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<title>Legendre-Polynom</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Legendre-Polynom</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">Die Legendre-Polynome (nach <a href="Adrien-Marie_Legendre" title="Adrien-Marie Legendre">Adrien-Marie Legendre</a>), auch zonale Kugelfunktionen genannt, sind spezielle <a href="Polynom" title="Polynom">Polynome</a>, die auf dem Intervall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-1,1]}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51e3b7f14a6f70e614728c583409a0b9a8b9de01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.461ex; height:2.843ex;" alt="{\displaystyle [-1,1]}" loading="lazy"> ein <a href="Orthogonale_Polynome" title="Orthogonale Polynome">orthogonales</a> Funktionensystem bilden. Sie sind die partikulären Lösungen der legendreschen Differentialgleichung. Eine wichtige Rolle spielen die Legendre-Polynome in der <a href="Theoretische_Physik" class="mw-redirect" title="Theoretische Physik">theoretischen Physik</a>, insbesondere in der <a href="Elektrodynamik" title="Elektrodynamik">Elektrodynamik</a> und in der <a href="Quantenmechanik" title="Quantenmechanik">Quantenmechanik</a>, sowie im Bereich der <a href="Filter_(Elektrotechnik)" title="Filter (Elektrotechnik)">Filtertechnik</a> bei den <a href="Legendre-Filter" title="Legendre-Filter">Legendre-Filtern</a>.

<div class="mw-heading mw-heading2"><h2 id="Differentialgleichung_und_Polynome">Differentialgleichung und Polynome</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Legendresche_Differentialgleichung">Legendresche Differentialgleichung</h3></div>

Die legendresche <a href="Differentialgleichung" title="Differentialgleichung">Differentialgleichung</a>

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<annotation encoding="application/x-tex">{\displaystyle \left(1-x^{2}\right)\,y''(x)-2x\,y'(x)+n(n+1)\,y(x)=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b391f9036bcb7fc924bf75a50ed82d8790b23cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:45.816ex; height:3.343ex;" alt="{\displaystyle \left(1-x^{2}\right)\,y''(x)-2x\,y'(x)+n(n+1)\,y(x)=0}" loading="lazy"></dd></dl>

kann als gewöhnliche <a href="Lineare_gew%C3%B6hnliche_Differentialgleichung" title="Lineare gewöhnliche Differentialgleichung">lineare Differentialgleichung</a> zweiter Ordnung auch in der Form

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\left[\left(1-x^{2}\right)\,y'(x)\right]+n(n+1)\,y(x)=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d8d5660d19479dff340689f286576cb5d0aaf67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:40.444ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\left[\left(1-x^{2}\right)\,y'(x)\right]+n(n+1)\,y(x)=0}" loading="lazy"></dd></dl>

für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in [-1,1]}">

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<annotation encoding="application/x-tex">{\displaystyle x\in [-1,1]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f9d0dda56ce3e01e14570ac9aef0021c6125722.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.631ex; height:2.843ex;" alt="{\displaystyle x\in [-1,1]}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} _{0}}">

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<annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} _{0}}</annotation>

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Sie ist ein Spezialfall der <a href="Sturm-Liouville-Problem" title="Sturm-Liouville-Problem">Sturm-Liouville-Differentialgleichung</a>

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<annotation encoding="application/x-tex">{\displaystyle -{\frac {\mathrm {d} }{\mathrm {d} x}}\left((1-x^{2}){\frac {\mathrm {d} y}{\mathrm {d} x}}\right)=n(n+1)y.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/27b1ddf1546fc10dc1458253b4d9e9ec9ce16d08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.232ex; height:6.176ex;" alt="{\displaystyle -{\frac {\mathrm {d} }{\mathrm {d} x}}\left((1-x^{2}){\frac {\mathrm {d} y}{\mathrm {d} x}}\right)=n(n+1)y.}" loading="lazy"></dd></dl>

Die allgemeine Lösung dieser Differentialgleichung lautet

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<annotation encoding="application/x-tex">{\displaystyle y(x)=A\,P_{n}(x)+B\,Q_{n}(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/def5c77c604cd2dc3e8054b3136a8f1687992e21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.56ex; height:2.843ex;" alt="{\displaystyle y(x)=A\,P_{n}(x)+B\,Q_{n}(x)}" loading="lazy"></dd></dl>

mit den beiden linear unabhängigen Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{n}(x)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/56b845e52aab1ff2164ab7ce026bf467823f32de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.85ex; height:2.843ex;" alt="{\displaystyle P_{n}(x)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{n}(x)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a5eab5a40e5daf0466f67130cc14c1436a28777.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.196ex; height:2.843ex;" alt="{\displaystyle Q_{n}(x)}" loading="lazy">. Man bezeichnet die Legendre-Polynome <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{n}(x)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/56b845e52aab1ff2164ab7ce026bf467823f32de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.85ex; height:2.843ex;" alt="{\displaystyle P_{n}(x)}" loading="lazy"> daher auch als Legendre-Funktionen 1. Art und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{n}(x)}">

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

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Darüber hinaus existiert noch eine verallgemeinerte Legendresche Differentialgleichung, deren Lösungen <a href="Zugeordnete_Legendrepolynome" title="Zugeordnete Legendrepolynome">zugeordnete Legendrepolynome</a> heißen.

<div class="mw-heading mw-heading3"><h3 id="Erste_Polynome">Erste Polynome</h3></div>

Die ersten Legendre-Polynome lauten:

