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variants of this functions
LegendreP




Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreP[nu,z] > Operations > Orthogonality, completeness, and Fourier expansions





http://functions.wolfram.com/07.07.25.0002.01









  


  








Input Form





Integrate[(Sqrt[(2 m + 1)/2] LegendreP[m, t]) (Sqrt[(2 n + 1)/2] LegendreP[n, t]), {t, -1, 1}] == KroneckerDelta[m, n]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "1"]], "1"], RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["2", "m"]], "+", "1"]], "2"]], RowBox[List["LegendreP", "[", RowBox[List["m", ",", "t"]], "]"]]]], ")"]], RowBox[List["(", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["2", "n"]], "+", "1"]], "2"]], RowBox[List["LegendreP", "[", RowBox[List["n", ",", "t"]], "]"]]]], ")"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List["KroneckerDelta", "[", RowBox[List["m", ",", "n"]], "]"]]]]]]








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> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 1 </mn> </msubsup> <mrow> <mrow> <mo> ( </mo> <mrow> <msqrt> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> m </mi> </msub> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </msqrt> <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> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> </msub> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> -1 </cn> </lowlimit> <uplimit> <cn type='integer'> 1 </cn> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> LegendreP </ci> <ci> m </ci> <ci> t </ci> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> LegendreP </ci> <ci> n </ci> <ci> t </ci> </apply> </apply> </apply> </apply> <apply> <ci> KroneckerDelta </ci> <ci> m </ci> <ci> n </ci> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "1"]], "1"], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "m_"]], "+", "1"]], ")"]]]]], " ", RowBox[List["LegendreP", "[", RowBox[List["m_", ",", "t_"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n_"]], "+", "1"]], ")"]]]]], " ", RowBox[List["LegendreP", "[", RowBox[List["n_", ",", "t_"]], "]"]]]], ")"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["KroneckerDelta", "[", RowBox[List["m", ",", "n"]], "]"]]]]]]


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





2001-10-29