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<title>Analytic function</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Analytic function</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Analytic_expression" class="mw-redirect" title="Analytic expression">analytic expression</a> or <a href="Analytic_signal" title="Analytic signal">analytic signal</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">This article is about both real and complex analytic functions. For analytic functions in complex analysis specifically, see <a href="Holomorphic_function" title="Holomorphic function">holomorphic function</a>. For analytic functions in SQL, see <a href="Window_function_(SQL)" title="Window function (SQL)">Window function (SQL)</a>.</div>

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</style><table class="sidebar nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle"><span style="font-size: 8pt; font-weight: none"><a href="Mathematical_analysis" title="Mathematical analysis">Mathematical analysis</a> → <b>Complex analysis</b></span></td></tr><tr><th class="sidebar-title-with-pretitle"><a href="Complex_analysis" title="Complex analysis">Complex analysis</a></th></tr><tr><td class="sidebar-image"><span class="skin-invert-image" typeof="mw:File"></span></td></tr><tr><th class="sidebar-heading">

<a href="Complex_number" title="Complex number">Complex numbers</a></th></tr><tr><td class="sidebar-content">

<ul><li><a href="Real_number" title="Real number">Real number</a></li>

<li><a href="Imaginary_number" title="Imaginary number">Imaginary number</a></li>

<li><a href="Complex_plane" title="Complex plane">Complex plane</a></li>

<li><a href="Complex_conjugate" title="Complex conjugate">Complex conjugate</a></li>

<li><a href="Euler's_formula" title="Euler's formula">Euler's formula</a></li></ul></td>

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<a href="Complex_analysis#Major_results" title="Complex analysis">Basic theory</a></th></tr><tr><td class="sidebar-content">

<ul><li><a href="Argument_principle" title="Argument principle">Argument principle</a></li>

<li><a href="Residue_(complex_analysis)" title="Residue (complex analysis)">Residue</a></li>

<li><a href="Essential_singularity" title="Essential singularity">Essential singularity</a></li>

<li><a href="Isolated_singularity" title="Isolated singularity">Isolated singularity</a></li>

<li><a href="Removable_singularity" title="Removable singularity">Removable singularity</a></li>

<li><a href="Zeros_and_poles" title="Zeros and poles">Zeros and poles</a></li></ul></td>

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<a href="Complex_analysis#Complex_functions" title="Complex analysis">Complex functions</a></th></tr><tr><td class="sidebar-content">

<ul><li><a href="Complex_analysis#Complex_functions" title="Complex analysis">Complex-valued function</a></li>

<li><a href="Antiderivative_(complex_analysis)" title="Antiderivative (complex analysis)">Antiderivative</a></li>

<li><a href="Entire_function" title="Entire function">Entire function</a></li>

<li><a href="Holomorphic_function" title="Holomorphic function">Holomorphic function</a></li>

<li><a href="Meromorphic_function" title="Meromorphic function">Meromorphic function</a></li>

<li><a href="Cauchy%E2%80%93Riemann_equations" title="Cauchy–Riemann equations">Cauchy–Riemann equations</a></li>

<li><a href="Formal_power_series" title="Formal power series">Formal power series</a></li>

<li><a href="Laurent_series" title="Laurent series">Laurent series</a></li></ul></td>

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<a href="Complex_analysis#Major_results" title="Complex analysis">Theorems</a></th></tr><tr><td class="sidebar-content">

<ul><li><a href="Analyticity_of_holomorphic_functions" title="Analyticity of holomorphic functions">Analyticity of holomorphic functions</a></li>

<li><a href="Cauchy's_integral_theorem" title="Cauchy's integral theorem">Cauchy's integral theorem</a></li>

<li><a href="Cauchy's_integral_formula" title="Cauchy's integral formula">Cauchy's integral formula</a></li>

<li><a href="Residue_theorem" title="Residue theorem">Residue theorem</a></li>

<li><a href="Liouville's_theorem_(complex_analysis)" title="Liouville's theorem (complex analysis)">Liouville's theorem</a></li>

<li><a href="Picard_theorem" title="Picard theorem">Picard theorem</a></li>

<li><a href="Weierstrass_factorization_theorem" title="Weierstrass factorization theorem">Weierstrass factorization theorem</a></li></ul></td>

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<a href="Geometric_function_theory" title="Geometric function theory">Geometric function theory</a></th></tr><tr><td class="sidebar-content">