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ae79f3c1f989d47802dddcb9a7d78846631e81a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.333ex; height:2.843ex;" alt="{\displaystyle P_{0}(x)=1\,}" loading="lazy"></dd>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5d62f681eb0aa3c05ca4fce7f2090daf24fa3c83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.501ex; height:2.843ex;" alt="{\displaystyle P_{1}(x)=x\,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle P_{2}(x)={\frac {1}{2}}(3x^{2}-1)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/36028b2e42a0eb4bf1ee15ac59f937932ff109d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.141ex; height:5.176ex;" alt="{\displaystyle P_{2}(x)={\frac {1}{2}}(3x^{2}-1)}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle P_{3}(x)={\frac {1}{2}}(5x^{3}-3x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9880b3eab467d0def0f61fcfc0c20cbe103a0b1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.471ex; height:5.176ex;" alt="{\displaystyle P_{3}(x)={\frac {1}{2}}(5x^{3}-3x)}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle P_{4}(x)={\frac {1}{8}}(35x^{4}-30x^{2}+3)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c1e5f6794123567cec1238a360349b9bfbdb9243.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.853ex; height:5.176ex;" alt="{\displaystyle P_{4}(x)={\frac {1}{8}}(35x^{4}-30x^{2}+3)}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle P_{5}(x)={\frac {1}{8}}(63x^{5}-70x^{3}+15x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ed8feb4bdb7301960dfd18bc5b360d55e23f7292.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.345ex; height:5.176ex;" alt="{\displaystyle P_{5}(x)={\frac {1}{8}}(63x^{5}-70x^{3}+15x)}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle P_{6}(x)={\frac {1}{16}}(231x^{6}-315x^{4}+105x^{2}-5)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0779918a6f000fc717f9b8bd223deb139b928b99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:41.052ex; height:5.176ex;" alt="{\displaystyle P_{6}(x)={\frac {1}{16}}(231x^{6}-315x^{4}+105x^{2}-5)}" loading="lazy"></dd></dl>

Das <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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<annotation encoding="application/x-tex">{\displaystyle P_{n}(x)=\sum _{k=0}^{\lfloor n/2\rfloor }(-1)^{k}{\frac {(2n-2k)!\ }{(n-k)!\ (n-2k)!\ k!\ 2^{n}}}x^{n-2k}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db8aa5b0bfc8a33f00e4d9204d91a6b83b294790.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:48.197ex; height:7.843ex;" alt="{\displaystyle P_{n}(x)=\sum _{k=0}^{\lfloor n/2\rfloor }(-1)^{k}{\frac {(2n-2k)!\ }{(n-k)!\ (n-2k)!\ k!\ 2^{n}}}x^{n-2k}}" loading="lazy"></dd></dl>

mit der <a href="Gau%C3%9F-Klammer" class="mw-redirect" title="Gauß-Klammer">Gauß-Klammer</a>

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<mtext>gerade</mtext>

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<mtext>ungerade</mtext>

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<annotation encoding="application/x-tex">{\displaystyle \left\lfloor {\frac {n}{2}}\right\rfloor ={\begin{cases}{\frac {n}{2}}&n\ {\text{gerade}}\\{\frac {n-1}{2}}&n\ {\text{ungerade}}\end{cases}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d05d0d0ac2a0d7ea8b63aa4ebc8f66cad6b0b006.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.809ex; height:7.509ex;" alt="{\displaystyle \left\lfloor {\frac {n}{2}}\right\rfloor ={\begin{cases}{\frac {n}{2}}&n\ {\text{gerade}}\\{\frac {n-1}{2}}&n\ {\text{ungerade}}\end{cases}}}" loading="lazy"></dd></dl>

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

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</math><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">-te Legendre-Polynom hat den Grad <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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</math><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"> und ist aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} [x]}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/da33ddef34ad0c677b6eb7e042f574ee0e7553f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.431ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} [x]}" loading="lazy">, d. h., es hat <a href="Rationale_Zahl" title="Rationale Zahl">rationale</a> Koeffizienten. Für die Legendre-Polynome gibt es mehrere Darstellungsformen.

<div class="mw-heading mw-heading3"><h3 id="Konstruktion_orthogonaler_Polynome">Konstruktion orthogonaler Polynome</h3></div>

Für ein Intervall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=[a,b]}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d6214bb3ce7f00e496c0706edd1464ac60b73b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.825ex; height:2.843ex;" alt="{\displaystyle I=[a,b]}" loading="lazy"> und eine darauf gegebene Gewichtsfunktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w(x)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/504b9ce86ec0b1d4267109ee950497126f04713c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.803ex; height:2.843ex;" alt="{\displaystyle w(x)}" loading="lazy"> ist eine Folge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (P_{n})}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ad27bbb029bbb304df9511d74fb883bb0f78337.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.52ex; height:2.843ex;" alt="{\displaystyle (P_{n})}" loading="lazy"> von reellen Polynomen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{n}\in \mathbb {R} [X]}">

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<annotation encoding="application/x-tex">{\displaystyle P_{n}\in \mathbb {R} [X]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee840442661e2d073eadbb276082fbb8b4b361e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.503ex; height:2.843ex;" alt="{\displaystyle P_{n}\in \mathbb {R} [X]}" loading="lazy"> <a href="Orthogonale_Polynome" title="Orthogonale Polynome">orthogonal</a>, wenn sie die Orthogonalitätsbedingung

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<annotation encoding="application/x-tex">{\displaystyle \int \limits _{a}^{b}w(x)\,P_{n}(x)\,P_{m}(x)\,\mathrm {d} x=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/96ec5b44ffd0fa9084f16964746efe5b9a17ab5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:27.584ex; height:9.176ex;" alt="{\displaystyle \int \limits _{a}^{b}w(x)\,P_{n}(x)\,P_{m}(x)\,\mathrm {d} x=0}" loading="lazy"></dd></dl>

für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m,n\in \mathbb {N} _{0}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m,n\in \mathbb {N} _{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5a780ac395a468ca213ec2bbee33e2d7a888cb7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.042ex; height:2.509ex;" alt="{\displaystyle m,n\in \mathbb {N} _{0}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\neq n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a50a2eaa8dcad2a8e19c9fd861a0fdd641bdfa46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.534ex; height:2.676ex;" alt="{\displaystyle m\neq n}" loading="lazy"> erfüllt.

Für das Intervall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=[-1,1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I=[-1,1]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7edb704e6cad4f0d3828511af70f70be8861ccab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.731ex; height:2.843ex;" alt="{\displaystyle I=[-1,1]}" loading="lazy"> zusammen mit der einfachsten aller Gewichtsfunktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w(x)=1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7c180b2dd805adc4a3afffa7d44bf3bbff94de42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.064ex; height:2.843ex;" alt="{\displaystyle w(x)=1}" loading="lazy"> können solche orthogonalen Polynome mit Hilfe des <a href="Gram-Schmidtsches_Orthogonalisierungsverfahren" title="Gram-Schmidtsches Orthogonalisierungsverfahren">Gram-Schmidtschen Orthogonalisierungsverfahrens</a> ausgehend von den <a href="Monom" title="Monom">Monomen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x^{n})_{n\in \mathbb {N} }}">

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x^{n})_{n\in \mathbb {N} }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/735f3f4d831a09e058c4ad6663548063aa604924.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.859ex; height:2.843ex;" alt="{\displaystyle (x^{n})_{n\in \mathbb {N} }}" loading="lazy"> iterativ erzeugt werden. Die Legendre-Polynome ergeben sich, wenn dabei zusätzlich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{n}(1)=1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7f402d2c8687d58c99012b2f32fe6e6d4119427.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.943ex; height:2.843ex;" alt="{\displaystyle P_{n}(1)=1}" loading="lazy"> gefordert wird.

