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variants of this functions
LegendreQ






Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreQ[nu,mu,3,z] > Integral representations > On the real axis > Of the direct function





http://functions.wolfram.com/07.12.07.0001.01









  


  










Input Form





LegendreQ[\[Nu], \[Mu], 3, z] == (Gamma[\[Mu] + \[Nu] + 1]/Gamma[\[Nu] + 1]) E^(Pi I \[Mu]) 2^(-\[Nu] - 1) (z^2 - 1)^(\[Mu]/2) Integrate[(1 - t^2)^\[Nu] (z - t)^(-\[Mu] - \[Nu] - 1), {t, -1, 1}] /; Re[\[Nu]] > -1 && !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]_", ",", "3", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Mu]", "+", "\[Nu]", "+", "1"]], "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Mu]"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["z", "2"], "-", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "1"]], "1"], RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["t", "2"]]], ")"]], "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "t"]], ")"]], RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]", "-", "1"]]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], ">", RowBox[List["-", "1"]]]], "&&", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", "1"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]]]










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





2001-10-29





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