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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Holomorphic function</span></span>
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</style><div role="note" class="hatnote navigation-not-searchable">For Zariski's theory of holomorphic functions on an algebraic variety, see <a href="Formal_holomorphic_function" title="Formal holomorphic function">formal holomorphic function</a>.</div>
<div role="note" class="hatnote navigation-not-searchable">"Holomorphism" redirects here; not to be confused with <a href="Homomorphism" title="Homomorphism">Homomorphism</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">
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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>holomorphic function</b> is a <a href="Complex-valued_function" class="mw-redirect" title="Complex-valued function">complex-valued function</a> of one or <a href="Function_of_several_complex_variables" title="Function of several complex variables">more</a> <a href="Complex_number" title="Complex number">complex</a> variables that is <a href="Differentiable_function#Differentiability_in_complex_analysis" title="Differentiable function">complex differentiable</a> in a <a href="Neighbourhood_(mathematics)" title="Neighbourhood (mathematics)">neighbourhood</a> of each point in a <a href="Domain_(mathematical_analysis)" title="Domain (mathematical analysis)">domain</a> in <a href="Function_of_several_complex_variables#The_complex_coordinate_space" title="Function of several complex variables">complex coordinate space</a> <span class="nowrap"><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 {C} ^{n}}">
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</math></span><img src="./a53b4e76242764d1bca004168353c380fef25258.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 {C} ^{n}}" loading="lazy"></span></span>. The existence of a complex derivative in a neighbourhood is a very strong condition: It implies that a holomorphic function is <a href="Infinitely_differentiable" class="mw-redirect" title="Infinitely differentiable">infinitely differentiable</a> and locally equal to its own <a href="Taylor_series" title="Taylor series">Taylor series</a> (is <i><a href="Analytic_function" title="Analytic function">analytic</a></i>). Holomorphic functions are the central objects of study in <a href="Complex_analysis" title="Complex analysis">complex analysis</a>.
</p><p>Though the term <i><a href="Analytic_function" title="Analytic function">analytic function</a></i> is often used interchangeably with "holomorphic function", the word "analytic" is defined in a broader sense to denote any function (real, complex, or of more general type) that can be written as a convergent <a href="Power_series" title="Power series">power series</a> in a neighbourhood of each point in its <a href="Domain_of_a_function" title="Domain of a function">domain</a>. That all holomorphic functions are complex analytic functions, and vice versa, is a <a href="Holomorphic_functions_are_analytic" class="mw-redirect" title="Holomorphic functions are analytic">major theorem in complex analysis</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p><p>Holomorphic functions are also sometimes referred to as <i>regular functions</i>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> A holomorphic function whose domain is the whole <a href="Complex_plane" title="Complex plane">complex plane</a> is called an <a href="Entire_function" title="Entire function">entire function</a>. The phrase "holomorphic at a point <span class="nowrap"><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 z_{0}}">
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</math></span><img src="./9e72d1d86e86355892b39b8eb32b964834e113bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{0}}" loading="lazy"></span></span> in the complex plane.
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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Given a complex-valued function <span class="nowrap"><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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</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></span> of a single complex variable, the <b>derivative</b> of <span class="nowrap"><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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</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></span> at a point <span class="nowrap"><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 z_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z_{0}}</annotation>
</semantics>
</math></span><img src="./9e72d1d86e86355892b39b8eb32b964834e113bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{0}}" loading="lazy"></span></span> in its domain is defined as the <a href="Limit_of_a_function" title="Limit of a function">limit</a><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>
</p>
<dl><dd><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'(z_{0})=\lim _{z\to z_{0}}{\frac {f(z)-f(z_{0})}{z-z_{0}}}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
<mo stretchy="false">→<!-- → --></mo>
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
</munder>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>z</mi>
<mo>−<!-- − --></mo>
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f'(z_{0})=\lim _{z\to z_{0}}{\frac {f(z)-f(z_{0})}{z-z_{0}}}.}</annotation>
</semantics>
</math></span><img src="./34df99111d0727236ade3f006e89f736e2a7222c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:27.167ex; height:6.009ex;" alt="{\displaystyle f'(z_{0})=\lim _{z\to z_{0}}{\frac {f(z)-f(z_{0})}{z-z_{0}}}.}" loading="lazy"></span></dd></dl>
<p>This is the same definition as for the <a href="Derivative" title="Derivative">derivative</a> of a <a href="Real_function" class="mw-redirect" title="Real function">real function</a>, except that all quantities are complex. In particular, the limit is taken as the complex number <span class="nowrap"><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 z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z}</annotation>
</semantics>
</math></span><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"></span></span> tends to <span class="nowrap"><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 z_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z_{0}}</annotation>
</semantics>
</math></span><img src="./9e72d1d86e86355892b39b8eb32b964834e113bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{0}}" loading="lazy"></span></span>, and this means that the same value is obtained for any sequence of complex values for <span class="nowrap"><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 z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z}</annotation>
</semantics>
</math></span><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"></span></span> that tends to <span class="nowrap"><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 z_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z_{0}}</annotation>
</semantics>
</math></span><img src="./9e72d1d86e86355892b39b8eb32b964834e113bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{0}}" loading="lazy"></span></span>. If the limit exists, <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./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></span> is said to be <b>complex differentiable</b> at <span class="nowrap"><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 z_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z_{0}}</annotation>
</semantics>
</math></span><img src="./9e72d1d86e86355892b39b8eb32b964834e113bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{0}}" loading="lazy"></span></span>. This concept of complex differentiability shares several properties with <a href="Derivative" title="Derivative">real differentiability</a>: It is <a href="Linear_transformation" class="mw-redirect" title="Linear transformation">linear</a> and obeys the <a href="Product_rule" title="Product rule">product rule</a>, <a href="Quotient_rule" title="Quotient rule">quotient rule</a>, and <a href="Chain_rule" title="Chain rule">chain rule</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><p>A function is <b>holomorphic</b> on an <a href="Open_set" title="Open set">open set</a> <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U}</annotation>
</semantics>
</math></span><img src="./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></span> if it is <i>complex differentiable</i> at <i>every</i> point of <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U}</annotation>
</semantics>
</math></span><img src="./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></span>. A function <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./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></span> is <i>holomorphic</i> at a point <span class="nowrap"><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 z_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z_{0}}</annotation>
</semantics>
</math></span><img src="./9e72d1d86e86355892b39b8eb32b964834e113bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{0}}" loading="lazy"></span></span> if it is holomorphic on some <a href="Neighbourhood_(mathematics)" title="Neighbourhood (mathematics)">neighbourhood</a> of <span class="nowrap"><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 z_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z_{0}}</annotation>
</semantics>
</math></span><img src="./9e72d1d86e86355892b39b8eb32b964834e113bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{0}}" loading="lazy"></span></span>.<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>
A function is <i>holomorphic</i> on some non-open set <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span></span> if it is holomorphic at every point of <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span></span>.
</p><p>A function may be complex differentiable at a point but not holomorphic at this point. For example, the 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 \textstyle f(z)=|z|{\vphantom {l}}^{2}=z{\bar {z}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mpadded width="0">
<mphantom>
<mi>l</mi>
</mphantom>
</mpadded>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>=</mo>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>z</mi>
<mo stretchy="false">¯<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle f(z)=|z|{\vphantom {l}}^{2}=z{\bar {z}}}</annotation>
</semantics>
</math></span><img src="./6044ccfae306aab708b7540997efbfbfbc412724.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.193ex; height:3.176ex;" alt="{\displaystyle \textstyle f(z)=|z|{\vphantom {l}}^{2}=z{\bar {z}}}" loading="lazy"></span> <i>is</i> complex differentiable at <span class="nowrap"><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 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>
</semantics>
</math></span><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"></span></span>, but <i>is not</i> complex differentiable anywhere else, esp. including in no place close to <span class="nowrap"><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 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>
</semantics>
</math></span><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"></span></span> (see the Cauchy–Riemann equations, below). So, it is <i>not</i> holomorphic at <span class="nowrap"><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 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>
</semantics>
</math></span><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"></span></span>.