<ul><li><a href="Complex_manifold" title="Complex manifold">Complex manifold</a></li>

<li><a href="Conformal_map" title="Conformal map">Conformal map</a></li>

<li><a href="Quasiconformal_mapping" title="Quasiconformal mapping">Quasiconformal mapping</a></li>

<li><a href="Riemann_surface" title="Riemann surface">Riemann surface</a></li>

<li><a href="Schwarz_lemma" title="Schwarz lemma">Schwarz lemma</a></li>

<li><a href="Winding_number" title="Winding number">Winding number</a></li>

<li><a href="Analytic_continuation" title="Analytic continuation">Analytic continuation</a></li>

<li><a href="Riemann's_mapping_theorem" class="mw-redirect" title="Riemann's mapping theorem">Riemann's mapping theorem</a></li></ul></td>

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<a href="Complex_analysis#History" title="Complex analysis">People</a></th></tr><tr><td class="sidebar-content">

<ul><li><a href="Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li>

<li><a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li>

<li><a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a></li>

<li><a href="Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a></li>

<li><a href="Karl_Weierstrass" title="Karl Weierstrass">Karl Weierstrass</a></li></ul></td>

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, an <b>analytic function</b> is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> that is locally given by a <a href="Convergent_series" title="Convergent series">convergent</a> <a href="Power_series" title="Power series">power series</a>. There exist both <b>real analytic functions</b> and <b>complex analytic functions</b>. Functions of each type are <a href="Smooth_function" class="mw-redirect" title="Smooth function">infinitely differentiable</a>, but complex analytic functions exhibit properties that do not generally hold for real analytic functions.

</p><p>A function is analytic if and only if for every <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

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</math></span><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"></span> in its <a href="Domain_of_a_function" title="Domain of a function">domain</a>, its <a href="Taylor_series" title="Taylor series">Taylor series</a> about <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

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</math></span><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"></span> converges to the function in some <a href="Neighborhood_(topology)" class="mw-redirect" title="Neighborhood (topology)">neighborhood</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

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</math></span><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"></span>. This is stronger than merely being <a href="Smoothness" title="Smoothness">infinitely differentiable</a> at <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

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</math></span><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"></span>, and therefore having a well-defined Taylor series; the <a href="Fabius_function" title="Fabius function">Fabius function</a> provides an example of a function that is infinitely differentiable but not analytic.

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<div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2></div>

<p>Formally, a function <span class="mwe-math-element mwe-math-element-inline"><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}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is <i>real analytic</i> on an <a href="Open_set" title="Open set">open set</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

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<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

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</math></span><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></span> in the <a href="Real_line" class="mw-redirect" title="Real line">real line</a> if for any <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\in D}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle x_{0}\in D}</annotation>

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</math></span><img src="./b107b130543b591ac3037f4bc152da314ec51586.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.149ex; height:2.509ex;" alt="{\displaystyle x_{0}\in D}" loading="lazy"></span> one can write

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<annotation encoding="application/x-tex">{\displaystyle f(x)=\sum _{n=0}^{\infty }a_{n}\left(x-x_{0}\right)^{n}=a_{0}+a_{1}(x-x_{0})+a_{2}(x-x_{0})^{2}+\cdots }</annotation>

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</p><p>in which the coefficients <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0},a_{1},\dots }">

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<annotation encoding="application/x-tex">{\displaystyle a_{0},a_{1},\dots }</annotation>

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</math></span><img src="./acda3c3b0612c2e0d03099a7a5b25e6a6206a475.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.359ex; height:2.009ex;" alt="{\displaystyle a_{0},a_{1},\dots }" loading="lazy"></span> are real numbers and the <a href="Series_(mathematics)" title="Series (mathematics)">series</a> is <a href="Convergent_series" title="Convergent series">convergent</a> to <span class="mwe-math-element mwe-math-element-inline"><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)}">

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<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

</math></span><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"></span>.