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

<div class="mw-heading mw-heading3"><h3 id="Rodrigues-Formel">Rodrigues-Formel</h3></div>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</mrow>

</msup>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/96c2a73da1d77159e06be9deeb7e299c76b3d1a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.786ex; height:6.176ex;" alt="{\displaystyle P_{n}(x)={\frac {1}{2^{n}\,n!}}\cdot {\frac {\mathrm {d} ^{n}}{\mathrm {d} x^{n}}}{\bigg (}(x^{2}-1)^{n}{\bigg )}}" loading="lazy"></dd></dl>

Die <a href="Olinde_Rodrigues" title="Olinde Rodrigues">Rodrigues</a>-Formel kann man mit der <a href="Formel_von_Fa%C3%A0_di_Bruno" title="Formel von Faà di Bruno">Formel von Faà di Bruno</a> auswerten und erhält wieder die explizite Form des <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

<div class="mw-heading mw-heading3"><h3 id="Integraldarstellung">Integraldarstellung</h3></div>

Für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {C} \setminus \{+1,-1\}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {C} \setminus \{+1,-1\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/91823bb7876551d90fae654b13d7312d82c363b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.343ex; height:2.843ex;" alt="{\displaystyle x\in \mathbb {C} \setminus \{+1,-1\}}" loading="lazy"> gilt

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

<mrow>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{n}(x)={\frac {1}{\pi }}\int _{0}^{\pi }\left(x+{\sqrt {x^{2}-1}}\cos \varphi \right)^{n}\,\mathrm {d} \varphi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b19a996951aa129a3d9b4ff0df8c108fd4b986ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:41.009ex; height:5.843ex;" alt="{\displaystyle P_{n}(x)={\frac {1}{\pi }}\int _{0}^{\pi }\left(x+{\sqrt {x^{2}-1}}\cos \varphi \right)^{n}\,\mathrm {d} \varphi }" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Rekursionsformeln">Rekursionsformeln</h3></div>

Für die Legendre-Polynome gelten folgende Rekursionsformeln:

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<mfrac>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(n+1)P_{n+1}(x)&=(2n+1)xP_{n}(x)-nP_{n-1}(x)\,\,\,\,\,\quad \quad (n=1,2,\ldots ;P_{0}=1;P_{1}=x)\\(x^{2}-1){\frac {\mathrm {d} }{\mathrm {d} x}}P_{n}(x)&=nxP_{n}(x)-nP_{n-1}(x)\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03c915f8a441c0cc976cafcfc2a0b511f0ca8d6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.427ex; margin-bottom: -0.244ex; width:85.324ex; height:8.509ex;" alt="{\displaystyle {\begin{aligned}(n+1)P_{n+1}(x)&=(2n+1)xP_{n}(x)-nP_{n-1}(x)\,\,\,\,\,\quad \quad (n=1,2,\ldots ;P_{0}=1;P_{1}=x)\\(x^{2}-1){\frac {\mathrm {d} }{\mathrm {d} x}}P_{n}(x)&=nxP_{n}(x)-nP_{n-1}(x)\end{aligned}}}" loading="lazy"></dd></dl>

Die erste rekursive Formel lässt sich mittels der Substitution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n'=n+1}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><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"> in folgender, häufig zu findender Weise darstellen:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle nP_{n}(x)=(2n-1)xP_{n-1}(x)-(n-1)P_{n-2}(x)\,\,\,\,\,\qquad (n=2,3,\ldots ;P_{0}=1;P_{1}=x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/280f259374e871afab7fb4d9f1f365632defdf9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:82.226ex; height:2.843ex;" alt="{\displaystyle nP_{n}(x)=(2n-1)xP_{n-1}(x)-(n-1)P_{n-2}(x)\,\,\,\,\,\qquad (n=2,3,\ldots ;P_{0}=1;P_{1}=x)}" loading="lazy"></dd></dl>

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{n}(x)=\sum _{i=0}^{n}a_{n,i}x^{i}=a_{n,n}x^{n}+\cdots +a_{n,1}x+a_{n,0}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{n}(x)=\sum _{i=0}^{n}a_{n,i}x^{i}=a_{n,n}x^{n}+\cdots +a_{n,1}x+a_{n,0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/886dd6dcc18ac738207ead63b06aec4b1226cf3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:47.861ex; height:6.843ex;" alt="{\displaystyle P_{n}(x)=\sum _{i=0}^{n}a_{n,i}x^{i}=a_{n,n}x^{n}+\cdots +a_{n,1}x+a_{n,0}}" loading="lazy"> die Darstellung des Legendre-Polynoms, dann gilt für die Koeffizienten:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/044e6c23ab2a0da31aef94e61f688ed5cf55c0f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:42.77ex; height:3.009ex;" alt="{\displaystyle na_{n,i}=(2n-1)a_{n-1,i-1}(x)-(n-1)a_{n-2,i}}" loading="lazy"></dd></dl>

Für alle geraden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n,i}=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14e36ec4145ee659182defceb2733ed7413a31db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.734ex; height:2.843ex;" alt="{\displaystyle a_{n,i}=0}" loading="lazy"> für alle ungeraden Indexe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> und für alle ungeraden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n,i}=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14e36ec4145ee659182defceb2733ed7413a31db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.734ex; height:2.843ex;" alt="{\displaystyle a_{n,i}=0}" loading="lazy"> für alle geraden Indexe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

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

</semantics>

Durch Anwendung der Ableitungsregel für Ausdrücke der Art <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=x^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6bfa4e2bf99e419336f5d0c2144ad3a3c0677994.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.802ex; height:2.676ex;" alt="{\displaystyle y=x^{n}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y'=nx^{n-1}=nx^{-1}y}">

<msup>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/57291e83ff8cc37471b334e1da7e884e49e4b347.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.298ex; height:3.009ex;" alt="{\displaystyle y'=nx^{n-1}=nx^{-1}y}" loading="lazy">, bzw.