</p><p>The relationship between real differentiability and complex differentiability is the following: If a complex function <span class="nowrap"><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+iy)=u(x,y)+i\,v(x,y)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mi>i</mi>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>u</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>i</mi>
<mspace width="thinmathspace"></mspace>
<mi>v</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x+iy)=u(x,y)+i\,v(x,y)}</annotation>
</semantics>
</math></span><img src="./fc5bc333ff95cf884422c1e04c0e459c5e07cc5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.458ex; height:2.843ex;" alt="{\displaystyle f(x+iy)=u(x,y)+i\,v(x,y)}" loading="lazy"></span></span> is holomorphic, then <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>u</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle u}</annotation>
</semantics>
</math></span><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.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 u}" loading="lazy"></span></span> and <span class="nowrap"><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 v}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>v</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle v}</annotation>
</semantics>
</math></span><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"></span></span> have first partial derivatives with respect to <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</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></span> and <span class="nowrap"><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 y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y}</annotation>
</semantics>
</math></span><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span></span>, and satisfy the <a href="Cauchy%E2%80%93Riemann_equations" title="Cauchy–Riemann equations">Cauchy–Riemann equations</a>:<sup id="cite_ref-Mark_6-0" class="reference"><a href="#cite_note-Mark-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>
</p>
<dl><dd><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 {\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}}\qquad {\mbox{and}}\qquad {\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}}\,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>u</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>v</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>y</mi>
</mrow>
</mfrac>
</mrow>
<mspace width="2em"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mtext>and</mtext>
</mstyle>
</mrow>
<mspace width="2em"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>u</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>y</mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>v</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mspace width="thinmathspace"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}}\qquad {\mbox{and}}\qquad {\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}}\,}</annotation>
</semantics>
</math></span><img src="./b2b7e5ead4b2cdcc7ccd78602fc988fc371d56ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.191ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}}\qquad {\mbox{and}}\qquad {\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}}\,}" loading="lazy"></span></dd></dl>
<p>or, equivalently, the <a href="Wirtinger_derivative" class="mw-redirect" title="Wirtinger derivative">Wirtinger derivative</a> of <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./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></span> with respect to <span class="nowrap"><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 {\bar {z}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>z</mi>
<mo stretchy="false">¯<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\bar {z}}}</annotation>
</semantics>
</math></span><img src="./52dd0599595d539f7d757ec21da6c6e6ac3ad427.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.296ex; height:2.009ex;" alt="{\displaystyle {\bar {z}}}" loading="lazy"></span></span>, the <a href="Complex_conjugate" title="Complex conjugate">complex conjugate</a> of <span class="nowrap"><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 z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z}</annotation>
</semantics>
</math></span><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"></span></span>, is zero:<sup id="cite_ref-Gunning_7-0" class="reference"><a href="#cite_note-Gunning-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>
</p>
<dl><dd><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 {\frac {\partial f}{\partial {\bar {z}}}}=0,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>f</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>z</mi>
<mo stretchy="false">¯<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial f}{\partial {\bar {z}}}}=0,}</annotation>
</semantics>
</math></span><img src="./090fb63c0b79ba8876f050090d9d16ca2ee444b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.358ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial f}{\partial {\bar {z}}}}=0,}" loading="lazy"></span></dd></dl>
<p>which is to say that, roughly, <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./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></span> is functionally independent from <span class="nowrap"><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 {\bar {z}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>z</mi>
<mo stretchy="false">¯<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\bar {z}}}</annotation>
</semantics>
</math></span><img src="./52dd0599595d539f7d757ec21da6c6e6ac3ad427.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.296ex; height:2.009ex;" alt="{\displaystyle {\bar {z}}}" loading="lazy"></span></span>, the complex conjugate of <span class="nowrap"><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 z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z}</annotation>
</semantics>
</math></span><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"></span></span>.
</p><p>If continuity is not given, the converse is not necessarily true. A simple converse is that if <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>u</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle u}</annotation>
</semantics>
</math></span><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.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 u}" loading="lazy"></span></span> and <span class="nowrap"><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 v}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>v</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle v}</annotation>
</semantics>
</math></span><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"></span></span> have <i>continuous</i> first partial derivatives and satisfy the Cauchy–Riemann equations, then <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./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></span> is holomorphic. A more satisfying converse, which is much harder to prove, is the <a href="Looman%E2%80%93Menchoff_theorem" title="Looman–Menchoff theorem">Looman–Menchoff theorem</a>: if <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./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></span> is continuous, <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>u</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle u}</annotation>
</semantics>
</math></span><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.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 u}" loading="lazy"></span></span> and <span class="nowrap"><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 v}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>v</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle v}</annotation>
</semantics>
</math></span><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"></span></span> have first partial derivatives (but not necessarily continuous), and they satisfy the Cauchy–Riemann equations, then <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./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></span> is holomorphic.<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><p>An immediate useful consequence of the Cauchy Riemann Equations above is that the complex derivative can be defined explicitly in terms of real partial derivatives. 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(z)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(z)}</annotation>
</semantics>
</math></span><img src="./d8dd568d570b390c337c0a911f0a1c5c214e8240.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.176ex; height:2.843ex;" alt="{\displaystyle f(z)}" loading="lazy"></span> is a complex function that is complex differentiable about 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 z=x+iy}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>z</mi>
<mo>=</mo>
<mi>x</mi>
<mo>+</mo>
<mi>i</mi>
<mi>y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z=x+iy}</annotation>
</semantics>
</math></span><img src="./08e90bb6b36fef59c6113eed2a08f10d77240741.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.315ex; height:2.509ex;" alt="{\displaystyle z=x+iy}" loading="lazy"></span> then (as we did earlier in the article) we can write <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(z)=f(x+iy)=u(x,y)+iv(x,y)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mi>i</mi>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>u</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>i</mi>
<mi>v</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(z)=f(x+iy)=u(x,y)+iv(x,y)}</annotation>
</semantics>
</math></span><img src="./07fa0c5ef6ae36c4b48104ead58d2949891ce19f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.346ex; height:2.843ex;" alt="{\displaystyle f(z)=f(x+iy)=u(x,y)+iv(x,y)}" loading="lazy"></span> and then the complex derivative of the function can be written as <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'(z)={\frac {\partial u}{\partial x}}+i{\frac {\partial v}{\partial x}}={\frac {\partial v}{\partial y}}-i{\frac {\partial u}{\partial y}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>u</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>+</mo>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>v</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>v</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>y</mi>
</mrow>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>u</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>y</mi>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f'(z)={\frac {\partial u}{\partial x}}+i{\frac {\partial v}{\partial x}}={\frac {\partial v}{\partial y}}-i{\frac {\partial u}{\partial y}}}</annotation>
</semantics>
</math></span><img src="./5d2ba4be8cfca7b07c4795c65e94d7085c5e58ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.147ex; height:6.009ex;" alt="{\displaystyle f'(z)={\frac {\partial u}{\partial x}}+i{\frac {\partial v}{\partial x}}={\frac {\partial v}{\partial y}}-i{\frac {\partial u}{\partial y}}}" loading="lazy"></span> <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>
</p>
<div class="mw-heading mw-heading2"><h2 id="Terminology">Terminology</h2></div>
<p>The term <i>holomorphic</i> was introduced in 1875 by <a href="Charles_Auguste_Briot" title="Charles Auguste Briot">Charles Briot</a> and <a href="Jean-Claude_Bouquet" class="mw-redirect" title="Jean-Claude Bouquet">Jean-Claude Bouquet</a>, two of <a href="Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a>'s students, and derives from the Greek <a href="https://en.wiktionary.org/wiki/%E1%BD%85%CE%BB%CE%BF%CF%82" class="extiw external" title="wikt:ὅλος">ὅλος</a> (<i>hólos</i>) meaning "whole", and <a href="https://en.wiktionary.org/wiki/%CE%BC%CE%BF%CF%81%CF%86%CE%AE" class="extiw external" title="wikt:μορφή">μορφή</a> (<i>morphḗ</i>) meaning "form" or "appearance" or "type", in contrast to the term <i><a href="Meromorphic_function" title="Meromorphic function">meromorphic</a></i> derived from <a href="https://en.wiktionary.org/wiki/%CE%BC%CE%AD%CF%81%CE%BF%CF%82" class="extiw external" title="wikt:μέρος">μέρος</a> (<i>méros</i>) meaning "part". A holomorphic function resembles an <a href="Entire_function" title="Entire function">entire function</a> ("whole") in a <a href="Domain_(mathematical_analysis)" title="Domain (mathematical analysis)">domain</a> of the complex plane while a meromorphic function (defined to mean holomorphic except at certain isolated <a href="Zeros_and_poles" title="Zeros and poles">poles</a>), resembles a rational fraction ("part") of entire functions in a domain of the complex plane.<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> Cauchy had instead used the term <i>synectic</i>.<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>
</p><p>Today, the term "holomorphic function" is sometimes preferred to "analytic function". An important result in complex analysis is that every holomorphic function is complex analytic, a fact that does not follow obviously from the definitions. The term "analytic" is however also in wide use.
