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http://functions.wolfram.com/07.11.07.0001.01
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LegendreQ[\[Nu], \[Mu], 2, z] ==
(Gamma[\[Nu] + 1]/(2 Gamma[\[Nu] - \[Mu] + 1]))
Integrate[((z + Cosh[t] Sqrt[z^2 - 1])^(-\[Nu] - 1)/E^((\[Mu] Pi I)/2) +
E^((\[Mu] Pi I)/2) (z - Cosh[t] Sqrt[z^2 - 1])^(-\[Nu] - 1))
Cosh[\[Mu] t], {t, 0, Infinity}] /;
-1 < z < 1 && Re[\[Nu] + \[Mu]] > -1 && Re[\[Nu] - \[Mu]] > -1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "2", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], RowBox[List[RowBox[List["2", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "-", "\[Mu]", "+", "1"]], "]"]]]], " "]]], RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[Mu]", " ", "\[Pi]", " ", "\[ImaginaryI]"]], "2"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", RowBox[List[RowBox[List["Cosh", "[", "t", "]"]], SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]]]]]], ")"]], RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "1"]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[Mu]", " ", "\[Pi]", " ", "\[ImaginaryI]"]], "2"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", " ", RowBox[List[RowBox[List["Cosh", "[", "t", "]"]], SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]]]]]], ")"]], RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "1"]]]]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["\[Mu]", " ", "t"]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "<", "z", "<", "1"]], " ", "\[And]", RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Nu]", "+", "\[Mu]"]], "]"]], ">", RowBox[List["-", "1"]]]], "\[And]", RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Nu]", "-", "\[Mu]"]], "]"]], ">", RowBox[List["-", "1"]]]]]]]]]]
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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> <msubsup> <semantics> <mi> Q </mi> <annotation encoding='Mathematica'> TagBox["Q", LegendreQ] </annotation> </semantics> <mi> ν </mi> <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> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <mi> μ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mi> ∞ </mi> </msubsup> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> μ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mrow> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mo> + </mo> <msup> <mrow> <msup> <mi> ⅇ </mi> <mfrac> <mrow> <mi> μ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mrow> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mtext> </mtext> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mtext> </mtext> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> ⁢ </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> < </mo> <mi> z </mi> <mo> < </mo> <mn> 1 </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> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <mi> μ </mi> </mrow> <mo> ) </mo> </mrow> <mo> > </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> LegendreQ </ci> <ci> ν </ci> <ci> μ </ci> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> μ </ci> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <cosh /> <ci> t </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> μ </ci> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <cosh /> <ci> t </ci> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <cosh /> <apply> <times /> <ci> μ </ci> <ci> t </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <lt /> <cn type='integer'> -1 </cn> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <gt /> <apply> <real /> <apply> <plus /> <ci> μ </ci> <ci> ν </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <gt /> <apply> <real /> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], " ", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[Mu]", " ", "\[Pi]", " ", "\[ImaginaryI]"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", RowBox[List[RowBox[List["Cosh", "[", "t", "]"]], " ", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]]]]]], ")"]], RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "1"]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[Mu]", " ", "\[Pi]", " ", "\[ImaginaryI]"]], "2"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", RowBox[List[RowBox[List["Cosh", "[", "t", "]"]], " ", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]]]]]], ")"]], RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "1"]]]]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["\[Mu]", " ", "t"]], "]"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], RowBox[List["2", " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "-", "\[Mu]", "+", "1"]], "]"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "<", "z", "<", "1"]], "&&", RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Nu]", "+", "\[Mu]"]], "]"]], ">", RowBox[List["-", "1"]]]], "&&", RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Nu]", "-", "\[Mu]"]], "]"]], ">", RowBox[List["-", "1"]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
