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http://functions.wolfram.com/07.12.27.0009.01
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LegendreQ[\[Nu], \[Mu], 3, z] == (E^(Pi I \[Mu])/Gamma[-\[Mu]])
(z^2 - 1)^(\[Mu]/2) Integrate[LegendreQ[\[Nu], 0, 3, t]
(t - z)^(-\[Mu] - 1), {t, z, Infinity}] /;
Re[\[Mu]] < 0 && Re[\[Mu] + \[Nu]] > -1 &&
!IntervalMemberQ[Interval[{-Infinity, 1}], z]
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Cell[BoxData[RowBox[List[" ", RowBox[List[RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Mu]"]]], RowBox[List["Gamma", "[", RowBox[List["-", "\[Mu]"]], "]"]]], SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["z", "2"], "-", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List[SubsuperscriptBox["\[Integral]", "z", "\[Infinity]"], " ", RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "0", ",", "3", ",", "t"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["t", "-", "z"]], ")"]], RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "1"]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Mu]", "]"]], "<", "0"]], "\[And]", RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Mu]", "+", "\[Nu]"]], "]"]], ">", RowBox[List["-", "1"]]]], "\[And]", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", "1"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]]]]]]]]]]
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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> <mi> z </mi> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> μ </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> μ </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msubsup> <mo> ∫ </mo> <mi> z </mi> <mi> ∞ </mi> </msubsup> <mrow> <mrow> <mstyle scriptlevel='0'> <msubsup> <semantics> <mi> 𝔔 </mi> <annotation encoding='Mathematica'> TagBox["\[GothicCapitalQ]", LegendreQ] </annotation> </semantics> <mi> ν </mi> <mn> 0 </mn> </msubsup> </mstyle> <mo> ( </mo> <mi> z </mi> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> t </mi> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> μ </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> μ </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> > </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> z </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <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> <mi> z </mi> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> μ </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> μ </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msubsup> <mo> ∫ </mo> <mi> z </mi> <mi> ∞ </mi> </msubsup> <mrow> <mrow> <mstyle scriptlevel='0'> <msubsup> <semantics> <mi> 𝔔 </mi> <annotation encoding='Mathematica'> TagBox["\[GothicCapitalQ]", LegendreQ] </annotation> </semantics> <mi> ν </mi> <mn> 0 </mn> </msubsup> </mstyle> <mo> ( </mo> <mi> z </mi> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> t </mi> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> μ </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> μ </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> > </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> z </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Mu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["z", "2"], "-", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List[SubsuperscriptBox["\[Integral]", "z", "\[Infinity]"], RowBox[List[RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "0", ",", "3", ",", "t"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["t", "-", "z"]], ")"]], RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "1"]]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], RowBox[List["Gamma", "[", RowBox[List["-", "\[Mu]"]], "]"]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Mu]", "]"]], "<", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Mu]", "+", "\[Nu]"]], "]"]], ">", RowBox[List["-", "1"]]]], "&&", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", "1"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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