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<title>Antiholomorphic function</title>
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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Antiholomorphic function</span></span>
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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, <b>antiholomorphic functions</b> (also called <b>antianalytic functions</b><sup id="cite_ref-math-encyclopedia_1-0" class="reference"><a href="#cite_note-math-encyclopedia-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>) are a family of <a href="Function_(mathematics)" title="Function (mathematics)">functions</a> closely related to but distinct from <a href="Holomorphic_function" title="Holomorphic function">holomorphic functions</a>.
</p><p>A function of the complex variable <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}">
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<mi>z</mi>
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</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> defined on an <a href="Open_set" title="Open set">open set</a> in the <a href="Complex_plane" title="Complex plane">complex plane</a> is said to be <b>antiholomorphic</b> if its <a href="Derivative" title="Derivative">derivative</a> with respect to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {z}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<mi>z</mi>
<mo stretchy="false">¯<!-- ¯ --></mo>
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<annotation encoding="application/x-tex">{\displaystyle {\bar {z}}}</annotation>
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</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> exists in the neighbourhood of each and every point in that set, where <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>
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</mrow>
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<annotation encoding="application/x-tex">{\displaystyle {\bar {z}}}</annotation>
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</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> is the <a href="Complex_conjugate" title="Complex conjugate">complex conjugate</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">
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<mi>z</mi>
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<annotation encoding="application/x-tex">{\displaystyle z}</annotation>
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</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>.
</p><p>
A definition of antiholomorphic function follows:<sup id="cite_ref-math-encyclopedia_1-1" class="reference"><a href="#cite_note-math-encyclopedia-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><blockquote><p>"[a] function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)=u+iv}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<mo stretchy="false">(</mo>
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<mo>=</mo>
<mi>u</mi>
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<mi>i</mi>
<mi>v</mi>
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<annotation encoding="application/x-tex">{\displaystyle f(z)=u+iv}</annotation>
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</math></span><img src="./2d88a646ebae416aa4c1aa22f04034cdca998abf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.375ex; height:2.843ex;" alt="{\displaystyle f(z)=u+iv}" loading="lazy"></span> of one or more complex variables <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=\left(z_{1},\dots ,z_{n}\right)\in \mathbb {C} ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>z</mi>
<mo>=</mo>
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<mo>(</mo>
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<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle z=\left(z_{1},\dots ,z_{n}\right)\in \mathbb {C} ^{n}}</annotation>
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</math></span><img src="./695a127481baae131c0353b667569aad066b60b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.346ex; height:2.843ex;" alt="{\displaystyle z=\left(z_{1},\dots ,z_{n}\right)\in \mathbb {C} ^{n}}" loading="lazy"></span> [is said to be anti-holomorphic if (and only if) it] is the complex conjugate of a holomorphic 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 {\overline {f\left(z\right)}}=u-iv}">
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<annotation encoding="application/x-tex">{\displaystyle {\overline {f\left(z\right)}}=u-iv}</annotation>
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</math></span><img src="./27502fce28fc7d11efe9dcfe998e3164ecc9b03f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.877ex; height:3.676ex;" alt="{\displaystyle {\overline {f\left(z\right)}}=u-iv}" loading="lazy"></span>."</p></blockquote>
<p>One can show that 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>
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</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 <a href="Holomorphic_function" title="Holomorphic function">holomorphic function</a> on an open set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>D</mi>
</mstyle>
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<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>, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f({\bar {z}})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>z</mi>
<mo stretchy="false">¯<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f({\bar {z}})}</annotation>
</semantics>
</math></span><img src="./2a7a0642ca0504ab12931595e6389dfd85264014.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.384ex; height:2.843ex;" alt="{\displaystyle f({\bar {z}})}" loading="lazy"></span> is an antiholomorphic function on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {D}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>D</mi>
<mo stretchy="false">¯<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\bar {D}}}</annotation>
</semantics>
</math></span><img src="./6a899b65dfb3ab3073b4f7f257be4491141bee64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.509ex;" alt="{\displaystyle {\bar {D}}}" loading="lazy"></span>, where <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 {D}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>D</mi>
<mo stretchy="false">¯<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\bar {D}}}</annotation>
</semantics>
</math></span><img src="./6a899b65dfb3ab3073b4f7f257be4491141bee64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.509ex;" alt="{\displaystyle {\bar {D}}}" loading="lazy"></span> is the reflection of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> across the real axis; in other words, <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 {D}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>D</mi>
<mo stretchy="false">¯<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\bar {D}}}</annotation>
</semantics>
</math></span><img src="./6a899b65dfb3ab3073b4f7f257be4491141bee64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.509ex;" alt="{\displaystyle {\bar {D}}}" loading="lazy"></span> is the set of complex conjugates of elements of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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>. Moreover, any antiholomorphic function can be obtained in this manner from a holomorphic function. This implies that a function is antiholomorphic <a href="If_and_only_if" title="If and only if">if and only if</a> it can be expanded in a <a href="Power_series" title="Power series">power series</a> in <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>
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</mstyle>
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<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> in a neighborhood of each point in its domain. Also, a function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(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>
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<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 antiholomorphic on an open set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>D</mi>
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<annotation encoding="application/x-tex">{\displaystyle D}</annotation>
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</math></span><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></span> if and only if 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 {\overline {f(z)}}}">
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<annotation encoding="application/x-tex">{\displaystyle {\overline {f(z)}}}</annotation>
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</math></span><img src="./3a1e2b1d29eaddd642266f1d49a0975110a93144.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.291ex; height:3.676ex;" alt="{\displaystyle {\overline {f(z)}}}" loading="lazy"></span> is holomorphic on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">
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<annotation encoding="application/x-tex">{\displaystyle D}</annotation>
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</p><p>If a function is both holomorphic and antiholomorphic, then it is constant on any <a href="Connected_space" title="Connected space">connected component</a> of its domain.
</p>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<li id="cite_note-math-encyclopedia-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-math-encyclopedia_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-math-encyclopedia_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Encyclopedia of Mathematics, Springer and The European Mathematical Society, <a rel="nofollow" class="external free" href="https://encyclopediaofmath.org/wiki/Anti-holomorphic_function">https://encyclopediaofmath.org/wiki/Anti-holomorphic_function</a>, As of 11 September 2020, This article was adapted from an original article by E. D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics, <style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1402006098</bdi>.</span>
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