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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 hypergeometric functions > Involving 2F1





http://functions.wolfram.com/07.11.26.0032.01









  


  








Input Form





LegendreQ[\[Nu], \[Mu], 2, z] == (2^(-1 - \[Nu])/(\[Mu] Gamma[1 - \[Mu] + \[Nu]])) (1 + z)^(\[Nu] - \[Mu]/2) (1 - z)^(\[Mu]/2) (Cos[Pi \[Mu]] Gamma[1 + \[Mu]] Gamma[1 - \[Mu] + \[Nu]] ((1 + z)^\[Mu]/(1 - z)^\[Mu]) Hypergeometric2F1[-\[Nu], -\[Mu] - \[Nu], 1 - \[Mu], (z - 1)/(z + 1)] - Gamma[1 + \[Mu] + \[Nu]] Gamma[1 - \[Mu]] Hypergeometric2F1[-\[Nu], \[Mu] - \[Nu], 1 + \[Mu], (z - 1)/(z + 1)]) /; !Element[\[Mu], Integers] && !IntervalMemberQ[Interval[{-Infinity, -1}], z]








Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["\[Nu]", "-", FractionBox["\[Mu]", "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], "\[Mu]"], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]], ",", RowBox[List["1", "-", "\[Mu]"]], ",", FractionBox[RowBox[List["z", "-", "1"]], RowBox[List["z", "+", "1"]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "\[Mu]"]], "-", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]"]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["\[Mu]", "-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Mu]"]], ",", FractionBox[RowBox[List["z", "-", "1"]], RowBox[List["z", "+", "1"]]]]], "]"]]]]]], ")"]]]], RowBox[List["\[Mu]", " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], "]"]]]]], "/;", RowBox[List[RowBox[List["!", RowBox[List["\[Mu]", "\[Element]", "Integers"]]]], "&&", 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)





2007-05-02