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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 > Classical cases involving Legendre P





http://functions.wolfram.com/07.12.26.0051.01









  


  








Input Form





(1/Sqrt[z + 1]) LegendreP[\[Mu] - 1/2, -\[Nu] + 1/2, 3, Sqrt[z + 1]] LegendreQ[\[Nu], \[Mu], 3, Sqrt[(z + 1)/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] /; !IntervalMemberQ[Interval[{-1, 0}], z]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[FractionBox["1", SqrtBox[RowBox[List["z", "+", "1"]]]], RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["\[Mu]", "-", FractionBox["1", "2"]]], ",", RowBox[List[RowBox[List["-", "\[Nu]"]], "+", FractionBox["1", "2"]]], ",", "3", ",", SqrtBox[RowBox[List["z", "+", "1"]]]]], "]"]], " ", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", SqrtBox[FractionBox[RowBox[List["z", "+", "1"]], "z"]]]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Mu]"]]], RowBox[List[SqrtBox["2"], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], "]"]]]]], " ", 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"]], "]"]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]]]]]]








MathML Form







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</mo> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </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["z_", "+", "1"]]]]], "]"]], " ", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", SqrtBox[FractionBox[RowBox[List["z_", "+", "1"]], "z_"]]]], "]"]]]], SqrtBox[RowBox[List["z_", "+", "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"]], "]"]]]], RowBox[List[SqrtBox["2"], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], "]"]]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]


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





2001-10-29