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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Theta-Operator (Differentialoperator)</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>Als <b>Theta-Operator</b> bezeichnet man in der <a href="Mathematik" title="Mathematik">Mathematik</a> den <a href="Differentialoperator" title="Differentialoperator">Differentialoperator</a>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =x{\frac {d}{dx}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>θ<!-- θ --></mi>
<mo>=</mo>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>d</mi>
<mrow>
<mi>d</mi>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \theta =x{\frac {d}{dx}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d472c2e2a6bd9d8d4ee8e2129efc4066ef0d6926.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.9ex; height:5.509ex;" alt="{\displaystyle \theta =x{\frac {d}{dx}}}" loading="lazy"></span></dd></dl>
<p>bzw. 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 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> Variablen
</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 \theta =x_{1}{\frac {\partial }{\partial x_{1}}}+\ldots +x_{n}{\frac {\partial }{\partial x_{n}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>θ<!-- θ --></mi>
<mo>=</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mo>+</mo>
<mo>…<!-- … --></mo>
<mo>+</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \theta =x_{1}{\frac {\partial }{\partial x_{1}}}+\ldots +x_{n}{\frac {\partial }{\partial x_{n}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e617066a054580439f93ae59493af72b6bb2b4a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.766ex; height:5.843ex;" alt="{\displaystyle \theta =x_{1}{\frac {\partial }{\partial x_{1}}}+\ldots +x_{n}{\frac {\partial }{\partial x_{n}}}}" loading="lazy"></span>.</dd></dl>
<p>Er wird auch als <b>Homogenitätsoperator</b> bezeichnet, weil seine <a href="Eigenfunktion" class="mw-redirect" title="Eigenfunktion">Eigenfunktionen</a> die <a href="Homogenes_Polynom" title="Homogenes Polynom">homogenen Polynome</a> sind. Es gilt 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 \theta (x^{k})=kx^{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>θ<!-- θ --></mi>
<mo stretchy="false">(</mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>k</mi>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \theta (x^{k})=kx^{k}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d785be654057a3a2f52937320c6d80c56e3e98a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.046ex; height:3.176ex;" alt="{\displaystyle \theta (x^{k})=kx^{k}}" loading="lazy"></span></dd></dl>
<p>bzw. 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 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> Variablen
</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 \theta (f)=kf}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>θ<!-- θ --></mi>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>k</mi>
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \theta (f)=kf}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dfc6f5da8907c2bbd0ccfdfac75b9cafa8fe86e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.767ex; height:2.843ex;" alt="{\displaystyle \theta (f)=kf}" loading="lazy"></span></dd></dl>
<p>für ein homogenes Polynom vom Grad <span class="mwe-math-element mwe-math-element-inline"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>
<ul><li><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ThetaOperator.html">Theta Operator</a> (Wolfram MathWorld)</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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