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http://functions.wolfram.com/07.12.27.0007.01
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LegendreQ[\[Nu], \[Mu], 3, Cosh[z]] ==
Sqrt[Pi/2] (Gamma[\[Mu] + \[Nu] + 1]/Sqrt[Sinh[z]]) E^(Pi I \[Mu])
LegendreP[-\[Mu] - 1/2, -\[Nu] - 1/2, 3, Coth[z]] /; Re[z] > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", RowBox[List["Cosh", "[", "z", "]"]]]], "]"]], "\[Equal]", RowBox[List[SqrtBox[FractionBox["\[Pi]", "2"]], FractionBox[RowBox[List[" ", RowBox[List["Gamma", "[", RowBox[List["\[Mu]", "+", "\[Nu]", "+", "1"]], "]"]]]], RowBox[List[" ", SqrtBox[RowBox[List["Sinh", "[", "z", "]"]]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Mu]"]]], RowBox[List["LegendreP", "[", RowBox[List[RowBox[List[RowBox[List["-", "\[Mu]"]], "-", FractionBox["1", "2"]]], ",", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", FractionBox["1", "2"]]], ",", "3", ",", RowBox[List["Coth", "[", "z", "]"]]]], "]"]]]]]], "/;", " ", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]
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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> <mrow> <mstyle scriptlevel='0'> <msubsup> <semantics> <mi> 𝔔 </mi> <annotation encoding='Mathematica'> TagBox["\[GothicCapitalQ]", LegendreQ] </annotation> </semantics> <mi> ν </mi> <mi> μ </mi> </msubsup> </mstyle> <mo> ( </mo> <mstyle scriptlevel='0'> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> <mo> ⩵ </mo> <mrow> <msqrt> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </msqrt> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> + </mo> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> </msup> <mo> ⁢ </mo> <mfrac> <mn> 1 </mn> <mrow> <msup> <mi> sinh </mi> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mtext> </mtext> <mrow> <mstyle scriptlevel='0'> <msubsup> <semantics> <mi> 𝔓 </mi> <annotation encoding='Mathematica'> TagBox["\[GothicCapitalP]", LegendreQ] </annotation> </semantics> <mrow> <mrow> <mo> - </mo> <mi> μ </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msubsup> </mstyle> <mo> ( </mo> <mstyle scriptlevel='0'> <mrow> <mi> coth </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreQ </ci> <ci> 𝔔 </ci> </apply> <ci> ν </ci> </apply> <ci> μ </ci> </apply> <apply> <times /> <ci> cosh </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> μ </ci> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <pi /> <imaginaryi /> <ci> μ </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <sinh /> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreQ </ci> <ci> 𝔓 </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <ci> coth </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <gt /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", RowBox[List["Cosh", "[", "z_", "]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SqrtBox[FractionBox["\[Pi]", "2"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Mu]", "+", "\[Nu]", "+", "1"]], "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Mu]"]]], " ", RowBox[List["LegendreP", "[", RowBox[List[RowBox[List[RowBox[List["-", "\[Mu]"]], "-", FractionBox["1", "2"]]], ",", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", FractionBox["1", "2"]]], ",", "3", ",", RowBox[List["Coth", "[", "z", "]"]]]], "]"]]]], SqrtBox[RowBox[List["Sinh", "[", "z", "]"]]]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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