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b12488bd7c3473041d80455b792ae413726f4ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.179ex; height:3.343ex;" alt="{\displaystyle y^{(m)}=(n-m+1)x^{-1}y^{(m-1)}}" loading="lazy"> ergibt sich folgende rekursive Darstellung der Legendre-Polynome, welche auch die Ableitungen dieser Polynome berücksichtigt:

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n-m)P_{n}^{(m)}(x)=(2n-1)xP_{n-1}^{(m)}(x)-(n-1+m)P_{n-2}^{(m)}(x)\,\,\,\,\,\qquad (n>1;\,\,m=0\ldots n-1)}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (n-m)P_{n}^{(m)}(x)=(2n-1)xP_{n-1}^{(m)}(x)-(n-1+m)P_{n-2}^{(m)}(x)\,\,\,\,\,\qquad (n>1;\,\,m=0\ldots n-1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4e4ea39c47bbe741717d8eac2d9ba063c544f03b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:90.676ex; height:3.843ex;" alt="{\displaystyle (n-m)P_{n}^{(m)}(x)=(2n-1)xP_{n-1}^{(m)}(x)-(n-1+m)P_{n-2}^{(m)}(x)\,\,\,\,\,\qquad (n>1;\,\,m=0\ldots n-1)}" loading="lazy"></dd></dl>

Die Anfangsbedingungen lauten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{m}^{(m)}(x)={\frac {(2m)!}{2^{m}m!}}}">

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{m}^{(m)}(x)={\frac {(2m)!}{2^{m}m!}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ebf39dfb787f13a58c9deac73c11151fbbc23461.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.508ex; height:5.843ex;" alt="{\displaystyle P_{m}^{(m)}(x)={\frac {(2m)!}{2^{m}m!}}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{k}^{(m)}(x)={0}\,\,\,\,\,\qquad (k<m)}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{k}^{(m)}(x)={0}\,\,\,\,\,\qquad (k<m)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bbdccf5d64628d5124e2a63de6961aefa91e0367.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.916ex; height:3.676ex;" alt="{\displaystyle P_{k}^{(m)}(x)={0}\,\,\,\,\,\qquad (k<m)}" loading="lazy"> .

Bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e57f21007575fd03e3be0da20af34d25829cc9a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=0}" loading="lazy"> ergibt sich wiederum die weiter oben angegebene Formel mit ihren Anfangsbedingungen.

<div class="mw-heading mw-heading3"><h3 id="Vollständiges_Orthogonalsystem">Vollständiges Orthogonalsystem</h3></div>

Man betrachte den <a href="Hilbertraum" title="Hilbertraum">Hilbertraum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V:=L^{2}([-1,1];\mathbb {R} )}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V:=L^{2}([-1,1];\mathbb {R} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8fb95919b12ba38e52bb364a5093cc0b0364657d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.152ex; height:3.176ex;" alt="{\displaystyle V:=L^{2}([-1,1];\mathbb {R} )}" loading="lazy"> der <a href="Quadratintegrierbar" class="mw-redirect" title="Quadratintegrierbar">quadratintegrierbaren</a> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-1,1]}">

</mstyle>

</mrow>

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

</semantics>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle f,g\rangle =\int _{-1}^{1}f(x)g(x)\mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b839564f704d5e23ceed13949faa136d5ee4bc5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.643ex; height:6.343ex;" alt="{\displaystyle \langle f,g\rangle =\int _{-1}^{1}f(x)g(x)\mathrm {d} x}" loading="lazy">.</dd></dl>

Die Familie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (P_{n})_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0d6b29b7acd063c66d9705ee1625850177da6a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.739ex; height:2.843ex;" alt="{\displaystyle (P_{n})_{n}}" loading="lazy"> der Legendre-Polynome bildet auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (V,\langle \cdot ,\cdot \rangle )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (V,\langle \cdot ,\cdot \rangle )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07d89cdf91d100d247a9b5e39f2b4249ad7134e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.767ex; height:2.843ex;" alt="{\displaystyle (V,\langle \cdot ,\cdot \rangle )}" loading="lazy"> ein vollständiges <a href="Orthogonalsystem" title="Orthogonalsystem">Orthogonalsystem</a>, sie sind also ein Spezialfall von <a href="Orthogonale_Polynome" title="Orthogonale Polynome">orthogonalen Polynomen</a>. Normiert man diese, so bilden sie ein vollständiges Orthonormalsystem auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">.

Es gilt

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int \limits _{-1}^{1}P_{n}(x)P_{m}(x)\,\mathrm {d} x={\frac {2}{2n+1}}\delta _{nm}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f98403edfd97dd9f23d59ca553f2b18e84a298b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; margin-left: -0.4ex; width:32.335ex; height:9.343ex;" alt="{\displaystyle \int \limits _{-1}^{1}P_{n}(x)P_{m}(x)\,\mathrm {d} x={\frac {2}{2n+1}}\delta _{nm}}" loading="lazy">,</dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{nm}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta _{nm}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f596bddc2ec62dcdd068006e870f2ed0324c7384.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.694ex; height:2.676ex;" alt="{\displaystyle \delta _{nm}}" loading="lazy"> das <a href="Kronecker-Delta" title="Kronecker-Delta">Kronecker-Delta</a> bezeichnet.

