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<title>Meromorphe Funktion</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Meromorphe Funktion</span></h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p><b>Meromorphie</b> ist eine Eigenschaft von bestimmten <a href="Komplexwertige_Funktion" title="Komplexwertige Funktion">komplexwertigen Funktionen</a>, die in der <a href="Funktionentheorie" title="Funktionentheorie">Funktionentheorie</a> (einem Teilgebiet der <a href="Mathematik" title="Mathematik">Mathematik</a>) behandelt werden.

</p><p>Für viele Fragestellungen der Funktionentheorie ist der Begriff der <a href="Holomorphe_Funktion" title="Holomorphe Funktion">holomorphen Funktion</a> zu speziell. Dies liegt daran, dass der <a href="Kehrwert" title="Kehrwert">Kehrwert</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{f}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{f}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14e22b329663dcafd279fbbf565b682bcfc22fb5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:1.74ex; height:4.009ex;" alt="{\displaystyle {\tfrac {1}{f}}}" loading="lazy"></span> einer holomorphen Funktion <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.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> an einer <a href="Nullstelle" title="Nullstelle">Nullstelle</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.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> eine <a href="Definitionsl%C3%BCcke" title="Definitionslücke">Definitionslücke</a> hat und somit <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}{f}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{f}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14e22b329663dcafd279fbbf565b682bcfc22fb5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:1.74ex; height:4.009ex;" alt="{\displaystyle {\tfrac {1}{f}}}" loading="lazy"></span> dort auch nicht komplex differenzierbar ist. Man führt daher den allgemeineren Begriff der <i>meromorphen Funktion</i> ein, die auch <a href="Isolierter_Punkt" title="Isolierter Punkt">isolierte</a> <a href="Polstelle" title="Polstelle">Polstellen</a> besitzen kann.

</p><p>Meromorphe Funktionen lassen sich lokal als <a href="Laurentreihe" class="mw-redirect" title="Laurentreihe">Laurentreihen</a> mit abbrechendem Hauptteil darstellen. Ist <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}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.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> ein <a href="Gebiet_(Mathematik)" title="Gebiet (Mathematik)">Gebiet</a> von <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} }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/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>, so bildet die Menge der auf <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}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.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> meromorphen Funktionen einen <a href="K%C3%B6rper_(Algebra)" title="Körper (Algebra)">Körper</a>.

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

<div class="mw-heading mw-heading3"><h3 id="Auf_den_komplexen_Zahlen">Auf den komplexen Zahlen</h3></div>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/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> eine nichtleere <a href="Offene_Menge" title="Offene Menge">offene</a> <a href="Teilmenge" title="Teilmenge">Teilmenge</a> der Menge <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} }">

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</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/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> der <a href="Komplexe_Zahl" title="Komplexe Zahl">komplexen Zahlen</a> und <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_{f}}">

<msub>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f2f5845a89e5c8d7b5f7a4cb01d79ccac71e318.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.629ex; height:2.843ex;" alt="{\displaystyle P_{f}}" loading="lazy"></span> eine weitere Teilmenge von <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} }">

</mrow>

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/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>, die nur aus isolierten Punkten besteht. Eine Funktion <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.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> heißt meromorph, wenn sie für Stellen aus <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\setminus P_{f}}">

<msub>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2fa34a22cbb5f22774289ae1862787b05c984bd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.748ex; height:3.009ex;" alt="{\displaystyle D\setminus P_{f}}" loading="lazy"></span> definiert und holomorph ist und für Stellen aus <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_{f}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f2f5845a89e5c8d7b5f7a4cb01d79ccac71e318.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.629ex; height:2.843ex;" alt="{\displaystyle P_{f}}" loading="lazy"></span> <a href="Polstelle" title="Polstelle">Pole</a> hat. <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_{f}}">

<msub>

</mrow>

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</mstyle>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f2f5845a89e5c8d7b5f7a4cb01d79ccac71e318.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.629ex; height:2.843ex;" alt="{\displaystyle P_{f}}" loading="lazy"></span> wird als <i>Polstellenmenge</i> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.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> bezeichnet.