</p>
<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>
<p>Because complex differentiation is linear and obeys the product, quotient, and chain rules, the sums, products and compositions of holomorphic functions are holomorphic, and the quotient of two holomorphic functions is holomorphic wherever the denominator is not zero.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> That is, if functions <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./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></span> and <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span></span> are holomorphic in a domain <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U}</annotation>
</semantics>
</math></span><img src="./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></span>, then so are <span class="nowrap"><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+g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>+</mo>
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f+g}</annotation>
</semantics>
</math></span><img src="./4d94a24abd865f6f9fd67a7df7e531cae1c769b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.235ex; height:2.509ex;" alt="{\displaystyle f+g}" loading="lazy"></span></span>, <span class="nowrap"><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-g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>−<!-- − --></mo>
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f-g}</annotation>
</semantics>
</math></span><img src="./e3f3019b383024c33e03b71a287d195f958ca89f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.235ex; height:2.509ex;" alt="{\displaystyle f-g}" loading="lazy"></span></span>, <span class="nowrap"><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 fg}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle fg}</annotation>
</semantics>
</math></span><img src="./06bac4638bb56f14688118ce88c188c7a021eb29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.395ex; height:2.509ex;" alt="{\displaystyle fg}" loading="lazy"></span></span>, and <span class="nowrap"><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\circ g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>∘<!-- ∘ --></mo>
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f\circ g}</annotation>
</semantics>
</math></span><img src="./b2f61ca7838709fbae07dce9c0d513770f10cfae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle f\circ g}" loading="lazy"></span></span>. Furthermore, <span class="nowrap"><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/g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f/g}</annotation>
</semantics>
</math></span><img src="./b5c6b7962d3532e248f07cd42b1bdc9e007b137d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.557ex; height:2.843ex;" alt="{\displaystyle f/g}" loading="lazy"></span></span> is holomorphic if <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span></span> has no zeros in <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U}</annotation>
</semantics>
</math></span><img src="./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></span>; otherwise it is <a href="Meromorphic" class="mw-redirect" title="Meromorphic">meromorphic</a>.
</p><p>If one identifies <span class="nowrap"><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 {C} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>
</semantics>
</math></span><img src="./f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy"></span></span> with the real <a href="Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a> <span class="nowrap"><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 \textstyle \mathbb {R} ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle \mathbb {R} ^{2}}</annotation>
</semantics>
</math></span><img src="./4f77a4c9424d3811db00e535695910dfc9e07db9.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 \textstyle \mathbb {R} ^{2}}" loading="lazy"></span></span>, then the holomorphic functions coincide with those functions of two real variables with continuous first derivatives which solve the <a href="Cauchy%E2%80%93Riemann_equations" title="Cauchy–Riemann equations">Cauchy–Riemann equations</a>, a set of two <a href="Partial_differential_equation" title="Partial differential equation">partial differential equations</a>.<sup id="cite_ref-Mark_6-1" class="reference"><a href="#cite_note-Mark-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>
</p><p>Every holomorphic function can be separated into its real and imaginary parts <span class="nowrap"><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+iy)=u(x,y)+i\,v(x,y)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mi>i</mi>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>u</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>i</mi>
<mspace width="thinmathspace"></mspace>
<mi>v</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x+iy)=u(x,y)+i\,v(x,y)}</annotation>
</semantics>
</math></span><img src="./fc5bc333ff95cf884422c1e04c0e459c5e07cc5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.458ex; height:2.843ex;" alt="{\displaystyle f(x+iy)=u(x,y)+i\,v(x,y)}" loading="lazy"></span></span>, and each of these is a <a href="Harmonic_function" title="Harmonic function">harmonic function</a> on <span class="nowrap"><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 \textstyle \mathbb {R} ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle \mathbb {R} ^{2}}</annotation>
</semantics>
</math></span><img src="./4f77a4c9424d3811db00e535695910dfc9e07db9.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 \textstyle \mathbb {R} ^{2}}" loading="lazy"></span></span> (each satisfies <a href="Laplace's_equation" title="Laplace's equation">Laplace's equation</a> <span class="nowrap"><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 \textstyle \nabla ^{2}u=\nabla ^{2}v=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<msup>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>u</mi>
<mo>=</mo>
<msup>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>v</mi>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle \nabla ^{2}u=\nabla ^{2}v=0}</annotation>
</semantics>
</math></span><img src="./00e759a73478b1ccbb5613f1fa3a760a9ca9b4fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.797ex; height:2.509ex;" alt="{\displaystyle \textstyle \nabla ^{2}u=\nabla ^{2}v=0}" loading="lazy"></span></span>), with <span class="nowrap"><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 v}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>v</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle v}</annotation>
</semantics>
</math></span><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"></span></span> the <a href="Harmonic_conjugate" title="Harmonic conjugate">harmonic conjugate</a> of <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>u</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle u}</annotation>
</semantics>
</math></span><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.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 u}" loading="lazy"></span></span>.<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>
Conversely, every harmonic function <span class="nowrap"><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(x,y)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>u</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle u(x,y)}</annotation>
</semantics>
</math></span><img src="./b57461332078df873c81bca2a395fd13f78bef50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.658ex; height:2.843ex;" alt="{\displaystyle u(x,y)}" loading="lazy"></span></span> on a <a href="Simply_connected_space" title="Simply connected space">simply connected</a> domain <span class="nowrap"><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 \textstyle \Omega \subset \mathbb {R} ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo>⊂<!-- ⊂ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle \Omega \subset \mathbb {R} ^{2}}</annotation>
</semantics>
</math></span><img src="./526ba705ce550d6924524d95eb501c8e3e7176e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.509ex; height:2.676ex;" alt="{\displaystyle \textstyle \Omega \subset \mathbb {R} ^{2}}" loading="lazy"></span></span> is the real part of a holomorphic function: If <span class="nowrap"><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 v}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>v</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle v}</annotation>
</semantics>
</math></span><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"></span></span> is the harmonic conjugate of <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>u</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle u}</annotation>
</semantics>
</math></span><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.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 u}" loading="lazy"></span></span>, unique up to a constant, then <span class="nowrap"><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+iy)=u(x,y)+i\,v(x,y)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mi>i</mi>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>u</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>i</mi>
<mspace width="thinmathspace"></mspace>
<mi>v</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x+iy)=u(x,y)+i\,v(x,y)}</annotation>
</semantics>
</math></span><img src="./fc5bc333ff95cf884422c1e04c0e459c5e07cc5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.458ex; height:2.843ex;" alt="{\displaystyle f(x+iy)=u(x,y)+i\,v(x,y)}" loading="lazy"></span></span> is holomorphic.
</p><p><a href="Cauchy's_integral_theorem" title="Cauchy's integral theorem">Cauchy's integral theorem</a> implies that the <a href="Contour_integral" class="mw-redirect" title="Contour integral">contour integral</a> of every holomorphic function along a <a href="Loop_(topology)" title="Loop (topology)">loop</a> vanishes:<sup id="cite_ref-Lang_14-0" class="reference"><a href="#cite_note-Lang-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>
</p>
<dl><dd><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 \oint _{\gamma }f(z)\,\mathrm {d} z=0.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mo>∮<!-- ∮ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>γ<!-- γ --></mi>
</mrow>
</msub>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mi>z</mi>
<mo>=</mo>
<mn>0.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \oint _{\gamma }f(z)\,\mathrm {d} z=0.}</annotation>
</semantics>
</math></span><img src="./0b3ccfab34a09ace37b76ba2214db9b1f5e09ed5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.656ex; height:6.009ex;" alt="{\displaystyle \oint _{\gamma }f(z)\,\mathrm {d} z=0.}" loading="lazy"></span></dd></dl>
<p>Here <span class="nowrap"><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 \gamma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>γ<!-- γ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>
</semantics>
</math></span><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"></span></span> is a <a href="Rectifiable_path" class="mw-redirect" title="Rectifiable path">rectifiable path</a> in a simply connected <a href="Domain_(mathematical_analysis)" title="Domain (mathematical analysis)">complex domain</a> <span class="nowrap"><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 {C} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
<mo>⊂<!-- ⊂ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U\subset \mathbb {C} }</annotation>
</semantics>
</math></span><img src="./1987e3c9fabe04333aa33cf0452f94a6daa71cbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.559ex; height:2.176ex;" alt="{\displaystyle U\subset \mathbb {C} }" loading="lazy"></span></span> whose start point is equal to its end point, and <span class="nowrap"><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\colon U\to \mathbb {C} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:<!-- : --></mo>
<mi>U</mi>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f\colon U\to \mathbb {C} }</annotation>
</semantics>
</math></span><img src="./97128887c8e8a0fac3a7a8a8775682f0680fcfb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.387ex; height:2.509ex;" alt="{\displaystyle f\colon U\to \mathbb {C} }" loading="lazy"></span></span> is a holomorphic function.
