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variants of this functions
LegendreP




Mathematica Notation

Traditional Notation









Polynomials > LegendreP[n,z] > Integration > Definite integration > Involving the direct function





http://functions.wolfram.com/05.03.21.0007.01









  


  








Input Form





Integrate[LegendreP[m, Cos[t]] LegendreP[n, Cos[t]] LegendreP[k, Cos[t]] Sin[t], {t, 0, Pi}] == 2 ThreeJSymbol[{m, 0}, {n, 0}, {k, 0}]^2








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Pi]"], RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["m", ",", RowBox[List["Cos", "[", "t", "]"]]]], "]"]], RowBox[List["LegendreP", "[", RowBox[List["n", ",", RowBox[List["Cos", "[", "t", "]"]]]], "]"]], RowBox[List["LegendreP", "[", RowBox[List["k", ",", RowBox[List["Cos", "[", "t", "]"]]]], "]"]], RowBox[List["Sin", "[", "t", "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List["2", " ", SuperscriptBox[RowBox[List["ThreeJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List["m", ",", "0"]], "}"]], ",", RowBox[List["{", RowBox[List["n", ",", "0"]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", "0"]], "}"]]]], "]"]], "2"]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mi> &#960; </mi> </msubsup> <mrow> <mrow> <mrow> <msub> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> m </mi> </msub> <mo> ( </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> k </mi> </msub> <mo> ( </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mrow> <semantics> <mo> ( </mo> <annotation encoding='Mathematica'> TagBox[StyleBox[&quot;(&quot;, Rule[SpanMaxSize, DirectedInfinity[1]]], ThreeJSymbol] </annotation> </semantics> <mtext> &#8287; </mtext> <mtable> <mtr> <mtd> <mi> m </mi> </mtd> <mtd> <mi> n </mi> </mtd> <mtd> <mi> k </mi> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mn> 0 </mn> </mtd> <mtd> <mn> 0 </mn> </mtd> </mtr> </mtable> <mtext> &#8287; </mtext> <semantics> <mo> ) </mo> <annotation encoding='Mathematica'> TagBox[StyleBox[&quot;)&quot;, Rule[SpanMaxSize, DirectedInfinity[1]]], ThreeJSymbol] </annotation> </semantics> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <pi /> </uplimit> <apply> <times /> <apply> <ci> LegendreP </ci> <ci> m </ci> <apply> <cos /> <ci> t </ci> </apply> </apply> <apply> <ci> LegendreP </ci> <ci> n </ci> <apply> <cos /> <ci> t </ci> </apply> </apply> <apply> <ci> LegendreP </ci> <ci> k </ci> <apply> <cos /> <ci> t </ci> </apply> </apply> <apply> <sin /> <ci> t </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <ci> ThreeJSymbol </ci> <list> <ci> m </ci> <cn type='integer'> 0 </cn> </list> <list> <ci> n </ci> <cn type='integer'> 0 </cn> </list> <list> <ci> k </ci> <cn type='integer'> 0 </cn> </list> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Pi]"], RowBox[List[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["m_", ",", RowBox[List["Cos", "[", "t_", "]"]]]], "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List["n_", ",", RowBox[List["Cos", "[", "t_", "]"]]]], "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List["k_", ",", RowBox[List["Cos", "[", "t_", "]"]]]], "]"]], " ", RowBox[List["Sin", "[", "t_", "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["2", " ", SuperscriptBox[RowBox[List["ThreeJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List["m", ",", "0"]], "}"]], ",", RowBox[List["{", RowBox[List["n", ",", "0"]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", "0"]], "}"]]]], "]"]], "2"]]]]]]]


Date Added to functions.wolfram.com (modification date)





2001-10-29