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variants of this functions
LegendreQ




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/07.12.26.0038.01









  


  








Input Form





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








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", SqrtBox[RowBox[List["z", "+", "1"]]]]], "]"]], " ", RowBox[List["LegendreQ", "[", RowBox[List[RowBox[List["\[Mu]", "-", FractionBox["1", "2"]]], ",", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ",", "3", ",", SqrtBox[RowBox[List[FractionBox["1", "z"], "+", "1"]]]]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]", "+", FractionBox["1", "2"]]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]", "+", "\[Nu]"]], "]"]]]], RowBox[List["2", " ", SqrtBox["2"]]]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "4"], ",", RowBox[List[FractionBox["1", "4"], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["5", "4"], "+", "\[Nu]"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", RowBox[List[FractionBox["1", "4"], "+", "\[Mu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "4"], "-", "\[Mu]"]], "}"]]]], "}"]], ",", "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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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", SqrtBox[RowBox[List["z_", "+", "1"]]]]], "]"]], " ", RowBox[List["LegendreQ", "[", RowBox[List[RowBox[List["\[Mu]_", "-", FractionBox["1", "2"]]], ",", RowBox[List["\[Nu]_", "+", FractionBox["1", "2"]]], ",", "3", ",", SqrtBox[RowBox[List[FractionBox["1", "z_"], "+", "1"]]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]", "+", FractionBox["1", "2"]]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]", "+", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "4"], ",", RowBox[List[FractionBox["1", "4"], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["5", "4"], "+", "\[Nu]"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", RowBox[List[FractionBox["1", "4"], "+", "\[Mu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "4"], "-", "\[Mu]"]], "}"]]]], "}"]], ",", "z"]], "]"]]]], RowBox[List["2", " ", SqrtBox["2"]]]], "/;", 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