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<div class="mw-heading mw-heading3"><h3 id="Auf_einer_riemannschen_Fläche"><span id="Auf_einer_riemannschen_Fl.C3.A4che"></span>Auf einer riemannschen Fläche</h3></div>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> eine <a href="Riemannsche_Fl%C3%A4che" title="Riemannsche Fläche">riemannsche Fläche</a> und <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}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> eine offene Teilmenge von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>. Unter einer meromorphen Funktion auf <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}">

</mstyle>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> verstehen wir eine holomorphe Funktion <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 Y'\rightarrow \mathbb {C} }">

<msup>

</msup>

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f\colon Y'\rightarrow \mathbb {C} }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dad41189540c87e30f1ebafb105ad8bb0c39f558.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.19ex; height:2.843ex;" alt="{\displaystyle f\colon Y'\rightarrow \mathbb {C} }" loading="lazy"></span>, wobei <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'\subset Y}">

<msup>

</msup>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bee901050145cf916120634c2edfbb3c939acfbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.457ex; height:2.509ex;" alt="{\displaystyle Y'\subset Y}" loading="lazy"></span> eine offene Teilmenge ist, so dass die folgenden Eigenschaften gelten:

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<ul><li>Die Menge <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_{f}:=Y\setminus Y'}">

<msub>

</mrow>

</msub>

<msup>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle P_{f}:=Y\setminus Y'}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9a88d8f1848dc12e70d506ba36951e00fff437f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.927ex; height:3.176ex;" alt="{\displaystyle P_{f}:=Y\setminus Y'}" loading="lazy"></span> hat nur isolierte Punkte.</li>

<li>Für jeden Punkt <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\in Y\setminus Y'}">

<msup>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle p\in Y\setminus Y'}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d6ef2484fef5ec6a36bd89a9deced41c781aaf02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:10.652ex; height:3.009ex;" alt="{\displaystyle p\in Y\setminus Y'}" loading="lazy"></span> gilt</li></ul>

</mrow>

</munder>

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<annotation encoding="application/x-tex">{\displaystyle \lim _{x\rightarrow p}|f(x)|=\infty }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cfcb09e9730675cb2ee42627a0e2397ac5d5ddf4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.931ex; height:4.176ex;" alt="{\displaystyle \lim _{x\rightarrow p}|f(x)|=\infty }" loading="lazy"></span>.</dd></dl>

<p>Die Punkte aus der Menge <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\setminus Y'}">

<msup>

</msup>

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/872764e11f35602cebe0b988fb3a509eac019209.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.553ex; height:3.009ex;" alt="{\displaystyle Y\setminus Y'}" loading="lazy"></span> werden Pole von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.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> genannt. Die Menge aller meromorphen Funktionen auf <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> wird mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}(Y,\mathbb {C} )}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}(Y,\mathbb {C} )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f7b154c5b18fb1546862ae64b5d079209d74af1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.085ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}}(Y,\mathbb {C} )}" loading="lazy"></span> bezeichnet und bildet, falls <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> zusammenhängend ist, einen <a href="K%C3%B6rper_(Algebra)" title="Körper (Algebra)">Körper</a>.

Diese Definition ist natürlich äquivalent zur Definition auf den komplexen Zahlen, falls <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> eine Teilmenge derer ist.

</p>

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

<ul><li>Alle holomorphen Funktionen sind auch meromorph, da ihre Polstellenmenge leer ist.</li>

<li>Die Kehrwertfunktion <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\mapsto {\tfrac {1}{z}}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z\mapsto {\tfrac {1}{z}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f547dbbfb26c4d20814119f72b1fb42d4d0dc70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.36ex; height:3.343ex;" alt="{\displaystyle z\mapsto {\tfrac {1}{z}}}" loading="lazy"></span> ist meromorph; ihre Polstellenmenge ist <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\}}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ff0df9ef65c0572eb676580ce1c02b8ec40f694.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{0\}}" loading="lazy"></span>. Allgemeiner sind alle <a href="Rationale_Funktion" title="Rationale Funktion">rationalen Funktionen</a>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z\mapsto {\frac {a_{m}z^{m}+\dotsb +a_{0}}{b_{n}z^{n}+\dotsb +b_{0}}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/26df573f9fe558e9db6994bae7d220cf4e4a5aff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.897ex; height:5.676ex;" alt="{\displaystyle z\mapsto {\frac {a_{m}z^{m}+\dotsb +a_{0}}{b_{n}z^{n}+\dotsb +b_{0}}}}" loading="lazy"></span></dd></dl></li></ul>

<dl><dd>meromorph. Die Polstellenmenge ist hier jeweils eine Teilmenge der <a href="Nullstellenmenge" title="Nullstellenmenge">Nullstellenmenge</a> des Nennerpolynoms.</dd></dl>

<ul><li>Für jede meromorphe Funktion <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\neq 0}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88725631c1c1c441d1b0db0aed0e22246f162b96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.54ex; height:2.676ex;" alt="{\displaystyle f\neq 0}" loading="lazy"></span> ist ihr Kehrwert <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}{f}}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{f}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14e22b329663dcafd279fbbf565b682bcfc22fb5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:1.74ex; height:4.009ex;" alt="{\displaystyle {\tfrac {1}{f}}}" loading="lazy"></span> ebenfalls meromorph.</li></ul>