</p><p><a href="Cauchy's_integral_formula" title="Cauchy's integral formula">Cauchy's integral formula</a> states that every function holomorphic inside a <a href="Disk_(mathematics)" title="Disk (mathematics)">disk</a> is completely determined by its values on the disk's boundary.<sup id="cite_ref-Lang_14-1" class="reference"><a href="#cite_note-Lang-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Furthermore: Suppose <span class="nowrap"><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 {C} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
<mo>⊂<!-- ⊂ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U\subset \mathbb {C} }</annotation>
</semantics>
</math></span><img src="./1987e3c9fabe04333aa33cf0452f94a6daa71cbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.559ex; height:2.176ex;" alt="{\displaystyle U\subset \mathbb {C} }" loading="lazy"></span></span> is a complex domain, <span class="nowrap"><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\colon U\to \mathbb {C} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:<!-- : --></mo>
<mi>U</mi>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f\colon U\to \mathbb {C} }</annotation>
</semantics>
</math></span><img src="./97128887c8e8a0fac3a7a8a8775682f0680fcfb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.387ex; height:2.509ex;" alt="{\displaystyle f\colon U\to \mathbb {C} }" loading="lazy"></span></span> is a holomorphic function and the closed disk <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\equiv \{z:|z-z_{0}|\leq r\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>D</mi>
<mo>≡<!-- ≡ --></mo>
<mo fence="false" stretchy="false">{</mo>
<mi>z</mi>
<mo>:</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>z</mi>
<mo>−<!-- − --></mo>
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>≤<!-- ≤ --></mo>
<mi>r</mi>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle D\equiv \{z:|z-z_{0}|\leq r\}}</annotation>
</semantics>
</math></span><img src="./95b394dadecab060d85a71a335047c86a40efe63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.878ex; height:2.843ex;" alt="{\displaystyle D\equiv \{z:|z-z_{0}|\leq r\}}" loading="lazy"></span> is <a href="Neighbourhood_(mathematics)#Neighbourhood_of_a_set" title="Neighbourhood (mathematics)">completely contained</a> in <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U}</annotation>
</semantics>
</math></span><img src="./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></span>. Let <span class="nowrap"><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 \gamma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>γ<!-- γ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>
</semantics>
</math></span><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"></span></span> be the circle forming the <a href="Boundary_(topology)" title="Boundary (topology)">boundary</a> of <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>D</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle D}</annotation>
</semantics>
</math></span><img src="./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></span>. Then for every <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a}</annotation>
</semantics>
</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span></span> in the <a href="Interior_(topology)" title="Interior (topology)">interior</a> of <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>D</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle D}</annotation>
</semantics>
</math></span><img src="./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></span>:
</p>
<dl><dd><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(a)={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {f(z)}{z-a}}\,\mathrm {d} z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<mi>π<!-- π --></mi>
<mi>i</mi>
</mrow>
</mfrac>
</mrow>
<msub>
<mo>∮<!-- ∮ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>γ<!-- γ --></mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>z</mi>
<mo>−<!-- − --></mo>
<mi>a</mi>
</mrow>
</mfrac>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mi>z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(a)={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {f(z)}{z-a}}\,\mathrm {d} z}</annotation>
</semantics>
</math></span><img src="./951fa1dacd15f5e5217b31159f3bb5202f12bbe4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.503ex; height:6.509ex;" alt="{\displaystyle f(a)={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {f(z)}{z-a}}\,\mathrm {d} z}" loading="lazy"></span></dd></dl>
<p>where the contour integral is taken <a href="Curve_orientation" title="Curve orientation">counter-clockwise</a>.
</p><p>The derivative <span class="nowrap"><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'}(a)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {f'}(a)}</annotation>
</semantics>
</math></span><img src="./091c036a9529371ae89fe04ae6e14173611a803e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.044ex; height:3.009ex;" alt="{\displaystyle {f'}(a)}" loading="lazy"></span></span> can be written as a contour integral<sup id="cite_ref-Lang_14-2" class="reference"><a href="#cite_note-Lang-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> using <a href="Cauchy's_differentiation_formula" class="mw-redirect" title="Cauchy's differentiation formula">Cauchy's differentiation formula</a>:
</p>
<dl><dd><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'\!(a)={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {f(z)}{(z-a)^{2}}}\,\mathrm {d} z,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mspace width="negativethinmathspace"></mspace>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<mi>π<!-- π --></mi>
<mi>i</mi>
</mrow>
</mfrac>
</mrow>
<msub>
<mo>∮<!-- ∮ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>γ<!-- γ --></mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo>−<!-- − --></mo>
<mi>a</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mi>z</mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f'\!(a)={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {f(z)}{(z-a)^{2}}}\,\mathrm {d} z,}</annotation>
</semantics>
</math></span><img src="./b81525fa41f9656e591ffe8964fb1a4a660ca735.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.353ex; height:6.509ex;" alt="{\displaystyle f'\!(a)={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {f(z)}{(z-a)^{2}}}\,\mathrm {d} z,}" loading="lazy"></span></dd></dl>
<p>for any simple loop positively winding once around <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a}</annotation>
</semantics>
</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span></span>, and
</p>
<dl><dd><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'\!(a)=\lim \limits _{\gamma \to a}{\frac {i}{2{\mathcal {A}}(\gamma )}}\oint _{\gamma }f(z)\,\mathrm {d} {\bar {z}},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mspace width="negativethinmathspace"></mspace>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munder>
<mo form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>γ<!-- γ --></mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>a</mi>
</mrow>
</munder>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>i</mi>
<mrow>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">A</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>γ<!-- γ --></mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<msub>
<mo>∮<!-- ∮ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>γ<!-- γ --></mi>
</mrow>
</msub>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>z</mi>
<mo stretchy="false">¯<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f'\!(a)=\lim \limits _{\gamma \to a}{\frac {i}{2{\mathcal {A}}(\gamma )}}\oint _{\gamma }f(z)\,\mathrm {d} {\bar {z}},}</annotation>
</semantics>
</math></span><img src="./be69f8a7c878ac0418664c3c50c1f4e6cacadadd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.512ex; height:6.009ex;" alt="{\displaystyle f'\!(a)=\lim \limits _{\gamma \to a}{\frac {i}{2{\mathcal {A}}(\gamma )}}\oint _{\gamma }f(z)\,\mathrm {d} {\bar {z}},}" loading="lazy"></span></dd></dl>
<p>for <a href="Infinitesimal" title="Infinitesimal">infinitesimal</a> positive loops <span class="nowrap"><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 \gamma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>γ<!-- γ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>
</semantics>
</math></span><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"></span></span> around <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a}</annotation>
</semantics>
</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span></span>.
</p><p>In regions where the first derivative is not zero, holomorphic functions are <a href="Conformal_map" title="Conformal map">conformal</a>: they preserve angles and the shape (but not size) of small figures.<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>
</p><p>Every <a href="Holomorphic_functions_are_analytic" class="mw-redirect" title="Holomorphic functions are analytic">holomorphic function is analytic</a>. That is, a holomorphic function <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./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></span> has derivatives of every order at each point <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a}</annotation>
</semantics>
</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span></span> in its domain, and it coincides with its own <a href="Taylor_series" title="Taylor series">Taylor series</a> at <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a}</annotation>
</semantics>
</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span></span> in a neighbourhood of <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a}</annotation>
</semantics>
</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span></span>. In fact, <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./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></span> coincides with its Taylor series at <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a}</annotation>
</semantics>
</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span></span> in any disk centred at that point and lying within the domain of the function.
