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variants of this functions
LegendreP






Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreP[nu,mu,3,z] > Representations through more general functions > Through Meijer G > Classical cases involving products of Legendre P





http://functions.wolfram.com/07.09.26.0038.01









  


  










Input Form





LegendreP[\[Nu], \[Mu], 3, Sqrt[1 + z]] LegendreP[\[Mu] - 1/2, \[Nu] + 1/2, 3, Sqrt[1 + 1/z]] == (1/(Sqrt[2] Pi Gamma[-\[Mu] - \[Nu]])) MeijerG[{{3/4, 5/4 + \[Nu]}, {1/4 - \[Nu]}}, {{1/4, 1/4 - \[Mu]}, {\[Mu] + 1/4}}, z] /; !IntervalMemberQ[Interval[{-Infinity, -1}], z]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", SqrtBox[RowBox[List["1", "+", "z"]]]]], "]"]], RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["\[Mu]", "-", FractionBox["1", "2"]]], ",", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ",", "3", ",", SqrtBox[RowBox[List["1", "+", FractionBox["1", "z"]]]]]], "]"]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List[SqrtBox["2"], "\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]], "]"]]]]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "4"], ",", RowBox[List[FractionBox["5", "4"], "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "4"], "-", "\[Nu]"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", RowBox[List[FractionBox["1", "4"], "-", "\[Mu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List["\[Mu]", "+", FractionBox["1", "4"]]], "}"]]]], "}"]], ",", "z"]], "]"]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", "1"]]]], "}"]], "]"]], ",", "z"]], "]"]], "]"]]]]]]










MathML Form







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</mo> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <mrow> <msubsup> <semantics> <mi> &#120083; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[GothicCapitalP]&quot;, LegendreP] </annotation> </semantics> <mi> &#957; </mi> <mi> &#956; </mi> </msubsup> <mo> ( </mo> <semantics> <msqrt> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <annotation encoding='Mathematica'> TagBox[SqrtBox[RowBox[List[&quot;z&quot;, &quot;+&quot;, &quot;1&quot;]]], HoldComplete[LegendreP, 3]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <semantics> <mi> &#120083; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[GothicCapitalP]&quot;, LegendreP] </annotation> </semantics> <mrow> <mi> &#956; </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mrow> <mi> &#957; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", SqrtBox[RowBox[List["1", "+", "z_"]]]]], "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["\[Mu]_", "-", FractionBox["1", "2"]]], ",", RowBox[List["\[Nu]_", "+", FractionBox["1", "2"]]], ",", "3", ",", SqrtBox[RowBox[List["1", "+", FractionBox["1", "z_"]]]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "4"], ",", RowBox[List[FractionBox["5", "4"], "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "4"], "-", "\[Nu]"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", RowBox[List[FractionBox["1", "4"], "-", "\[Mu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List["\[Mu]", "+", FractionBox["1", "4"]]], "}"]]]], "}"]], ",", "z"]], "]"]], RowBox[List[SqrtBox["2"], " ", "\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]], "]"]]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", "1"]]]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]










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





2001-10-29





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