Introduction to Partial Differential Equations</i>. Springer Science & Business Media. p. 182. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-02099-0" title="Special:BookSources/978-3-319-02099-0"><bdi>978-3-319-02099-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Partial+Differential+Equations&rft.pages=182&rft.pub=Springer+Science+%26+Business+Media&rft.date=2013&rft.isbn=978-3-319-02099-0&rft.au=Peter+Olver&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinite+difference" class="Z3988"></span></span> </li> <li id="cite_note-Chaudhry2007-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Chaudhry2007_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Chaudhry2007_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Chaudhry2007_3-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFM_Hanif_Chaudhry2007" class="citation book cs1">M Hanif Chaudhry (2007). <i>Open-Channel Flow</i>. Springer. p. 369. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-68648-6" title="Special:BookSources/978-0-387-68648-6"><bdi>978-0-387-68648-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Open-Channel+Flow&rft.pages=369&rft.pub=Springer&rft.date=2007&rft.isbn=978-0-387-68648-6&rft.au=M+Hanif+Chaudhry&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinite+difference" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Jordán, op. cit., p. 1 and Milne-Thomson, p. xxi. Milne-Thomson, Louis Melville (2000): <i>The Calculus of Finite Differences</i> (Chelsea Pub Co, 2000) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0821821077" title="Special:BookSources/978-0821821077">978-0821821077</a></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://divisbyzero.com/2018/02/13/finite-differences-of-polynomials/">"Finite differences of polynomials"</a>. 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(January 1, 1909). <a rel="nofollow" class="external text" href="https://archive.org/stream/journal43instuoft#page/236/mode/2up">"On the Graphic Delineation of Interpolation Formulæ"</a>. <i>Journal of the Institute of Actuaries</i>. <b>43</b> (2): <span class="nowrap">235–</span>241. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS002026810002494X">10.1017/S002026810002494X</a><span class="reference-accessdate">. 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" on YouTube</a></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Newton, Isaac, (1687). <a rel="nofollow" class="external text" href="https://archive.org/details/bub_gb_KaAIAAAAIAAJ/page/n459"><i>Principia</i>, Book III, Lemma V, Case 1</a></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFIaroslav_V._Blagouchine2018" class="citation journal cs1">Iaroslav V. Blagouchine (2018). <a rel="nofollow" class="external text" href="http://math.colgate.edu/~integers/sjs3/sjs3.pdf">"Three notes on Ser's and Hasse's representations for the zeta-functions"</a> <span class="cs1-format">(PDF)</span>. <i>Integers (Electronic Journal of Combinatorial Number Theory)</i>. <b>18A</b>: <span class="nowrap">1–</span>45. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1606.02044">1606.02044</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5281%2Fzenodo.10581385">10.5281/zenodo.10581385</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Integers+%28Electronic+Journal+of+Combinatorial+Number+Theory%29&rft.atitle=Three+notes+on+Ser%27s+and+Hasse%27s+representations+for+the+zeta-functions&rft.volume=18A&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E45&rft.date=2018&rft_id=info%3Aarxiv%2F1606.02044&rft_id=info%3Adoi%2F10.5281%2Fzenodo.10581385&rft.au=Iaroslav+V.+Blagouchine&rft_id=http%3A%2F%2Fmath.colgate.edu%2F~integers%2Fsjs3%2Fsjs3.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinite+difference" class="Z3988"></span></span> </li> <li id="cite_note-Hucht-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hucht_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKönigHucht2021" class="citation journal cs1">König, Jürgen; Hucht, Fred (2021). <a rel="nofollow" class="external text" href="https://doi.org/10.21468%2FSciPostPhys.10.1.007">"Newton series expansion of bosonic operator functions"</a>. <i>SciPost Physics</i>. <b>10</b> (1): 007. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2008.11139">2008.11139</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2021ScPP...10....7K">2021ScPP...10....7K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.21468%2FSciPostPhys.10.1.007">10.21468/SciPostPhys.10.1.007</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:221293056">221293056</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SciPost+Physics&rft.atitle=Newton+series+expansion+of+bosonic+operator+functions&rft.volume=10&rft.issue=1&rft.pages=007&rft.date=2021&rft_id=info%3Aarxiv%2F2008.11139&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A221293056%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.21468%2FSciPostPhys.10.1.007&rft_id=info%3Abibcode%2F2021ScPP...10....7K&rft.aulast=K%C3%B6nig&rft.aufirst=J%C3%BCrgen&rft.au=Hucht%2C+Fred&rft_id=https%3A%2F%2Fdoi.org%2F10.21468%252FSciPostPhys.10.1.007&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinite+difference" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a href="/wiki/Robert_D._Richtmyer" title="Robert D. Richtmyer">Richtmeyer, D.</a> and Morton, K.W., (1967). <i>Difference Methods for Initial Value Problems</i>, 2nd ed., Wiley, New York.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBoole1872" class="citation book cs1"><a href="/wiki/George_Boole" title="George Boole">Boole, George</a> (1872). <a rel="nofollow" class="external text" href="https://archive.org/details/cu31924031240934"><i>A Treatise on the Calculus of Finite Differences</i></a> (2nd ed.). Macmillan and Company – via <a href="/wiki/Internet_Archive" title="Internet Archive">Internet Archive</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Treatise+on+the+Calculus+of+Finite+Differences&rft.edition=2nd&rft.pub=Macmillan+and+Company&rft.date=1872&rft.aulast=Boole&rft.aufirst=George&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcu31924031240934&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinite+difference" class="Z3988"></span> Also, a Dover reprint edition, 1960.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJordan1965" class="citation book cs1">Jordan, Charles (1965) [1939]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3RfZOsDAyQsC&pg=PA1"><i>Calculus of Finite Differences</i></a>. 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"Umbral deformations on discrete space-time". <i>International Journal of Modern Physics A</i>. <b>23</b> (13): <span class="nowrap">200–</span>214. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0710.2306">0710.2306</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008IJMPA..23.2005Z">2008IJMPA..23.2005Z</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS0217751X08040548">10.1142/S0217751X08040548</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16797959">16797959</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Modern+Physics+A&rft.atitle=Umbral+deformations+on+discrete+space-time&rft.volume=23&rft.issue=13&rft.pages=%3Cspan+class%3D%22nowrap%22%3E200-%3C%2Fspan%3E214&rft.date=2008&rft_id=info%3Aarxiv%2F0710.2306&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A16797959%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1142%2FS0217751X08040548&rft_id=info%3Abibcode%2F2008IJMPA..23.2005Z&rft.aulast=Zachos&rft.aufirst=C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinite+difference" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCurtrightZachos2013" class="citation journal cs1">Curtright, T. L.; Zachos, C. K. (2013). <a rel="nofollow" class="external text" href="https://doi.org/10.3389%2Ffphy.2013.00015">"Umbral Vade Mecum"</a>. <i>Frontiers in Physics</i>. <b>1</b>: 15. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1304.0429">1304.0429</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013FrP.....1...15C">2013FrP.....1...15C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.3389%2Ffphy.2013.00015">10.3389/fphy.2013.00015</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14106142">14106142</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Frontiers+in+Physics&rft.atitle=Umbral+Vade+Mecum&rft.volume=1&rft.pages=15&rft.date=2013&rft_id=info%3Aarxiv%2F1304.0429&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14106142%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.3389%2Ffphy.2013.00015&rft_id=info%3Abibcode%2F2013FrP.....1...15C&rft.aulast=Curtright&rft.aufirst=T.