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Micron Document

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<title>Hermitesches Polynom</title>

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

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<p>Die <b>Hermiteschen Polynome</b> (nach <a href="Charles_Hermite" title="Charles Hermite">Charles Hermite</a>) sind <a href="Polynom" title="Polynom">Polynome</a> mit folgenden äquivalenten Darstellungen:

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<annotation encoding="application/x-tex">{\displaystyle H_{n}(x)=(-1)^{n}e^{x^{2}}{\frac {\mathrm {d} ^{n}}{\mathrm {d} x^{n}}}e^{-x^{2}}\,,}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9bacb5e669aa3fc255b293d317583083421ff991.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.55ex; height:5.509ex;" alt="{\displaystyle H_{n}(x)=(-1)^{n}e^{x^{2}}{\frac {\mathrm {d} ^{n}}{\mathrm {d} x^{n}}}e^{-x^{2}}\,,}" loading="lazy"></span></dd></dl>

<p>bzw.

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<annotation encoding="application/x-tex">{\displaystyle H_{n}(x)=e^{x^{2}/2}\,\left(x-{\frac {\mathrm {d} }{\mathrm {d} x}}\right)^{n}\,e^{-x^{2}/2}\,.}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/36d94cbd71046d8a9d2cbcb4a4e777c991d1b35d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.205ex; height:6.176ex;" alt="{\displaystyle H_{n}(x)=e^{x^{2}/2}\,\left(x-{\frac {\mathrm {d} }{\mathrm {d} x}}\right)^{n}\,e^{-x^{2}/2}\,.}" loading="lazy"></span>

</p><p>Die Hermiteschen Polynome (mit einem festen <span class="mwe-math-element mwe-math-element-inline"><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}">

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

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</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>) sind Lösungen der <b>Hermiteschen Differentialgleichung</b>, einer <a href="Lineare_gew%C3%B6hnliche_Differentialgleichung" title="Lineare gewöhnliche Differentialgleichung">linearen Differentialgleichung</a> zweiter Ordnung:

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<annotation encoding="application/x-tex">{\displaystyle H_{n}''(x)-2\,x\cdot H_{n}'(x)+2\,n\cdot H_{n}(x)=0\qquad (n=0,1,2,\dots ).}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2adea68d27105b7bb4ade216fa3cfcb5e77a6f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.987ex; height:2.843ex;" alt="{\displaystyle H_{n}''(x)-2\,x\cdot H_{n}'(x)+2\,n\cdot H_{n}(x)=0\qquad (n=0,1,2,\dots ).}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Explizite_Darstellung">Explizite Darstellung</h2></div>

<p>Aus der ersten Darstellung erhält man mit der <a href="Formel_von_Fa%C3%A0_di_Bruno" title="Formel von Faà di Bruno">Formel von Faà di Bruno</a> die explizite Darstellung

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<annotation encoding="application/x-tex">{\displaystyle H_{n}(x)=(-1)^{n}\sum _{k_{1}+2k_{2}=n}{\frac {n!}{k_{1}!k_{2}!}}(-1)^{k_{1}+k_{2}}(2x)^{k_{1}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c6460dddc70632a9787fba327eeb78b51eadc896.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:46.45ex; height:6.843ex;" alt="{\displaystyle H_{n}(x)=(-1)^{n}\sum _{k_{1}+2k_{2}=n}{\frac {n!}{k_{1}!k_{2}!}}(-1)^{k_{1}+k_{2}}(2x)^{k_{1}}}" loading="lazy"></span></dd></dl>

<p>also

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<annotation encoding="application/x-tex">{\displaystyle H_{0}(x)=1}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04fafa99666c4042ad1c8cd34a590c11204066f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.385ex; height:2.843ex;" alt="{\displaystyle H_{0}(x)=1}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle H_{1}(x)=2x}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/55b10fadd301caa4fabe887de3173dbd6c0e7333.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.715ex; height:2.843ex;" alt="{\displaystyle H_{1}(x)=2x}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle H_{2}(x)=(2x)^{2}-2=4x^{2}-2}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9eaa29cfb55b3df5d57d7549aca3fd4e7894a9b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.229ex; height:3.176ex;" alt="{\displaystyle H_{2}(x)=(2x)^{2}-2=4x^{2}-2}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle H_{3}(x)=(2x)^{3}-6(2x)=8x^{3}-12x}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1aadf8c2d33100d0ad9b72c104e2d299919b3751.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.023ex; height:3.176ex;" alt="{\displaystyle H_{3}(x)=(2x)^{3}-6(2x)=8x^{3}-12x}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle H_{4}(x)=(2x)^{4}-12(2x)^{2}+12=16x^{4}-48x^{2}+12}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eefaf1adafd9f8c9390c1ea89450ae3fed308fbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.787ex; height:3.176ex;" alt="{\displaystyle H_{4}(x)=(2x)^{4}-12(2x)^{2}+12=16x^{4}-48x^{2}+12}" loading="lazy"></span></dd></dl>

