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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Lagrangesche Inversionsformel</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>Die <b>Lagrangesche Inversionsformel</b> in der <a href="Mathematik" title="Mathematik">Mathematik</a> entwickelt zu einer gegebenen <a href="Analytische_Funktion" title="Analytische Funktion">analytischen Funktion</a> die Potenzreihe der <a href="Umkehrfunktion" title="Umkehrfunktion">Umkehrfunktion</a>.
</p>

<div class="mw-heading mw-heading2"><h2 id="Aussage">Aussage</h2></div>
<p>Gegeben sei eine Gleichung
</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 z=f(w)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>z</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z=f(w)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca1f41b383da8c58725e5510b7dce0aa9d094726.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.939ex; height:2.843ex;" alt="{\displaystyle z=f(w)}" loading="lazy"></span></dd></dl>
<p>mit einer am 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 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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/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> <a href="Analytische_Funktion" title="Analytische Funktion">analytischen Funktion</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 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="./_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> und <span style="white-space: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^{\prime }(a)\neq 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f^{\prime }(a)\neq 0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa7e483876f2797b346276debd6255a7af685bed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.305ex; height:3.009ex;" alt="{\displaystyle f^{\prime }(a)\neq 0}" loading="lazy"></span>.</span>
Dann ist es möglich, <span class="mwe-math-element mwe-math-element-inline"><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="./_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> zu invertieren, also die Gleichung nach <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>w</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle w}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" loading="lazy"></span> in Form einer <a href="Formale_Potenzreihe" title="Formale Potenzreihe">formalen Potenzreihe</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 w=g(z)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>w</mi>
<mo>=</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle w=g(z)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/53969a6b25293f37fd5706480e5958bf391c8f9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.776ex; height:2.843ex;" alt="{\displaystyle w=g(z)}" loading="lazy"></span> aufzulösen:<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>
<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 g(z)=a+\sum _{n=1}^{\infty }g_{n}{\frac {(z-f(a))^{n}}{n!}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>a</mi>
<mo>+</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo>−<!-- − --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mrow>
<mrow>
<mi>n</mi>
<mo>!</mo>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g(z)=a+\sum _{n=1}^{\infty }g_{n}{\frac {(z-f(a))^{n}}{n!}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e90d994ad4280d10e54da43aefe4ec45e51ceeaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.362ex; height:6.843ex;" alt="{\displaystyle g(z)=a+\sum _{n=1}^{\infty }g_{n}{\frac {(z-f(a))^{n}}{n!}}}" loading="lazy"></span></dd></dl>
<p>mit
</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 g_{n}=\lim _{w\to a}\left[{\frac {d^{n-1}}{dw^{n-1}}}\left({\frac {w-a}{f(w)-f(a)}}\right)^{n}\right].}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>=</mo>
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>w</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>a</mi>
</mrow>
</munder>
<mrow>
<mo>[</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mrow>
<mi>d</mi>
<msup>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>w</mi>
<mo>−<!-- − --></mo>
<mi>a</mi>
</mrow>
<mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mrow>
<mo>]</mo>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g_{n}=\lim _{w\to a}\left[{\frac {d^{n-1}}{dw^{n-1}}}\left({\frac {w-a}{f(w)-f(a)}}\right)^{n}\right].}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/753fb3461cb2be9c14ef2558b6c29086b5ab31c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.412ex; height:6.509ex;" alt="{\displaystyle g_{n}=\lim _{w\to a}\left[{\frac {d^{n-1}}{dw^{n-1}}}\left({\frac {w-a}{f(w)-f(a)}}\right)^{n}\right].}" loading="lazy"></span></dd></dl>
<p>Die Potenzreihe <span class="mwe-math-element mwe-math-element-inline"><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(z)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g(z)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8eb6fb9c0f0d402f5ebbebfd5e34bedd39a4a52b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.013ex; height:2.843ex;" alt="{\displaystyle g(z)}" loading="lazy"></span> hat einen von 0 verschiedenen <a href="Konvergenzradius" title="Konvergenzradius">Konvergenzradius</a>, d.&nbsp;h., sie ist eine analytische Funktion in einer Umgebung des Punktes <span style="white-space: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=f(a)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>z</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z=f(a)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/67278de25211717708353ca117b99b2c035ba917.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.504ex; height:2.843ex;" alt="{\displaystyle z=f(a)}" loading="lazy"></span>.</span>
Die Formel invertiert <span class="mwe-math-element mwe-math-element-inline"><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="./_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> als <a href="Formale_Potenzreihe" title="Formale Potenzreihe">formale Potenzreihe</a> in <span style="white-space: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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/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> Sie kann zu einer Formel für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(g(z))}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>H</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H(g(z))}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40b3dbedf45d71c691b8a148b6101184c518b7dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.886ex; height:2.843ex;" alt="{\displaystyle H(g(z))}" loading="lazy"></span> mit einer beliebigen formalen Potenzreihe <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>H</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"></span> erweitert und in vielen Fällen 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 f^{\prime }(a)=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f^{\prime }(a)=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec9fe44f9a66a9043a6c906dd438d5206d71bb8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.305ex; height:3.009ex;" alt="{\displaystyle f^{\prime }(a)=0}" loading="lazy"></span> (dann eine „mehrwertige“ Funktion) verallgemeinert werden.
</p><p>Der Satz wurde von <a href="Joseph_Louis_Lagrange" class="mw-redirect" title="Joseph Louis Lagrange">Lagrange</a><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> bewiesen und von <a href="Hans_Heinrich_B%C3%BCrmann" title="Hans Heinrich Bürmann">Hans Heinrich Bürmann</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><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><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> verallgemeinert, beides im späten 18. Jahrhundert. Es gibt Weiterentwicklungen in Richtung <a href="Komplexe_Analysis" class="mw-redirect" title="Komplexe Analysis">komplexe Analysis</a> und <a href="Kurvenintegral" title="Kurvenintegral">Kurvenintegrale</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Taylorreihe">Taylorreihe</h2></div>
<p>Die obige Formel gibt für eine formale Potenzreihe <span class="mwe-math-element mwe-math-element-inline"><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="./_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> nicht direkt die Koeffizienten der <a href="Formale_Potenzreihe#Inverses_der_Komposition_(Umkehrfunktion)" title="Formale Potenzreihe">formalen Umkehrfunktion</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 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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/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> ausgedrückt in den Koeffizienten von <span style="white-space: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="./_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>.</span> Kann man die Funktionen <span class="mwe-math-element mwe-math-element-inline"><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="./_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> 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 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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/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> als formale Potenzreihe
</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(w)=\sum _{k=0}^{\infty }b_{k}{\frac {w^{k}}{k!}}\qquad \mathrm {und} \qquad g(z)=\sum _{k=0}^{\infty }c_{k}{\frac {z^{k}}{k!}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msup>
<mrow>
<mi>k</mi>
<mo>!</mo>
</mrow>
</mfrac>
</mrow>
<mspace width="2em"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">u</mi>
<mi mathvariant="normal">n</mi>
<mi mathvariant="normal">d</mi>
</mrow>
<mspace width="2em"></mspace>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msup>
<mrow>
<mi>k</mi>
<mo>!</mo>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(w)=\sum _{k=0}^{\infty }b_{k}{\frac {w^{k}}{k!}}\qquad \mathrm {und} \qquad g(z)=\sum _{k=0}^{\infty }c_{k}{\frac {z^{k}}{k!}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1c585cacfdc35847f714b7fddb0ac91872ed563b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.4ex; height:7.009ex;" alt="{\displaystyle f(w)=\sum _{k=0}^{\infty }b_{k}{\frac {w^{k}}{k!}}\qquad \mathrm {und} \qquad g(z)=\sum _{k=0}^{\infty }c_{k}{\frac {z^{k}}{k!}}}" loading="lazy"></span></dd></dl>
<p>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 b_{0}=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b_{0}=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ecd3a5bc848cc6a60f444229ad2781c71ed528a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.313ex; height:2.509ex;" alt="{\displaystyle b_{0}=0}" loading="lazy"></span> 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 b_{1}\neq 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b_{1}\neq 0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/20a6ccd9b713bd30d8f7d54284eb6d425cac5e55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.313ex; height:2.676ex;" alt="{\displaystyle b_{1}\neq 0}" loading="lazy"></span> ausdrücken,
dann können die Koeffizienten der Inversen <span class="mwe-math-element mwe-math-element-inline"><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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/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> mithilfe von <a href="Bell-Polynom" title="Bell-Polynom">Bell-Polynomen</a> angegeben werden:<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>
</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 c_{n}={\frac {1}{b_{1}^{n}}}\sum _{k=1}^{n-1}(-1)^{k}n^{(k)}B_{n-1,k}(a_{1},a_{2},\ldots ,a_{n-k})\qquad (n\geq 2)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msubsup>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msubsup>
</mfrac>
</mrow>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</munderover>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msup>
<msup>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msub>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>,</mo>
<mi>k</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mi>k</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mspace width="2em"></mspace>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>≥<!-- ≥ --></mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle c_{n}={\frac {1}{b_{1}^{n}}}\sum _{k=1}^{n-1}(-1)^{k}n^{(k)}B_{n-1,k}(a_{1},a_{2},\ldots ,a_{n-k})\qquad (n\geq 2)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0adfc9bf5858f30dcb8647f3e11df9f0bb02cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:57.429ex; height:7.343ex;" alt="{\displaystyle c_{n}={\frac {1}{b_{1}^{n}}}\sum _{k=1}^{n-1}(-1)^{k}n^{(k)}B_{n-1,k}(a_{1},a_{2},\ldots ,a_{n-k})\qquad (n\geq 2)}" loading="lazy"></span> ,</dd></dl>
<p>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 a_{k}={\frac {b_{k+1}}{(k+1)b_{1}}},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mrow>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a_{k}={\frac {b_{k+1}}{(k+1)b_{1}}},}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d1b83174333799b952065ce81153fdd3311bfa6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.975ex; height:6.176ex;" alt="{\displaystyle a_{k}={\frac {b_{k+1}}{(k+1)b_{1}}},}" loading="lazy"></span> &nbsp;&nbsp; <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1}={\frac {1}{b_{1}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle c_{1}={\frac {1}{b_{1}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1c6e47a3f60629b69d651b88de77189f3e5d292d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.047ex; height:5.676ex;" alt="{\displaystyle c_{1}={\frac {1}{b_{1}}}}" loading="lazy"></span> &nbsp; und &nbsp; <span class="mwe-math-element mwe-math-element-inline"><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^{(k)}=n(n+1)\cdots (n+k-1)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo>=</mo>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>⋯<!-- ⋯ --></mo>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>+</mo>
<mi>k</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n^{(k)}=n(n+1)\cdots (n+k-1)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f510074b8c31c4eee7ea7d1af6bbb38cac3ec420.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.219ex; height:3.343ex;" alt="{\displaystyle n^{(k)}=n(n+1)\cdots (n+k-1)}" loading="lazy"></span>&nbsp; als <a href="Fallende_und_steigende_Faktorielle" title="Fallende und steigende Faktorielle">steigender Faktorielle</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Explizite_Formel">Explizite Formel</h2></div>
<p>Die folgende explizite Formel gilt nicht nur für analytische Funktionen (über <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.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 {R} }" loading="lazy"></span> oder <span class="mwe-math-element mwe-math-element-inline"><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="./_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>), sondern für alle <a href="Formale_Potenzreihe" title="Formale Potenzreihe">formalen Potenzreihen</a> über einem <a href="Ring_(Algebra)#Kommutativer_Ring_mit_Eins" title="Ring (Algebra)">Ring</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 R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> mit Eins.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>Anm 1<span class="cite-bracket">]</span></a></sup>
Ist nämlich
</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 A(X)=\sum _{k=1}^{\infty }a_{k}X^{k}\in R[[X]]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msup>
<mo>∈<!-- ∈ --></mo>
<mi>R</mi>
<mo stretchy="false">[</mo>
<mo stretchy="false">[</mo>
<mi>X</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A(X)=\sum _{k=1}^{\infty }a_{k}X^{k}\in R[[X]]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a9bd6cfb380f962d838f8ad75a295d59bd69884.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.949ex; height:6.843ex;" alt="{\displaystyle A(X)=\sum _{k=1}^{\infty }a_{k}X^{k}\in R[[X]]}" loading="lazy"></span></dd></dl>
<p>eine formale Potenzreihe, dann 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 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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.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> genau dann eine (formale) <a href="Umkehrfunktion" title="Umkehrfunktion">Umkehrfunktion</a> (ein formales kompositionelles Inverses)
</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 B(X)=\sum _{n=1}^{\infty }b_{n}X^{n}\in R[[X]]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo>∈<!-- ∈ --></mo>
<mi>R</mi>
<mo stretchy="false">[</mo>
<mo stretchy="false">[</mo>
<mi>X</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B(X)=\sum _{n=1}^{\infty }b_{n}X^{n}\in R[[X]]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c9abacdc32f6f2bcee4121e898fdf1658d41d0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.997ex; height:6.843ex;" alt="{\displaystyle B(X)=\sum _{n=1}^{\infty }b_{n}X^{n}\in R[[X]]}" loading="lazy"></span> ,</dd></dl>
<p>wenn der Koeffizient <span class="mwe-math-element mwe-math-element-inline"><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_{1}=A^{\prime }(0)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a_{1}=A^{\prime }(0)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d39c2043182be61100d23d4b88e1c753beaa3348.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.782ex; height:3.009ex;" alt="{\displaystyle a_{1}=A^{\prime }(0)}" loading="lazy"></span> invertierbar (eine <a href="Einheit_(Mathematik)" title="Einheit (Mathematik)">Einheit</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 R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> ist.