+L.&rft.au=Zachos%2C+C.+K.&rft_id=https%3A%2F%2Fdoi.org%2F10.3389%252Ffphy.2013.00015&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinite+difference" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLevyLessman1992" class="citation book cs1">Levy, H.; Lessman, F. (1992). <i>Finite Difference Equations</i>. Dover. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-67260-3" title="Special:BookSources/0-486-67260-3"><bdi>0-486-67260-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Finite+Difference+Equations&rft.pub=Dover&rft.date=1992&rft.isbn=0-486-67260-3&rft.aulast=Levy&rft.aufirst=H.&rft.au=Lessman%2C+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinite+difference" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAmes1977" class="citation book cs1">Ames, W.F. (1977). <i>Numerical Methods for Partial Differential Equations</i>. New York, NY: Academic Press. Section 1.6. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-056760-1" title="Special:BookSources/0-12-056760-1"><bdi>0-12-056760-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+Methods+for+Partial+Differential+Equations&rft.place=New+York%2C+NY&rft.pages=Section-1.6&rft.pub=Academic+Press&rft.date=1977&rft.isbn=0-12-056760-1&rft.aulast=Ames&rft.aufirst=W.F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinite+difference" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHildebrand1968" class="citation book cs1"><a href="/wiki/Francis_B._Hildebrand" title="Francis B. Hildebrand">Hildebrand, F.B.</a> (1968). <i>Finite-Difference Equations and Simulations</i>. Englewood Cliffs, NJ: Prentice-Hall. Section 2.2.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Finite-Difference+Equations+and+Simulations&rft.place=Englewood+Cliffs%2C+NJ&rft.pages=Section-2.2&rft.pub=Prentice-Hall&rft.date=1968&rft.aulast=Hildebrand&rft.aufirst=F.B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinite+difference" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFFlajoletSedgewick1995" class="citation journal cs1">Flajolet, Philippe; <a href="/wiki/Robert_Sedgewick_(computer_scientist)" title="Robert Sedgewick (computer scientist)">Sedgewick, Robert</a> (1995). <a rel="nofollow" class="external text" href="http://algo.inria.fr/flajolet/Publications/FlSe95.pdf">"Mellin transforms and asymptotics: Finite differences and Rice's integrals"</a> <span class="cs1-format">(PDF)</span>. <i>Theoretical Computer Science</i>. <b>144</b> (<span class="nowrap">1–</span>2): <span class="nowrap">101–</span>124. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0304-3975%2894%2900281-M">10.1016/0304-3975(94)00281-M</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Theoretical+Computer+Science&rft.atitle=Mellin+transforms+and+asymptotics%3A+Finite+differences+and+Rice%27s+integrals&rft.volume=144&rft.issue=%3Cspan+class%3D%22nowrap%22%3E1%E2%80%93%3C%2Fspan%3E2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E101-%3C%2Fspan%3E124&rft.date=1995&rft_id=info%3Adoi%2F10.1016%2F0304-3975%2894%2900281-M&rft.aulast=Flajolet&rft.aufirst=Philippe&rft.au=Sedgewick%2C+Robert&rft_id=http%3A%2F%2Falgo.inria.fr%2Fflajolet%2FPublications%2FFlSe95.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinite+difference" class="Z3988"></span></span> </li> </ol></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li>Richardson, C. H. (1954): <i>An Introduction to the Calculus of Finite Differences</i> (Van Nostrand (1954) <a rel="nofollow" class="external text" href="http://babel.hathitrust.org/cgi/pt?id=mdp.39015000982945;view=1up;seq=5">online copy</a></li> <li>Mickens, R. E. (1991): <i>Difference Equations: Theory and Applications</i> (Chapman and Hall/CRC) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0442001360" title="Special:BookSources/978-0442001360">978-0442001360</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_difference&action=edit&section=19" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Finite-difference_calculus">"Finite-difference