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variants of this functions
LegendreQ






Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreQ[nu,mu,3,z] > Series representations > Generalized power series > Expansions at z==1





http://functions.wolfram.com/07.12.06.0004.01









  


  










Input Form





LegendreQ[\[Nu], \[Mu], 3, z] == (Pi/2) Csc[\[Mu] Pi] E^(I Pi \[Mu]) ((1/Gamma[1 - \[Mu]]) ((z + 1)^(\[Mu]/2)/(z - 1)^(\[Mu]/2)) (1 + ((\[Nu] (1 + \[Nu]))/(2 (1 - \[Mu]))) (z - 1) - (((1 - \[Nu]) \[Nu] (1 + \[Nu]) (2 + \[Nu]))/(8 (1 - \[Mu]) (2 - \[Mu]))) (z - 1)^2 + \[Ellipsis]) - (Pochhammer[\[Nu] - \[Mu] + 1, 2 \[Mu]]/Gamma[1 + \[Mu]]) ((z - 1)^(\[Mu]/2)/(z + 1)^(\[Mu]/2)) (1 + ((\[Nu] (1 + \[Nu]))/(2 (1 + \[Mu]))) (z - 1) - (((1 - \[Nu]) \[Nu] (1 + \[Nu]) (2 + \[Nu]))/(8 (1 + \[Mu]) (2 + \[Mu]))) (z - 1)^2 + \[Ellipsis])) /; Abs[(1 - z)/2] < 1 && !Element[\[Mu], Integers]










Standard Form





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MathML Form







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</ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["Csc", "[", RowBox[List["\[Mu]", " ", "\[Pi]"]], "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Mu]"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "-", "\[Mu]"]], ")"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[Nu]"]], ")"]], " ", "\[Nu]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "2"]]], RowBox[List["8", " ", RowBox[List["(", RowBox[List["1", "-", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "-", "\[Mu]"]], ")"]]]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]]]]], "-", FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "-", "\[Mu]", "+", "1"]], ",", RowBox[List["2", " ", "\[Mu]"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "\[Mu]"]], ")"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[Nu]"]], ")"]], " ", "\[Nu]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "2"]]], RowBox[List["8", " ", RowBox[List["(", RowBox[List["1", "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "\[Mu]"]], ")"]]]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", FractionBox[RowBox[List["1", "-", "z"]], "2"], "]"]], "<", "1"]], "&&", RowBox[List["!", RowBox[List["\[Mu]", "\[Element]", "Integers"]]]]]]]]]]]]










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





2001-10-29