<p>Hermitesche Polynome lassen sich durch folgende Rekursionsformeln berechnen <span class="mwe-math-element mwe-math-element-inline"><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\in \mathbb {N} _{0},H_{-1}(x):=0)}">

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<annotation encoding="application/x-tex">{\displaystyle (n\in \mathbb {N} _{0},H_{-1}(x):=0)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle H_{n+1}(x)=2\,x\,H_{n}(x)-2\,n\,H_{n-1}(x)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8de419fe450ef9b8efc2c9f6330eb41b76267d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.604ex; height:2.843ex;" alt="{\displaystyle H_{n+1}(x)=2\,x\,H_{n}(x)-2\,n\,H_{n-1}(x)}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle H_{n}'(x)=2\,n\,H_{n-1}(x)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c9323020a55ba271b1d6c6826b3824afd6818e4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.108ex; height:2.843ex;" alt="{\displaystyle H_{n}'(x)=2\,n\,H_{n-1}(x)}" loading="lazy"></span></dd></dl>

<p>Da bei jedem Iterationsschritt ein <span class="mwe-math-element mwe-math-element-inline"><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}">

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<annotation encoding="application/x-tex">{\displaystyle H_{n}(-x)=(-1)^{n}\cdot H_{n}(x)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa07c879d803ed0726988e85583f4dead6c6a637.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.161ex; height:2.843ex;" alt="{\displaystyle H_{n}(-x)=(-1)^{n}\cdot H_{n}(x)}" loading="lazy"></span></dd></dl>

<p>ausdrücken lässt.

</p><p>Die rekursive Darstellung der o. g. Hermiteschen Polynome lässt sich durch die einfache 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 n'=n+1}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84a852ac978212e21c29fa7cc5e21c97daed28f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.575ex; height:2.676ex;" alt="{\displaystyle n'=n+1}" loading="lazy"></span> auch wie folgt schreiben:

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<annotation encoding="application/x-tex">{\displaystyle H_{n}(x)=2xH_{n-1}(x)-2(n-1)H_{n-2}(x)\,\,\,\,\,\quad \quad (n=1,2\ldots )}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c0a80a9c140652fbdc396815af764e0743a7a02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:59.22ex; height:2.843ex;" alt="{\displaystyle H_{n}(x)=2xH_{n-1}(x)-2(n-1)H_{n-2}(x)\,\,\,\,\,\quad \quad (n=1,2\ldots )}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Orthogonalität"><span id="Orthogonalit.C3.A4t"></span>Orthogonalität</h2></div>

<p>Die Hermiteschen Polynome erfüllen bezüglich der Gewichtsfunktion <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-x^{2}}}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/da5decb0035215bdd45d3d40b4b2c3a158d00fc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.366ex; height:3.009ex;" alt="{\displaystyle e^{-x^{2}}}" loading="lazy"></span> die <a href="Orthogonale_Polynome" title="Orthogonale Polynome">Orthogonalitätsrelation</a>

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<annotation encoding="application/x-tex">{\displaystyle \int \limits _{-\infty }^{+\infty }e^{-x^{2}}\cdot H_{n}(x)\cdot H_{m}(x)\,dx=2^{n}\cdot n!\cdot {\sqrt {\pi }}\cdot \delta _{nm}.}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38eee93a8f79a19ee12ef695da31c8cb1c706b70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; margin-left: -0.447ex; width:48.199ex; height:9.009ex;" alt="{\displaystyle \int \limits _{-\infty }^{+\infty }e^{-x^{2}}\cdot H_{n}(x)\cdot H_{m}(x)\,dx=2^{n}\cdot n!\cdot {\sqrt {\pi }}\cdot \delta _{nm}.}" loading="lazy"></span></dd></dl>

<p>Das heißt, dass bestimmte reelle Funktionen nach den Hermiteschen Polynomen in eine <a href="Reihenentwicklung" title="Reihenentwicklung">Reihe</a> entwickelt werden können.