calculus"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Finite-difference+calculus&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DFinite-difference_calculus&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinite+difference" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://reference.wolfram.com/mathematica/tutorial/NDSolvePDE.html#c:4">Table of useful finite difference formula generated using Mathematica</a></li> <li>D. Gleich (2005), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090419132601/http://www.stanford.edu/~dgleich/publications/finite-calculus.pdf"><i>Finite Calculus: A Tutorial for Solving Nasty Sums</i></a></li> <li><a rel="nofollow" class="external text" href="http://mathformeremortals.wordpress.com/2013/01/12/a-numerical-second-derivative-from-three-points/">Discrete Second Derivative from Unevenly Spaced Points</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output 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href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">Free variables and bound variables</a></li> <li><a href="/wiki/Graph_of_a_function" title="Graph of a function">Graph of a function</a></li> <li><a href="/wiki/Linear_function" title="Linear function">Linear function</a></li> <li><a href="/wiki/Radian" title="Radian">Radian</a></li> <li><a href="/wiki/Rolle%27s_theorem" title="Rolle's theorem">Rolle's theorem</a></li> <li><a href="/wiki/Secant_line" title="Secant line">Secant</a></li> <li><a href="/wiki/Slope" title="Slope">Slope</a></li> <li><a href="/wiki/Tangent" title="Tangent">Tangent</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">Limits</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Indeterminate_form" title="Indeterminate form">Indeterminate form</a></li> <li><a href="/wiki/Limit_of_a_function" title="Limit of a function">Limit of a function</a> <ul><li><a href="/wiki/One-sided_limit" title="One-sided limit">One-sided limit</a></li></ul></li> <li><a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">Limit of a sequence</a></li> <li><a href="/wiki/Order_of_approximation" title="Order of approximation">Order of approximation</a></li> <li><a href="/wiki/(%CE%B5,_%CE%B4)-definition_of_limit" class="mw-redirect" title="(ε, δ)-definition of limit">(ε, δ)-definition of limit</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Differential_calculus" title="Differential calculus">Differential calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Derivative" title="Derivative">Derivative</a></li> <li><a href="/wiki/Second_derivative" title="Second derivative">Second derivative</a></li> <li><a href="/wiki/Partial_derivative" title="Partial derivative">Partial derivative</a></li> <li><a href="/wiki/Differential_(mathematics)" title="Differential (mathematics)">Differential</a></li> <li><a href="/wiki/Differential_operator" title="Differential operator">Differential operator</a></li> <li><a href="/wiki/Mean_value_theorem" title="Mean value theorem">Mean value theorem</a></li> <li><a href="/wiki/Notation_for_differentiation" title="Notation for differentiation">Notation</a> <ul><li><a href="/wiki/Leibniz%27s_notation" title="Leibniz's notation">Leibniz's notation</a></li> <li><a href="/wiki/Newton%27s_notation_for_differentiation" class="mw-redirect" title="Newton's notation for differentiation">Newton's notation</a></li></ul></li> <li><a href="/wiki/Differentiation_rules" title="Differentiation rules">Rules of differentiation</a> <ul><li><a href="/wiki/Linearity_of_differentiation" title="Linearity of differentiation">linearity</a></li> <li><a href="/wiki/Power_rule" title="Power rule">Power</a></li> <li><a href="/wiki/Sum_rule_in_differentiation" class="mw-redirect" title="Sum rule in differentiation">Sum</a></li> <li><a href="/wiki/Chain_rule" title="Chain rule">Chain</a></li> <li><a href="/wiki/L%27H%C3%B4pital%27s_rule" title="L'Hôpital's rule">L'Hôpital's</a></li> <li><a href="/wiki/Product_rule" title="Product rule">Product</a> <ul><li><a href="/wiki/General_Leibniz_rule" title="General Leibniz rule">General Leibniz's rule</a></li></ul></li> <li><a href="/wiki/Quotient_rule" title="Quotient rule">Quotient</a></li></ul></li> <li>Other techniques <ul><li><a href="/wiki/Implicit_differentiation" class="mw-redirect" title="Implicit differentiation">Implicit differentiation</a></li> <li><a href="/wiki/Inverse_functions_and_differentiation" class="mw-redirect" title="Inverse