</p><p>Alternatively, a real analytic function is an <a href="Smooth_function" class="mw-redirect" title="Smooth function">infinitely differentiable function</a> such that the <a href="Taylor_series" title="Taylor series">Taylor series</a> at any point <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

</math></span><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"></span> in its domain

</p><p><span class="mwe-math-element mwe-math-element-block"><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 T(x)=\sum _{n=0}^{\infty }{\frac {f^{(n)}(x_{0})}{n!}}(x-x_{0})^{n}}">

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T(x)=\sum _{n=0}^{\infty }{\frac {f^{(n)}(x_{0})}{n!}}(x-x_{0})^{n}}</annotation>

</semantics>

</math></span></span>

</p><p>converges to <span class="mwe-math-element mwe-math-element-inline"><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)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math></span><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"></span> for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> in a neighborhood of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

</math></span><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"></span> <a href="Pointwise_convergence" title="Pointwise convergence">pointwise</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> The set of all real analytic functions on a given set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math></span><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></span> is often denoted by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}^{\,\omega }(D)}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}^{\,\omega }(D)}</annotation>

</semantics>

</math></span><img src="./525b611457deda24f0fe6097595ca18dc62d2091.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.619ex; height:2.843ex;" alt="{\displaystyle {\mathcal {C}}^{\,\omega }(D)}" loading="lazy"></span>, or just by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}^{\,\omega }}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}^{\,\omega }}</annotation>

</semantics>

</math></span><img src="./6d9a9ccd07c2679dc3fe5fac55ad2c8271c7a2ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.885ex; height:2.343ex;" alt="{\displaystyle {\mathcal {C}}^{\,\omega }}" loading="lazy"></span> if the domain is understood.

</p><p>A function <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> defined on some subset of the real line is said to be real analytic at a point <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> if there is a neighborhood <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math></span><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></span> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> on which <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is real analytic.

</p><p>The definition of a <i>complex analytic function</i> is obtained by replacing, in the definitions above, "real" with "complex" and "real line" with "complex plane". A function is complex analytic if and only if it is <a href="Holomorphic_function" title="Holomorphic function">holomorphic</a> i.e. it is complex differentiable. For this reason the terms "holomorphic" and "analytic" are often used interchangeably for such functions.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>In complex analysis, a function is called analytic in an open set "U" if it is (complex) differentiable at each point in "U" and its complex derivative is continuous on "U".<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<p>Typical examples of analytic functions are

</p>

<ul><li>The following <a href="Elementary_function" title="Elementary function">elementary functions</a>:

<ul><li>All <a href="Polynomial" title="Polynomial">polynomials</a>: if a polynomial has degree <i>n</i>, any terms of degree larger than <i>n</i> in its Taylor series expansion must immediately vanish to 0, and so this series will be trivially convergent. Furthermore, every polynomial is its own <a href="Maclaurin_series" class="mw-redirect" title="Maclaurin series">Maclaurin series</a>.</li>

<li>The <a href="Exponential_function" title="Exponential function">exponential function</a> is analytic. Any Taylor series for this function converges not only for <i>x</i> close enough to <i>x</i><sub>0</sub> (as in the definition) but for all values of <i>x</i> (real or complex).</li>

<li>The <a href="Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric functions</a>, <a href="Logarithm" title="Logarithm">logarithm</a>, and the <a href="Exponentiation" title="Exponentiation">power functions</a> are analytic on any open set of their domain.</li></ul></li>

<li>Most <a href="Special_function" class="mw-redirect" title="Special function">special functions</a> (at least in some range of the complex plane):

<ul><li><a href="Hypergeometric_function" title="Hypergeometric function">hypergeometric functions</a></li>

<li><a href="Bessel_function" title="Bessel function">Bessel functions</a></li>

<li><a href="Gamma_function" title="Gamma function">gamma functions</a></li></ul></li></ul>

<p>Typical examples of functions that are not analytic are

</p>

<ul><li>The <a href="Absolute_value" title="Absolute value">absolute value</a> function when defined on the set of real numbers or <a href="Complex_number" title="Complex number">complex numbers</a> is not everywhere analytic because it is not differentiable at 0.</li>

<li><a href="Piecewise" class="mw-redirect" title="Piecewise">Piecewise defined</a> functions (functions given by different formulae in different regions) are typically not analytic where the pieces meet.</li>

<li>The <a href="Complex_conjugate" title="Complex conjugate">complex conjugate</a> function <i>z</i> → <i>z</i>* is not complex analytic, although its restriction to the real line is the identity function and therefore real analytic, and it is real analytic as a function from <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation>

</semantics>

</math></span><img src="./e150115ab9f63023215109595b76686a1ff890fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}" loading="lazy"></span> to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation>

</semantics>

</math></span><img src="./e150115ab9f63023215109595b76686a1ff890fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}" loading="lazy"></span>.</li>