</p><p>Der einfacheren Rechnung halber substituieren wir <span class="mwe-math-element mwe-math-element-inline"><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="./_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> durch <span class="mwe-math-element mwe-math-element-inline"><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_{1}^{-1}Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msubsup>
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a_{1}^{-1}Y}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad7d5c937fb1652f4b46700baba835877af0c25d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.336ex; height:3.343ex;" alt="{\displaystyle a_{1}^{-1}Y}" loading="lazy"></span> und schreiben
</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 C(Y):=A(a_{1}^{-1}Y)=:Y+\sum _{k=2}^{\infty }c_{k}Y^{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
<mo stretchy="false">(</mo>
<mi>Y</mi>
<mo stretchy="false">)</mo>
<mo>:=</mo>
<mi>A</mi>
<mo stretchy="false">(</mo>
<msubsup>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msubsup>
<mi>Y</mi>
<mo stretchy="false">)</mo>
<mo>=:</mo>
<mi>Y</mi>
<mo>+</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C(Y):=A(a_{1}^{-1}Y)=:Y+\sum _{k=2}^{\infty }c_{k}Y^{k}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c316689e087e452c64ee70657ec58324f8401ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.168ex; height:6.843ex;" alt="{\displaystyle C(Y):=A(a_{1}^{-1}Y)=:Y+\sum _{k=2}^{\infty }c_{k}Y^{k}}" loading="lazy"></span></dd></dl>
<p>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 c_{k}:=a_{1}^{-k}a_{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>:=</mo>
<msubsup>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>k</mi>
</mrow>
</msubsup>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle c_{k}:=a_{1}^{-k}a_{k}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/624ac7c5b991fc69c2556fe88e67413403d3ffc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.756ex; height:3.343ex;" alt="{\displaystyle c_{k}:=a_{1}^{-k}a_{k}}" loading="lazy"></span> für <span style="white-space: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 k\geq 2}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
<mo>≥<!-- ≥ --></mo>
<mn>2</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k\geq 2}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c797a67c0a51167d373c013a9a020f4568a11754.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.472ex; height:2.343ex;" alt="{\displaystyle k\geq 2}" loading="lazy"></span>.</span>
</p><p>Die zugehörige formale Umkehrfunktion sei
</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 D(Y):=Y+\sum _{n=2}^{\infty }d_{n}Y^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>D</mi>
<mo stretchy="false">(</mo>
<mi>Y</mi>
<mo stretchy="false">)</mo>
<mo>:=</mo>
<mi>Y</mi>
<mo>+</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>=</mo>
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle D(Y):=Y+\sum _{n=2}^{\infty }d_{n}Y^{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c13d339f6e7a0ecb68f3f95f4cceb6f3b686f66a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.154ex; height:6.843ex;" alt="{\displaystyle D(Y):=Y+\sum _{n=2}^{\infty }d_{n}Y^{n}}" loading="lazy"></span>,</dd></dl>
<p>so dass <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C{\bigl (}D(Y){\bigr )}=D(Y)+\sum _{k=2}^{\infty }c_{k}{\bigl (}D(Y){\bigr )}^{k}=Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="1.2em" minsize="1.2em">(</mo>
</mrow>
</mrow>
<mi>D</mi>
<mo stretchy="false">(</mo>
<mi>Y</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>
<mo>=</mo>
<mi>D</mi>
<mo stretchy="false">(</mo>
<mi>Y</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="1.2em" minsize="1.2em">(</mo>
</mrow>
</mrow>
<mi>D</mi>
<mo stretchy="false">(</mo>
<mi>Y</mi>
<mo stretchy="false">)</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="1.2em" minsize="1.2em">)</mo>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msup>
<mo>=</mo>
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C{\bigl (}D(Y){\bigr )}=D(Y)+\sum _{k=2}^{\infty }c_{k}{\bigl (}D(Y){\bigr )}^{k}=Y}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8044d184a5e39b274d90b9a1a14d7139474897ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:40.284ex; height:6.843ex;" alt="{\displaystyle C{\bigl (}D(Y){\bigr )}=D(Y)+\sum _{k=2}^{\infty }c_{k}{\bigl (}D(Y){\bigr )}^{k}=Y}" loading="lazy"></span> ist.
Die Koeffizienten 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 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="./_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> lassen sich durch Koeffizientenvergleich in der Gleichung
</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 D(Y)=Y-\sum _{k=2}^{\infty }c_{k}{\bigl (}D(Y){\bigr )}^{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>D</mi>
<mo stretchy="false">(</mo>
<mi>Y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>Y</mi>
<mo>−<!-- − --></mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="1.2em" minsize="1.2em">(</mo>
</mrow>
</mrow>
<mi>D</mi>
<mo stretchy="false">(</mo>
<mi>Y</mi>
<mo stretchy="false">)</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="1.2em" minsize="1.2em">)</mo>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle D(Y)=Y-\sum _{k=2}^{\infty }c_{k}{\bigl (}D(Y){\bigr )}^{k}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d8e86331b9c334dcb3958e4ec4205524a40b5f3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.782ex; height:6.843ex;" alt="{\displaystyle D(Y)=Y-\sum _{k=2}^{\infty }c_{k}{\bigl (}D(Y){\bigr )}^{k}}" loading="lazy"></span></dd></dl>
<p>für <span class="mwe-math-element mwe-math-element-inline"><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\geq 2}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>≥<!-- ≥ --></mo>
<mn>2</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\geq 2}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6bf67f9d06ca3af619657f8d20ee1322da77174.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 2}" loading="lazy"></span> sofort zu
</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 {\begin{array}{llll}d_{n}\!\!\!\!&amp;=-\left[Y^{n}\right]&amp;\displaystyle \sum _{k=2}^{\infty }c_{k}{\bigl (}D(Y){\bigr )}^{k}\end{array}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="left left left left" rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mspace width="negativethinmathspace"></mspace>
<mspace width="negativethinmathspace"></mspace>
<mspace width="negativethinmathspace"></mspace>
<mspace width="negativethinmathspace"></mspace>
</mtd>
<mtd>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow>
<mo>[</mo>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo>]</mo>
</mrow>
</mtd>
<mtd>
<mstyle displaystyle="true" scriptlevel="0">
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="1.2em" minsize="1.2em">(</mo>
</mrow>
</mrow>
<mi>D</mi>
<mo stretchy="false">(</mo>
<mi>Y</mi>
<mo stretchy="false">)</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="1.2em" minsize="1.2em">)</mo>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msup>
</mstyle>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{llll}d_{n}\!\!\!\!&amp;=-\left[Y^{n}\right]&amp;\displaystyle \sum _{k=2}^{\infty }c_{k}{\bigl (}D(Y){\bigr )}^{k}\end{array}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af3532b5c76e3c0afb0f2e34ca33445a6396f39e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:29.9ex; height:6.843ex;" alt="{\displaystyle {\begin{array}{llll}d_{n}\!\!\!\!&amp;=-\left[Y^{n}\right]&amp;\displaystyle \sum _{k=2}^{\infty }c_{k}{\bigl (}D(Y){\bigr )}^{k}\end{array}}}" loading="lazy"></span></dd>
<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 \left.{\begin{array}{llllrr}\;&amp;=-\left[Y^{n}\right]&amp;{\Bigl (}&amp;c_{2}(1\;Y^{1}&amp;+\,d_{2}Y^{2}&amp;+\dotsb +&amp;d_{n-1}Y^{n-1})^{2}\\&amp;&amp;+&amp;c_{3}&amp;(1\;Y^{1}&amp;+\dotsb +&amp;d_{n-2}Y^{n-2})^{3}\\&amp;&amp;\,\vdots \\&amp;&amp;+&amp;c_{n-1}&amp;&amp;(1\;Y^{1}+&amp;d_{2}\,Y^{2})^{n-1}\\&amp;&amp;+&amp;c_{n}&amp;&amp;&amp;(1\;Y^{1})^{n}\;\;\;\\&amp;&amp;{\Bigr )}\end{array}}\qquad \qquad \right\rbrace {\text{ (RF)}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="left left left left right right" rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mspace width="thickmathspace"></mspace>
</mtd>
<mtd>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow>
<mo>[</mo>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo>]</mo>
</mrow>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="1.623em" minsize="1.623em">(</mo>
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<mtd>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msub>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mspace width="thickmathspace"></mspace>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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</msup>
</mtd>
<mtd>
<mo>+</mo>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msub>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mtd>
<mtd>
<mo>+</mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>+</mo>
</mtd>
<mtd>
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
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<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
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<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<mtd></mtd>
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<mo>+</mo>
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<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
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<mtd>
<mo stretchy="false">(</mo>
<mn>1</mn>
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<mi>Y</mi>
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<mn>1</mn>
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<mo>+</mo>
<mo>⋯<!-- ⋯ --></mo>
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<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>2</mn>
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<mi>Y</mi>
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<mi>n</mi>
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<mo stretchy="false">)</mo>
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<mn>3</mn>
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</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd></mtd>
<mtd>
<mspace width="thinmathspace"></mspace>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd></mtd>
<mtd>
<mo>+</mo>
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<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
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<mo stretchy="false">(</mo>
<mn>1</mn>
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<mi>Y</mi>
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<mn>1</mn>
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<mi>d</mi>
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<mn>2</mn>
</mrow>
</msub>
<mspace width="thinmathspace"></mspace>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
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</mtr>
<mtr>
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<mtd></mtd>
<mtd>
<mo>+</mo>
</mtd>
<mtd>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mtd>
<mtd></mtd>
<mtd></mtd>
<mtd>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mspace width="thickmathspace"></mspace>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mspace width="thickmathspace"></mspace>
<mspace width="thickmathspace"></mspace>
<mspace width="thickmathspace"></mspace>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd></mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="1.623em" minsize="1.623em">)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mspace width="2em"></mspace>
<mspace width="2em"></mspace>
</mrow>
<mo>}</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mtext>&nbsp;(RF)</mtext>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \left.{\begin{array}{llllrr}\;&amp;=-\left[Y^{n}\right]&amp;{\Bigl (}&amp;c_{2}(1\;Y^{1}&amp;+\,d_{2}Y^{2}&amp;+\dotsb +&amp;d_{n-1}Y^{n-1})^{2}\\&amp;&amp;+&amp;c_{3}&amp;(1\;Y^{1}&amp;+\dotsb +&amp;d_{n-2}Y^{n-2})^{3}\\&amp;&amp;\,\vdots \\&amp;&amp;+&amp;c_{n-1}&amp;&amp;(1\;Y^{1}+&amp;d_{2}\,Y^{2})^{n-1}\\&amp;&amp;+&amp;c_{n}&amp;&amp;&amp;(1\;Y^{1})^{n}\;\;\;\\&amp;&amp;{\Bigr )}\end{array}}\qquad \qquad \right\rbrace {\text{ (RF)}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/08f30050cdeb283d559cbea0b4e94b1eb7e4f6ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.671ex; margin-top: -0.393ex; width:77.885ex; height:24.676ex;" alt="{\displaystyle \left.{\begin{array}{llllrr}\;&amp;=-\left[Y^{n}\right]&amp;{\Bigl (}&amp;c_{2}(1\;Y^{1}&amp;+\,d_{2}Y^{2}&amp;+\dotsb +&amp;d_{n-1}Y^{n-1})^{2}\\&amp;&amp;+&amp;c_{3}&amp;(1\;Y^{1}&amp;+\dotsb +&amp;d_{n-2}Y^{n-2})^{3}\\&amp;&amp;\,\vdots \\&amp;&amp;+&amp;c_{n-1}&amp;&amp;(1\;Y^{1}+&amp;d_{2}\,Y^{2})^{n-1}\\&amp;&amp;+&amp;c_{n}&amp;&amp;&amp;(1\;Y^{1})^{n}\;\;\;\\&amp;&amp;{\Bigr )}\end{array}}\qquad \qquad \right\rbrace {\text{ (RF)}}}" loading="lazy"></span></dd></dl>
<p>ausrechnen mit dem Operator <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[Y^{n}\right]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow>
<mo>[</mo>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo>]</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \left[Y^{n}\right]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59f8c29fbc47141901f9312ab88c77470d6c6680.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.412ex; height:2.843ex;" alt="{\displaystyle \left[Y^{n}\right]}" loading="lazy"></span> für <a href="Formale_Potenzreihe#Koeffizientenextraktion" title="Formale Potenzreihe">Koeffizientenextraktion</a>.