</p><p>From an algebraic point of view, the set of holomorphic functions on an open set is a <a href="Commutative_ring" title="Commutative ring">commutative ring</a> and a <a href="Complex_vector_space" class="mw-redirect" title="Complex vector space">complex vector space</a>. Additionally, the set of holomorphic functions in an open set <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U}</annotation>
</semantics>
</math></span><img src="./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></span> is an <a href="Integral_domain" title="Integral domain">integral domain</a> <a href="If_and_only_if" title="If and only if">if and only if</a> the open set <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U}</annotation>
</semantics>
</math></span><img src="./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></span> is connected. <sup id="cite_ref-Gunning_7-1" class="reference"><a href="#cite_note-Gunning-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> In fact, it is a <a href="Locally_convex_topological_vector_space" title="Locally convex topological vector space">locally convex topological vector space</a>, with the <a href="Norm_(mathematics)" title="Norm (mathematics)">seminorms</a> being the <a href="Suprema" class="mw-redirect" title="Suprema">suprema</a> on <a href="Compact_subset" class="mw-redirect" title="Compact subset">compact subsets</a>.
</p><p>From a geometric perspective, a function <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./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></span> is holomorphic at <span class="nowrap"><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 z_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z_{0}}</annotation>
</semantics>
</math></span><img src="./9e72d1d86e86355892b39b8eb32b964834e113bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{0}}" loading="lazy"></span></span> if and only if its <a href="Exterior_derivative" title="Exterior derivative">exterior derivative</a> <span class="nowrap"><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 \mathrm {d} f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} f}</annotation>
</semantics>
</math></span><img src="./9f07af1376e3e468900fa251f847b5c0755e7dc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.571ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} f}" loading="lazy"></span></span> in a neighbourhood <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U}</annotation>
</semantics>
</math></span><img src="./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></span> of <span class="nowrap"><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 z_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z_{0}}</annotation>
</semantics>
</math></span><img src="./9e72d1d86e86355892b39b8eb32b964834e113bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{0}}" loading="lazy"></span></span> is equal to <span class="nowrap"><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'(z)\,\mathrm {d} z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mi>z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f'(z)\,\mathrm {d} z}</annotation>
</semantics>
</math></span><img src="./330dda3552f439d193aa773314e187d2e9220bf6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.67ex; height:3.009ex;" alt="{\displaystyle f'(z)\,\mathrm {d} z}" loading="lazy"></span></span> for some continuous function <span class="nowrap"><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'}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f'}</annotation>
</semantics>
</math></span><img src="./258eaada38956fb69b8cb1a2eef46bcb97d3126b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.005ex; height:2.843ex;" alt="{\displaystyle f'}" loading="lazy"></span></span>. It follows from
</p>
<dl><dd><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 0=\mathrm {d} ^{2}f=\mathrm {d} (f'\,\mathrm {d} z)=\mathrm {d} f'\wedge \mathrm {d} z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>0</mn>
<mo>=</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>f</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo>∧<!-- ∧ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mi>z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 0=\mathrm {d} ^{2}f=\mathrm {d} (f'\,\mathrm {d} z)=\mathrm {d} f'\wedge \mathrm {d} z}</annotation>
</semantics>
</math></span><img src="./23b61dc210362bcda257a0fcd78ce494f46f951d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.219ex; height:3.176ex;" alt="{\displaystyle 0=\mathrm {d} ^{2}f=\mathrm {d} (f'\,\mathrm {d} z)=\mathrm {d} f'\wedge \mathrm {d} z}" loading="lazy"></span></dd></dl>
<p>that <span class="nowrap"><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 \mathrm {d} f'}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} f'}</annotation>
</semantics>
</math></span><img src="./c56980acdd88b889d6bef7cf391b09ad52d3c922.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.298ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} f'}" loading="lazy"></span></span> is also proportional to <span class="nowrap"><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 \mathrm {d} z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mi>z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} z}</annotation>
</semantics>
</math></span><img src="./ea3c148df96cf0581719e9fef9f7062633e1533f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.381ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} z}" loading="lazy"></span></span>, implying that the derivative <span class="nowrap"><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 \mathrm {d} f'}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} f'}</annotation>
</semantics>
</math></span><img src="./c56980acdd88b889d6bef7cf391b09ad52d3c922.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.298ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} f'}" loading="lazy"></span></span> is itself holomorphic and thus that <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./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></span> is infinitely differentiable. Similarly, <span class="nowrap"><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 \mathrm {d} (f\,\mathrm {d} z)=f'\,\mathrm {d} z\wedge \mathrm {d} z=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mi>z</mi>
<mo>∧<!-- ∧ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mi>z</mi>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} (f\,\mathrm {d} z)=f'\,\mathrm {d} z\wedge \mathrm {d} z=0}</annotation>
</semantics>
</math></span><img src="./afa04fd7f08b7644d90697b6449a54c18bcc2b3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.244ex; height:3.009ex;" alt="{\displaystyle \mathrm {d} (f\,\mathrm {d} z)=f'\,\mathrm {d} z\wedge \mathrm {d} z=0}" loading="lazy"></span></span> implies that any function <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./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></span> that is holomorphic on the simply connected region <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U}</annotation>
</semantics>
</math></span><img src="./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></span> is also integrable on <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U}</annotation>
</semantics>
</math></span><img src="./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></span>.
</p><p>(For a path <span class="nowrap"><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 \gamma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>γ<!-- γ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>
</semantics>
</math></span><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"></span></span> from <span class="nowrap"><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 z_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z_{0}}</annotation>
</semantics>
</math></span><img src="./9e72d1d86e86355892b39b8eb32b964834e113bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{0}}" loading="lazy"></span></span> to <span class="nowrap"><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 z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z}</annotation>
</semantics>
</math></span><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"></span></span> lying entirely in <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U}</annotation>
</semantics>
</math></span><img src="./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></span>, define <span class="nowrap"><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_{\gamma }(z)=F(0)+\int _{\gamma }f\,\mathrm {d} z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>γ<!-- γ --></mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>+</mo>
<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>γ<!-- γ --></mi>
</mrow>
</msub>
<mi>f</mi>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mi>z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F_{\gamma }(z)=F(0)+\int _{\gamma }f\,\mathrm {d} z}</annotation>
</semantics>
</math></span><img src="./8a3d0692c0fb64ca12b46f33659de45aa13b9a12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.019ex; height:6.009ex;" alt="{\displaystyle F_{\gamma }(z)=F(0)+\int _{\gamma }f\,\mathrm {d} z}" loading="lazy"></span></span>; in light of the <a href="Jordan_curve_theorem" title="Jordan curve theorem">Jordan curve theorem</a> and the <a href="Stokes'_theorem" title="Stokes' theorem">generalized Stokes' theorem</a>, <span class="nowrap"><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_{\gamma }(z)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>γ<!-- γ --></mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F_{\gamma }(z)}</annotation>
</semantics>
</math></span><img src="./7a709a06adc83549fec43dd10f564f7a43c57b30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.517ex; height:3.009ex;" alt="{\displaystyle F_{\gamma }(z)}" loading="lazy"></span></span> is independent of the particular choice of path <span class="nowrap"><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 \gamma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>γ<!-- γ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>
</semantics>
</math></span><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"></span></span>, and thus <span class="nowrap"><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(z)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F(z)}</annotation>
</semantics>
</math></span><img src="./5cf298d777120df944559c5e985b88a824debb80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.638ex; height:2.843ex;" alt="{\displaystyle F(z)}" loading="lazy"></span></span> is a well-defined function on <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U}</annotation>
</semantics>
</math></span><img src="./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></span> having <span class="nowrap"><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 \mathrm {d} F=f\,\mathrm {d} z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mi>F</mi>
<mo>=</mo>
<mi>f</mi>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mi>z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} F=f\,\mathrm {d} z}</annotation>
</semantics>
</math></span><img src="./7338359b4357512dfa3160d1c7188613a49493b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.178ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} F=f\,\mathrm {d} z}" loading="lazy"></span></span> or <span class="nowrap"><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={\frac {\mathrm {d} F}{\mathrm {d} z}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mi>F</mi>
</mrow>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mi>z</mi>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f={\frac {\mathrm {d} F}{\mathrm {d} z}}}</annotation>
</semantics>
</math></span><img src="./2e3c9fe91f13b855572891ea0f8e49d77cf419af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.246ex; height:5.509ex;" alt="{\displaystyle f={\frac {\mathrm {d} F}{\mathrm {d} z}}}" loading="lazy"></span></span>.)
