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variants of this functions
LegendreQ






Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreQ[nu,mu,2,z] > Representations through more general functions > Through other functions > Involving some hypergeometric-type functions





http://functions.wolfram.com/07.11.26.0027.01









  


  










Input Form





LegendreQ[\[Nu], \[Mu], 2, z] == ((2^(-1 - \[Mu]) Sin[Pi (\[Mu] - \[Nu])])/ Sqrt[Pi]) ((1 - z)^(\[Mu]/2)/(1 + z)^(\[Mu]/2)) (2^(2 \[Mu]) Pi Csc[Pi \[Mu]] Csc[Pi (\[Mu] + \[Nu])] Gamma[1/2 + \[Mu]] (1 + z)^\[Mu] GegenbauerC[-1 - \[Mu] - \[Nu], 1/2 + \[Mu], z] + (Cot[Pi \[Mu]] Gamma[\[Mu] + \[Nu] + 1] Gamma[1/2 - \[Mu]] Gamma[\[Mu] - \[Nu]] GegenbauerC[\[Mu] + \[Nu], 1/2 - \[Mu], z])/ (1 - z)^\[Mu])










Standard Form





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MathML Form







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</mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#956; </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> cot </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> &#956; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> &#956; 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</ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#956; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> C </ci> <apply> <plus /> <ci> &#956; </ci> <ci> &#957; </ci> </apply> </apply> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> </apply> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#956; </ci> </apply> </apply> <pi /> <apply> <csc /> <apply> <times /> <pi /> <ci> &#956; </ci> </apply> </apply> <apply> <csc /> <apply> <times /> <pi /> <apply> <plus /> <ci> &#956; </ci> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#956; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> z </ci> </apply> <ci> &#956; </ci> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> C </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <ci> &#956; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "\[Mu]"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]"]], ")"]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["2", " ", "\[Mu]"]]], " ", "\[Pi]", " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Mu]"]], "]"]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], "\[Mu]"], " ", RowBox[List["GegenbauerC", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "-", "\[Mu]", "-", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]"]], ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Mu]", "+", "\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Mu]", "-", "\[Nu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["-", "\[Mu]"]]], " ", RowBox[List["GegenbauerC", "[", RowBox[List[RowBox[List["\[Mu]", "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]"]], ",", "z"]], "]"]]]]]], ")"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]]]]]]]]]










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





2001-10-29





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