Dabei bedeutet die Vollständigkeit, dass sich jede Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in V}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3fd9aeefd10ac4d735121b2f9f0a195a4894dba6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.906ex; height:2.509ex;" alt="{\displaystyle f\in V}" loading="lazy"> in der von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \cdot ,\cdot \rangle }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle \cdot ,\cdot \rangle }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9a50080b735975d8001c9552ac2134b49ad534c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle \langle \cdot ,\cdot \rangle }" loading="lazy"> erzeugten <a href="Normtopologie" title="Normtopologie">Normtopologie</a> nach Legendre-Polynomen „entwickeln“ lässt:

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=\sum _{n=0}^{\infty }c_{n}\,P_{n}(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35007215c13cbd0462e6c7f2e4962becbe824241.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.72ex; height:6.843ex;" alt="{\displaystyle f(x)=\sum _{n=0}^{\infty }c_{n}\,P_{n}(x)}" loading="lazy"></dd></dl>

mit den Entwicklungskoeffizienten

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{n}={\frac {2\,n+1}{2}}\,\int \limits _{-1}^{1}f(x)\,P_{n}(x)\,\mathrm {d} x.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eae2c88d4e912f6e4432dcd27aaa09a3ced80d11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:30.773ex; height:9.343ex;" alt="{\displaystyle c_{n}={\frac {2\,n+1}{2}}\,\int \limits _{-1}^{1}f(x)\,P_{n}(x)\,\mathrm {d} x.}" loading="lazy"></dd></dl>

In der physikalischen oder technischen Literatur wird die Vollständigkeit gern wie folgt als <a href="Distribution_(Mathematik)" title="Distribution (Mathematik)">Distributionsgleichung</a> geschrieben:

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</msup>

<msub>

</mrow>

</msub>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\frac {2\,n+1}{2}}\,P_{n}(x')\,P_{n}(x)=\delta (x'-x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df2ff62d82d250a4dbb6477e44f76b4c683659a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.825ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\frac {2\,n+1}{2}}\,P_{n}(x')\,P_{n}(x)=\delta (x'-x)}" loading="lazy">,</dd></dl>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy"> die diracsche <a href="Delta-Distribution" title="Delta-Distribution">Delta-Distribution</a> ist.

Eine solche Distributionsgleichung ist immer so zu lesen, dass beide Seiten dieser Gleichung auf Testfunktionen anzuwenden sind.

Wendet man die rechte Seite auf eine solche Testfunktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto f(x)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/56846d99e780ebee609728e76389fa84f6cbac81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.361ex; height:2.843ex;" alt="{\displaystyle x\mapsto f(x)}" loading="lazy"> an, so erhält man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x')}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8f2a352e80e95a54cd90e4f21b64dc6d44581acc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.102ex; height:3.009ex;" alt="{\displaystyle f(x')}" loading="lazy">.

Zur Anwendung der linken Seite muss man definitionsgemäß mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"> multiplizieren und anschließend über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> integrieren. Dann erhält man aber genau obige Entwicklungsformel (mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x'}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ac74959896052e160a5953102e4bc3850fe93b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.509ex;" alt="{\displaystyle x'}" loading="lazy"> an Stelle von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

Orthogonalität und Vollständigkeit lassen sich daher kurz und prägnant wie folgt schreiben:

<ul><li>Orthogonalität: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle P_{n},P_{m}\rangle =0}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle P_{n},P_{m}\rangle =0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7e97b73004a0fd2656c387b31e861c3f1dd62c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.982ex; height:2.843ex;" alt="{\displaystyle \langle P_{n},P_{m}\rangle =0}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\neq n}">

</mstyle>

</mrow>

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

</semantics>

<li>Vollständigkeit: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\sum _{n=0}^{\infty }{\frac {2n+1}{2}}\langle f,P_{n}\rangle \,P_{n}(x)}">

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=\sum _{n=0}^{\infty }{\frac {2n+1}{2}}\langle f,P_{n}\rangle \,P_{n}(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c92a4bc8b72f5384ef5d35e9f804f3fc5e655dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.724ex; height:6.843ex;" alt="{\displaystyle f(x)=\sum _{n=0}^{\infty }{\frac {2n+1}{2}}\langle f,P_{n}\rangle \,P_{n}(x)}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in L^{2}([-1,1];\mathbb {R} )}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\in L^{2}([-1,1];\mathbb {R} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d85daf21c61d8c5dac8a5c524c19de9e27cc4f2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.738ex; height:3.176ex;" alt="{\displaystyle f\in L^{2}([-1,1];\mathbb {R} )}" loading="lazy"> (im Sinne der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba162c66ca85776c83557af5088cc6f8584d1912.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}" loading="lazy">-Konvergenz).</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Nullstellen">Nullstellen</h3></div>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/56b845e52aab1ff2164ab7ce026bf467823f32de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.85ex; height:2.843ex;" alt="{\displaystyle P_{n}(x)}" loading="lazy"> hat auf dem Intervall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=[-1,1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I=[-1,1]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7edb704e6cad4f0d3828511af70f70be8861ccab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.731ex; height:2.843ex;" alt="{\displaystyle I=[-1,1]}" loading="lazy"> genau <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> einfache Nullstellen. Sie liegen symmetrisch zum Nullpunkt der Abszisse, da Legendre-Polynome entweder gerade oder ungerade sind. Zwischen zwei benachbarten Nullstellen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{n}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/56b845e52aab1ff2164ab7ce026bf467823f32de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.85ex; height:2.843ex;" alt="{\displaystyle P_{n}(x)}" loading="lazy"> liegt genau eine <a href="Nullstelle" title="Nullstelle">Nullstelle</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{n-1}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/57457d4501af595e315f2e066d0ede59a3feb855.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.95ex; height:2.843ex;" alt="{\displaystyle P_{n-1}(x)}" loading="lazy">. In welchem Verhältnis eine Nullstelle von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{n-1}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/57457d4501af595e315f2e066d0ede59a3feb855.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.95ex; height:2.843ex;" alt="{\displaystyle P_{n-1}(x)}" loading="lazy"> das Intervall zwischen zwei Nullstellen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{n}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/56b845e52aab1ff2164ab7ce026bf467823f32de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.85ex; height:2.843ex;" alt="{\displaystyle P_{n}(x)}" loading="lazy"> teilt, oder auch umgekehrt bis auf die äußeren von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{n}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