functions and differentiation">Inverse functions and differentiation</a></li> <li><a href="/wiki/Logarithmic_derivative" title="Logarithmic derivative">Logarithmic derivative</a></li> <li><a href="/wiki/Related_rates" title="Related rates">Related rates</a></li></ul></li> <li><a href="/wiki/Stationary_point" title="Stationary point">Stationary points</a> <ul><li><a href="/wiki/First_derivative_test" class="mw-redirect" title="First derivative test">First derivative test</a></li> <li><a href="/wiki/Second_derivative_test" class="mw-redirect" title="Second derivative test">Second derivative test</a></li> <li><a href="/wiki/Extreme_value_theorem" title="Extreme value theorem">Extreme value theorem</a></li> <li><a href="/wiki/Maximum_and_minimum" title="Maximum and minimum">Maximum and minimum</a></li></ul></li> <li>Further applications <ul><li><a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a></li> <li><a href="/wiki/Taylor%27s_theorem" title="Taylor's theorem">Taylor's theorem</a></li></ul></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential equation</a> <ul><li><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary differential equation</a></li> <li><a href="/wiki/Partial_differential_equation" title="Partial differential equation">Partial differential equation</a></li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Integral_calculus" class="mw-redirect" title="Integral calculus">Integral calculus</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antiderivative" title="Antiderivative">Antiderivative</a></li> <li><a href="/wiki/Arc_length" title="Arc length">Arc length</a></li> <li><a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a></li> <li><a href="/wiki/Integral#Properties" title="Integral">Basic properties</a></li> <li><a href="/wiki/Constant_of_integration" title="Constant of integration">Constant of integration</a></li> <li><a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem of calculus</a> <ul><li><a href="/wiki/Leibniz_integral_rule" title="Leibniz integral rule">Differentiating under the integral sign</a></li></ul></li> <li><a href="/wiki/Integration_by_parts" title="Integration by parts">Integration by parts</a></li> <li><a href="/wiki/Integration_by_substitution" title="Integration by substitution">Integration by substitution</a> <ul><li><a href="/wiki/Trigonometric_substitution" title="Trigonometric substitution">trigonometric</a></li> <li><a href="/wiki/Euler_substitution" title="Euler substitution">Euler</a></li> <li><a href="/wiki/Tangent_half-angle_substitution" title="Tangent half-angle substitution">Tangent half-angle substitution</a></li></ul></li> <li><a href="/wiki/Partial_fractions_in_integration" class="mw-redirect" title="Partial fractions in integration">Partial fractions in integration</a> <ul><li><a href="/wiki/Quadratic_integral" title="Quadratic integral">Quadratic integral</a></li></ul></li> <li><a href="/wiki/Trapezoidal_rule" title="Trapezoidal rule">Trapezoidal rule</a></li> <li>Volumes <ul><li><a href="/wiki/Disc_integration" title="Disc integration">Washer method</a></li> <li><a href="/wiki/Shell_integration" title="Shell integration">Shell method</a></li></ul></li> <li><a href="/wiki/Integral_equation" title="Integral equation">Integral equation</a></li> <li><a href="/wiki/Integro-differential_equation" title="Integro-differential equation">Integro-differential equation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Vector_calculus" title="Vector calculus">Vector calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Derivatives <ul><li><a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">Curl</a></li> <li><a href="/wiki/Directional_derivative" title="Directional derivative">Directional derivative</a></li> <li><a href="/wiki/Divergence" title="Divergence">Divergence</a></li> <li><a href="/wiki/Gradient" title="Gradient">Gradient</a></li> <li><a href="/wiki/Laplace_operator" title="Laplace operator">Laplacian</a></li></ul></li> <li>Basic theorems <ul><li><a href="/wiki/Fundamental_Theorem_of_Line_Integrals" class="mw-redirect" title="Fundamental Theorem of Line Integrals">Line integrals</a></li> <li><a href="/wiki/Green%27s_theorem" title="Green's theorem">Green's</a></li> <li><a