<li>Other <a href="Non-analytic_smooth_function" title="Non-analytic smooth function">non-analytic smooth functions</a>, and in particular any smooth function <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> with compact support, i.e. <span class="mwe-math-element mwe-math-element-inline"><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\in {\mathcal {C}}_{0}^{\infty }(\mathbb {R} ^{n})}">

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\in {\mathcal {C}}_{0}^{\infty }(\mathbb {R} ^{n})}</annotation>

</semantics>

</math></span><img src="./e2db2c0dc133e7e1e0a113d49d88898ca657f7cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.944ex; height:2.843ex;" alt="{\displaystyle f\in {\mathcal {C}}_{0}^{\infty }(\mathbb {R} ^{n})}" loading="lazy"></span>, cannot be analytic on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

</semantics>

</math></span><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"></span>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Alternative_characterizations">Alternative characterizations</h2></div>

<p>The following conditions are equivalent:

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is real analytic on an open set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math></span><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></span>.</li>

<li>There is a complex analytic extension of <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> to an open set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\subset \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\subset \mathbb {C} }</annotation>

</semantics>

</math></span><img src="./ecbf949b480c4f7be87bfc6792e6588bd2f70b9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.603ex; height:2.176ex;" alt="{\displaystyle G\subset \mathbb {C} }" loading="lazy"></span> which contains <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math></span><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></span>.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is smooth and for every <a href="Compact_set" class="mw-redirect" title="Compact set">compact set</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\subset D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K\subset D}</annotation>

</semantics>

</math></span><img src="./90efac9fc3511c40b565f673076670c1a924dd4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.089ex; height:2.176ex;" alt="{\displaystyle K\subset D}" loading="lazy"></span> there exists a constant <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math></span><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></span> such that for every <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in K}</annotation>

</semantics>

</math></span><img src="./f9075ef2df706bf2764f76182a1ddf55ff231a92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.236ex; height:2.176ex;" alt="{\displaystyle x\in K}" loading="lazy"></span> and every non-negative integer <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> the following bound holds<sup id="cite_ref-FOOTNOTEKrantzParks200215_5-0" class="reference"><a href="#cite_note-FOOTNOTEKrantzParks200215-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element mwe-math-element-block"><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 \left|{\frac {d^{k}f}{dx^{k}}}(x)\right|\leq C^{k+1}k!}">

<mrow>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left|{\frac {d^{k}f}{dx^{k}}}(x)\right|\leq C^{k+1}k!}</annotation>

</semantics>

</math></span></span></li></ol>

<p>Complex analytic functions are exactly equivalent to <a href="Holomorphic_function" title="Holomorphic function">holomorphic functions</a>, and are thus much more easily characterized.

</p><p>For the case of an analytic function with several variables (see below), the real analyticity can be characterized using the <a href="Fourier%E2%80%93Bros%E2%80%93Iagolnitzer_transform" title="Fourier–Bros–Iagolnitzer transform">Fourier–Bros–Iagolnitzer transform</a>.

</p><p>In the multivariable case, real analytic functions satisfy a direct generalization of the third characterization.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subset \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U\subset \mathbb {R} ^{n}}</annotation>

</semantics>

</math></span><img src="./c1caefb347c86337ea7cd0c354acd2294bd7d81d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.778ex; height:2.343ex;" alt="{\displaystyle U\subset \mathbb {R} ^{n}}" loading="lazy"></span> be an open set, and let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:U\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:U\to \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./05f481901ff501baa824d1eab35eba6d9410ba57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.29ex; height:2.509ex;" alt="{\displaystyle f:U\to \mathbb {R} }" loading="lazy"></span>.

Then <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is real analytic on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math></span><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"></span> if and only if <span class="mwe-math-element mwe-math-element-inline"><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\in C^{\infty }(U)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\in C^{\infty }(U)}</annotation>

</semantics>

</math></span><img src="./9271f99825877470cdcf8b2b36c5ac3bc2000bf1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.384ex; height:2.843ex;" alt="{\displaystyle f\in C^{\infty }(U)}" loading="lazy"></span> and for every compact <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\subseteq U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K\subseteq U}</annotation>