</p>

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

<p>Eine <a href="Erzeugende_Funktion" title="Erzeugende Funktion">erzeugende Funktion</a> für die Hermite-Polynome ist

</p>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(x,t)=e^{2xt-t^{2}}=\sum _{n=0}^{\infty }{\frac {t^{n}}{n!}}H_{n}(x)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/357386204da3e9444eaad51b9f2a0f7e88e62cb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.251ex; height:6.843ex;" alt="{\displaystyle F(x,t)=e^{2xt-t^{2}}=\sum _{n=0}^{\infty }{\frac {t^{n}}{n!}}H_{n}(x)}" loading="lazy"></span> .</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Andere_Darstellung_der_Hermiteschen_Polynome">Andere Darstellung der Hermiteschen Polynome</h2></div>

<p>Eine andere Definitionsmöglichkeit der Hermiteschen Polynome (Statistiker-Konvention) ist

</p>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

<msqrt>

</msqrt>

</mrow>

<msup>

</mrow>

</msup>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

<mfrac>

<msup>

</mrow>

</mrow>

</msup>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle He_{n}(x)=2^{-n/2}H_{n}(x/{\sqrt {2}})=(-1)^{n}e^{x^{2}/2}{\frac {\mathrm {d} ^{n}}{\mathrm {d} x^{n}}}e^{-x^{2}/2}.}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/36f863e617d35591c6d61ac3748ba8a68823f096.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:51.618ex; height:5.509ex;" alt="{\displaystyle He_{n}(x)=2^{-n/2}H_{n}(x/{\sqrt {2}})=(-1)^{n}e^{x^{2}/2}{\frac {\mathrm {d} ^{n}}{\mathrm {d} x^{n}}}e^{-x^{2}/2}.}" loading="lazy"></span></dd></dl>

<p>Sie sind bezüglich der Gewichtsfunktion <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-x^{2}/2}}">

<msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{-x^{2}/2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88d971ebcdea40e4177fe2d6feaa463cf1aa00b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.01ex; height:3.009ex;" alt="{\displaystyle e^{-x^{2}/2}}" loading="lazy"></span> orthogonal

</p>

</mrow>

</mrow>

</munderover>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msqrt>

</msqrt>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int \limits _{-\infty }^{\infty }e^{-x^{2}/2}\,He_{n}(x)\,He_{m}(x)\,dx={\sqrt {2\,\pi }}\,n!\,\delta _{mn}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/55726a064123d80472d6c2c079246f411bf0efa9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; margin-left: -0.447ex; width:43.31ex; height:8.843ex;" alt="{\displaystyle \int \limits _{-\infty }^{\infty }e^{-x^{2}/2}\,He_{n}(x)\,He_{m}(x)\,dx={\sqrt {2\,\pi }}\,n!\,\delta _{mn}}" loading="lazy"></span></dd></dl>

<p>und erfüllen die Differentialgleichung

</p>

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y''-x\,y'+n\,y=0.}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d6bb35734bcbccf0910ac8786b5d4b561055d3ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.385ex; height:2.843ex;" alt="{\displaystyle y''-x\,y'+n\,y=0.}" loading="lazy"></span></dd></dl>

<p>Sie lassen sich rekursiv durch

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle He_{n+1}(x)=x\,He_{n}(x)-n\,He_{n-1}(x)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34ac86835caef3f946e7ea31a2b0f3919c839bee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.152ex; height:2.843ex;" alt="{\displaystyle He_{n+1}(x)=x\,He_{n}(x)-n\,He_{n-1}(x)}" loading="lazy"></span></dd></dl>

<p>bestimmen.

</p>

<div class="mw-heading mw-heading2"><h2 id="Binomischer_Lehrsatz">Binomischer Lehrsatz</h2></div>

<p>Für die Hermiteschen Polynome gilt eine Formel, die eine ähnliche Gestalt hat wie der <a href="Binomische_Formel#Verallgemeinerungen" class="mw-redirect" title="Binomische Formel">binomische Lehrsatz</a>. 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 a^{2}+b^{2}=1}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a^{2}+b^{2}=1}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6f2a8a741afa000c8ab3745dc3d57320f3402085.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.437ex; height:2.843ex;" alt="{\displaystyle a^{2}+b^{2}=1}" loading="lazy"></span> ist

</p>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{n}(ax+by)=\sum _{k=0}^{n}{\binom {n}{k}}a^{k}b^{n-k}H_{k}(x)H_{n-k}(y).}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1c7988b56c848558e6619ba657131aaeb96b3040.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:45.893ex; height:7.009ex;" alt="{\displaystyle H_{n}(ax+by)=\sum _{k=0}^{n}{\binom {n}{k}}a^{k}b^{n-k}H_{k}(x)H_{n-k}(y).}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Index_mit_negativem_Wert">Index mit negativem Wert</h2></div>