Da die Formel <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{(RF)}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtext>(RF)</mtext>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\text{(RF)}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/94e39c7760bf45ac59e357f648e0bb446a304ea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.038ex; height:2.843ex;" alt="{\displaystyle {\text{(RF)}}}" loading="lazy"></span> auf ihrer rechten Seite nur Koeffizienten <span class="mwe-math-element mwe-math-element-inline"><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_{j}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d_{j}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3fa3426b07cfa37c76382ddbecfb4c880889657f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.119ex; height:2.843ex;" alt="{\displaystyle d_{j}}" loading="lazy"></span> mit Indizes <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j<n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>j</mi>
<mo>&lt;</mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle j&lt;n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/630dfff2636f9bec687bb23bed642f608c4f6921.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:5.478ex; height:2.509ex;" alt="{\displaystyle j<n}" loading="lazy"></span> enthält, stellt sie eine <a href="Rekursion#Rekursion_in_der_Mathematik" title="Rekursion">rekursive Spezifikation</a> der <span class="mwe-math-element mwe-math-element-inline"><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_{n}(n\geq 2)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>≥<!-- ≥ --></mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d_{n}(n\geq 2)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dff7e2b6492957eef7a3b4c7e13907500f115e58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.892ex; height:2.843ex;" alt="{\displaystyle d_{n}(n\geq 2)}" loading="lazy"></span> dar.
</p>
<dl><dt>Bemerkung</dt>
<dd>Da die Formel <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{(RF)}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtext>(RF)</mtext>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\text{(RF)}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/94e39c7760bf45ac59e357f648e0bb446a304ea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.038ex; height:2.843ex;" alt="{\displaystyle {\text{(RF)}}}" loading="lazy"></span> nur <a href="Ring_(Algebra)#Ring" title="Ring (Algebra)">Ringoperationen</a> (nur Additionen und Multiplikationen und keine Division) enthält, sind die Koeffizienten <span class="mwe-math-element mwe-math-element-inline"><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_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/858aa7b9662c90d0e01c9615aac9b1fdd84a02dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.427ex; height:2.509ex;" alt="{\displaystyle d_{n}}" loading="lazy"></span> ganzzahlige <a href="Polynom" title="Polynom">Polynome</a> in den <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle c_{k}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d2f8052630e67b00d04e3487e1d68ed7070470b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.096ex; height:2.009ex;" alt="{\displaystyle c_{k}}" loading="lazy"></span>; das hat zur Folge, dass <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{(RF)}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtext>(RF)</mtext>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\text{(RF)}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/94e39c7760bf45ac59e357f648e0bb446a304ea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.038ex; height:2.843ex;" alt="{\displaystyle {\text{(RF)}}}" loading="lazy"></span> über allen kommutativen unitären Ringen unabhängig von der <a href="Charakteristik_(Algebra)" title="Charakteristik (Algebra)">Charakteristik</a> und also gewissermaßen universell gültig ist.</dd></dl>
<p>Eine Herleitung der expliziten Auflösung
</p>
<dl><dd><table style="text-align:left">

<tbody><tr>
<td><span class="mwe-math-element mwe-math-element-inline"><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_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/858aa7b9662c90d0e01c9615aac9b1fdd84a02dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.427ex; height:2.509ex;" alt="{\displaystyle d_{n}}" loading="lazy"></span></td>
<td style="text-align:right"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>=</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle =}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum (-1)^{i_{2}+i_{3}+\cdots }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>∑<!-- ∑ --></mo>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>+</mo>
<mo>⋯<!-- ⋯ --></mo>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sum (-1)^{i_{2}+i_{3}+\cdots }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e0020e11f5b2b766bd36a2aeeb358c30da22110b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.648ex; height:3.843ex;" alt="{\displaystyle \sum (-1)^{i_{2}+i_{3}+\cdots }}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><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 {(n-1+i_{2}+i_{3}+\cdots )!}{n!\;\;i_{2}!\,i_{3}!\,\cdots }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>+</mo>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>+</mo>
<mo>⋯<!-- ⋯ --></mo>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
<mrow>
<mi>n</mi>
<mo>!</mo>
<mspace width="thickmathspace"></mspace>
<mspace width="thickmathspace"></mspace>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>!</mo>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>!</mo>
<mspace width="thinmathspace"></mspace>
<mo>⋯<!-- ⋯ --></mo>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {(n-1+i_{2}+i_{3}+\cdots )!}{n!\;\;i_{2}!\,i_{3}!\,\cdots }}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b27c2b117dcdd30fc000cf8e6ff69049e3721acd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.648ex; height:6.176ex;" alt="{\displaystyle {\frac {(n-1+i_{2}+i_{3}+\cdots )!}{n!\;\;i_{2}!\,i_{3}!\,\cdots }}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{2}^{\,i_{2}}c_{3}^{\,i_{3}}\cdots }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
</msubsup>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
</mrow>
</msubsup>
<mo>⋯<!-- ⋯ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle c_{2}^{\,i_{2}}c_{3}^{\,i_{3}}\cdots }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9dd7758127f881e2789eaa33d951c997506298da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.161ex; height:3.509ex;" alt="{\displaystyle c_{2}^{\,i_{2}}c_{3}^{\,i_{3}}\cdots }" loading="lazy"></span>
</td></tr>
<tr>
<td></td>
<td style="text-align:right"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>=</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle =}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum (-1)^{\sum _{\nu =2}^{\infty }i_{\nu }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>∑<!-- ∑ --></mo>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>ν<!-- ν --></mi>
<mo>=</mo>
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>ν<!-- ν --></mi>
</mrow>
</msub>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sum (-1)^{\sum _{\nu =2}^{\infty }i_{\nu }}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/36a5fd24a1d573f663cc06a29b383580198dc69b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:14.505ex; height:3.843ex;" alt="{\displaystyle \sum (-1)^{\sum _{\nu =2}^{\infty }i_{\nu }}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><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 {(n-1+\sum _{\nu =2}^{\infty }i_{\nu })!}{\;n!\quad \prod _{\nu =2}^{\infty }i_{\nu }!}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>+</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>ν<!-- ν --></mi>
<mo>=</mo>
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>ν<!-- ν --></mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
<mrow>
<mspace width="thickmathspace"></mspace>
<mi>n</mi>
<mo>!</mo>
<mspace width="1em"></mspace>
<munderover>
<mo>∏<!-- ∏ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>ν<!-- ν --></mi>
<mo>=</mo>
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>ν<!-- ν --></mi>
</mrow>
</msub>
<mo>!</mo>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {(n-1+\sum _{\nu =2}^{\infty }i_{\nu })!}{\;n!\quad \prod _{\nu =2}^{\infty }i_{\nu }!}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/247fcd0382c3a96e0fca301dc8e4f67b89e9accd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.481ex; height:6.843ex;" alt="{\displaystyle {\frac {(n-1+\sum _{\nu =2}^{\infty }i_{\nu })!}{\;n!\quad \prod _{\nu =2}^{\infty }i_{\nu }!}}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{\nu =2}^{\infty }c_{\nu }^{\,i_{\nu }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munderover>
<mo>∏<!-- ∏ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>ν<!-- ν --></mi>
<mo>=</mo>
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>ν<!-- ν --></mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>ν<!-- ν --></mi>
</mrow>
</msub>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \prod _{\nu =2}^{\infty }c_{\nu }^{\,i_{\nu }}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65adc87f19623df3a7c394c77ea3247d8dc70353.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:6.424ex; height:6.843ex;" alt="{\displaystyle \prod _{\nu =2}^{\infty }c_{\nu }^{\,i_{\nu }}}" loading="lazy"></span> ,
</td></tr></tbody></table></dd></dl>
<p>bei der über alle Kombinationen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{2},i_{3},\ldots \in \mathbb {N} _{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>∈<!-- ∈ --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">N</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i_{2},i_{3},\ldots \in \mathbb {N} _{0}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d38bb0d0043cdf2475415092ab9afd8ca672db2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.077ex; height:2.509ex;" alt="{\displaystyle i_{2},i_{3},\ldots \in \mathbb {N} _{0}}" loading="lazy"></span> mit &nbsp; <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\,i_{2}+2\,i_{3}+\cdots =\textstyle \sum _{\nu =2}^{\infty }(\nu -1)\,i_{\nu }=n-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>+</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>+</mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>=</mo>
<mstyle displaystyle="false" scriptlevel="0">
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>ν<!-- ν --></mi>
<mo>=</mo>
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<mo stretchy="false">(</mo>
<mi>ν<!-- ν --></mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>ν<!-- ν --></mi>
</mrow>
</msub>
<mo>=</mo>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1\,i_{2}+2\,i_{3}+\cdots =\textstyle \sum _{\nu =2}^{\infty }(\nu -1)\,i_{\nu }=n-1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34b0ba085bba80c03e7e54c7e7f4871d489cd87b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:41.806ex; height:3.176ex;" alt="{\displaystyle 1\,i_{2}+2\,i_{3}+\cdots =\textstyle \sum _{\nu =2}^{\infty }(\nu -1)\,i_{\nu }=n-1}" loading="lazy"></span><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>Anm 2<span class="cite-bracket">]</span></a></sup> &nbsp; zu summieren ist,
findet sich bei Morse und Feshbach.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>
</p><p>Die ersten paar Koeffizienten 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 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="./_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> sind:
</p>
<dl><dd><table style="text-align:left">

<tbody><tr>
<td><span class="mwe-math-element mwe-math-element-inline"><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_{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d_{1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4cccb5a6a2f1acab4ca255e0be86c224ed82282a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{1}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\;1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mspace width="thickmathspace"></mspace>
<mn>1</mn>
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<annotation encoding="application/x-tex">{\displaystyle =\;1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a32bd1de36c2c74926bd110cb35cbbbb5680848.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.261ex; height:2.176ex;" alt="{\displaystyle =\;1}" loading="lazy"></span>
</td></tr>
<tr>
<td><span class="mwe-math-element mwe-math-element-inline"><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_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d_{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9276f8f68c5c23329de74ad76e69f6801358fb1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{2}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\,-c_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>=</mo>
<mspace width="thinmathspace"></mspace>
<mo>−<!-- − --></mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle =\,-c_{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/90674ff712838b7d9aa369b8b690e418eb8e4ae8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.71ex; height:2.343ex;" alt="{\displaystyle =\,-c_{2}}" loading="lazy"></span>
</td></tr>
<tr>
<td><span class="mwe-math-element mwe-math-element-inline"><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_{3}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle d_{3}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f051bd12b5413014eeb4944816cb5672edfcff4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{3}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\,-c_{3}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>=</mo>
<mspace width="thinmathspace"></mspace>