</semantics>

</math></span><img src="./03e72a341c3ea869b7cd9a149f349b972a854a32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.947ex; height:2.343ex;" alt="{\displaystyle K\subseteq U}" loading="lazy"></span> there exists a constant <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math></span><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></span> such that for every multi-index <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in \mathbb {Z} _{\geq 0}^{n}}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha \in \mathbb {Z} _{\geq 0}^{n}}</annotation>

</semantics>

</math></span><img src="./5972a2f39837231103325e911d944d7cd31717bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.211ex; height:3.009ex;" alt="{\displaystyle \alpha \in \mathbb {Z} _{\geq 0}^{n}}" loading="lazy"></span> the following bound holds<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p><p><span class="mwe-math-element mwe-math-element-block"><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 \sup _{x\in K}\left|{\frac {\partial ^{\alpha }f}{\partial x^{\alpha }}}(x)\right|\leq C^{|\alpha |+1}\alpha !}">

</mrow>

</munder>

<mrow>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sup _{x\in K}\left|{\frac {\partial ^{\alpha }f}{\partial x^{\alpha }}}(x)\right|\leq C^{|\alpha |+1}\alpha !}</annotation>

</semantics>

</math></span></span>

</p>

<div class="mw-heading mw-heading2"><h2 id="Properties_of_analytic_functions">Properties of analytic functions</h2></div>

<ul><li>The sums, products, and <a href="Function_composition" title="Function composition">compositions</a> of analytic functions are analytic.</li>

<li>The <a href="Multiplicative_inverse" title="Multiplicative inverse">reciprocal</a> of an analytic function that is nowhere zero is analytic, as is the inverse of an invertible analytic function whose <a href="Derivative" title="Derivative">derivative</a> is nowhere zero. (See also the <a href="Lagrange_inversion_theorem" title="Lagrange inversion theorem">Lagrange inversion theorem</a>.)</li>

<li>Any analytic function is <a href="Smooth_function" class="mw-redirect" title="Smooth function">smooth</a>, that is, infinitely differentiable. The converse is not true for real functions; in fact, in a certain sense, the real analytic functions are sparse compared to all real infinitely differentiable functions. For the complex numbers, the converse does hold, and in fact any function differentiable <i>once</i> on an open set is analytic on that set (see "analyticity and differentiability" below).</li>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega \subseteq \mathbb {C} }</annotation>

</semantics>

</math></span><img src="./fd24e469a556af8609cf27aea06f0d2a885b35c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.455ex; height:2.343ex;" alt="{\displaystyle \Omega \subseteq \mathbb {C} }" loading="lazy"></span>, the set <i>A</i>(Ω) of all analytic functions <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u:\Omega \to \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u:\Omega \to \mathbb {C} }</annotation>

</semantics>

</math></span><img src="./518ff61da040dae470184e91a2c23b9635026ee0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.237ex; height:2.176ex;" alt="{\displaystyle u:\Omega \to \mathbb {C} }" loading="lazy"></span> is a <a href="Fr%C3%A9chet_space" title="Fréchet space">Fréchet space</a> with respect to the uniform convergence on compact sets. The fact that uniform limits on compact sets of analytic functions are analytic is an easy consequence of <a href="Morera's_theorem" title="Morera's theorem">Morera's theorem</a>. The set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\infty }(\Omega )}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{\infty }(\Omega )}</annotation>

</semantics>

</math></span><img src="./d726ca589415b69a7dd8cde7596f8ecbe8375916.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.106ex; height:2.843ex;" alt="{\displaystyle A_{\infty }(\Omega )}" loading="lazy"></span> of all <a href="Bounded_function" title="Bounded function">bounded</a> analytic functions with the <a href="Supremum_norm" class="mw-redirect" title="Supremum norm">supremum norm</a> is a <a href="Banach_space" title="Banach space">Banach space</a>.</li></ul>

<p>A polynomial cannot be zero at too many points unless it is the zero polynomial (more precisely, the number of zeros is at most the degree of the polynomial). A similar but weaker statement holds for analytic functions. If the set of zeros of an analytic function ƒ has an <a href="Accumulation_point" title="Accumulation point">accumulation point</a> inside its <a href="Domain_of_a_function" title="Domain of a function">domain</a>, then ƒ is zero everywhere on the <a href="Connected_space" title="Connected space">connected component</a> containing the accumulation point. In other words, if (<i>r<sub>n</sub></i>) is a <a href="Sequence" title="Sequence">sequence</a> of distinct numbers such that ƒ(<i>r</i><sub><i>n</i></sub>) = 0 for all <i>n</i> and this sequence <a href="Limit_of_a_sequence" title="Limit of a sequence">converges</a> to a point <i>r</i> in the domain of <i>D</i>, then ƒ is identically zero on the connected component of <i>D</i> containing <i>r</i>. This is known as the <a href="Identity_theorem" title="Identity theorem">identity theorem</a>.