href="/wiki/Stokes%27_theorem" title="Stokes' theorem">Stokes'</a></li> <li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Gauss'</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Multivariable_calculus" title="Multivariable calculus">Multivariable calculus</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Divergence theorem</a></li> <li><a href="/wiki/Geometric_calculus" title="Geometric calculus">Geometric</a></li> <li><a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian matrix</a></li> <li><a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian matrix and determinant</a></li> <li><a href="/wiki/Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a></li> <li><a href="/wiki/Line_integral" title="Line integral">Line integral</a></li> <li><a href="/wiki/Matrix_calculus" title="Matrix calculus">Matrix</a></li> <li><a href="/wiki/Multiple_integral" title="Multiple integral">Multiple integral</a></li> <li><a href="/wiki/Partial_derivative" title="Partial derivative">Partial derivative</a></li> <li><a href="/wiki/Surface_integral" title="Surface integral">Surface integral</a></li> <li><a href="/wiki/Volume_integral" title="Volume integral">Volume integral</a></li> <li>Advanced topics <ul><li><a href="/wiki/Differential_form" title="Differential form">Differential forms</a></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Generalized_Stokes%27_theorem" class="mw-redirect" title="Generalized Stokes' theorem">Generalized Stokes' theorem</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sequences and series</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetico-geometric_sequence" title="Arithmetico-geometric sequence">Arithmetico-geometric sequence</a></li> <li>Types of series <ul><li><a href="/wiki/Alternating_series" title="Alternating series">Alternating</a></li> <li><a href="/wiki/Binomial_series" title="Binomial series">Binomial</a></li> <li><a href="/wiki/Fourier_series" title="Fourier series">Fourier</a></li> <li><a href="/wiki/Geometric_series" title="Geometric series">Geometric</a></li> <li><a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">Harmonic</a></li> <li><a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">Infinite</a></li> <li><a href="/wiki/Power_series" title="Power series">Power</a> <ul><li><a href="/wiki/Maclaurin_series" class="mw-redirect" title="Maclaurin series">Maclaurin</a></li> <li><a href="/wiki/Taylor_series" title="Taylor series">Taylor</a></li></ul></li> <li><a href="/wiki/Telescoping_series" title="Telescoping series">Telescoping</a></li></ul></li> <li>Tests of convergence <ul><li><a href="/wiki/Abel%27s_test" title="Abel's test">Abel's</a></li> <li><a href="/wiki/Alternating_series_test" title="Alternating series test">Alternating series</a></li> <li><a href="/wiki/Cauchy_condensation_test" title="Cauchy condensation test">Cauchy condensation</a></li> <li><a href="/wiki/Direct_comparison_test" title="Direct comparison test">Direct comparison</a></li> <li><a href="/wiki/Dirichlet%27s_test" title="Dirichlet's test">Dirichlet's</a></li> <li><a href="/wiki/Integral_test_for_convergence" title="Integral test for convergence">Integral</a></li> <li><a href="/wiki/Limit_comparison_test" title="Limit comparison test">Limit comparison</a></li> <li><a href="/wiki/Ratio_test" title="Ratio test">Ratio</a></li> <li><a href="/wiki/Root_test" title="Root test">Root</a></li> <li><a href="/wiki/Term_test" class="mw-redirect" title="Term test">Term</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Special functions<br />and numbers</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bernoulli_number" title="Bernoulli number">Bernoulli numbers</a></li> <li><a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e (mathematical constant)</a></li> <li><a href="/wiki/Exponential_function" title="Exponential function">Exponential function</a></li> <li><a href="/wiki/Natural_logarithm" title="Natural logarithm">Natural logarithm</a></li> <li><a href="/wiki/Stirling%27s_approximation" title="Stirling's approximation">Stirling's approximation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/History_of_calculus" title="History of calculus">History of calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adequality" title="Adequality">Adequality</a></li> <li><a