Die Bestimmung der Nullstellen der Legendre-Polynome ist in der <a href="Numerische_Mathematik" title="Numerische Mathematik">numerischen Mathematik</a> eine häufige Aufgabe, da sie eine zentrale Rolle bei der <a href="Gau%C3%9F-Quadratur#Gauß-Legendre-Integration" title="Gauß-Quadratur">Gauß-Legendre-Quadratur</a> oder der unter <a href="#Vollständiges_Orthogonalsystem">„Vollständiges Orthogonalsystem“</a> erwähnten Entwicklung „beliebiger“ Funktionen nach Polynomen spielen. Es gibt zwar zahlreiche Tabellenwerke dafür, aber oft ist ihr Gebrauch mit Unannehmlichkeiten verbunden, weil man für eine flexible Reaktion eine Vielzahl an Tabellen in geeigneten Genauigkeiten vorhalten müsste. Bei der Nullstellensuche ist die Kenntnis des Intervalls nur von beschränktem Wert bei der Wahl eines Iterationsanfangs, zumal auch noch die Kenntnis der Nullstellen eines anderen Polynoms erforderlich ist. Eine mit zunehmendem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> genauer werdende Näherung der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math><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">-ten Nullstelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d2b88c64c76a03611549fb9b4cf4ed060b56002.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.418ex; height:2.009ex;" alt="{\displaystyle x_{k}}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{n}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/56b845e52aab1ff2164ab7ce026bf467823f32de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.85ex; height:2.843ex;" alt="{\displaystyle P_{n}(x)}" loading="lazy"> ist gegeben durch:<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{k}\approx \cos \left(\pi \,{\frac {4k-1}{4n+2}}\right),\quad k=1,\ldots ,n.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cfc6ca11c1d3234fdb6b82844e1374e9027bfef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.213ex; height:6.176ex;" alt="{\displaystyle x_{k}\approx \cos \left(\pi \,{\frac {4k-1}{4n+2}}\right),\quad k=1,\ldots ,n.}" loading="lazy"></dd></dl>

Für beispielsweise <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{10}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{10}(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/feafad702857cb49a8d901aceb0702899c63b414.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.507ex; height:2.843ex;" alt="{\displaystyle P_{10}(x)}" loading="lazy"> werden so alle Nullstellen auf wenigstens zwei Dezimalstellen genau abgeschätzt, mit Fehlern zwischen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0{,}00102}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0{,}00102}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eddda74d6d4d645ac619c110dd2ee4af9d4f62cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.622ex; height:2.509ex;" alt="{\displaystyle 0{,}00102}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0{,}00016}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0{,}00016}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/edfb33802ea61230d5187cf4664876dca46be124.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.622ex; height:2.509ex;" alt="{\displaystyle 0{,}00016}" loading="lazy">, während das kleinste Nullstellenintervall von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{9}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{9}(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/90f77d22eb435e91e8bd9da00ae73c50d5c6f3fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.685ex; height:2.843ex;" alt="{\displaystyle P_{9}(x)}" loading="lazy"> nur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0{,}13}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0{,}13}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c30c08b0f4c7188b1677b3e69ba6b3a4ef12034f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.134ex; height:2.509ex;" alt="{\displaystyle 0{,}13}" loading="lazy"> ist.

Bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{20}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{20}(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d6d389d917a8206e6ba49ad5c95e28bd3fa3e2cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.507ex; height:2.843ex;" alt="{\displaystyle P_{20}(x)}" loading="lazy"> sind bereits drei Dezimalstellen sicher, mit Fehlern zwischen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0{,}00028}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0{,}00028}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99d06ce0c5b143eb028cc12af35ad19cc71319fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.622ex; height:2.509ex;" alt="{\displaystyle 0{,}00028}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0{,}00002}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0{,}00002}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/101a615409a751e5f35b794473b74e27947f40cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.622ex; height:2.509ex;" alt="{\displaystyle 0{,}00002}" loading="lazy">, während die beste Einschachtelung durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{19}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{19}(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/991f8a99f6fb5113aa5c66c4aef1d32eab76e882.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.507ex; height:2.843ex;" alt="{\displaystyle P_{19}(x)}" loading="lazy"> nur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0{,}032}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0{,}032}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba6666ba2d02b1f6345f53502d3f87fef48efc67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.297ex; height:2.509ex;" alt="{\displaystyle 0{,}032}" loading="lazy"> ist.

Der maximale Schätzfehler für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{200}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{200}(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99433c60967066a1a54c4324317228739924d88a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.329ex; height:2.843ex;" alt="{\displaystyle P_{200}(x)}" loading="lazy"> ist nur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0{,}0000031}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0{,}0000031}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de13c9a5ed2f0d28f1d04ec839c54e5aa1e12af0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.946ex; height:2.509ex;" alt="{\displaystyle 0{,}0000031}" loading="lazy"> bei den beiden fünften Nullstellen von außen, deren exakter Betrag mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0{,}99722851428\ldots }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0{,}99722851428\ldots }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d7eed0fe39f613ee1a66a452c6bf14574d99a3de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.707ex; height:2.509ex;" alt="{\displaystyle 0{,}99722851428\ldots }" loading="lazy"> beginnt.

Mit einem solchen Startwert und den beiden ersten <a href="#Rekursionsformeln">„Rekursionsformeln“</a> lassen sich mit einem Rechengang sowohl der Funktionswert als auch dessen Ableitung bestimmen. Mithilfe des <a href="Newton-Verfahren" class="mw-redirect" title="Newton-Verfahren">Newton-Verfahrens</a> lassen sich alle Nullstellen bis auf die beiden äußeren mit mehr als quadratischer Konvergenz finden, da sich die Nullstellen in unmittelbarer Nähe der Wendestellen befinden. Die beiden äußeren Nullstellen konvergieren „nur“ quadratisch, d. h. ein anfänglicher Abstand zur Nullstelle von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0{,}00102}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0{,}00102}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eddda74d6d4d645ac619c110dd2ee4af9d4f62cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.622ex; height:2.509ex;" alt="{\displaystyle 0{,}00102}" loading="lazy"> verkleinert sich nach einer Iteration zunächst auf ungefähr <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0{,}00102^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0{,}00102^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b7a9e6ff09e8ab3f634a0f99e625c2aada1fd9da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.676ex; height:3.009ex;" alt="{\displaystyle 0{,}00102^{2}}" loading="lazy">, dann auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0{,}00102^{4},0{,}00102^{8}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0{,}00102^{4},0{,}00102^{8}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/159c78571bdcc50dd5263e62fa734a55faa69bc9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.386ex; height:3.009ex;" alt="{\displaystyle 0{,}00102^{4},0{,}00102^{8}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0{,}00102^{16}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0{,}00102^{16}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f6a284428b51cbd873ab244d0960bf7de96ec5e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.498ex; height:3.009ex;" alt="{\displaystyle 0{,}00102^{16}}" loading="lazy">.

Die angegebene Abschätzung ist Teil eines sehr kurzen <a href="Algorithmus" title="Algorithmus">Algorithmus</a>, die sowohl alle Nullstellen eines Legendre-Polynoms als auch die passenden Gewichte für die Gauß-Legendre-Quadratur liefert.