<mo>−<!-- − --></mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle =\,-c_{3}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c27a71e587d1c342cb723bfc813176dcbb487cdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.71ex; height:2.343ex;" alt="{\displaystyle =\,-c_{3}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +2\,c_{2}^{\,2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>+</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle +2\,c_{2}^{\,2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ee81c23b3da79fe53cc3cf9af1fe172e9796c93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.806ex; height:3.176ex;" alt="{\displaystyle +2\,c_{2}^{\,2}}" loading="lazy"></span></td>
<td></td>
<td></td>
<td colspan="7" style="text-align:right"><span class="mwe-math-element mwe-math-element-inline"><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 {(2+1+0)!}{3!\;1!\;0!}}c_{3}^{\,1}c_{2}^{\,0}+\!{\tfrac {(2+0+2)!}{3!\;0!\;2!}}c_{3}^{\,0}c_{2}^{\,2})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mo>=</mo>
<mspace width="thinmathspace"></mspace>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mo>+</mo>
<mn>1</mn>
<mo>+</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
<mrow>
<mn>3</mn>
<mo>!</mo>
<mspace width="thickmathspace"></mspace>
<mn>1</mn>
<mo>!</mo>
<mspace width="thickmathspace"></mspace>
<mn>0</mn>
<mo>!</mo>
</mrow>
</mfrac>
</mstyle>
</mrow>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>1</mn>
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</msubsup>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
</mrow>
</msubsup>
<mo>+</mo>
<mspace width="negativethinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mo>+</mo>
<mn>0</mn>
<mo>+</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
<mrow>
<mn>3</mn>
<mo>!</mo>
<mspace width="thickmathspace"></mspace>
<mn>0</mn>
<mo>!</mo>
<mspace width="thickmathspace"></mspace>
<mn>2</mn>
<mo>!</mo>
</mrow>
</mfrac>
</mstyle>
</mrow>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
</mrow>
</msubsup>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
</msubsup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (=\,-{\tfrac {(2+1+0)!}{3!\;1!\;0!}}c_{3}^{\,1}c_{2}^{\,0}+\!{\tfrac {(2+0+2)!}{3!\;0!\;2!}}c_{3}^{\,0}c_{2}^{\,2})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7bde629339d2e4337b59e9b6992ea822fc3eb5f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:33.896ex; height:4.343ex;" alt="{\displaystyle (=\,-{\tfrac {(2+1+0)!}{3!\;1!\;0!}}c_{3}^{\,1}c_{2}^{\,0}+\!{\tfrac {(2+0+2)!}{3!\;0!\;2!}}c_{3}^{\,0}c_{2}^{\,2})}" loading="lazy"></span><sup id="cite_ref-exp0_11-0" class="reference"><a href="#cite_note-exp0-11"><span class="cite-bracket">[</span>Anm 3<span class="cite-bracket">]</span></a></sup> &nbsp;
</td></tr>
<tr>
<td><span class="mwe-math-element mwe-math-element-inline"><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_{4}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d_{4}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d43e86e828c6871d10a70e4c86a6cea2415991f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{4}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\,-c_{4}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>=</mo>
<mspace width="thinmathspace"></mspace>
<mo>−<!-- − --></mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle =\,-c_{4}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc5b183eb194325a0750f6fa1667f4e786429834.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.71ex; height:2.343ex;" alt="{\displaystyle =\,-c_{4}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +5\,c_{3}c_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>+</mo>
<mn>5</mn>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle +5\,c_{3}c_{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5686ccdc7a150a374ea59baebbcd9ff30608f9f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.48ex; height:2.509ex;" alt="{\displaystyle +5\,c_{3}c_{2}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -5\,c_{2}^{\,3}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>−<!-- − --></mo>
<mn>5</mn>
<mspace width="thinmathspace"></mspace>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle -5\,c_{2}^{\,3}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e790a86d2ea45ecf07ba404eb27feaf0257fbced.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.806ex; height:3.176ex;" alt="{\displaystyle -5\,c_{2}^{\,3}}" loading="lazy"></span></td>
<td></td>
<td colspan="7"><span class="mwe-math-element mwe-math-element-inline"><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 {(3+1+0+0)!}{4!\;1!\;0!\;0!}}c_{4}^{\,1}c_{3}^{\,0}c_{2}^{\,0}+\!{\tfrac {(3+0+1+1)!}{4!\;0!\;1!\;1!}}\,c_{4}^{\,0}c_{3}^{\,1}c_{2}^{\,1}-\!{\tfrac {(3+0+0+3)!}{4!\;0!\;0!\;3!}}\,c_{4}^{\,0}c_{3}^{\,0}c_{2}^{\,3})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mo>=</mo>
<mspace width="thinmathspace"></mspace>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mo>+</mo>
<mn>1</mn>
<mo>+</mo>
<mn>0</mn>
<mo>+</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
<mrow>
<mn>4</mn>
<mo>!</mo>
<mspace width="thickmathspace"></mspace>
<mn>1</mn>
<mo>!</mo>
<mspace width="thickmathspace"></mspace>
<mn>0</mn>
<mo>!</mo>
<mspace width="thickmathspace"></mspace>
<mn>0</mn>
<mo>!</mo>
</mrow>
</mfrac>
</mstyle>
</mrow>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>1</mn>
</mrow>
</msubsup>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
</mrow>
</msubsup>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
</mrow>
</msubsup>
<mo>+</mo>
<mspace width="negativethinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mo>+</mo>
<mn>0</mn>
<mo>+</mo>
<mn>1</mn>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
<mrow>
<mn>4</mn>
<mo>!</mo>
<mspace width="thickmathspace"></mspace>
<mn>0</mn>
<mo>!</mo>
<mspace width="thickmathspace"></mspace>
<mn>1</mn>
<mo>!</mo>
<mspace width="thickmathspace"></mspace>
<mn>1</mn>
<mo>!</mo>
</mrow>
</mfrac>
</mstyle>
</mrow>
<mspace width="thinmathspace"></mspace>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
</mrow>
</msubsup>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>1</mn>
</mrow>
</msubsup>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>1</mn>
</mrow>
</msubsup>
<mo>−<!-- − --></mo>
<mspace width="negativethinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mo>+</mo>
<mn>0</mn>
<mo>+</mo>
<mn>0</mn>
<mo>+</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
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<mrow>
<mn>4</mn>
<mo>!</mo>
<mspace width="thickmathspace"></mspace>
<mn>0</mn>
<mo>!</mo>
<mspace width="thickmathspace"></mspace>
<mn>0</mn>
<mo>!</mo>
<mspace width="thickmathspace"></mspace>
<mn>3</mn>
<mo>!</mo>
</mrow>
</mfrac>
</mstyle>
</mrow>
<mspace width="thinmathspace"></mspace>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
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</msubsup>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
</mrow>
</msubsup>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
</mrow>
</msubsup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (=\,-{\tfrac {(3+1+0+0)!}{4!\;1!\;0!\;0!}}c_{4}^{\,1}c_{3}^{\,0}c_{2}^{\,0}+\!{\tfrac {(3+0+1+1)!}{4!\;0!\;1!\;1!}}\,c_{4}^{\,0}c_{3}^{\,1}c_{2}^{\,1}-\!{\tfrac {(3+0+0+3)!}{4!\;0!\;0!\;3!}}\,c_{4}^{\,0}c_{3}^{\,0}c_{2}^{\,3})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f58fb2be96d39264b29bdda08670a9aad295094.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:63.262ex; height:4.343ex;" alt="{\displaystyle (=\,-{\tfrac {(3+1+0+0)!}{4!\;1!\;0!\;0!}}c_{4}^{\,1}c_{3}^{\,0}c_{2}^{\,0}+\!{\tfrac {(3+0+1+1)!}{4!\;0!\;1!\;1!}}\,c_{4}^{\,0}c_{3}^{\,1}c_{2}^{\,1}-\!{\tfrac {(3+0+0+3)!}{4!\;0!\;0!\;3!}}\,c_{4}^{\,0}c_{3}^{\,0}c_{2}^{\,3})}" loading="lazy"></span><sup id="cite_ref-exp0_11-1" class="reference"><a href="#cite_note-exp0-11"><span class="cite-bracket">[</span>Anm 3<span class="cite-bracket">]</span></a></sup>
</td></tr>
<tr>
<td><span class="mwe-math-element mwe-math-element-inline"><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_{5}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d_{5}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72322fe1165d07456c47dc500bf73d00c1ca703f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{5}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\,-c_{5}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>=</mo>
<mspace width="thinmathspace"></mspace>
<mo>−<!-- − --></mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle =\,-c_{5}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3ac106ffcb87aafb3cc6541cdbe5d6abd46a7d6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.71ex; height:2.343ex;" alt="{\displaystyle =\,-c_{5}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +6\,c_{4}c_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>+</mo>
<mn>6</mn>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msub>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle +6\,c_{4}c_{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c7959ad4cba7aac4a1b9e3ddf61071840322f7e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.48ex; height:2.509ex;" alt="{\displaystyle +6\,c_{4}c_{2}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +3\,c_{3}^{\,2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>+</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle +3\,c_{3}^{\,2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/56a56ff2f92155b322422c5c2605bfc57981352e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.806ex; height:3.176ex;" alt="{\displaystyle +3\,c_{3}^{\,2}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -21\,c_{3}c_{2}^{\,2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>−<!-- − --></mo>
<mn>21</mn>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle -21\,c_{3}c_{2}^{\,2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8adf54ac622a45def1fe1100f20401bf01fe52f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.029ex; height:3.176ex;" alt="{\displaystyle -21\,c_{3}c_{2}^{\,2}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +14\,c_{2}^{\,4}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>+</mo>
<mn>14</mn>
<mspace width="thinmathspace"></mspace>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>4</mn>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle +14\,c_{2}^{\,4}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4dc80a093ce4f2d161e7963f346f18eb6ab9196.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.968ex; height:3.176ex;" alt="{\displaystyle +14\,c_{2}^{\,4}}" loading="lazy"></span>
</td></tr>
<tr>
<td><span class="mwe-math-element mwe-math-element-inline"><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_{6}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>6</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d_{6}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f81b45b8ed71a5bd76f3e54f75dcee544ef60a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{6}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\,-c_{6}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>=</mo>
<mspace width="thinmathspace"></mspace>
<mo>−<!-- − --></mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>6</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle =\,-c_{6}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8f0ebcfb2582e051a2ccedec4fb90c502b75a423.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.71ex; height:2.343ex;" alt="{\displaystyle =\,-c_{6}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +7\,c_{5}c_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>+</mo>
<mn>7</mn>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
</msub>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle +7\,c_{5}c_{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05427f758b6527fa0e555925f79044a83ee78635.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.48ex; height:2.509ex;" alt="{\displaystyle +7\,c_{5}c_{2}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +7\,c_{4}c_{3}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>+</mo>
<mn>7</mn>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msub>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle +7\,c_{4}c_{3}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/42b568fe0720f7a68e230fb944c8490df3bfc659.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.48ex; height:2.509ex;" alt="{\displaystyle +7\,c_{4}c_{3}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -28\,c_{4}c_{2}^{\,2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>−<!-- − --></mo>
<mn>28</mn>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msub>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle -28\,c_{4}c_{2}^{\,2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7df8ed6bd05d3ef996cb63d40ea36033a387525c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.029ex; height:3.176ex;" alt="{\displaystyle -28\,c_{4}c_{2}^{\,2}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -28\,c_{3}^{\,2}c_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>−<!-- − --></mo>
<mn>28</mn>
<mspace width="thinmathspace"></mspace>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
</msubsup>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle -28\,c_{3}^{\,2}c_{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f30b4d74f17c9127749f1827d66611a198bce05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.029ex; height:3.176ex;" alt="{\displaystyle -28\,c_{3}^{\,2}c_{2}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +84\,c_{3}c_{2}^{\,3}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>+</mo>
<mn>84</mn>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle +84\,c_{3}c_{2}^{\,3}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/50346e7b3a3dce154e55444db8ed262c2cadb7c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.029ex; height:3.176ex;" alt="{\displaystyle +84\,c_{3}c_{2}^{\,3}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -42\,c_{2}^{\,5}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>−<!-- − --></mo>