</p><p>Also, if all the derivatives of an analytic function at a point are zero, the function is constant on the corresponding connected component.

</p><p>These statements imply that while analytic functions do have more <a href="Degrees_of_freedom_(physics_and_chemistry)" title="Degrees of freedom (physics and chemistry)">degrees of freedom</a> than polynomials, they are still quite rigid.

</p>

<div class="mw-heading mw-heading2"><h2 id="Analyticity_and_differentiability">Analyticity and differentiability</h2></div>

<p>As noted above, any analytic function (real or complex) is infinitely differentiable (also known as smooth, or <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}^{\infty }}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}^{\infty }}</annotation>

</semantics>

</math></span><img src="./a5ed72bb2fb83c421d84887d252bbee98aa6eae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.119ex; height:2.343ex;" alt="{\displaystyle {\mathcal {C}}^{\infty }}" loading="lazy"></span>). (Note that this differentiability is in the sense of real variables; compare complex derivatives below.) There exist smooth real functions that are not analytic: see <a href="Non-analytic_smooth_function" title="Non-analytic smooth function">non-analytic smooth function</a>. In fact there are many such functions.

</p><p>The situation is quite different when one considers complex analytic functions and complex derivatives. It can be proved that <a href="Proof_that_holomorphic_functions_are_analytic" class="mw-redirect" title="Proof that holomorphic functions are analytic">any complex function differentiable (in the complex sense) in an open set is analytic</a>. Consequently, in <a href="Complex_analysis" title="Complex analysis">complex analysis</a>, the term <i>analytic function</i> is synonymous with <i><a href="Holomorphic_function" title="Holomorphic function">holomorphic function</a></i>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Real_versus_complex_analytic_functions">Real versus complex analytic functions</h2></div>

<p>Real and complex analytic functions have important differences (one could notice that even from their different relationship with differentiability). Analyticity of complex functions is a more restrictive property, as it has more restrictive necessary conditions and complex analytic functions have more structure than their real-line counterparts.<sup id="cite_ref-FOOTNOTEKrantzParks2002_8-0" class="reference"><a href="#cite_note-FOOTNOTEKrantzParks2002-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p><p>According to <a href="Liouville's_theorem_(complex_analysis)" title="Liouville's theorem (complex analysis)">Liouville's theorem</a>, any bounded complex analytic function defined on the whole complex plane is constant. The corresponding statement for real analytic functions, with the complex plane replaced by the real line, is clearly false; this is illustrated by

</p><p><span class="mwe-math-element mwe-math-element-block"><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)={\frac {1}{x^{2}+1}}.}">

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {1}{x^{2}+1}}.}</annotation>

</semantics>

</math></span></span>

</p><p>Also, if a complex analytic function is defined in an open <a href="Ball_(mathematics)" title="Ball (mathematics)">ball</a> around a point <i>x</i><sub>0</sub>, its power series expansion at <i>x</i><sub>0</sub> is convergent in the whole open ball (<a href="Analyticity_of_holomorphic_functions" title="Analyticity of holomorphic functions">holomorphic functions are analytic</a>). This statement for real analytic functions (with open ball meaning an open <a href="Interval_(mathematics)" title="Interval (mathematics)">interval</a> of the real line rather than an open <a href="Disk_(mathematics)" title="Disk (mathematics)">disk</a> of the complex plane) is not true in general; the function of the example above gives an example for <i>x</i><sub>0</sub> = 0 and a ball of radius exceeding 1, since the power series <span class="nowrap">1 − <i>x</i><sup>2</sup> + <i>x</i><sup>4</sup> − <i>x</i><sup>6</sup>...</span> diverges for |<i>x</i>| ≥ 1.