href="/wiki/Brook_Taylor" title="Brook Taylor">Brook Taylor</a></li> <li><a href="/wiki/Colin_Maclaurin" title="Colin Maclaurin">Colin Maclaurin</a></li> <li><a href="/wiki/Generality_of_algebra" title="Generality of algebra">Generality of algebra</a></li> <li><a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a></li> <li><a href="/wiki/Infinitesimal" title="Infinitesimal">Infinitesimal</a></li> <li><a href="/wiki/Infinitesimal_calculus" class="mw-redirect" title="Infinitesimal calculus">Infinitesimal calculus</a></li> <li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li> <li><a href="/wiki/Fluxion" title="Fluxion">Fluxion</a></li> <li><a href="/wiki/Law_of_Continuity" class="mw-redirect" title="Law of Continuity">Law of Continuity</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><i><a href="/wiki/Method_of_Fluxions" title="Method of Fluxions">Method of Fluxions</a></i></li> <li><i><a href="/wiki/The_Method_of_Mechanical_Theorems" title="The Method of Mechanical Theorems">The Method of Mechanical Theorems</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Lists</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Integrals32" scope="row" class="navbox-group" style="width:1%;text-align:left"><a href="/wiki/Lists_of_integrals" title="Lists of integrals">Integrals</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_integrals_of_rational_functions" title="List of integrals of rational functions">rational functions</a></li> <li><a href="/wiki/List_of_integrals_of_irrational_functions" title="List of integrals of irrational functions">irrational functions</a></li> <li><a href="/wiki/List_of_integrals_of_exponential_functions" title="List of integrals of exponential functions">exponential functions</a></li> <li><a href="/wiki/List_of_integrals_of_logarithmic_functions" title="List of integrals of logarithmic functions">logarithmic functions</a></li> <li><a href="/wiki/List_of_integrals_of_hyperbolic_functions" title="List of integrals of hyperbolic functions">hyperbolic functions</a> <ul><li><a href="/wiki/List_of_integrals_of_inverse_hyperbolic_functions" title="List of integrals of inverse hyperbolic functions">inverse</a></li></ul></li> <li><a href="/wiki/List_of_integrals_of_trigonometric_functions" title="List of integrals of trigonometric functions">trigonometric functions</a> <ul><li><a href="/wiki/List_of_integrals_of_inverse_trigonometric_functions" title="List of integrals of inverse trigonometric functions">inverse</a></li> <li><a href="/wiki/Integral_of_the_secant_function" title="Integral of the secant function">Secant</a></li> <li><a href="/wiki/Integral_of_secant_cubed" title="Integral of secant cubed">Secant cubed</a></li></ul></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_limits" title="List of limits">List of limits</a></li> <li><a href="/wiki/Differentiation_rules" title="Differentiation rules">List of derivatives</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous topics</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Complex calculus <ul><li><a href="/wiki/Contour_integral" class="mw-redirect" title="Contour integral">Contour integral</a></li></ul></li> <li>Differential geometry <ul><li><a href="/wiki/Manifold" title="Manifold">Manifold</a></li> <li><a href="/wiki/Curvature" title="Curvature">Curvature</a></li> <li><a href="/wiki/Differential_geometry_of_curves" class="mw-redirect" title="Differential geometry of curves">of curves</a></li> <li><a href="/wiki/Differential_geometry_of_surfaces" title="Differential geometry of surfaces">of surfaces</a></li> <li><a href="/wiki/Tensor" title="Tensor">Tensor</a></li></ul></li> <li><a href="/wiki/Euler%E2%80%93Maclaurin_formula" title="Euler–Maclaurin formula">Euler–Maclaurin formula</a></li> <li><a href="/wiki/Gabriel%27s_horn" title="Gabriel's horn">Gabriel's horn</a></li> <li><a href="/wiki/Integration_Bee" title="Integration Bee">Integration Bee</a></li> <li><a href="/wiki/Proof_that_22/7_exceeds_%CF%80" title="Proof that 22/7 exceeds π">Proof that 22/7 exceeds π</a></li> <li><a href="/wiki/Regiomontanus%27_angle_maximization_problem" title="Regiomontanus' angle maximization problem">Regiomontanus' angle 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