<p>Die Ableitung der komplementären <a href="Fehlerfunktion" title="Fehlerfunktion">Fehlerfunktion</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 1-\operatorname {erf} (x)=\operatorname {erfc} (x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1-\operatorname {erf} (x)=\operatorname {erfc} (x)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/28d2168f1a0efe2867010a3cb8fa111f4d5b1ed3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.723ex; height:2.843ex;" alt="{\displaystyle 1-\operatorname {erf} (x)=\operatorname {erfc} (x)}" loading="lazy"></span> ist

</p>

<mfrac>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {erfc} (x)=-{\frac {2}{\sqrt {\pi }}}e^{-x^{2}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9a3abb5553d4866dcb23550beee9dd37a8ab4bbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:24.049ex; height:6.343ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {erfc} (x)=-{\frac {2}{\sqrt {\pi }}}e^{-x^{2}}}" loading="lazy"></span>.</dd></dl>

<p>Damit kann die Darstellung der Hermiteschen Polynome auch folgendermaßen geschrieben werden:<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>

<msub>

</mrow>

</msub>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

<mfrac>

<msup>

</mrow>

</mrow>

</msup>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{n}(x)={\frac {\sqrt {\pi }}{2}}(-1)^{(n+1)}e^{x^{2}}{\frac {\mathrm {d} ^{n+1}}{\mathrm {d} x^{n+1}}}\operatorname {erfc} (x)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ef3541e823f9c1b5884683eff137b1f07a50222.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:39.949ex; height:6.176ex;" alt="{\displaystyle H_{n}(x)={\frac {\sqrt {\pi }}{2}}(-1)^{(n+1)}e^{x^{2}}{\frac {\mathrm {d} ^{n+1}}{\mathrm {d} x^{n+1}}}\operatorname {erfc} (x)}" loading="lazy"></span>,</dd></dl>

<p>sodass 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 n=-1}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/29e4adfef8131b59aa818f2877c061297f01272c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.464ex; height:2.343ex;" alt="{\displaystyle n=-1}" loading="lazy"></span> findet:

</p>

<msub>

</mrow>

</msub>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{-1}(x)={\frac {\sqrt {\pi }}{2}}e^{x^{2}}\operatorname {erfc} (x)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/be454376e55f8812458e99afbd66e90fe4d9c1b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.907ex; height:5.843ex;" alt="{\displaystyle H_{-1}(x)={\frac {\sqrt {\pi }}{2}}e^{x^{2}}\operatorname {erfc} (x)}" loading="lazy"></span>.</dd></dl>

<p>Die Funktionen höherer Indizes berechnen sich als:

</p>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</mrow>

</msup>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{n-1}(x)={\frac {(-1)^{n}}{2^{-n}(-n)!}}{\frac {\mathrm {d} ^{-n}}{\mathrm {d} x^{-n}}}H_{-1}(x)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7e824b915b5950ce0130a7330d63aeb3bd111da7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:35.001ex; height:6.676ex;" alt="{\displaystyle H_{n-1}(x)={\frac {(-1)^{n}}{2^{-n}(-n)!}}{\frac {\mathrm {d} ^{-n}}{\mathrm {d} x^{-n}}}H_{-1}(x)}" loading="lazy"></span> oder rekursiv <span class="mwe-math-element mwe-math-element-inline"><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_{n-1}(x)={\frac {1}{2n}}H_{n}'(x)}">

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{n-1}(x)={\frac {1}{2n}}H_{n}'(x)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eab91a1fe50f86889ee98f03bedd4eca51798d21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.17ex; height:5.176ex;" alt="{\displaystyle H_{n-1}(x)={\frac {1}{2n}}H_{n}'(x)}" loading="lazy"></span> 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=(-1,-2,-3,\dotsc )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=(-1,-2,-3,\dotsc )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/16a949ce5b38ba7e49f53d8c1d6e4aaea4caff0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.039ex; height:2.843ex;" alt="{\displaystyle n=(-1,-2,-3,\dotsc )}" loading="lazy"></span>.</dd></dl>

<p>Die so erhaltenen Funktionen genügen wie die Polynome mit positivem Index der hermiteschen Differentialgleichung.

