Integrate[LegendreP[n, m, 2, t] LegendreP[l, m, 2, t], {t, -1, 1}] == ((2 (n + m)!)/((2 n + 1) (n - m)!)) KroneckerDelta[n, l] /; Element[l, Integers] && l >= 0 && Element[m, Integers] && m >= 0

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "1"]], "1"], RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["n", ",", "m", ",", "2", ",", "t"]], "]"]], RowBox[List["LegendreP", "[", RowBox[List["l", ",", "m", ",", "2", ",", "t"]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["2", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "m"]], ")"]], "!"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], ")"]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "m"]], ")"]], "!"]]]]], RowBox[List["KroneckerDelta", "[", RowBox[List["n", ",", "l"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["l", "\[Element]", "Integers"]], "\[And]", RowBox[List["l", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <mo> &#8747; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 1 </mn> </msubsup> <mrow> <mrow> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <semantics> <mi> t </mi> <annotation encoding='Mathematica'> TagBox[&quot;t&quot;, HoldComplete[LegendreP, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> l </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <semantics> <mi> t </mi> <annotation encoding='Mathematica'> TagBox[&quot;t&quot;, HoldComplete[LegendreP, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> n </mi> <mo> , </mo> <mi> l </mi> </mrow> </msub> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> l </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <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> <ci> LegendreP </ci> <ci> n </ci> <ci> m </ci> <cn type='integer'> 2 </cn> <ci> t </ci> </apply> <apply> <ci> LegendreP </ci> <ci> l </ci> <ci> m </ci> <cn type='integer'> 2 </cn> <ci> t </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <ci> n </ci> </apply> </apply> <apply> <ci> KroneckerDelta </ci> <ci> n </ci> <ci> l </ci> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> l </ci> <ci> &#8469; </ci> </apply> <apply> <in /> <ci> m </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "1"]], "1"], RowBox[List[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["n_", ",", "m_", ",", "2", ",", "t_"]], "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List["l_", ",", "m_", ",", "2", ",", "t_"]], "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "m"]], ")"]], "!"]]]], ")"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List["n", ",", "l"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "m"]], ")"]], "!"]]]]], "/;", RowBox[List[RowBox[List["l", "\[Element]", "Integers"]], "&&", RowBox[List["l", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]]]

2001-10-29