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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/07.09.26.0034.01









  


  








Input Form





(1/Sqrt[1 + z]) LegendreP[\[Nu], \[Mu], 3, Sqrt[(1 + z)/z]] LegendreP[-\[Nu], \[Mu], 3, Sqrt[(1 + z)/z]] == (1/(Sqrt[Pi] Gamma[1 - \[Mu] + \[Nu]] Gamma[1 - \[Mu] - \[Nu]])) MeijerG[{{1/2 + \[Mu]}, {1/2, 1/2 - \[Mu]}}, {{1/2 - \[Nu], 1/2 + \[Nu], 0}, {}}, z] /; !IntervalMemberQ[Interval[{-1, 0}], z]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[FractionBox["1", SqrtBox[RowBox[List["1", "+", "z"]]]], RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", SqrtBox[FractionBox[RowBox[List["1", "+", "z"]], "z"]]]], "]"]], RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "\[Mu]", ",", "3", ",", SqrtBox[FractionBox[RowBox[List["1", "+", "z"]], "z"]]]], "]"]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List[SqrtBox["\[Pi]"], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "-", "\[Nu]"]], "]"]]]]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", "0"]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]]]]]]








MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", FractionBox[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", SqrtBox[FractionBox[RowBox[List["1", "+", "z_"]], "z_"]]]], "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["-", "\[Nu]_"]], ",", "\[Mu]_", ",", "3", ",", SqrtBox[FractionBox[RowBox[List["1", "+", "z_"]], "z_"]]]], "]"]]]], SqrtBox[RowBox[List["1", "+", "z_"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", "0"]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "-", "\[Nu]"]], "]"]]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]


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





2001-10-29