<mn>42</mn>
<mspace width="thinmathspace"></mspace>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>5</mn>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle -42\,c_{2}^{\,5}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51945e55286b4dc6b440f60c858f1b3cc4b82956.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.968ex; height:3.176ex;" alt="{\displaystyle -42\,c_{2}^{\,5}}" loading="lazy"></span>
</td></tr>
<tr>
<td><span class="mwe-math-element mwe-math-element-inline"><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_{7}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>7</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d_{7}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d8869156ed1de6956bed42d31da36c3568ac69b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{7}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\,-c_{7}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>=</mo>
<mspace width="thinmathspace"></mspace>
<mo>−<!-- − --></mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>7</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle =\,-c_{7}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/721b67665921262b7f301b45724a622b5f7f3ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.71ex; height:2.343ex;" alt="{\displaystyle =\,-c_{7}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +8\,c_{6}c_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>+</mo>
<mn>8</mn>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>6</mn>
</mrow>
</msub>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle +8\,c_{6}c_{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c79e1591b33c78e2c8ab1091c137f7d2b89771f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.48ex; height:2.509ex;" alt="{\displaystyle +8\,c_{6}c_{2}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +8\,c_{5}c_{3}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>+</mo>
<mn>8</mn>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
</msub>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle +8\,c_{5}c_{3}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b40ee0f23ee9df1639b6361cba1246d31cd0d44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.48ex; height:2.509ex;" alt="{\displaystyle +8\,c_{5}c_{3}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -36\,c_{5}c_{2}^{\,2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>−<!-- − --></mo>
<mn>36</mn>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
</msub>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle -36\,c_{5}c_{2}^{\,2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/096d9d647f447386a95ac886bc14daca7ca859fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.029ex; height:3.176ex;" alt="{\displaystyle -36\,c_{5}c_{2}^{\,2}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +4\,c_{4}^{\,2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>+</mo>
<mn>4</mn>
<mspace width="thinmathspace"></mspace>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle +4\,c_{4}^{\,2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d44bcbdd6feb51165e84a1b82b2b2722bd63461.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.806ex; height:3.176ex;" alt="{\displaystyle +4\,c_{4}^{\,2}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -72\,c_{4}c_{3}c_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>−<!-- − --></mo>
<mn>72</mn>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msub>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle -72\,c_{4}c_{3}c_{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0057b7b5e2452ea1644a194737c2eab4be4a377b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.703ex; height:2.509ex;" alt="{\displaystyle -72\,c_{4}c_{3}c_{2}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +120\,c_{4}c_{2}^{\,3}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>+</mo>
<mn>120</mn>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msub>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle +120\,c_{4}c_{2}^{\,3}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd335926774ae28ce4c631169dff4de04953a04b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.192ex; height:3.176ex;" alt="{\displaystyle +120\,c_{4}c_{2}^{\,3}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -12\,c_{3}^{\,3}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>−<!-- − --></mo>
<mn>12</mn>
<mspace width="thinmathspace"></mspace>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle -12\,c_{3}^{\,3}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/578c2d5dfffc9f2b696c1a65c3ce90a7d63e5a36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.968ex; height:3.176ex;" alt="{\displaystyle -12\,c_{3}^{\,3}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +180\,c_{3}^{\,2}c_{2}^{\,2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>+</mo>
<mn>180</mn>
<mspace width="thinmathspace"></mspace>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
</msubsup>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle +180\,c_{3}^{\,2}c_{2}^{\,2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/53de6a5a5c5eb03bceabc0a84a07fd55179e33f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.579ex; height:3.176ex;" alt="{\displaystyle +180\,c_{3}^{\,2}c_{2}^{\,2}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -330\,c_{3}c_{2}^{\,4}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>−<!-- − --></mo>
<mn>330</mn>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>4</mn>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle -330\,c_{3}c_{2}^{\,4}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/73814fcedeca7d839883b3db915f498e8c27b985.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.192ex; height:3.176ex;" alt="{\displaystyle -330\,c_{3}c_{2}^{\,4}}" loading="lazy"></span></td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +132\,c_{2}^{\,6}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>+</mo>
<mn>132</mn>
<mspace width="thinmathspace"></mspace>
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>6</mn>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle +132\,c_{2}^{\,6}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b868a0e9be1f7c59505829b4b1bcf26bbec10f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.131ex; height:3.176ex;" alt="{\displaystyle +132\,c_{2}^{\,6}}" loading="lazy"></span>
</td></tr></tbody></table></dd></dl>
<p>Die <a href="Monom" title="Monom">Monome</a> sind hier in den Zeilen <a href="Lexikographisch" class="mw-redirect" title="Lexikographisch">lexikographisch</a> absteigend geordnet, d.&nbsp;h., <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{3}^{\,2}=c_{3}c_{3}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
</msubsup>
<mo>=</mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle c_{3}^{\,2}=c_{3}c_{3}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/28f9a005a5490cba67a3a3b11c779fdc3161e3d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.669ex; height:3.176ex;" alt="{\displaystyle c_{3}^{\,2}=c_{3}c_{3}}" loading="lazy"></span> kommt vor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{3}c_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle c_{3}c_{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/09042a53244a2f8c27416e1f277de0ae044e6ff9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.122ex; height:2.009ex;" alt="{\displaystyle c_{3}c_{2}}" loading="lazy"></span> kommt vor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{3}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle c_{3}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1dc52bfbaf6e577fbed72a716068f4533700bd3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{3}}" loading="lazy"></span> kommt vor <span style="white-space: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 c_{2}^{\,2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle c_{2}^{\,2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf5f42eb2c51b0c8e852818a752f91f4b3d48560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.448ex; height:3.176ex;" alt="{\displaystyle c_{2}^{\,2}}" loading="lazy"></span>.</span>
Die (ganzzahligen) Koeffizienten dieser Polynome sind in dieser Anordnung zusammengestellt in der Folge <a href="https://oeis.org/A304462" class="extiw external" title="oeis:A304462">A304462</a> in <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>.
Die Folge <a href="https://oeis.org/A000041" class="extiw external" title="oeis:A000041">A000041</a> in <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a> enthält die Anzahl der Monome in der <span class="mwe-math-element mwe-math-element-inline"><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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/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>-ten Zeile (= <a href="Partitionsfunktion" title="Partitionsfunktion">Anzahl der Partitionen</a> einer <span style="white-space: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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/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>-elementigen</span> Menge).
</p><p>Mit der Substitution <span class="mwe-math-element mwe-math-element-inline"><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(X):=a_{1}\,B(X)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>D</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>:=</mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mspace width="thinmathspace"></mspace>
<mi>B</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle D(X):=a_{1}\,B(X)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/896ec4cd731878d3f2e49ca5c6272bf20b357ea4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.683ex; height:2.843ex;" alt="{\displaystyle D(X):=a_{1}\,B(X)}" loading="lazy"></span> ergibt sich
</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 X=C{\bigl (}D(X){\bigr )}=A{\bigl (}a_{1}^{-1}D(X){\bigr )}=A{\bigl (}B(X){\bigr )}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>=</mo>
<mi>C</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="1.2em" minsize="1.2em">(</mo>
</mrow>
</mrow>
<mi>D</mi>
<mo stretchy="false">(</mo>
<mi>X</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>
<mo>=</mo>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="1.2em" minsize="1.2em">(</mo>
</mrow>
</mrow>
<msubsup>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msubsup>
<mi>D</mi>
<mo stretchy="false">(</mo>
<mi>X</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>
<mo>=</mo>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="1.2em" minsize="1.2em">(</mo>
</mrow>
</mrow>
<mi>B</mi>
<mo stretchy="false">(</mo>
<mi>X</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 X=C{\bigl (}D(X){\bigr )}=A{\bigl (}a_{1}^{-1}D(X){\bigr )}=A{\bigl (}B(X){\bigr )}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4caf49b8949460fa242aa50e34d0a6bc3bca9119.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.46ex; height:3.343ex;" alt="{\displaystyle X=C{\bigl (}D(X){\bigr )}=A{\bigl (}a_{1}^{-1}D(X){\bigr )}=A{\bigl (}B(X){\bigr )}}" loading="lazy"></span>,</dd></dl>
<p>so dass <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(X)=a_{1}^{-1}D(X)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msubsup>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msubsup>
<mi>D</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B(X)=a_{1}^{-1}D(X)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/abcb736cbafe9d4636b604a14d8077305f2b6637.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.928ex; height:3.343ex;" alt="{\displaystyle B(X)=a_{1}^{-1}D(X)}" loading="lazy"></span> die gesuchte Umkehrfunktion 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 A(X)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A(X)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8563aa1d8928dc8c7a94f613be7bbcdaf1604908.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.532ex; height:2.843ex;" alt="{\displaystyle A(X)}" loading="lazy"></span> ist.
Sie hat die Koeffizienten
</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 b_{k}=a_{1}^{-1}d_{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>=</mo>
<msubsup>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msubsup>
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b_{k}=a_{1}^{-1}d_{k}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2fa8b44f6a918f0b6cebf2be6bdd0de091a9f30e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.045ex; height:3.343ex;" alt="{\displaystyle b_{k}=a_{1}^{-1}d_{k}}" loading="lazy"></span>,</dd></dl>
<p>die allesamt ganzzahlige Polynome 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 a_{1}^{-1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a_{1}^{-1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eac43c15969120a96d205fd5beee9ff578edda5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.563ex; height:3.343ex;" alt="{\displaystyle a_{1}^{-1}}" loading="lazy"></span> und den <span class="mwe-math-element mwe-math-element-inline"><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_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/790f9209748c2dca7ed7b81932c37c02af1dbc31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}" loading="lazy"></span> (<span class="mwe-math-element mwe-math-element-inline"><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\geq 2}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>≥<!-- ≥ --></mo>
<mn>2</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\geq 2}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6bf67f9d06ca3af619657f8d20ee1322da77174.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 2}" loading="lazy"></span>) sind.