</p><p>Any real analytic function on some <a href="Open_set" title="Open set">open set</a> on the real line can be extended to a complex analytic function on some open set of the complex plane. However, not every real analytic function defined on the whole real line can be extended to a complex function defined on the whole complex plane. The function <i>f</i>(<i>x</i>) defined in the paragraph above is a counterexample, as it is not defined for <i>x</i> = ±i. This explains why the Taylor series of <i>f</i>(<i>x</i>) diverges for |<i>x</i>| > 1, i.e., the <a href="Radius_of_convergence" title="Radius of convergence">radius of convergence</a> is 1 because the complexified function has a <a href="Complex_pole" class="mw-redirect" title="Complex pole">pole</a> at distance 1 from the evaluation point 0 and no further poles within the open disc of radius 1 around the evaluation point.

</p>

<div class="mw-heading mw-heading2"><h2 id="Analytic_functions_of_several_variables">Analytic functions of several variables</h2></div>

<p>One can define analytic functions in several variables by means of power series in those variables (see <a href="Power_series" title="Power series">power series</a>). Analytic functions of several variables have some of the same properties as analytic functions of one variable. However, especially for complex analytic functions, new and interesting phenomena show up in 2 or more complex dimensions:

</p>

<ul><li>Zero sets of complex analytic functions in more than one variable are never <a href="Discrete_space" title="Discrete space">discrete</a>. This can be proved by <a href="Hartogs's_extension_theorem" title="Hartogs's extension theorem">Hartogs's extension theorem</a>.</li>

<li><a href="Domain_of_holomorphy" title="Domain of holomorphy">Domains of holomorphy</a> for single-valued functions consist of arbitrary (connected) open sets. In several complex variables, however, only some connected open sets are domains of holomorphy. The characterization of domains of holomorphy leads to the notion of <a href="Pseudoconvexity" title="Pseudoconvexity">pseudoconvexity</a>.</li></ul>

<ul><li><a href="Cauchy%E2%80%93Riemann_equations" title="Cauchy–Riemann equations">Cauchy–Riemann equations</a></li>

<li><a href="Holomorphic_function" title="Holomorphic function">Holomorphic function</a></li>

<li><a href="Paley%E2%80%93Wiener_theorem" title="Paley–Wiener theorem">Paley–Wiener theorem</a></li>

<li><a href="Quasi-analytic_function" title="Quasi-analytic function">Quasi-analytic function</a></li>

<li><a href="Infinite_compositions_of_analytic_functions" title="Infinite compositions of analytic functions">Infinite compositions of analytic functions</a></li>

<li><a href="Non-analytic_smooth_function" title="Non-analytic smooth function">Non-analytic smooth function</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">This implies <a href="Uniform_convergence" title="Uniform convergence">uniform convergence</a> as well in a (possibly smaller) neighborhood of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

</math></span><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"></span>.</span>

</li>

</ol></div></div>

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</style><cite id="CITEREFChurchillBrownVerhey1948" class="citation book cs1">Churchill; Brown; Verhey (1948). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/complexvariable00chur/page/46"><i>Complex Variables and Applications</i></a></span>. McGraw-Hill. p. <a rel="nofollow" class="external text" href="https://archive.org/details/complexvariable00chur/page/46">46</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-010855-2</bdi>. <q>A function <i>f</i> of the complex variable <i>z</i> is <i>analytic</i> at point <i>z</i><sub>0</sub> if its derivative exists not only at <i>z</i> but at each point <i>z</i> in some neighborhood of <i>z</i><sub>0</sub>. It is analytic in a region <i>R</i> if it is analytic at every point in <i>R</i>. The term <i>holomorphic</i> is also used in the literature to denote analyticity</q></cite> <span class="cs1-hidden-error citation-comment"><code class="cs1-code">{{cite book}}</code>: </span><span class="cs1-hidden-error citation-comment">ISBN / Date incompatibility (help)</span></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFGamelin2004" class="citation book cs1">Gamelin, Theodore W. (2004). <i>Complex Analysis</i>. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9788181281142</bdi>.</cite></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"><cite id="CITEREFStrichartz,_Robert_S.1994" class="citation book cs1">Strichartz, Robert S. (1994). <i>A guide to distribution theory and Fourier transforms</i>. Boca Raton: CRC Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8493-8273-4</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/28890674">28890674</a>.</cite></span>