</p><p>Sie lauten:

</p>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

</msqrt>

</mrow>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{-1}(x)={\tfrac {1}{2}}{\sqrt {\pi }}e^{x^{2}}\operatorname {erfc} (x)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05ab2398881565a298a7806c514fb24ee2d8d746.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:25.729ex; height:3.843ex;" alt="{\displaystyle H_{-1}(x)={\tfrac {1}{2}}{\sqrt {\pi }}e^{x^{2}}\operatorname {erfc} (x)}" loading="lazy"></span></dd>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

</msqrt>

</mrow>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{-2}(x)={\tfrac {1}{2}}(1-x{\sqrt {\pi }}e^{x^{2}}\operatorname {erfc} (x))}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61705a56cf6e230d3e94079f788cc00e4c5b42c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:32.871ex; height:3.843ex;" alt="{\displaystyle H_{-2}(x)={\tfrac {1}{2}}(1-x{\sqrt {\pi }}e^{x^{2}}\operatorname {erfc} (x))}" loading="lazy"></span></dd>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msup>

</mrow>

</msup>

<msqrt>

</msqrt>

</mrow>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{-3}(x)={\tfrac {1}{8}}(-2x+(1+2x^{2}){\sqrt {\pi }}e^{x^{2}}\operatorname {erfc} (x))}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc503edc4d1ab5c3eb2681fcf295bf7d242cf9f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:44.038ex; height:4.009ex;" alt="{\displaystyle H_{-3}(x)={\tfrac {1}{8}}(-2x+(1+2x^{2}){\sqrt {\pi }}e^{x^{2}}\operatorname {erfc} (x))}" loading="lazy"></span></dd>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b8619532e44ee1ccae3ab03405a6885260d09ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.723ex; height:0.843ex;" alt="{\displaystyle \ldots }" loading="lazy"></span></dd></dl>

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

<p>Ihre Bedeutung erhalten die Hermite-Polynome durch ihre vielseitige Anwendbarkeit in der Physik. Zum Beispiel werden sie zur Konstruktion der <a href="Orthonormalit%C3%A4t" class="mw-redirect" title="Orthonormalität">orthonormierten</a> Lösungsfunktionen des

<a href="Quantenmechanik" title="Quantenmechanik">quantenmechanischen</a>

<a href="Harmonischer_Oszillator_(Quantenmechanik)" title="Harmonischer Oszillator (Quantenmechanik)">harmonischen Oszillators</a> benötigt.

Diese entsprechen den <a href="Hermitesche_Funktion" title="Hermitesche Funktion">Hermiteschen Funktionen</a>, die man

durch Multiplikation mit der <a href="Normalverteilung" title="Normalverteilung">gaußschen Normalverteilung</a>

und geeigneter Normierung erhält.

</p><p>Eine weitere Anwendung finden sie in der <a href="Methode_der_finiten_Elemente" class="mw-redirect" title="Methode der finiten Elemente">Finite-Elemente-Methode</a> als Formfunktionen.

</p><p>Die Wahrscheinlichkeitsdichte der <a href="Studentsche_t-Verteilung#Nichtzentrale_t-Verteilung" title="Studentsche t-Verteilung">nicht-zentralen Studentschen t-Verteilung</a> lässt sich ausdrücken mittels Hermitescher Polynomfunktionen, deren Index negative Werte hat.

</p>

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Bell-Polynom" title="Bell-Polynom">Bell-Polynom</a></li>

<li><a href="Formel_von_Fa%C3%A0_di_Bruno" title="Formel von Faà di Bruno">Formel von Faà di Bruno</a></li>

<li><a href="Asymptotische_Entwicklungen_vom_Plancherel-Rotach-Typ" title="Asymptotische Entwicklungen vom Plancherel-Rotach-Typ">Asymptotische Entwicklungen vom Plancherel-Rotach-Typ</a></li></ul>

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

<ul><li>I.N. Bronstein u. a.: <i><a href="Taschenbuch_der_Mathematik" title="Taschenbuch der Mathematik">Taschenbuch der Mathematik</a></i>. 5. Auflage. Verlag Harri Deutsch, Frankfurt am Main / Thun 2001, ISBN 3-8171-2005-2</li>

<li><a href="Milton_Abramowitz" title="Milton Abramowitz">Milton Abramowitz</a>, <a href="Irene_Stegun" title="Irene Stegun">Irene Stegun</a>: <i><a href="Abramowitz-Stegun" title="Abramowitz-Stegun">Handbook of Mathematical Functions</a></i></li>

<li>Murray R. Spiegel: <i>Höhere Mathematik für Ingenieure und Naturwissenschaftler</i>. McGraw-Hill</li></ul>

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

<ul><li>Eric W. Weisstein: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/HermitePolynomial.html"><i>Hermite Polynomial</i>.</a> <a href="MathWorld" title="MathWorld">MathWorld</a>.</li></ul>

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

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><a href="Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/HermitePolynomial.html"><i>Hermite Polynomial</i>.</a> In: <i><a href="MathWorld" title="MathWorld">MathWorld</a></i> (englisch). </span>

</li>

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