<div class="mw-heading mw-heading3"><h3 id="Allgemeine_Eigenschaften">Allgemeine Eigenschaften</h3></div>

Für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }" loading="lazy"> und jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in [-1,1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in [-1,1]}</annotation>

</semantics>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&P_{n}(1)=1\\&P_{n}(-x)=(-1)^{n}\,P_{n}(x)\\&P_{2n}(0)=(-1)^{n}{\frac {1\cdot 3\cdot \ldots \cdot (2n-1)}{2\cdot 4\cdot \ldots \cdot 2n}}\\&P_{2n+1}(0)=0\\&P'_{n}(0)=nP_{n-1}(0)\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd14f5f511d4acca41212dbdfd000f86d70455ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.505ex; width:35.644ex; height:18.176ex;" alt="{\displaystyle {\begin{aligned}&P_{n}(1)=1\\&P_{n}(-x)=(-1)^{n}\,P_{n}(x)\\&P_{2n}(0)=(-1)^{n}{\frac {1\cdot 3\cdot \ldots \cdot (2n-1)}{2\cdot 4\cdot \ldots \cdot 2n}}\\&P_{2n+1}(0)=0\\&P'_{n}(0)=nP_{n-1}(0)\end{aligned}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Erzeugende_Funktion">Erzeugende Funktion</h3></div>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a9c6d458566aec47a7259762034790c8981aefab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle x\in \mathbb {R} }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/169fae60c23a2027ece2aa7fd4b5047492887e91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:2.176ex;" alt="{\displaystyle z\in \mathbb {C} }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|<1}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1c0fa57b899b653a3823f85f43fd666309c09b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.643ex; height:2.843ex;" alt="{\displaystyle |z|<1}" loading="lazy"> gilt

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (1-2xz+z^{2})^{-1/2}=\sum _{n=0}^{\infty }P_{n}(x)z^{n}\ .}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e00e1d70f0b71a57345ee73ff3222229acd95c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.581ex; height:6.843ex;" alt="{\displaystyle (1-2xz+z^{2})^{-1/2}=\sum _{n=0}^{\infty }P_{n}(x)z^{n}\ .}" loading="lazy"></dd></dl>

Dabei hat die <a href="Potenzreihe" title="Potenzreihe">Potenzreihe</a> auf der rechten Seite für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1\leq x\leq +1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83ddcce80ca957c5ef40bc23b953397f840a1910.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.468ex; height:2.343ex;" alt="{\displaystyle -1\leq x\leq +1}" loading="lazy"> den

<a href="Konvergenzradius" title="Konvergenzradius">Konvergenzradius</a> 1.

Die Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\mapsto (1-2xz+z^{2})^{-1/2}}">

<msup>

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</msup>

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle z\mapsto (1-2xz+z^{2})^{-1/2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f608e8261b78c6468c1bc26cde83be3bf3e5013.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.056ex; height:3.343ex;" alt="{\displaystyle z\mapsto (1-2xz+z^{2})^{-1/2}}" loading="lazy"> wird daher als <a href="Erzeugende_Funktion" title="Erzeugende Funktion">erzeugende Funktion</a> der Legendre-Polynome <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{n}}">

<msub>

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</msub>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5949c8b1de44005a1af3a11188361f2a830842d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.711ex; height:2.509ex;" alt="{\displaystyle P_{n}}" loading="lazy"> bezeichnet.

Der in der Physik oft auftretende Term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/|{\vec {x}}-{\vec {x}}\,'|}">

</mrow>

</mrow>

<mover>

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<mover>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle 1/|{\vec {x}}-{\vec {x}}\,'|}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a1efb25864d1f1445170da31185854261581fbdf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.19ex; height:3.009ex;" alt="{\displaystyle 1/|{\vec {x}}-{\vec {x}}\,'|}" loading="lazy"> (z. B. in den <a href="Potential_(Physik)" title="Potential (Physik)">Potentialen</a> der <a href="Newtonsches_Gravitationsgesetz" title="Newtonsches Gravitationsgesetz">newtonschen Gravitation</a> oder der <a href="Elektrostatik" title="Elektrostatik">Elektrostatik</a>; <a href="Multipolentwicklung" title="Multipolentwicklung">Multipolentwicklung</a>) lässt sich damit in eine Potenzreihe entwickeln für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {|{\vec {x}}\,'|}{|{\vec {x}}|}}={\tfrac {r\,'}{r}}<1}">

<mfrac>

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<mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {|{\vec {x}}\,'|}{|{\vec {x}}|}}={\tfrac {r\,'}{r}}<1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87daa57dd8c78b23ae5a5b9e689f1448316824d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.465ex; height:5.509ex;" alt="{\displaystyle {\tfrac {|{\vec {x}}\,'|}{|{\vec {x}}|}}={\tfrac {r\,'}{r}}<1}" loading="lazy">:

<mtr>

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<mtd>

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<msqrt>

<mover>

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</msup>

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<msup>

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<mover>

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<msup>

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<mfrac>

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</mfrac>

</mrow>

<mfrac>

<mrow>

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</mrow>

<mfrac>

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</msqrt>

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</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<msup>

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {1}{|{\vec {x}}-{\vec {x}}\,'|}}&={\frac {1}{\sqrt {{\vec {x}}\,^{2}-2{\vec {x}}\cdot {\vec {x}}\,'+{\vec {x}}\,'^{2}}}}={\frac {1}{\sqrt {r^{2}-2rr\,'\cos \alpha +r\,'^{2}}}}={\frac {1}{r{\sqrt {1-2{\frac {r'}{r}}\cos \alpha +({\frac {r'}{r}})^{2}}}}}\\&={\frac {1}{r}}\sum _{n=0}^{\infty }\left({\frac {r\,'}{r}}\right)^{n}P_{n}(\cos \alpha )\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12ffc5e3ed0e7fce1b0ac194c4b78363d93cd7d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.005ex; width:88.416ex; height:15.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {1}{|{\vec {x}}-{\vec {x}}\,'|}}&={\frac {1}{\sqrt {{\vec {x}}\,^{2}-2{\vec {x}}\cdot {\vec {x}}\,'+{\vec {x}}\,'^{2}}}}={\frac {1}{\sqrt {r^{2}-2rr\,'\cos \alpha +r\,'^{2}}}}={\frac {1}{r{\sqrt {1-2{\frac {r'}{r}}\cos \alpha +({\frac {r'}{r}})^{2}}}}}\\&={\frac {1}{r}}\sum _{n=0}^{\infty }\left({\frac {r\,'}{r}}\right)^{n}P_{n}(\cos \alpha )\end{aligned}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Legendre-Funktionen_2._Art">Legendre-Funktionen 2. Art</h2></div>

