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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 for powers of Legendre Q





http://functions.wolfram.com/07.12.26.0061.01









  


  








Input Form





LegendreQ[\[Nu], \[Mu], 3, Sqrt[z^2 + 1]/z]^2 == ((Sqrt[Pi] E^(2 Pi I \[Mu]) Gamma[1 + \[Mu] + \[Nu]])/ (2 Gamma[1 - \[Mu] + \[Nu]])) MeijerG[{{1, 1 - \[Mu], 1 + \[Mu]}, {}}, {{1 + \[Nu]}, {1/2, -\[Nu]}}, z, 1/2] /; !IntervalMemberQ[Interval[{-Infinity, 0}], z]








Standard Form





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MathML Form







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</mo> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <msup> <mrow> <msubsup> <semantics> <mi> &#120084; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[GothicCapitalQ]&quot;, LegendreQ] </annotation> </semantics> <mi> &#957; </mi> <mi> &#956; </mi> </msubsup> <mo> ( </mo> <semantics> <mfrac> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mi> z </mi> </mfrac> <annotation encoding='Mathematica'> TagBox[FractionBox[SqrtBox[RowBox[List[SuperscriptBox[&quot;z&quot;, &quot;2&quot;], &quot;+&quot;, &quot;1&quot;]]], &quot;z&quot;], HoldComplete[LegendreQ, 3]] </annotation> </semantics> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <msup> <mi> &#8519; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", FractionBox[SqrtBox[RowBox[List[SuperscriptBox["z_", "2"], "+", "1"]]], "z_"]]], "]"]], "2"], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["\[Pi]"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "\[Mu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]", "+", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "-", "\[Mu]"]], ",", RowBox[List["1", "+", "\[Mu]"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "+", "\[Nu]"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["-", "\[Nu]"]]]], "}"]]]], "}"]], ",", "z", ",", FractionBox["1", "2"]]], "]"]]]], RowBox[List["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["-", "\[Infinity]"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]


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





2001-10-29