</p>
<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>
<p>All <a href="Polynomial" title="Polynomial">polynomial</a> functions in <span class="nowrap"><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 z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z}</annotation>
</semantics>
</math></span><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"></span></span> with complex <a href="Coefficient" title="Coefficient">coefficients</a> are <a href="Entire_function" title="Entire function">entire functions</a> (holomorphic in the whole complex plane <span class="nowrap"><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 {C} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>
</semantics>
</math></span><img src="./f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy"></span></span>), and so are the <a href="Exponential_function#Complex_plane" title="Exponential function">exponential function</a> <span class="nowrap"><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 \exp z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>exp</mi>
<mo><!-- --></mo>
<mi>z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \exp z}</annotation>
</semantics>
</math></span><img src="./21a108271d26f82467f50e0bd74ec9f68be34a3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.028ex; height:2.009ex;" alt="{\displaystyle \exp z}" loading="lazy"></span></span> and the <a href="Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a> <span class="nowrap"><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 \cos {z}={\tfrac {1}{2}}{\bigl (}\exp(+iz)+\exp(-iz){\bigr )}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>cos</mi>
<mo><!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="1.2em" minsize="1.2em">(</mo>
</mrow>
</mrow>
<mi>exp</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mo>+</mo>
<mi>i</mi>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>exp</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mi>i</mi>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="1.2em" minsize="1.2em">)</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \cos {z}={\tfrac {1}{2}}{\bigl (}\exp(+iz)+\exp(-iz){\bigr )}}</annotation>
</semantics>
</math></span><img src="./9dc3e31029d6ae2555ff9913c31d794cdf96b9e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:32.821ex; height:3.509ex;" alt="{\displaystyle \cos {z}={\tfrac {1}{2}}{\bigl (}\exp(+iz)+\exp(-iz){\bigr )}}" loading="lazy"></span></span> and <span class="nowrap"><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 \sin {z}=-{\tfrac {1}{2}}i{\bigl (}\exp(+iz)-\exp(-iz){\bigr )}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>sin</mi>
<mo><!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="1.2em" minsize="1.2em">(</mo>
</mrow>
</mrow>
<mi>exp</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mo>+</mo>
<mi>i</mi>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>exp</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mi>i</mi>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="1.2em" minsize="1.2em">)</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sin {z}=-{\tfrac {1}{2}}i{\bigl (}\exp(+iz)-\exp(-iz){\bigr )}}</annotation>
</semantics>
</math></span><img src="./3a47f5ece6644e502f79de70082aab74a1b7fa23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:35.176ex; height:3.509ex;" alt="{\displaystyle \sin {z}=-{\tfrac {1}{2}}i{\bigl (}\exp(+iz)-\exp(-iz){\bigr )}}" loading="lazy"></span></span> (cf. <a href="Euler's_formula" title="Euler's formula">Euler's formula</a>). The <a href="Principal_branch" title="Principal branch">principal branch</a> of the <a href="Complex_logarithm" title="Complex logarithm">complex logarithm</a> function <span class="nowrap"><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 \log z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>log</mi>
<mo><!-- --></mo>
<mi>z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \log z}</annotation>
</semantics>
</math></span><img src="./3d208820c31babc9dfabefde37b4309cfceecc6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.447ex; height:2.509ex;" alt="{\displaystyle \log z}" loading="lazy"></span></span> is holomorphic on the domain <span class="nowrap"><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 {C} \smallsetminus \{z\in \mathbb {R} :z\leq 0\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<mo>∖<!-- ∖ --></mo>
<mo fence="false" stretchy="false">{</mo>
<mi>z</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>:</mo>
<mi>z</mi>
<mo>≤<!-- ≤ --></mo>
<mn>0</mn>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} \smallsetminus \{z\in \mathbb {R} :z\leq 0\}}</annotation>
</semantics>
</math></span><img src="./42febecda42157146e17fa832f1fa9dff096722b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.736ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} \smallsetminus \{z\in \mathbb {R} :z\leq 0\}}" loading="lazy"></span></span>. The <a href="Square_root#Principal_square_root_of_a_complex_number" title="Square root">square root</a> function can be defined as <span class="nowrap"><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 {\sqrt {z}}\equiv \exp {\bigl (}{\tfrac {1}{2}}\log z{\bigr )}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>z</mi>
</msqrt>
</mrow>
<mo>≡<!-- ≡ --></mo>
<mi>exp</mi>
<mo><!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="1.2em" minsize="1.2em">(</mo>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
<mi>log</mi>
<mo><!-- --></mo>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="1.2em" minsize="1.2em">)</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\sqrt {z}}\equiv \exp {\bigl (}{\tfrac {1}{2}}\log z{\bigr )}}</annotation>
</semantics>
</math></span><img src="./164d864ad5834b19d3b96ecca015ac6e425b1233.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.684ex; height:3.509ex;" alt="{\displaystyle {\sqrt {z}}\equiv \exp {\bigl (}{\tfrac {1}{2}}\log z{\bigr )}}" loading="lazy"></span></span> and is therefore holomorphic wherever the logarithm <span class="nowrap"><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 \log z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>log</mi>
<mo><!-- --></mo>
<mi>z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \log z}</annotation>
</semantics>
</math></span><img src="./3d208820c31babc9dfabefde37b4309cfceecc6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.447ex; height:2.509ex;" alt="{\displaystyle \log z}" loading="lazy"></span></span> is. The <a href="Multiplicative_inverse#Complex_numbers" title="Multiplicative inverse">reciprocal function</a> <span class="nowrap"><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 {\tfrac {1}{z}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mi>z</mi>
</mfrac>
</mstyle>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{z}}}</annotation>
</semantics>
</math></span><img src="./fa61046938315847ced83e826ab00e87a333ecb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.658ex; height:3.343ex;" alt="{\displaystyle {\tfrac {1}{z}}}" loading="lazy"></span></span> is holomorphic on <span class="nowrap"><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 {C} \smallsetminus \{0\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<mo>∖<!-- ∖ --></mo>
<mo fence="false" stretchy="false">{</mo>
<mn>0</mn>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} \smallsetminus \{0\}}</annotation>
</semantics>
</math></span><img src="./af3d432a24ec7a840831623118623c8308019105.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.006ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} \smallsetminus \{0\}}" loading="lazy"></span></span>. (The reciprocal function, and any other <a href="Rational_function" title="Rational function">rational function</a>, is <a href="Meromorphic_function" title="Meromorphic function">meromorphic</a> on <span class="nowrap"><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 {C} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>
</semantics>
</math></span><img src="./f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy"></span></span>.)
</p><p>As a consequence of the <a href="Cauchy%E2%80%93Riemann_equations" title="Cauchy–Riemann equations">Cauchy–Riemann equations</a>, any real-valued holomorphic function must be <a href="Constant_function" title="Constant function">constant</a>. Therefore, the <a href="Absolute_value#Complex_numbers" title="Absolute value">absolute value</a> <span class="nowrap"><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 |z|}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |z|}</annotation>
</semantics>
</math></span><img src="./28fd4d7dcabf618d707c21bd08306c7b3aa8b68e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.382ex; height:2.843ex;" alt="{\displaystyle |z|}" loading="lazy"></span>,</span> the <a href="Argument_(complex_analysis)" title="Argument (complex analysis)">argument</a> <span class="nowrap"><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 \arg z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>arg</mi>
<mo><!-- --></mo>
<mi>z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \arg z}</annotation>
</semantics>
</math></span><img src="./3b42890eec05c72e9d06b055e4f5d0587f07067a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.712ex; height:2.009ex;" alt="{\displaystyle \arg z}" loading="lazy"></span></span>, the <a href="Complex_number#Notation" title="Complex number">real part</a> <span class="nowrap"><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 \operatorname {Re} (z)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Re</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (z)}</annotation>
</semantics>
</math></span><img src="./0011b805a1992a94966191f7fdf2e2307beebd9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.64ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (z)}" loading="lazy"></span></span> and the <a href="Complex_number#Notation" title="Complex number">imaginary part</a> <span class="nowrap"><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 \operatorname {Im} (z)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Im</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Im} (z)}</annotation>
</semantics>
</math></span><img src="./2f33f1c2b5cf909df09edd814d3ef3b8f9de9f95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.673ex; height:2.843ex;" alt="{\displaystyle \operatorname {Im} (z)}" loading="lazy"></span></span> are not holomorphic. Another typical example of a <a href="Continuous_function" title="Continuous function">continuous function</a> which is not holomorphic is the <a href="Complex_conjugate" title="Complex conjugate">complex conjugate</a> <span class="nowrap"><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 {\bar {z}}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>z</mi>
<mo stretchy="false">¯<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\bar {z}}.}</annotation>
</semantics>
</math></span><img src="./2f343995da9d35ee3e2b74dae50ef9be5726982a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.943ex; height:2.009ex;" alt="{\displaystyle {\bar {z}}.}" loading="lazy"></span></span> (The complex conjugate is <a href="Antiholomorphic_function" title="Antiholomorphic function">antiholomorphic</a>.)
