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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 > Classical cases involving algebraic functions





http://functions.wolfram.com/07.11.26.0015.01









  


  








Input Form





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








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["\[Nu]", "/", "2"]]], " ", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "2", ",", FractionBox["1", SqrtBox[RowBox[List["z", "+", "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["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["\[Nu]", "-", FractionBox["\[Mu]", "2"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["\[Mu]", "2"], ",", RowBox[List["-", FractionBox["\[Mu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["\[Nu]", "-", FractionBox["\[Mu]", "2"]]], "}"]]]], "}"]], ",", "z"]], "]"]], "+", 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[RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["-", FractionBox["\[Mu]", "2"]]], "}"]], ",", RowBox[List["{", FractionBox["\[Mu]", "2"], "}"]]]], "}"]], ",", "z"]], "]"]]]]]], ")"]]]]]], "/;", 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["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]]]]]]]]








MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z_"]], ")"]], FractionBox["\[Nu]_", "2"]], " ", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "2", ",", FractionBox["1", SqrtBox[RowBox[List["z_", "+", "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["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["\[Nu]", "-", FractionBox["\[Mu]", "2"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["\[Mu]", "2"], ",", RowBox[List["-", FractionBox["\[Mu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["\[Nu]", "-", FractionBox["\[Mu]", "2"]]], "}"]]]], "}"]], ",", "z"]], "]"]], "+", 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[RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["-", FractionBox["\[Mu]", "2"]]], "}"]], ",", RowBox[List["{", FractionBox["\[Mu]", "2"], "}"]]]], "}"]], ",", "z"]], "]"]]]]]], ")"]]]], 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["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]]]


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





2001-10-29