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variants of this functions
LegendreQ




Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreQ[nu,mu,2,z] > Representations through more general functions > Through Meijer G > Generalized cases involving algebraic functions





http://functions.wolfram.com/07.11.26.0022.01









  


  








Input Form





(1 + z^2)^(\[Nu]/2) LegendreQ[\[Nu], \[Mu], 2, z/Sqrt[z^2 + 1]] == (-(Gamma[1 + \[Mu] + \[Nu]]/(2^(\[Nu] + 2) Sqrt[Pi]))) (MeijerG[{{1 + (\[Nu] - \[Mu])/2, 1 + (\[Mu] + \[Nu])/2}, {1 + (\[Mu] - \[Nu])/2}}, {{0, 1/2}, {1 + (\[Mu] - \[Nu])/2}}, z, 1/2] + Cos[Pi (\[Mu] + \[Nu])] MeijerG[{{1 + (\[Mu] + \[Nu])/2}, {1 + (\[Nu] - \[Mu])/2}}, {{0, 1/2}, {}}, z, 1/2]) /; !Element[\[Nu], Integers] && !Element[\[Mu], Integers] && Re[z] > 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["\[Nu]", "/", "2"]]], " ", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "2", ",", FractionBox["z", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "1"]]]]]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]", "+", "\[Nu]"]], "]"]], RowBox[List[SuperscriptBox["2", RowBox[List["\[Nu]", "+", "2"]]], " ", SqrtBox["\[Pi]"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", FractionBox[RowBox[List["\[Nu]", "-", "\[Mu]"]], "2"]]], ",", RowBox[List["1", "+", FractionBox[RowBox[List["\[Mu]", "+", "\[Nu]"]], "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "+", FractionBox[RowBox[List["\[Mu]", "-", "\[Nu]"]], "2"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "+", FractionBox[RowBox[List["\[Mu]", "-", "\[Nu]"]], "2"]]], "}"]]]], "}"]], ",", "z", ",", FractionBox["1", "2"]]], "]"]], "+", RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "+", FractionBox[RowBox[List["\[Mu]", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "+", FractionBox[RowBox[List["\[Nu]", "-", "\[Mu]"]], "2"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z", ",", FractionBox["1", "2"]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "Integers"]], "]"]], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Mu]", ",", "Integers"]], "]"]], "]"]], "\[And]", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]]]








MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["z_", "2"]]], ")"]], FractionBox["\[Nu]_", "2"]], " ", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "2", ",", FractionBox["z_", SqrtBox[RowBox[List[SuperscriptBox["z_", "2"], "+", "1"]]]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]", "+", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", FractionBox[RowBox[List["\[Nu]", "-", "\[Mu]"]], "2"]]], ",", RowBox[List["1", "+", FractionBox[RowBox[List["\[Mu]", "+", "\[Nu]"]], "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "+", FractionBox[RowBox[List["\[Mu]", "-", "\[Nu]"]], "2"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "+", FractionBox[RowBox[List["\[Mu]", "-", "\[Nu]"]], "2"]]], "}"]]]], "}"]], ",", "z", ",", FractionBox["1", "2"]]], "]"]], "+", RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "+", FractionBox[RowBox[List["\[Mu]", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "+", FractionBox[RowBox[List["\[Nu]", "-", "\[Mu]"]], "2"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z", ",", FractionBox["1", "2"]]], "]"]]]]]], ")"]]]], RowBox[List[SuperscriptBox["2", RowBox[List["\[Nu]", "+", "2"]]], " ", SqrtBox["\[Pi]"]]]]]], "/;", RowBox[List[RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]], "&&", RowBox[List["!", RowBox[List["\[Mu]", "\[Element]", "Integers"]]]], "&&", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]]]]]


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





2001-10-29