</p>
<div class="mw-heading mw-heading2"><h2 id="Several_variables">Several variables</h2></div>
<p>The definition of a holomorphic function generalizes to several complex variables in a straightforward way. A function <span class="nowrap"><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\colon (z_{1},z_{2},\ldots ,z_{n})\mapsto f(z_{1},z_{2},\ldots ,z_{n})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:<!-- : --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">↦<!-- ↦ --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f\colon (z_{1},z_{2},\ldots ,z_{n})\mapsto f(z_{1},z_{2},\ldots ,z_{n})}</annotation>
</semantics>
</math></span><img src="./fc62b4da60d87e4c5c84bda38a2d8489f0f1fe1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.389ex; height:2.843ex;" alt="{\displaystyle f\colon (z_{1},z_{2},\ldots ,z_{n})\mapsto f(z_{1},z_{2},\ldots ,z_{n})}" loading="lazy"></span></span> in <span class="nowrap"><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 n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span></span> complex variables is <b>analytic</b> at a point <span class="nowrap"><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 p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
</semantics>
</math></span><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span></span> if there exists a neighbourhood of <span class="nowrap"><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 p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
</semantics>
</math></span><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span></span> in which <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./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></span> is equal to a convergent power series in <span class="nowrap"><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 n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span></span> complex variables;<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>
the function <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./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></span> is <b>holomorphic</b> in an open subset <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U}</annotation>
</semantics>
</math></span><img src="./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></span> of <span class="nowrap"><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 {C} ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation>
</semantics>
</math></span><img src="./a53b4e76242764d1bca004168353c380fef25258.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 {C} ^{n}}" loading="lazy"></span></span> if it is analytic at each point in <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
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<annotation encoding="application/x-tex">{\displaystyle U}</annotation>
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</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></span>. <a href="Osgood's_lemma" title="Osgood's lemma">Osgood's lemma</a> shows (using the multivariate Cauchy integral formula) that, for a continuous function <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
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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></span>, this is equivalent to <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<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></span> being holomorphic in each variable separately (meaning that if any <span class="nowrap"><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 n-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle n-1}</annotation>
</semantics>
</math></span><img src="./fbd0b0f32b28f51962943ee9ede4fb34198a2521.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}" loading="lazy"></span></span> coordinates are fixed, then the restriction of <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./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></span> is a holomorphic function of the remaining coordinate). The much deeper <a href="Hartogs'_theorem" class="mw-redirect" title="Hartogs' theorem">Hartogs' theorem</a> proves that the continuity assumption is unnecessary: <span class="nowrap"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./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></span> is holomorphic if and only if it is holomorphic in each variable separately.
</p><p>More generally, a function of several complex variables that is <a href="Square_integrable" class="mw-redirect" title="Square integrable">square integrable</a> over every <a href="Compact_set" class="mw-redirect" title="Compact set">compact subset</a> of its domain is analytic if and only if it satisfies the Cauchy–Riemann equations in the sense of distributions.
</p><p>Functions of several complex variables are in some basic ways more complicated than functions of a single complex variable. For example, the region of convergence of a power series is not necessarily an open ball; these regions are logarithmically-convex <a href="Reinhardt_domain" class="mw-redirect" title="Reinhardt domain">Reinhardt domains</a>, the simplest example of which is a <a href="Polydisk" class="mw-redirect" title="Polydisk">polydisk</a>. However, they also come with some fundamental restrictions. Unlike functions of a single complex variable, the possible domains on which there are holomorphic functions that cannot be extended to larger domains are highly limited. Such a set is called a <a href="Domain_of_holomorphy" title="Domain of holomorphy">domain of holomorphy</a>.
</p><p>A <a href="Complex_differential_form#Holomorphic_forms" title="Complex differential form">complex differential <span class="nowrap"><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 (p,0)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo>,</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (p,0)}</annotation>
</semantics>
</math></span><img src="./608b63cd35d0b2ba89c90f43af1ffe40b0132c88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.175ex; height:2.843ex;" alt="{\displaystyle (p,0)}" loading="lazy"></span></span>-form</a> <span class="nowrap"><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 }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>
</semantics>
</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span></span> is holomorphic if and only if its antiholomorphic <a href="Complex_differential_form#The_Dolbeault_operators" title="Complex differential form">Dolbeault derivative</a> is zero: <span class="nowrap"><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 {\bar {\partial }}\alpha =0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mo stretchy="false">¯<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
<mi>α<!-- α --></mi>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\bar {\partial }}\alpha =0}</annotation>
</semantics>
</math></span><img src="./27b6fe278f26391dfee4049e694e95949bc7bbef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.237ex; height:2.676ex;" alt="{\displaystyle {\bar {\partial }}\alpha =0}" loading="lazy"></span></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Extension_to_functional_analysis">Extension to functional analysis</h2></div>
<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Infinite-dimensional_holomorphy" title="Infinite-dimensional holomorphy">infinite-dimensional holomorphy</a></div>
<p>The concept of a holomorphic function can be extended to the infinite-dimensional spaces of <a href="Functional_analysis" title="Functional analysis">functional analysis</a>. For instance, the <a href="Fr%C3%A9chet_derivative" title="Fréchet derivative">Fréchet</a> or <a href="Gateaux_derivative" title="Gateaux derivative">Gateaux derivative</a> can be used to define a notion of a holomorphic function on a <a href="Banach_space" title="Banach space">Banach space</a> over the field of complex numbers.