Die Rekursionsformeln der Legendre-Polynome gelten auch für die Legendre-Funktionen 2. Art, so dass diese sich iterativ mit der Angabe der ersten bestimmen lassen:

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<mrow>

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</mrow>

<mi>artanh</mi>

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<annotation encoding="application/x-tex">{\displaystyle Q_{0}(x)={\frac {1}{2}}\,\ln \left({\frac {1+x}{1-x}}\right)=\operatorname {artanh} (x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03e9daf74a896cc1d831e1a6e3d01704e0a3b64e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.396ex; height:6.176ex;" alt="{\displaystyle Q_{0}(x)={\frac {1}{2}}\,\ln \left({\frac {1+x}{1-x}}\right)=\operatorname {artanh} (x)}" loading="lazy"></dd>

<msub>

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<mrow>

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<mi>artanh</mi>

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<annotation encoding="application/x-tex">{\displaystyle Q_{1}(x)={\frac {x}{2}}\,\ln \left({\frac {1+x}{1-x}}\right)-1=x\operatorname {artanh} (x)-1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/512e3e4bcf8ef074a687d15157238d147557a1e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:46.285ex; height:6.176ex;" alt="{\displaystyle Q_{1}(x)={\frac {x}{2}}\,\ln \left({\frac {1+x}{1-x}}\right)-1=x\operatorname {artanh} (x)-1}" loading="lazy"></dd>

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<mrow>

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</mfrac>

</mrow>

<mfrac>

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</mrow>

<mrow>

<mrow>

<mrow>

<msup>

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</msup>

<mfrac>

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</mrow>

</mrow>

<mi>artanh</mi>

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</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q_{2}(x)={\frac {3\,x^{2}-1}{4}}\,\ln \left({\frac {1+x}{1-x}}\right)-{\frac {3\,x}{2}}={\frac {3}{2}}\left(\left(x^{2}-{\frac {1}{3}}\right)\operatorname {artanh} (x)-x\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e119b60a87f1ce9b7803228943587b05d2d38d9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:70.733ex; height:6.343ex;" alt="{\displaystyle Q_{2}(x)={\frac {3\,x^{2}-1}{4}}\,\ln \left({\frac {1+x}{1-x}}\right)-{\frac {3\,x}{2}}={\frac {3}{2}}\left(\left(x^{2}-{\frac {1}{3}}\right)\operatorname {artanh} (x)-x\right)}" loading="lazy"></dd>

<msub>

</mrow>

</msub>

<mfrac>

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<mfrac>

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<msup>

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<mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q_{3}(x)={\frac {5\,x^{3}-3\,x}{4}}\,\ln \left({\frac {1+x}{1-x}}\right)-{\frac {5\,x^{2}}{2}}+{\frac {2}{3}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05efb149c4515059f5a036141f18cdebd1061ebe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.372ex; height:6.343ex;" alt="{\displaystyle Q_{3}(x)={\frac {5\,x^{3}-3\,x}{4}}\,\ln \left({\frac {1+x}{1-x}}\right)-{\frac {5\,x^{2}}{2}}+{\frac {2}{3}}}" loading="lazy"></dd></dl>

Hierbei ist für den <a href="Logarithmus" title="Logarithmus">Logarithmus</a> der <a href="Logarithmus#Komplexer_Logarithmus" title="Logarithmus">Hauptzweig</a> zu verwenden, wodurch sich Singularitäten bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\pm 1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ae75026eda1ba1ac961f5003150288e9d2f3049b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.399ex; height:2.176ex;" alt="{\displaystyle x=\pm 1}" loading="lazy"> und in der komplexen Ebene Verzweigungsschnitte<a href="#cite_note-3">[3]</a> entlang <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\infty ,-1)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eecbbaf03fa846f2e0c57bf5f65e760d2f7e484d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.946ex; height:2.843ex;" alt="{\displaystyle (-\infty ,-1)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,\infty )}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d510c7f8d14cbdabeab5f79521273e9fb8e236a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.329ex; height:2.843ex;" alt="{\displaystyle (1,\infty )}" loading="lazy"> ergeben.

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

Unter anderem wird das Legendre-Polynom für Simulationen von Kugelsphären verwendet, so zum Beispiel zur Ermittlung des Taylor-Winkels im <a href="Taylor-Kegel" title="Taylor-Kegel">Taylor-Kegel</a>, welcher beim <a href="Elektrospinnen" title="Elektrospinnen">Elektrospinnen</a> der Geometrie zu Grunde liegt.

<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/LegendrePolynomial.html">Legendre Polynomial.</a> In: <a href="MathWorld" title="MathWorld">MathWorld</a> (englisch).</li>

<li>J. B. Calvert: <a rel="nofollow" class="external text" href="http://mysite.du.edu/~jcalvert/math/legendre.htm">Legendre Polynomials.</a> (englisch)</li></ul>

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

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> <a href="Numerical_Recipes" title="Numerical Recipes">Numerical Recipes</a>: Codeausschnitt aus <a rel="nofollow" class="external text" href="http://apps.nrbook.com/c/index.html">Numerical Recipes in C, Seite 152</a>: „z=cos(3.141592654*(i-0.25)/(n+0.5));“

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> <a href="Abramowitz-Stegun" title="Abramowitz-Stegun">Abramowitz-Stegun</a>: Handbook of Mathematical Functions. Asymptotische Entwicklung der Nullstellen in <a rel="nofollow" class="external text" href="http://people.math.sfu.ca/~cbm/aands/page_787.htm">Formel 22.16.6, Seite 787</a>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/BranchCut.html">Branch Cut.</a> Wolfram Research, abgerufen am 19. September 2018.

</li>

</ol></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2025-08-09" href="https://de.wikipedia.org/wiki/?title=Legendre-Polynom&oldid=258713256">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

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