Assosiated Legendre function of the second kind of type 2: Representations through equivalent functions (formula 07.11.27.0006)

LegendreQ

$Mathematica Notation$

Hypergeometric Functions

LegendreQ[nu,mu,2,z]

Representations through equivalent functions

With related functions

http://functions.wolfram.com/07.11.27.0006.01

Input Form

LegendreQ[\[Nu], \[Mu], 2, x] == ((1/2) (E^((Pi I \[Mu])/2) Limit[LegendreQ[\[Nu], \[Mu], 3, z], z -> x, Direction -> I] + Limit[LegendreQ[\[Nu], \[Mu], 3, z], z -> x, Direction -> -I]/E^((Pi I \[Mu])/2)))/E^(Pi I \[Mu]) /; -1 < x < 1

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "2", ",", "x"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", "\[Mu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Mu]"]], "2"]], " ", RowBox[List["Limit", "[", RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", "z"]], "]"]], ",", RowBox[List["z", "\[Rule]", "x"]], ",", RowBox[List["Direction", "\[Rule]", "I"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Mu]"]], "2"]]]], " ", RowBox[List["Limit", "[", RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", "z"]], "]"]], ",", RowBox[List["z", "\[Rule]", "x"]], ",", RowBox[List["Direction", "\[Rule]", RowBox[List["-", "I"]]]]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["-", "1"]], "<", "x", "<", "1"]]]]]]

MathML Form

<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> x </mi> <annotation encoding='Mathematica'> TagBox["x", HoldComplete[LegendreQ, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> π </mi> </mrow> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> z </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <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'> <mi> z </mi> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mi> ⅇ </mi> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> ⁢ </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> z </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mrow> <mi> x </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <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'> <mi> z </mi> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> < </mo> <mi> x </mi> <mo> < </mo> <mn> 1 </mn> </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> x </ci> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> <imaginaryi /> <ci> μ </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <imaginaryi /> <ci> μ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <limit /> <bvar> <ci> z </ci> </bvar> <condition> <apply> <tendsto /> <ci> z </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </condition> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreQ </ci> <ci> 𝔔 </ci> </apply> <ci> ν </ci> </apply> <ci> μ </ci> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <pi /> <imaginaryi /> <ci> μ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <limit /> <bvar> <ci> z </ci> </bvar> <condition> <apply> <tendsto /> <ci> z </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </condition> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreQ </ci> <ci> 𝔔 </ci> </apply> <ci> ν </ci> </apply> <ci> μ </ci> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <cn type='integer'> -1 </cn> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "2", ",", "x_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", "\[Mu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Mu]"]], "2"]], " ", RowBox[List["Limit", "[", RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", "z"]], "]"]], ",", RowBox[List["z", "\[Rule]", "x"]], ",", RowBox[List["Direction", "\[Rule]", "\[ImaginaryI]"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Mu]"]], ")"]]]]], " ", RowBox[List["Limit", "[", RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", "z"]], "]"]], ",", RowBox[List["z", "\[Rule]", "x"]], ",", RowBox[List["Direction", "\[Rule]", RowBox[List["-", "\[ImaginaryI]"]]]]]], "]"]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["-", "1"]], "<", "x", "<", "1"]]]]]]]]

Date Added to functions.wolfram.com (modification date)

2001-10-29

LegendreQ[nu,z]
LegendreQ[nu,mu,z]
LegendreQ[nu,mu,3,z]