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
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<ul><li><a href="Antiderivative_(complex_analysis)" title="Antiderivative (complex analysis)">Antiderivative (complex analysis)</a></li>
<li><a href="Antiholomorphic_function" title="Antiholomorphic function">Antiholomorphic function</a></li>
<li><a href="Biholomorphy" class="mw-redirect" title="Biholomorphy">Biholomorphy</a></li>
<li><a href="Cauchy's_estimate" title="Cauchy's estimate">Cauchy's estimate</a></li>
<li><a href="Harmonic_map" title="Harmonic map">Harmonic maps</a></li>
<li><a href="Harmonic_morphism" title="Harmonic morphism">Harmonic morphisms</a></li>
<li><a href="Holomorphic_separability" title="Holomorphic separability">Holomorphic separability</a></li>
<li><a href="Meromorphic_function" title="Meromorphic function">Meromorphic function</a></li>
<li><a href="Quadrature_domains" title="Quadrature domains">Quadrature domains</a></li>
<li><a href="Wirtinger_derivatives" title="Wirtinger derivatives">Wirtinger derivatives</a></li></ul>
</div>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">
<style data-mw-deduplicate="TemplateStyles:r1238218222">
/* start https://en.wikipedia.org/ */
.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("./mw/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("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/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}}
/* end https://en.wikipedia.org/ */
</style><cite class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php/Analytic_function#Analytic_functions_of_one_complex_variable">"Analytic functions of one complex variable"</a>. <i>Encyclopedia of Mathematics</i>. European Mathematical Society / Springer. 2015 – via encyclopediaofmath.org.</cite></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><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]<span class="reference-accessdate">, retrieved <span class="nowrap">February 26,</span> 2021</span></cite></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"><a href="Lars_Ahlfors" title="Lars Ahlfors">Ahlfors, L.</a>, <i>Complex Analysis, 3 ed.</i> (McGraw-Hill, 1979).</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="CITEREFHenrici1986" class="citation book cs1"><a href="Peter_Henrici_(mathematician)" title="Peter Henrici (mathematician)">Henrici, P.</a> (1986) [1974, 1977]. <i>Applied and Computational Complex Analysis</i>. Wiley.</cite> Three volumes, publ.: 1974, 1977, 1986.</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">
<cite id="CITEREFEbenfeltHungerbühlerKohnMok2011" class="citation book cs1">Ebenfelt, Peter; Hungerbühler, Norbert; Kohn, Joseph J.; Mok, Ngaiming; Straube, Emil J. (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3GeUgafFRgMC&q=holomorphic"><i>Complex Analysis</i></a>. Science & Business Media. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-0346-0009-5</bdi> – via Google.</cite></span>
</li>
<li id="cite_note-Mark-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-Mark_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Mark_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">
<cite id="CITEREFMarkushevich1965" class="citation book cs1">Markushevich, A.I. (1965). <i>Theory of Functions of a Complex Variable</i>. Prentice-Hall.</cite> [In three volumes.]</span>
</li>
<li id="cite_note-Gunning-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-Gunning_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Gunning_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">
<cite id="CITEREFGunningRossi1965" class="citation book cs1"><a href="Robert_Gunning_(mathematician)" class="mw-redirect" title="Robert Gunning (mathematician)">Gunning, Robert C.</a>; Rossi, Hugo (1965). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=L0zJmamx5AAC"><i>Analytic Functions of Several Complex Variables</i></a>. Modern Analysis. Englewood Cliffs, NJ: <a href="Prentice-Hall" class="mw-redirect" title="Prentice-Hall">Prentice-Hall</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780821869536</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0180696">0180696</a>. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0141.08601">0141.08601</a> – via Google.</cite></span>
</li>
<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">
<cite id="CITEREFGrayMorris1978" class="citation journal cs1">Gray, J.D.; Morris, S.A. (April 1978). "When is a function that satisfies the Cauchy-Riemann equations analytic?". <i><a href="The_American_Mathematical_Monthly" title="The American Mathematical Monthly">The American Mathematical Monthly</a></i>. <b>85</b> (4): <span class="nowrap">246–</span>256. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2321164">10.2307/2321164</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2321164">2321164</a>.</cite></span>
</li>
<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">
<cite id="CITEREFPonce_Campuzano2021" class="citation web cs1">Ponce Campuzano, Juan Carlos (14 August 2021). <a rel="nofollow" class="external text" href="https://math.libretexts.org/Bookshelves/Analysis/Complex_Analysis_-_A_Visual_and_Interactive_Introduction_(Ponce_Campuzano)/02%3A_Chapter_2/2.03%3A_Complex_Differentiation">"2.3: Complex Differentiation"</a>. <i>Complex Analysis – A Visual and Interactive Introduction</i>. LibreTexts<span class="reference-accessdate">. Retrieved <span class="nowrap">15 June</span> 2025</span>.</cite> </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">The original French terms were <i>holomorphe</i> and <i>méromorphe</i>. <div class="paragraphbreak" style="margin-top:0.5em"></div>
<cite id="CITEREFBriotBouquet1875" class="citation book cs1"><a href="Charles_Auguste_Briot" title="Charles Auguste Briot">Briot, Charles Auguste</a>; <a href="Jean-Claude_Bouquet" class="mw-redirect" title="Jean-Claude Bouquet">Bouquet, Jean-Claude</a> (1875). <a rel="nofollow" class="external text" href="https://archive.org/details/thoriedesfonct00briouoft/page/14/">"§15 fonctions holomorphes"</a>. <i>Théorie des fonctions elliptiques</i> (2nd ed.). Gauthier-Villars. pp. <span class="nowrap">14–</span>15. <q>Lorsqu'une fonction est continue, monotrope, et a une dérivée, quand la variable se meut dans une certaine partie du plan, nous dirons qu'elle est <i>holomorphe</i> dans cette partie du plan. Nous indiquons par cette dénomination qu'elle est semblable aux fonctions entières qui jouissent de ces propriétés dans toute l'étendue du plan. [...] ¶ Une fraction rationnelle admet comme pôles les racines du dénominateur; c'est une fonction holomorphe dans toute partie du plan qui ne contient aucun de ses pôles. ¶ Lorsqu'une fonction est holomorphe dans une partie du plan, excepté en certains pôles, nous dirons qu'elle est <i>méromorphe</i> dans cette partie du plan, c'est-à-dire semblable aux fractions rationnelles.</q> [When a function is continuous, <a href="Monodromy" title="Monodromy">monotropic</a>, and has a derivative, when the variable moves in a certain part of the [complex] plane, we say that it is <i>holomorphic</i> in that part of the plane. We mean by this name that it resembles <a href="Entire_function" title="Entire function">entire functions</a> which enjoy these properties in the full extent of the plane. [...] ¶ A rational fraction admits as <a href="Zeros_and_poles" title="Zeros and poles">poles</a> the <a href="Zeros_and_poles" title="Zeros and poles">roots</a> of the denominator; it is a holomorphic function in all that part of the plane which does not contain any poles. ¶ When a function is holomorphic in part of the plane, except at certain poles, we say that it is <i>meromorphic</i> in that part of the plane, that is to say it resembles rational fractions.]</cite> <div class="paragraphbreak" style="margin-top:0.5em"></div> <cite id="CITEREFHarknessMorley1893" class="citation book cs1"><a href="James_Harkness_(mathematician)" title="James Harkness (mathematician)">Harkness, James</a>; <a href="Frank_Morley" title="Frank Morley">Morley, Frank</a> (1893). <a rel="nofollow" class="external text" href="https://archive.org/details/treatiseontheory00harkrich/page/n176/">"5. Integration"</a>. <i>A Treatise on the Theory of Functions</i>. Macmillan. p. 161.</cite></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">Briot & Bouquet had previously also adopted Cauchy’s term <i>synectic</i> (<i>synectique</i> in French), in the 1859 first edition of their book. <div class="paragraphbreak" style="margin-top:0.5em"></div> <cite id="CITEREFBriotBouquet1859" class="citation book cs1"><a href="Charles_Auguste_Briot" title="Charles Auguste Briot">Briot, Charles Auguste</a>; <a href="Jean-Claude_Bouquet" class="mw-redirect" title="Jean-Claude Bouquet">Bouquet, Jean-Claude</a> (1859). <a rel="nofollow" class="external text" href="https://archive.org/details/fonctsdoublement00briorich/page/n37/">"§10"</a>. <i>Théorie des fonctions doublement périodiques</i>. Mallet-Bachelier. p. 11.</cite></span>
</li>
<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">
<cite id="CITEREFHenrici1993" class="citation book cs1"><a href="Peter_Henrici_(mathematician)" title="Peter Henrici (mathematician)">Henrici, Peter</a> (1993) [1986]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vKZPsjaXuF4C"><i>Applied and Computational Complex Analysis</i></a>. Wiley Classics Library. Vol. 3 (Reprint ed.). New York - Chichester - Brisbane - Toronto - Singapore: <a href="John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-58986-1</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0822470">0822470</a>. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1107.30300">1107.30300</a> – via Google.</cite></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">
<cite id="CITEREFEvans1998" class="citation book cs1"><a href="Lawrence_C._Evans" title="Lawrence C. Evans">Evans, L.C.</a> (1998). <i>Partial Differential Equations</i>. American Mathematical Society.</cite></span>
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<li id="cite_note-Lang-14"><span class="mw-cite-backlink">^ <a href="#cite_ref-Lang_14-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Lang_14-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Lang_14-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">
<cite id="CITEREFLang2003" class="citation book cs1"><a href="Serge_Lang" title="Serge Lang">Lang, Serge</a> (2003). <i>Complex Analysis</i>. Springer Verlag GTM. <a href="Springer_Verlag" class="mw-redirect" title="Springer Verlag">Springer Verlag</a>.</cite></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">
<cite id="CITEREFRudin1987" class="citation book cs1"><a href="Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1987). <i>Real and Complex Analysis</i> (3rd ed.). New York: McGraw–Hill Book Co. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-054234-1</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0924157">0924157</a>.</cite></span>
</li>
<li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">
<cite id="CITEREFGunningRossi" class="citation book cs1">Gunning and Rossi. <i>Analytic Functions of Several Complex Variables</i>. p. 2.</cite></span>
</li>
</ol></div>
<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>
<ul><li><cite id="CITEREFBlakey1958" class="citation book cs1">Blakey, Joseph (1958). <i>University Mathematics</i> (2nd ed.). London, UK: Blackie and Sons. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/2370110">2370110</a>.</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></ul>
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