</li>

<li id="cite_note-FOOTNOTEKrantzParks200215-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKrantzParks200215_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKrantzParks2002">Krantz & Parks 2002</a>, p. 15.</span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFKomatsu1960" class="citation journal cs1">Komatsu, Hikosaburo (1960). <a rel="nofollow" class="external text" href="https://projecteuclid.org/euclid.pja/1195524081">"A characterization of real analytic functions"</a>. <i>Proceedings of the Japan Academy</i>. <b>36</b> (3): <span class="nowrap">90–</span>93. <a href="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/10.3792%2Fpja%2F1195524081">10.3792/pja/1195524081</a></span>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0021-4280">0021-4280</a>.</cite></span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Gevrey_class#References">"Gevrey class - Encyclopedia of Mathematics"</a>. <i>encyclopediaofmath.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-30</span></span>.</cite></span>

</li>

<li id="cite_note-FOOTNOTEKrantzParks2002-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKrantzParks2002_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKrantzParks2002">Krantz & Parks 2002</a>.</span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFConway1978" class="citation book cs1"><a href="John_B._Conway" title="John B. Conway">Conway, John B.</a> (1978). <i>Functions of One Complex Variable I</i>. <a href="Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a> 11 (2nd ed.). Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-90328-6</bdi>.</cite></li>

<li><cite id="CITEREFKrantzParks2002" class="citation book cs1"><a href="Steven_G._Krantz" title="Steven G. Krantz">Krantz, Steven</a>; <a href="Harold_R._Parks" title="Harold R. Parks">Parks, Harold R.</a> (2002). <i>A Primer of Real Analytic Functions</i> (2nd ed.). Birkhäuser. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8176-4264-1</bdi>.</cite></li>

<li><cite id="CITEREFGamelin2004" class="citation book cs1">Gamelin, Theodore W. (2004). <i>Complex Analysis</i>. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9788181281142</bdi>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Analytic_function">"Analytic function"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li>

<li><span class="citation mathworld" id="Reference-Mathworld-Analytic_Function"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/AnalyticFunction.html">"Analytic Function"</a>. <i><a href="MathWorld" title="MathWorld">MathWorld</a></i>.</cite></span></li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20130615052245/http://ivisoft.org/index.php/software/8-soft/6-zersol">Solver for all zeros of a complex analytic function that lie within a rectangular region by Ivan B. Ivanov</a></li></ul>

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</style></div><div role="navigation" class="navbox" aria-labelledby="Function330" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Function330" style="font-size:114%;margin:0 4em"><a href="Function_(mathematics)" title="Function (mathematics)">Function</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="History_of_the_function_concept" title="History of the function concept">History</a></li>

<li><a href="List_of_mathematical_functions" title="List of mathematical functions">List of specific functions</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types by domain and codomain</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classes/properties</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="Constant_function" title="Constant function">Constant</a></li>

<li><a href="Identity_function" title="Identity function">Identity</a></li>

<li><a href="Linear_map" title="Linear map">Linear</a></li>

<li><a href="Polynomial" title="Polynomial">Polynomial</a></li>

<li><a href="Rational_function" title="Rational function">Rational</a></li>

<li><a href="Algebraic_function" title="Algebraic function">Algebraic</a></li>

<li><a href="Smooth_function" class="mw-redirect" title="Smooth function">Smooth</a></li>

<li><a href="Continuous_function" title="Continuous function">Continuous</a></li>

<li><a href="Measurable_function" title="Measurable function">Measurable</a></li>

<li><a href="Injective_function" title="Injective function">Injective</a></li>

<li><a href="Surjective_function" title="Surjective function">Surjective</a></li>

<li><a href="Bijection" title="Bijection">Bijective</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constructions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Restriction_(mathematics)" title="Restriction (mathematics)">Restriction</a></li>

<li><a href="Function_composition" title="Function composition">Composition</a></li>

<li><a href="Inverse_function" title="Inverse function">Inverse</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</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="Relation_(mathematics)" title="Relation (mathematics)">Relation</a> (<a href="Binary_relation" title="Binary relation">Binary relation</a>)</li>

<li><a href="Set-valued_function" title="Set-valued function">Set-valued</a></li>

<li><a href="Multivalued_function" title="Multivalued function">Multivalued</a></li>

<li><a href="Partial_function" title="Partial function">Partial</a></li>

<li><a href="Implicit_function" title="Implicit function">Implicit</a></li>

<li><a href="Function_space" title="Function space">Space</a></li>

<li><a href="Higher-order_function" title="Higher-order function">Higher-order</a></li>

<li><a href="Morphism" title="Morphism">Morphism</a></li>

<li><a href="Functor" title="Functor">Functor</a></li></ul>

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<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> Category</li></ul>

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