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http://functions.wolfram.com/07.12.03.0018.01
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LegendreQ[1, \[Mu], 3, z] ==
((-E^(I Pi \[Mu])) ((Pi Csc[Pi \[Mu]])/(2 Gamma[2 - \[Mu]]))
((z + 1)^\[Mu] (-z + \[Mu]) + (z - 1)^\[Mu] (z + \[Mu])))/
((z - 1)^(\[Mu]/2) (z + 1)^(\[Mu]/2))
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Cell[BoxData[RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["1", ",", "\[Mu]", ",", "3", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Mu]"]]]]], " ", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Mu]"]], "]"]]]], RowBox[List["2", " ", RowBox[List["Gamma", "[", RowBox[List["2", "-", "\[Mu]"]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "\[Mu]"]], "/", "2"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List[RowBox[List["-", "\[Mu]"]], "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "\[Mu]"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "z"]], "+", "\[Mu]"]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "\[Mu]"], " ", RowBox[List["(", RowBox[List["z", "+", "\[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> <mstyle scriptlevel='0'> <msubsup> <semantics> <mi> 𝔔 </mi> <annotation encoding='Mathematica'> TagBox["\[GothicCapitalQ]", LegendreQ] </annotation> </semantics> <mn> 1 </mn> <mi> μ </mi> </msubsup> </mstyle> <mo> ( </mo> <mstyle scriptlevel='0'> <mi> z </mi> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> </msup> </mrow> <mo> ⁢ </mo> <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> 2 </mn> <mo> - </mo> <mi> μ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> μ </mi> </mrow> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> μ </mi> </mrow> <mo> / </mo> <mn> 2 </mn> </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> <mo> - </mo> <mi> z </mi> </mrow> <mo> + </mo> <mi> μ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <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> <mi> z </mi> <mo> + </mo> <mi> μ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreQ </ci> <ci> 𝔔 </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> μ </ci> </apply> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> μ </ci> </apply> </apply> </apply> <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'> 2 </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 /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </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'> -1 </cn> <ci> z </ci> </apply> <ci> μ </ci> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <ci> μ </ci> </apply> <apply> <plus /> <ci> z </ci> <ci> μ </ci> </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["1", ",", "\[Mu]_", ",", "3", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Mu]"]]], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Mu]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["-", FractionBox["\[Mu]", "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["-", FractionBox["\[Mu]", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "\[Mu]"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "z"]], "+", "\[Mu]"]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "\[Mu]"], " ", RowBox[List["(", RowBox[List["z", "+", "\[Mu]"]], ")"]]]]]], ")"]]]], RowBox[List["2", " ", RowBox[List["Gamma", "[", RowBox[List["2", "-", "\[Mu]"]], "]"]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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