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http://functions.wolfram.com/07.11.03.0018.01
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LegendreQ[4, \[Mu], 2, z] == (((Pi Csc[Pi \[Mu]])/(2 Gamma[5 - \[Mu]]))
((-(1 - z)^\[Mu]) (9 + 105 z^4 + 105 z^3 \[Mu] - 10 \[Mu]^2 + \[Mu]^4 +
45 z^2 (-2 + \[Mu]^2) + 5 z \[Mu] (-11 + 2 \[Mu]^2)) +
(1 + z)^\[Mu] (9 + 105 z^4 - 105 z^3 \[Mu] - 10 \[Mu]^2 + \[Mu]^4 +
45 z^2 (-2 + \[Mu]^2) + z (55 \[Mu] - 10 \[Mu]^3)) Cos[Pi \[Mu]]))/
(1 - z^2)^(\[Mu]/2)
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Cell[BoxData[RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["4", ",", "\[Mu]", ",", "2", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Mu]"]], "]"]]]], RowBox[List["2", " ", RowBox[List["Gamma", "[", RowBox[List["5", "-", "\[Mu]"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[RowBox[List["-", "\[Mu]"]], "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "\[Mu]"]]], " ", RowBox[List["(", RowBox[List["9", "+", RowBox[List["105", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["105", " ", SuperscriptBox["z", "3"], " ", "\[Mu]"]], "-", RowBox[List["10", " ", SuperscriptBox["\[Mu]", "2"]]], "+", SuperscriptBox["\[Mu]", "4"], "+", RowBox[List["45", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", SuperscriptBox["\[Mu]", "2"]]], ")"]]]], "+", RowBox[List["5", " ", "z", " ", "\[Mu]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "11"]], "+", RowBox[List["2", " ", SuperscriptBox["\[Mu]", "2"]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], "\[Mu]"], " ", RowBox[List["(", RowBox[List["9", "+", RowBox[List["105", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["105", " ", SuperscriptBox["z", "3"], " ", "\[Mu]"]], "-", RowBox[List["10", " ", SuperscriptBox["\[Mu]", "2"]]], "+", SuperscriptBox["\[Mu]", "4"], "+", RowBox[List["45", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", SuperscriptBox["\[Mu]", "2"]]], ")"]]]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["55", " ", "\[Mu]"]], "-", RowBox[List["10", " ", SuperscriptBox["\[Mu]", "3"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Mu]"]], "]"]]]]]], ")"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msubsup> <semantics> <mi> Q </mi> <annotation encoding='Mathematica'> TagBox["Q", LegendreQ] </annotation> </semantics> <mn> 4 </mn> <mi> μ </mi> </msubsup> <mo> ( </mo> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox["z", HoldComplete[LegendreQ, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> csc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 5 </mn> <mo> - </mo> <mi> μ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mfrac> <mi> μ </mi> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> μ </mi> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 105 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 105 </mn> <mo> ⁢ </mo> <mi> μ </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 45 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 55 </mn> <mo> ⁢ </mo> <mi> μ </mi> </mrow> <mo> - </mo> <mrow> <mn> 10 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 3 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <msup> <mi> μ </mi> <mn> 4 </mn> </msup> <mo> - </mo> <mrow> <mn> 10 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 9 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> μ </mi> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 105 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 105 </mn> <mo> ⁢ </mo> <mi> μ </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 45 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <mi> μ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 11 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <msup> <mi> μ </mi> <mn> 4 </mn> </msup> <mo> - </mo> <mrow> <mn> 10 </mn> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 9 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> LegendreQ </ci> <cn type='integer'> 4 </cn> <ci> μ </ci> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <pi /> <apply> <csc /> <apply> <times /> <pi /> <ci> μ </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 5 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> μ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <ci> μ </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 105 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 105 </cn> <ci> μ </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 45 </cn> <apply> <plus /> <apply> <power /> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 55 </cn> <ci> μ </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 10 </cn> <apply> <power /> <ci> μ </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <ci> z </ci> </apply> <apply> <power /> <ci> μ </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 10 </cn> <apply> <power /> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 9 </cn> </apply> <apply> <cos /> <apply> <times /> <pi /> <ci> μ </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <ci> μ </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 105 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 105 </cn> <ci> μ </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 45 </cn> <apply> <plus /> <apply> <power /> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 5 </cn> <ci> μ </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -11 </cn> </apply> <ci> z </ci> </apply> <apply> <power /> <ci> μ </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 10 </cn> <apply> <power /> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 9 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreQ", "[", RowBox[List["4", ",", "\[Mu]_", ",", "2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Pi]", " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Mu]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", FractionBox["\[Mu]", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "\[Mu]"]]], " ", RowBox[List["(", RowBox[List["9", "+", RowBox[List["105", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["105", " ", SuperscriptBox["z", "3"], " ", "\[Mu]"]], "-", RowBox[List["10", " ", SuperscriptBox["\[Mu]", "2"]]], "+", SuperscriptBox["\[Mu]", "4"], "+", RowBox[List["45", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", SuperscriptBox["\[Mu]", "2"]]], ")"]]]], "+", RowBox[List["5", " ", "z", " ", "\[Mu]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "11"]], "+", RowBox[List["2", " ", SuperscriptBox["\[Mu]", "2"]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], "\[Mu]"], " ", RowBox[List["(", RowBox[List["9", "+", RowBox[List["105", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["105", " ", SuperscriptBox["z", "3"], " ", "\[Mu]"]], "-", RowBox[List["10", " ", SuperscriptBox["\[Mu]", "2"]]], "+", SuperscriptBox["\[Mu]", "4"], "+", RowBox[List["45", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", SuperscriptBox["\[Mu]", "2"]]], ")"]]]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["55", " ", "\[Mu]"]], "-", RowBox[List["10", " ", SuperscriptBox["\[Mu]", "3"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Mu]"]], "]"]]]]]], ")"]]]], RowBox[List["2", " ", RowBox[List["Gamma", "[", RowBox[List["5", "-", "\[Mu]"]], "]"]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