</p>
<div class="mw-heading mw-heading2"><h2 id="Formel_von_Lagrange-Bürmann"><span id="Formel_von_Lagrange-B.C3.BCrmann"></span>Formel von Lagrange-Bürmann</h2></div>
<p>Ein Sonderfall der Lagrangeschen Inversionsformel, die in der <a href="Kombinatorik" title="Kombinatorik">Kombinatorik</a> benutzt wird, gilt für <span class="mwe-math-element mwe-math-element-inline"><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(w)=w/\phi (w)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(w)=w/\phi (w)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/285cb1a540ea669d25e8bd6ec1145855ae171af8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.536ex; height:2.843ex;" alt="{\displaystyle f(w)=w/\phi (w)}" loading="lazy"></span> mit analytischem <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (w)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi (w)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0fd45987b81297beba5a5b7ccac52eda35100db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.859ex; height:2.843ex;" alt="{\displaystyle \phi (w)}" loading="lazy"></span> 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 \phi (0)\neq 0.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>≠<!-- ≠ --></mo>
<mn>0.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi (0)\neq 0.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ece5b7021cd867c2b611d0a8e3598c7200c7f56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.265ex; height:2.843ex;" alt="{\displaystyle \phi (0)\neq 0.}" loading="lazy"></span> Durch die Setzung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/90d476e5e765a5d77bbcff32e4584579207ec7d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=0}" loading="lazy"></span> wird <span class="mwe-math-element mwe-math-element-inline"><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)=f(0)=0.}">
<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>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(a)=f(0)=0.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb49073671684ab9965d7cbee248b39866847131.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.574ex; height:2.843ex;" alt="{\displaystyle f(a)=f(0)=0.}" loading="lazy"></span> Dann ist für die Inverse <span class="mwe-math-element mwe-math-element-inline"><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(z):=f^{-1}(z)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>:=</mo>
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g(z):=f^{-1}(z)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5797c44537f1c1eddd1216b59fb01061799fd174.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.309ex; height:3.176ex;" alt="{\displaystyle g(z):=f^{-1}(z)}" loading="lazy"></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 {\begin{aligned}g(z)&amp;=\sum _{n=1}^{\infty }\left(\lim _{w\to 0}\left({\frac {\operatorname {d} ^{n-1}}{\operatorname {d} w^{n-1}}}\left({\frac {w}{w/\phi (w)}}\right)^{n}\right){\frac {z^{n}}{n!}}\right)\\&amp;=\sum _{n=1}^{\infty }{\frac {1}{n}}\left({\frac {1}{(n-1)!}}\lim _{w\to 0}\left({\frac {\operatorname {d} ^{n-1}}{\operatorname {d} w^{n-1}}}\phi (w)^{n}\right)\right)z^{n},\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<mrow>
<mo>(</mo>
<mrow>
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>w</mi>
<mo stretchy="false">→<!-- → --></mo>
<mn>0</mn>
</mrow>
</munder>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi mathvariant="normal">d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mrow>
<mi mathvariant="normal">d</mi>
<mo>⁡<!-- ⁡ --></mo>
<msup>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>w</mi>
<mrow>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mrow>
<mi>n</mi>
<mo>!</mo>
</mrow>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>n</mi>
</mfrac>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
</mfrac>
</mrow>
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>w</mi>
<mo stretchy="false">→<!-- → --></mo>
<mn>0</mn>
</mrow>
</munder>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi mathvariant="normal">d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mrow>
<mi mathvariant="normal">d</mi>
<mo>⁡<!-- ⁡ --></mo>
<msup>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<mi>w</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo>,</mo>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}g(z)&amp;=\sum _{n=1}^{\infty }\left(\lim _{w\to 0}\left({\frac {\operatorname {d} ^{n-1}}{\operatorname {d} w^{n-1}}}\left({\frac {w}{w/\phi (w)}}\right)^{n}\right){\frac {z^{n}}{n!}}\right)\\&amp;=\sum _{n=1}^{\infty }{\frac {1}{n}}\left({\frac {1}{(n-1)!}}\lim _{w\to 0}\left({\frac {\operatorname {d} ^{n-1}}{\operatorname {d} w^{n-1}}}\phi (w)^{n}\right)\right)z^{n},\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb9500a22eaf4f9e94a87071b9d033aa63623830.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:51.091ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}g(z)&amp;=\sum _{n=1}^{\infty }\left(\lim _{w\to 0}\left({\frac {\operatorname {d} ^{n-1}}{\operatorname {d} w^{n-1}}}\left({\frac {w}{w/\phi (w)}}\right)^{n}\right){\frac {z^{n}}{n!}}\right)\\&amp;=\sum _{n=1}^{\infty }{\frac {1}{n}}\left({\frac {1}{(n-1)!}}\lim _{w\to 0}\left({\frac {\operatorname {d} ^{n-1}}{\operatorname {d} w^{n-1}}}\phi (w)^{n}\right)\right)z^{n},\end{aligned}}}" loading="lazy"></span></dd></dl>
<p>welches auch als
</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 [z^{n}]g(z)={\frac {1}{n}}[w^{n-1}]\phi (w)^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo stretchy="false">]</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>n</mi>
</mfrac>
</mrow>
<mo stretchy="false">[</mo>
<msup>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">]</mo>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<mi>w</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [z^{n}]g(z)={\frac {1}{n}}[w^{n-1}]\phi (w)^{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e930bf63cc9c299ea2d3fec6a2318b21f61ff76b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.299ex; height:5.176ex;" alt="{\displaystyle [z^{n}]g(z)={\frac {1}{n}}[w^{n-1}]\phi (w)^{n}}" loading="lazy"></span></dd></dl>
<p>geschrieben werden kann mit dem <a href="Formale_Potenzreihe#Koeffizientenextraktion" title="Formale Potenzreihe">Operator</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 [w^{r}]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<msup>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
</mrow>
</msup>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [w^{r}]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ce2885e02f3ac6762327f4c1ff160679cc1b6fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.932ex; height:2.843ex;" alt="{\displaystyle [w^{r}]}" loading="lazy"></span>, der den Koeffizienten des Terms <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w^{r}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle w^{r}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b05a4d2162bac07375e3dba409b811ef88ae9bc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.638ex; height:2.343ex;" alt="{\displaystyle w^{r}}" loading="lazy"></span> in der rechts davon stehenden formalen Potenzreihe 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 w}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>w</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle w}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" loading="lazy"></span> extrahiert.
</p><p>Eine nützliche Verallgemeinerung ist bekannt als <b>Formel von Lagrange-Bürmann</b>:
</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 [z^{n}]H(g(z))={\frac {1}{n}}[w^{n-1}]{\bigl (}H^{\prime }(w)\phi (w)^{n}{\bigr )}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo stretchy="false">]</mo>
<mi>H</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>n</mi>
</mfrac>
</mrow>
<mo stretchy="false">[</mo>
<msup>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">]</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="1.2em" minsize="1.2em">(</mo>
</mrow>
</mrow>
<msup>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<mi>w</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<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 [z^{n}]H(g(z))={\frac {1}{n}}[w^{n-1}]{\bigl (}H^{\prime }(w)\phi (w)^{n}{\bigr )}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df970873cf13d0bc3131bb8b01148df80d7224dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.564ex; height:5.176ex;" alt="{\displaystyle [z^{n}]H(g(z))={\frac {1}{n}}[w^{n-1}]{\bigl (}H^{\prime }(w)\phi (w)^{n}{\bigr )}}" loading="lazy"></span></dd></dl>
<p>mit einer beliebigen analytischen 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 H}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>H</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"></span>.
</p><p>Die Ableitung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{\prime }(w)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H^{\prime }(w)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84b17f0a61470e7cb2962271c3351f98a6264703.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.262ex; height:3.009ex;" alt="{\displaystyle H^{\prime }(w)}" loading="lazy"></span> kann eine sehr komplizierte Form annehmen, wann es durch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(w)(1-\phi ^{\prime }(w)/\phi (w))}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>H</mi>
<mo stretchy="false">(</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<msup>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H(w)(1-\phi ^{\prime }(w)/\phi (w))}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40c674b24d021fff606d6df699a12e58598464ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.914ex; height:3.009ex;" alt="{\displaystyle H(w)(1-\phi ^{\prime }(w)/\phi (w))}" loading="lazy"></span> ersetzt werden kann, um
</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 [z^{n}]H(g(z))=[w^{n}]H(w)\phi (w)^{n-1}{\bigl (}\phi (w)-w\phi ^{\prime }(w){\bigr )}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo stretchy="false">]</mo>
<mi>H</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">[</mo>
<msup>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo stretchy="false">]</mo>
<mi>H</mi>
<mo stretchy="false">(</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<mi>w</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="1.2em" minsize="1.2em">(</mo>
</mrow>
</mrow>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>w</mi>
<msup>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>w</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 [z^{n}]H(g(z))=[w^{n}]H(w)\phi (w)^{n-1}{\bigl (}\phi (w)-w\phi ^{\prime }(w){\bigr )}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5fc7a95b09731b54613829d8f02135a3b7084c49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:49.515ex; height:3.343ex;" alt="{\displaystyle [z^{n}]H(g(z))=[w^{n}]H(w)\phi (w)^{n-1}{\bigl (}\phi (w)-w\phi ^{\prime }(w){\bigr )}}" loading="lazy"></span></dd></dl>
<p>zu erhalten, welches
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 \phi ^{\prime }(w)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi ^{\prime }(w)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/29728a7dc2106879ad6cada7a15b990694716137.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.544ex; height:3.009ex;" alt="{\displaystyle \phi ^{\prime }(w)}" loading="lazy"></span> anstelle 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 H^{\prime }(w)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H^{\prime }(w)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84b17f0a61470e7cb2962271c3351f98a6264703.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.262ex; height:3.009ex;" alt="{\displaystyle H^{\prime }(w)}" loading="lazy"></span> Bezug nimmt.
</p>
<div class="mw-heading mw-heading2"><h2 id="Anwendungen">Anwendungen</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Die_Lambertsche_W-Funktion">Die Lambertsche W-Funktion</h3></div>
<div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→&nbsp;</span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="Lambertsche_W-Funktion" title="Lambertsche W-Funktion">Lambertsche W-Funktion</a></i></div>
<p>Die <a href="Lambertsche_W-Funktion" title="Lambertsche W-Funktion">Lambertsche W-Funktion</a> ist die durch die implizite Gleichung
</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 W(z)e^{W(z)}=z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>W</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>W</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo>=</mo>
<mi>z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle W(z)e^{W(z)}=z}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/280af73eda3831d684b91cb916eb41272ee13c3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.606ex; height:3.343ex;" alt="{\displaystyle W(z)e^{W(z)}=z}" loading="lazy"></span></dd></dl>
<p>definierte 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 W(z)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>W</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle W(z)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0880c10e0f49b5e9a27a905eaeb9178985aba890.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.333ex; height:2.843ex;" alt="{\displaystyle W(z)}" loading="lazy"></span> .
