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variants of this functions
LegendreQ






Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreQ[nu,mu,3,z] > Representations through more general functions > Through Meijer G > Generalized cases involving Legendre P





http://functions.wolfram.com/07.12.26.0076.01









  


  










Input Form





(1/Sqrt[z^2 + 1]) LegendreP[\[Mu] - 1/2, -\[Nu] + 1/2, 3, Sqrt[z^2 + 1]] LegendreQ[\[Nu], \[Mu], 3, Sqrt[1 + z^2]/z] == (E^(Pi I \[Mu])/(Sqrt[2] Gamma[1 - \[Mu] + \[Nu]])) MeijerG[{{1/4, 1/4 - \[Mu], 1/4 + \[Mu]}, {}}, {{1/4 + \[Nu]}, {-(1/4), 1/4 - \[Nu]}}, z, 1/2] /; Re[z] > 0










Standard Form





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MathML Form







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</ci> </apply> </apply> <apply> <plus /> <ci> &#956; </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </list> <list /> </list> <list> <list> <apply> <plus /> <ci> &#957; </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </list> <list> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <plus /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </list> </list> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <gt /> <apply> <times /> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", FractionBox[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["\[Mu]_", "-", FractionBox["1", "2"]]], ",", RowBox[List[RowBox[List["-", "\[Nu]_"]], "+", FractionBox["1", "2"]]], ",", "3", ",", SqrtBox[RowBox[List[SuperscriptBox["z_", "2"], "+", "1"]]]]], "]"]], " ", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", FractionBox[SqrtBox[RowBox[List["1", "+", SuperscriptBox["z_", "2"]]]], "z_"]]], "]"]]]], SqrtBox[RowBox[List[SuperscriptBox["z_", "2"], "+", "1"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Mu]"]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", RowBox[List[FractionBox["1", "4"], "-", "\[Mu]"]], ",", RowBox[List[FractionBox["1", "4"], "+", "\[Mu]"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "4"], "+", "\[Nu]"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], ",", RowBox[List[FractionBox["1", "4"], "-", "\[Nu]"]]]], "}"]]]], "}"]], ",", "z", ",", FractionBox["1", "2"]]], "]"]]]], RowBox[List[SqrtBox["2"], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], "]"]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]]]










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





2001-10-29