<ul><li>Die <a href="Tangens" class="mw-redirect" title="Tangens">Tangens</a>- bzw. die <a href="Kotangens" class="mw-redirect" title="Kotangens">Kotangens</a>-Funktion ist meromorph.</li></ul>

<ul><li>Die Funktion <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\mapsto e^{1/z}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z\mapsto e^{1/z}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/931f4060272cc6cde49c7416dfde40cd98d48dd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.431ex; height:2.843ex;" alt="{\displaystyle z\mapsto e^{1/z}}" loading="lazy"></span> ist nicht auf ganz <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} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/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> (und auf keiner Umgebung von <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.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>) meromorph, da <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.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> keine Polstelle, sondern eine <a href="Wesentliche_Singularit%C3%A4t" class="mw-redirect" title="Wesentliche Singularität">wesentliche Singularität</a> dieser Funktion ist.</li></ul>

<ul><li>Weitere Beispiele sind: <a href="Elliptische_Funktion" title="Elliptische Funktion">Elliptische Funktionen</a>, <a href="Gammafunktion" title="Gammafunktion">Gammafunktion</a>, <a href="Hurwitzsche_Zeta-Funktion" title="Hurwitzsche Zeta-Funktion">Hurwitzsche Zeta-Funktion</a>, <a href="Modulform" class="mw-redirect" title="Modulform">Modulformen</a>, <a href="Riemannsche_%CE%B6-Funktion" class="mw-redirect" title="Riemannsche ζ-Funktion">Riemannsche ζ-Funktion</a>, <a href="Spezielle_Funktionen" class="mw-redirect" title="Spezielle Funktionen">Spezielle Funktionen</a>.</li></ul>

<p>Wichtige Sätze über meromorphe Funktionen sind: <a href="Satz_von_Mittag-Leffler" title="Satz von Mittag-Leffler">Satz von Mittag-Leffler</a>, <a href="Residuensatz" title="Residuensatz">Residuensatz</a>, <a href="Satz_von_Riemann-Roch" title="Satz von Riemann-Roch">Satz von Riemann-Roch</a>.

</p>

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

<p>Im Gegensatz zu meromorphen Funktionen, die nur isolierte Pole aufweisen, gibt es in der komplexen Analysis keinen einheitlich etablierten Begriff für Funktionen mit wesentlichen Singularitäten. Während meromorphe Funktionen durch ihre „gutartigen“ Singularitäten charakterisiert sind, bei denen die Funktion gegen unendlich divergiert, zeigen Funktionen mit <a href="Isolierte_Singularit%C3%A4t" title="Isolierte Singularität"> wesentlichen Singularitäten</a> ein weitaus komplexeres Verhalten.

</p><p>Eine Funktion mit isolierten wesentlichen Singularitäten kann als aporomorph (von griechisch ἄπορος aporos „ausweglos, rätselhaft“) bezeichnet werden, wobei dieser Begriff jedoch nicht in der mathematischen Literatur etabliert ist. Diese Bezeichnung würde das unvorhersagbare und chaotische Verhalten solcher Funktionen in der Nähe ihrer Singularitäten reflektieren, wie es durch den Satz von Casorati-Weierstraß beschrieben wird.

</p>

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

<ul><li>E. Freitag & R. Busam – <i>Funktionentheorie 1</i>, Springer-Verlag, 4. Auflage, ISBN 3-540-67641-4</li>

<li>Otto Forster – <i>Riemannsche Flächen</i>, Springer-Verlag, 1977, ISBN 0-387-08034-1</li>

<li>E.M. Chirka: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Meromorphic function</cite>. In: <a href="Michiel_Hazewinkel" title="Michiel Hazewinkel">Michiel Hazewinkel</a> (Hrsg.): <cite class="lang" lang="en" dir="auto" style="font-style:italic"><a href="Encyclopedia_of_Mathematics" class="mw-redirect" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></cite>. Springer-Verlag und <a href="European_Mathematical_Society" title="European Mathematical Society">EMS</a> Press, Berlin 2002, ISBN 1-55608-010-7 (englisch, <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/meromorphic_function">encyclopediaofmath.org</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Meromorphe+Funktion&rft.atitle=Meromorphic+function&rft.au=E.M.+Chirka&rft.btitle=Encyclopedia+of+Mathematics&rft.date=2002&rft.genre=book&rft.isbn=1556080107&rft.place=Berlin&rft.pub=Springer-Verlag+und+EMS+Press" style="display:none"> </span></li></ul></div><div><div class="zim-footer">

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