</p><p>Mithilfe der Lagrangesche Inversionsformel errechnet man für die <a href="Taylor-Reihe" class="mw-redirect" title="Taylor-Reihe">Taylor-Reihe</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 W(z)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>W</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle W(z)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0880c10e0f49b5e9a27a905eaeb9178985aba890.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.333ex; height:2.843ex;" alt="{\displaystyle W(z)}" loading="lazy"></span> am 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 z=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>z</mi>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b92bfc06485cc90286474b14a516a68d8bfdd7b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.349ex; height:2.176ex;" alt="{\displaystyle z=0}" loading="lazy"></span>
wegen <span class="mwe-math-element mwe-math-element-inline"><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(w)=w\mathrm {e} ^{w}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>w</mi>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">e</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>w</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(w)=w\mathrm {e} ^{w}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7088474bd1eb2ab9199e79e7017728315ee45d3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.956ex; height:2.843ex;" alt="{\displaystyle f(w)=w\mathrm {e} ^{w}}" loading="lazy"></span> 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 a=b=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>=</mo>
<mi>b</mi>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a=b=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a9c78ffbb4fc115e8e2a630de5f90fa625effc69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.587ex; height:2.176ex;" alt="{\displaystyle a=b=0}" loading="lazy"></span> zuerst
</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 {\operatorname {d} ^{n}}{\operatorname {d} x^{n}}}\ \mathrm {e} ^{\alpha \,x}\,=\,\alpha ^{n}\,\mathrm {e} ^{\alpha \,x}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi mathvariant="normal">d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mrow>
<mi mathvariant="normal">d</mi>
<mo>⁡<!-- ⁡ --></mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mtext>&nbsp;</mtext>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">e</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</msup>
<mspace width="thinmathspace"></mspace>
<mo>=</mo>
<mspace width="thinmathspace"></mspace>
<msup>
<mi>α<!-- α --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mspace width="thinmathspace"></mspace>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">e</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {d} ^{n}}{\operatorname {d} x^{n}}}\ \mathrm {e} ^{\alpha \,x}\,=\,\alpha ^{n}\,\mathrm {e} ^{\alpha \,x}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7893d28a4426b6bdb8316fe8a9954d4003676f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.898ex; height:5.509ex;" alt="{\displaystyle {\frac {\operatorname {d} ^{n}}{\operatorname {d} x^{n}}}\ \mathrm {e} ^{\alpha \,x}\,=\,\alpha ^{n}\,\mathrm {e} ^{\alpha \,x}}" loading="lazy"></span> ,</dd></dl>
<p>woraus
</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 W(z)=\sum _{n=1}^{\infty }\lim _{w\to 0}\left({\frac {\operatorname {d} ^{\,n-1}}{\operatorname {d} w^{\,n-1}}}\ \mathrm {e} ^{-nw}\right){\frac {z^{n}}{n!}}\,=\,\sum _{n=1}^{\infty }(-n)^{n-1}\,{\frac {z^{n}}{n!}}=x-x^{2}+{\frac {3}{2}}x^{3}-{\frac {8}{3}}x^{4}+{\frac {125}{24}}x^{5}-\dotsb .}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>W</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>w</mi>
<mo stretchy="false">→<!-- → --></mo>
<mn>0</mn>
</mrow>
</munder>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi mathvariant="normal">d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mrow>
<mi mathvariant="normal">d</mi>
<mo>⁡<!-- ⁡ --></mo>
<msup>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mtext>&nbsp;</mtext>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">e</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>n</mi>
<mi>w</mi>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mrow>
<mi>n</mi>
<mo>!</mo>
</mrow>
</mfrac>
</mrow>
<mspace width="thinmathspace"></mspace>
<mo>=</mo>
<mspace width="thinmathspace"></mspace>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mi>n</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mrow>
<mi>n</mi>
<mo>!</mo>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mi>x</mi>
<mo>−<!-- − --></mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>8</mn>
<mn>3</mn>
</mfrac>
</mrow>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>125</mn>
<mn>24</mn>
</mfrac>
</mrow>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle W(z)=\sum _{n=1}^{\infty }\lim _{w\to 0}\left({\frac {\operatorname {d} ^{\,n-1}}{\operatorname {d} w^{\,n-1}}}\ \mathrm {e} ^{-nw}\right){\frac {z^{n}}{n!}}\,=\,\sum _{n=1}^{\infty }(-n)^{n-1}\,{\frac {z^{n}}{n!}}=x-x^{2}+{\frac {3}{2}}x^{3}-{\frac {8}{3}}x^{4}+{\frac {125}{24}}x^{5}-\dotsb .}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/beea9af4e7d250cd564be63d1bd17596fc61f3df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:95.663ex; height:6.843ex;" alt="{\displaystyle W(z)=\sum _{n=1}^{\infty }\lim _{w\to 0}\left({\frac {\operatorname {d} ^{\,n-1}}{\operatorname {d} w^{\,n-1}}}\ \mathrm {e} ^{-nw}\right){\frac {z^{n}}{n!}}\,=\,\sum _{n=1}^{\infty }(-n)^{n-1}\,{\frac {z^{n}}{n!}}=x-x^{2}+{\frac {3}{2}}x^{3}-{\frac {8}{3}}x^{4}+{\frac {125}{24}}x^{5}-\dotsb .}" loading="lazy"></span></dd></dl>
<p>Der <a href="Konvergenzradius" title="Konvergenzradius">Konvergenzradius</a> dieser Reihe 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 \mathrm {e} ^{-1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">e</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {e} ^{-1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68084fdc2a1cbcdb5286667552daac72aeda6621.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.365ex; height:2.676ex;" alt="{\displaystyle \mathrm {e} ^{-1}}" loading="lazy"></span> .
</p><p>Einen größeren Konvergenzradius erhält man auf ähnliche Weise:<br>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 f(z)=W(e^{z})-1\,}">
<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>W</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mspace width="thinmathspace"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(z)=W(e^{z})-1\,}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/607c39c9b1a7ef924d16d209ba4d5fa9face8731.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.994ex; height:2.843ex;" alt="{\displaystyle f(z)=W(e^{z})-1\,}" loading="lazy"></span> erfüllt die Gleichung
</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 1+f(z)+\ln(1+f(z))=z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
<mo>+</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1+f(z)+\ln(1+f(z))=z}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/37de0a17490af5422ed1afdb4ff53b110fa6fa7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.133ex; height:2.843ex;" alt="{\displaystyle 1+f(z)+\ln(1+f(z))=z}" loading="lazy"></span> .</dd></dl>
<p>Entwickelt man <span class="mwe-math-element mwe-math-element-inline"><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+\ln(1+z)\,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>z</mi>
<mo>+</mo>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z+\ln(1+z)\,}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/78dab9b58805fd15f9f9a2fffdb9199699038207.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.155ex; height:2.843ex;" alt="{\displaystyle z+\ln(1+z)\,}" loading="lazy"></span> in eine Potenzreihe und invertiert, dann erhält man für <span class="mwe-math-element mwe-math-element-inline"><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+1)=W(e^{z+1})-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>W</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(z+1)=W(e^{z+1})-1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45025da700f792e32e42d648bc83ee15e83f8565.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.71ex; height:3.176ex;" alt="{\displaystyle f(z+1)=W(e^{z+1})-1}" loading="lazy"></span>&nbsp;:
</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 W(e^{1+z})=1+{\frac {z}{2}}+{\frac {z^{2}}{16}}-{\frac {z^{3}}{192}}-{\frac {z^{4}}{3072}}+{\frac {13z^{5}}{61440}}-{\frac {47z^{6}}{1474560}}-{\frac {73z^{7}}{41287680}}+{\frac {2447z^{8}}{1321205760}}+{\mathcal {O}}(x^{9}).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>W</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>+</mo>
<mi>z</mi>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>z</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mn>16</mn>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<mn>192</mn>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mn>3072</mn>
</mfrac>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>13</mn>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
</msup>
</mrow>
<mn>61440</mn>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>47</mn>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>6</mn>
</mrow>
</msup>
</mrow>
<mn>1474560</mn>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>73</mn>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>7</mn>
</mrow>
</msup>
</mrow>
<mn>41287680</mn>
</mfrac>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>2447</mn>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>8</mn>
</mrow>
</msup>
</mrow>
<mn>1321205760</mn>
</mfrac>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">O</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>9</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle W(e^{1+z})=1+{\frac {z}{2}}+{\frac {z^{2}}{16}}-{\frac {z^{3}}{192}}-{\frac {z^{4}}{3072}}+{\frac {13z^{5}}{61440}}-{\frac {47z^{6}}{1474560}}-{\frac {73z^{7}}{41287680}}+{\frac {2447z^{8}}{1321205760}}+{\mathcal {O}}(x^{9}).}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d72efb431dd266f699b719b9f899ff0507784bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:98.132ex; height:5.843ex;" alt="{\displaystyle W(e^{1+z})=1+{\frac {z}{2}}+{\frac {z^{2}}{16}}-{\frac {z^{3}}{192}}-{\frac {z^{4}}{3072}}+{\frac {13z^{5}}{61440}}-{\frac {47z^{6}}{1474560}}-{\frac {73z^{7}}{41287680}}+{\frac {2447z^{8}}{1321205760}}+{\mathcal {O}}(x^{9}).}" loading="lazy"></span></dd></dl>
<p>Man kann daraus <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>W</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle W(x)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86ead36b42ec68c542b267f9e6bb62cf911a764b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.574ex; height:2.843ex;" alt="{\displaystyle W(x)}" loading="lazy"></span> ableiten, indem man <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln x-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \ln x-1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd4c46ad9dd52002e009d728e67b95768c86e7b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.659ex; height:2.343ex;" alt="{\displaystyle \ln x-1}" loading="lazy"></span> durch <span class="mwe-math-element mwe-math-element-inline"><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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/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> in dieser Reihe substituiert. Bspw. findet man <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W(1)=0{,}567143\dotsc }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>W</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0,567</mn>
<mn>143</mn>
<mo>…<!-- … --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle W(1)=0{,}567143\dotsc }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b385f317e77fbfdd9e60524bd68e97cdbd37703d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.4ex; height:2.843ex;" alt="{\displaystyle W(1)=0{,}567143\dotsc }" loading="lazy"></span> bei <span style="white-space: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=-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>z</mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z=-1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af5d92c041c1ddaa688b8f7f68d333e107e93709.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.157ex; height:2.343ex;" alt="{\displaystyle z=-1}" loading="lazy"></span>.</span>
</p>
<div class="mw-heading mw-heading3"><h3 id="Binärbäume"><span id="Bin.C3.A4rb.C3.A4ume"></span>Binärbäume</h3></div>
<p>Sei <span class="mwe-math-element mwe-math-element-inline"><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 {B}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">B</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e5622de88a69f68340f8dcb43d0b8bd443ba9e13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.543ex; height:2.176ex;" alt="{\displaystyle {\mathcal {B}}}" loading="lazy"></span> die Menge der <a href="Bin%C3%A4rer_Suchbaum#Unterschiedliche_Sprechweisen" title="Binärer Suchbaum">Binärbäume mit NIL-Knoten</a>.
Ein Baum 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 {\mathcal {B}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">B</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e5622de88a69f68340f8dcb43d0b8bd443ba9e13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.543ex; height:2.176ex;" alt="{\displaystyle {\mathcal {B}}}" loading="lazy"></span> ist entweder ein NIL-Knoten oder ein Knoten mit zwei Teilbäumen.
</p><p>Die Anzahl solcher Binärbäume 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 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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/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> (echten) Knoten sei 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 B_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f568bf6d34e97b9fdda0dc7e276d6c4501d2045.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.982ex; height:2.509ex;" alt="{\displaystyle B_{n}}" loading="lazy"></span> bezeichnet.
</p><p>Die Entfernung der Wurzel spaltet den Binärbaum in zwei kleinere Teilbäume.
Daraus folgt für die erzeugende 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 B(z)=\sum _{n=0}^{\infty }B_{n}z^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<msub>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B(z)=\sum _{n=0}^{\infty }B_{n}z^{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f84d0afc81d28fe38a137b4e1c4816c59d71945b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.793ex; height:6.843ex;" alt="{\displaystyle B(z)=\sum _{n=0}^{\infty }B_{n}z^{n}}" loading="lazy"></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 B(z)=1+zB(z)^{2}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mi>z</mi>
<mi>B</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B(z)=1+zB(z)^{2}.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79f35c169590336edfcc75752625715f6064a5ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.213ex; height:3.176ex;" alt="{\displaystyle B(z)=1+zB(z)^{2}.}" loading="lazy"></span></dd></dl>
<p>Nun sei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(z)=B(z)-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>B</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C(z)=B(z)-1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c0a2835276f735d7412f6a3e50f7386d1bb840f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.427ex; height:2.843ex;" alt="{\displaystyle C(z)=B(z)-1}" loading="lazy"></span>, und damit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(z)=z(C(z)+1)^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>z</mi>
<mo stretchy="false">(</mo>
<mi>C</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C(z)=z(C(z)+1)^{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ae885a2d5a6f9c6899e290a89ba371f0e5f8a86b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.381ex; height:3.176ex;" alt="{\displaystyle C(z)=z(C(z)+1)^{2}}" loading="lazy"></span> .
</p><p>Die Anwendung der Lagrangeschen Inversionsformel 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 \phi (w)=(w+1)^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>w</mi>
<mo>+</mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi (w)=(w+1)^{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/793fdfb9b2bb28e523a3127d6a0712b4b084baf2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.488ex; height:3.176ex;" alt="{\displaystyle \phi (w)=(w+1)^{2}}" loading="lazy"></span> ergibt:
</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 B_{n}=[z^{n}]C(z)={\frac {1}{n}}[w^{n-1}](w+1)^{2n}={\frac {1}{n}}{\binom {2n}{n-1}}={\frac {1}{n+1}}{\binom {2n}{n}}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>=</mo>
<mo stretchy="false">[</mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo stretchy="false">]</mo>
<mi>C</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>n</mi>
</mfrac>
</mrow>
<mo stretchy="false">[</mo>
<msup>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">]</mo>
<mo stretchy="false">(</mo>
<mi>w</mi>
<mo>+</mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>n</mi>
</mrow>
</msup>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>n</mi>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mrow>
<mn>2</mn>
<mi>n</mi>
</mrow>
<mrow>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mrow>
<mn>2</mn>
<mi>n</mi>
</mrow>
<mi>n</mi>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B_{n}=[z^{n}]C(z)={\frac {1}{n}}[w^{n-1}](w+1)^{2n}={\frac {1}{n}}{\binom {2n}{n-1}}={\frac {1}{n+1}}{\binom {2n}{n}}.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8ba46e7a5afc13aa8efe81f91c521b2a342fd73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:65.575ex; height:6.176ex;" alt="{\displaystyle B_{n}=[z^{n}]C(z)={\frac {1}{n}}[w^{n-1}](w+1)^{2n}={\frac {1}{n}}{\binom {2n}{n-1}}={\frac {1}{n+1}}{\binom {2n}{n}}.}" loading="lazy"></span></dd></dl>
<p>und das ist die <span class="mwe-math-element mwe-math-element-inline"><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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/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>-te <a href="Catalan-Zahl" title="Catalan-Zahl">Catalan-Zahl</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Anmerkungen">Anmerkungen</h2></div>
<ol class="references" data-mw-group="Anm">
<li id="cite_note-8"><span class="mw-cite-backlink"><a href="#cite_ref-8">↑</a></span> <span class="reference-text">Sie konvergieren im Ring <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R[[X]]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
<mo stretchy="false">[</mo>
<mo stretchy="false">[</mo>
<mi>X</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R[[X]]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc00b68cd3f2c0346dea47d276cea5331fa55159.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.331ex; height:2.843ex;" alt="{\displaystyle R[[X]]}" loading="lazy"></span> der formalen Potenzreihen unter der dortigen <a href="Formale_Potenzreihe#Konvergenz" title="Formale Potenzreihe">Krulltopologie</a>.<br>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 R=\mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
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<annotation encoding="application/x-tex">{\displaystyle R=\mathbb {R} }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2d656c1f5b101209ef2af1a7151ab945f3193fad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.541ex; height:2.176ex;" alt="{\displaystyle R=\mathbb {R} }" loading="lazy"></span> oder <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=\mathbb {C} }">
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<mi>R</mi>
<mo>=</mo>
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<mi mathvariant="double-struck">C</mi>
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<annotation encoding="application/x-tex">{\displaystyle R=\mathbb {C} }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58b370c20a4beca14023fa0e7557c3091d750805.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.541ex; height:2.176ex;" alt="{\displaystyle R=\mathbb {C} }" loading="lazy"></span> oder ein anderer <a href="Vollst%C3%A4ndiger_Raum" title="Vollständiger Raum">vollständiger Ring</a>, dann zieht die <a href="Konvergenzradius" title="Konvergenzradius">analytische</a> Konvergenz die formale nach sich, nicht aber umgekehrt.</span>
</li>
<li id="cite_note-9"><span class="mw-cite-backlink"><a href="#cite_ref-9">↑</a></span> <span class="reference-text">Diese Bedingung erzwingt das Verschwinden <a href="Fast_alle" title="Fast alle">fast aller</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 i_{\nu }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<mi>i</mi>
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<mi>ν<!-- ν --></mi>
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</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle i_{\nu }}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f59d253e4a8c526c88e1c11cafbaacec1045c51d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.906ex; height:2.509ex;" alt="{\displaystyle i_{\nu }}" loading="lazy"></span>, beschränkt also <span class="mwe-math-element mwe-math-element-inline"><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 \sum _{\nu =2}^{\infty }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>ν<!-- ν --></mi>
<mo>=</mo>
<mn>2</mn>
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<mi mathvariant="normal">∞<!-- ∞ --></mi>
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</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle \textstyle \sum _{\nu =2}^{\infty }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f758790147920709b3286a03c52bde2136c0de1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.658ex; height:3.176ex;" alt="{\displaystyle \textstyle \sum _{\nu =2}^{\infty }}" loading="lazy"></span> auf endlich viele effektive Summanden bzw. <span class="mwe-math-element mwe-math-element-inline"><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 \prod _{\nu =2}^{\infty }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<munderover>
<mo>∏<!-- ∏ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>ν<!-- ν --></mi>
<mo>=</mo>
<mn>2</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \textstyle \prod _{\nu =2}^{\infty }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04bf56801301629548e607f3270455d2c115bc23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.398ex; height:3.176ex;" alt="{\displaystyle \textstyle \prod _{\nu =2}^{\infty }}" loading="lazy"></span> auf endlich viele effektive Faktoren.</span>
</li>
<li id="cite_note-exp0-11"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-exp0_11-0">a</a></sup> <sup><a href="#cite_ref-exp0_11-1">b</a></sup></span> <span class="reference-text">(verschwindende <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{\nu }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>ν<!-- ν --></mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i_{\nu }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f59d253e4a8c526c88e1c11cafbaacec1045c51d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.906ex; height:2.509ex;" alt="{\displaystyle i_{\nu }}" loading="lazy"></span> ausgeschrieben)</span>
</li>
</ol>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">M. Abramowitz, I. A. Stegun (Hrsg.): <cite style="font-style:italic">Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</cite>. Dover, New York 1972, 3.6.6. Lagrange's Expansion, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em">&nbsp;</span>14</span> (<a rel="nofollow" class="external text" href="http://people.math.sfu.ca/~cbm/aands/page_14.htm">sfu.ca</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&amp;rfr_id=info:sid/de.wikipedia.org:Lagrangesche+Inversionsformel&amp;rft.atitle=3.6.6.+Lagrange%27s+Expansion&amp;rft.btitle=Handbook+of+Mathematical+Functions+with+Formulas%2C+Graphs%2C+and+Mathematical+Tables&amp;rft.date=1972&amp;rft.genre=bookitem&amp;rft.pages=14&amp;rft.place=New+York&amp;rft.pub=Dover" style="display:none">&nbsp;</span></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Lagrange, Joseph-Louis: <cite style="font-style:italic">Nouvelle méthode pour résoudre les équations littérales par le moyen des séries</cite>. In: <cite style="font-style:italic">Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin</cite>. 24. Jahrgang, 1770, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em">&nbsp;</span>251–326</span> (<a rel="nofollow" class="external text" href="https://archive.today/20120630182344/http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=41070">gdz.sub.uni-goettingen.de</a> (<span class="webarchiv-memento"><a href="Webarchivierung#Begrifflichkeiten" title="Webarchivierung">Memento</a></span> des <style data-mw-deduplicate="TemplateStyles:r250917974">
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</style><span class="dewiki-iconexternal"><a class="external text" href="https://redirecter.toolforge.org/?url=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Fno_cache%2Fdms%2Fload%2Fimg%2F%3FIDDOC%3D41070">Originals</a></span> vom 30.&nbsp;Juni 2012 im Webarchiv <i><a href="Archive.today" title="Archive.today">archive.today</a></i>) [abgerufen am 8.&nbsp;Mai 2018]).<span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rfr_id=info:sid/de.wikipedia.org:Lagrangesche+Inversionsformel&amp;rft.atitle=Nouvelle+m%C3%A9thode+pour+r%C3%A9soudre+les+%C3%A9quations+litt%C3%A9rales+par+le+moyen+des+s%C3%A9ries&amp;rft.au=Lagrange%2C+Joseph-Louis&amp;rft.btitle=M%C3%A9moires+de+l%27Acad%C3%A9mie+Royale+des+Sciences+et+Belles-Lettres+de+Berlin&amp;rft.date=1770&amp;rft.genre=book&amp;rft.pages=251-326&amp;rft.volume=24.+Jahrgang" style="display:none">&nbsp;</span> (Bemerkung: Obwohl Lagrange den Artikel im Jahr 1768 eingereicht hat, wurde er nicht vor 1770 veröffentlicht.)</span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Bürmann, Hans Heinrich, "Essai de calcul fonctionnaire aux constantes ad-libitum," eingereicht im Jahr 1796 beim Institut National de France. Für eine Zusammenfassung dieses Artikels siehe: Hindenburg, Carl Friedrich (Hrsg.): <cite style="font-style:italic">Archiv der reinen und angewandten Mathematik</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em">&nbsp;</span>2</span>. Schäferischen Buchhandlung, Leipzig 1798, Versuch einer vereinfachten Analysis; ein Auszug eines Auszuges von Herrn Bürmann, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em">&nbsp;</span>495–499</span> (<a rel="nofollow" class="external text" href="https://books.google.com/books?id=jj4DAAAAQAAJ&amp;pg=495#v=onepage&amp;q&amp;f=false">google.com</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&amp;rfr_id=info:sid/de.wikipedia.org:Lagrangesche+Inversionsformel&amp;rft.atitle=Versuch+einer+vereinfachten+Analysis%3B+ein+Auszug+eines+Auszuges+von+Herrn+B%C3%BCrmann&amp;rft.btitle=Archiv+der+reinen+und+angewandten+Mathematik&amp;rft.date=1798&amp;rft.genre=bookitem&amp;rft.pages=495-499&amp;rft.place=Leipzig&amp;rft.pub=Sch%C3%A4ferischen+Buchhandlung&amp;rft.volume=2" style="display:none">&nbsp;</span></span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">Bürmann, Hans Heinrich, "Formules du développement, de retour et d'integration," eingereicht an das Institut National de France. Bürmanns Manuskript überlebt in den Archiven der École Nationale des Ponts et Chaussées in Paris. (See ms. 1715.)</span>
</li>
<li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">Ein Bericht von Joseph-Louis Lagrange und Adrien-Marie Legendre über Bürmanns Theorem erscheint in: <a rel="nofollow" class="external text" href="https://gallica.bnf.fr/ark:/12148/bpt6k3217h.image.f22.langFR.pagination">"Rapport sur deux mémoires d'analyse du professeur Burmann,"</a> <i>Mémoires de l'Institut National des Sciences et Arts: Sciences Mathématiques et Physiques</i>, vol. 2, S. 13–17 (1799).</span>
</li>
<li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text"><a href="Edmund_Taylor_Whittaker" title="Edmund Taylor Whittaker">Edmund Taylor Whittaker</a> und <a href="George_Neville_Watson" title="George Neville Watson">George Neville Watson</a>. <i>A Course of Modern Analysis</i>. Cambridge University Press; 4th edition (January 2, 1927), S. 129–130</span>
</li>
<li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text">Eqn (11.43), p. 437, C.A. Charalambides, <i>Enumerative Combinatorics,</i> Chapman &amp; Hall / CRC, 2002</span>
</li>
<li id="cite_note-10"><span class="mw-cite-backlink"><a href="#cite_ref-10">↑</a></span> <span class="reference-text">Morse, P. M. and Feshbach, H. <i>Methods of Theoretical Physics, Part I</i>. New York: McGraw-Hill, pp. 411–413, 1953 (englisch). Zitiert nach <a href="Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/SeriesReversion.html"><i>Series Reversion</i>.</a> In: <i><a href="MathWorld" title="MathWorld">MathWorld</a></i> (englisch). </span>
</li>
</ol>
<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>
<ul><li><a href="Formel_von_Fa%C3%A0_di_Bruno" title="Formel von Faà di Bruno">Formel von Faà di Bruno</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>
<ul><li><a href="Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/LagrangeInversionTheorem.html"><i>Lagrangesche Inversionsformel</i>.</a> In: <i><a href="MathWorld" title="MathWorld">MathWorld</a></i> (englisch).</li>
<li><a href="Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/BuermannsTheorem.html"><i>Bürmann's Theorem</i>.</a> In: <i><a href="MathWorld" title="MathWorld">MathWorld</a></i> (englisch).</li>
<li><a href="Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/SeriesReversion.html"><i>Series Reversion</i>.</a> In: <i><a href="MathWorld" title="MathWorld">MathWorld</a></i> (englisch).</li>
<li><a href="https://en.wikipedia.org/wiki/Kepler%27s_equation#Inverse_Kepler_equation" class="extiw external" title="en:Kepler's equation">en:Kepler's equation#Inverse Kepler equation</a> Umkehrung der Kepler-Gleichung mithilfe der Lagrangeschen Inversionsformel</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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