Assosiated Legendre function of the second kind of type 3: Representations through equivalent functions (formula 07.12.27.0010)

LegendreQ

$Mathematica Notation$

Hypergeometric Functions

LegendreQ[nu,mu,3,z]

Representations through equivalent functions

With related functions

http://functions.wolfram.com/07.12.27.0010.01

Input Form

LegendreQ[\[Nu], \[Mu], 3, z] == (Pi/2) Csc[Pi \[Mu]] E^(Pi I \[Mu]) ((1 - z)^(\[Mu]/2)/(z - 1)^(\[Mu]/2)) (LegendreP[\[Nu], \[Mu], 2, z] - Pochhammer[\[Nu] - \[Mu] + 1, 2 \[Mu]] ((z - 1)^\[Mu]/(1 - z)^\[Mu]) LegendreP[\[Nu], -\[Mu], 2, z]) /; !Element[\[Mu], Integers]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["\[Pi]", "2"], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Mu]"]], "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Mu]"]]], " ", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]]], RowBox[List["(", RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "2", ",", "z"]], "]"]], "-", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "-", "\[Mu]", "+", "1"]], ",", RowBox[List["2", " ", "\[Mu]"]]]], "]"]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "\[Mu]"], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "\[Mu]"]], RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", RowBox[List["-", "\[Mu]"]], ",", "2", ",", "z"]], "]"]]]]]], ")"]]]]]], "/;", " ", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Mu]", ",", "Integers"]], "]"]], "]"]]]]]]

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["\[GothicCapitalQ]", LegendreQ] </annotation> </semantics> <mi> ν </mi> <mi> μ </mi> </msubsup> <mo> ( </mo> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox["z", HoldComplete[LegendreQ, 3]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mi> csc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> μ </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> </msup> <mo> ⁢ </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> μ </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> μ </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox["P", LegendreP] </annotation> </semantics> <mi> ν </mi> <mi> μ </mi> </msubsup> <mo> ( </mo> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox["z", HoldComplete[LegendreP, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <mi> μ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> μ </mi> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", "\[Mu]", "+", "1"]], ")"]], RowBox[List["2", " ", "\[Mu]"]]], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> μ </mi> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> μ </mi> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox["P", LegendreP] </annotation> </semantics> <mi> ν </mi> <mrow> <mo> - </mo> <mi> μ </mi> </mrow> </msubsup> <mo> ( </mo> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox["z", HoldComplete[LegendreP, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> μ </mi> <mo> ∉ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </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> <apply> <ci> HoldComplete </ci> <ci> LegendreQ </ci> <cn type='integer'> 3 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <csc /> <apply> <times /> <ci> μ </ci> <pi /> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> μ </ci> <pi /> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <ci> μ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <ci> μ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> LegendreP </ci> <ci> ν </ci> <ci> μ </ci> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <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> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <ci> μ </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> LegendreP </ci> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <notin /> <ci> μ </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Mu]"]], "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Mu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "2", ",", "z"]], "]"]], "-", FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "-", "\[Mu]", "+", "1"]], ",", RowBox[List["2", " ", "\[Mu]"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "\[Mu]"], " ", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", RowBox[List["-", "\[Mu]"]], ",", "2", ",", "z"]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "\[Mu]"]]]], ")"]]]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]]]]], "/;", RowBox[List["!", RowBox[List["\[Mu]", "\[Element]", "Integers"]]]]]]]]]]

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

2002-12-18

LegendreQ[nu,z]
LegendreQ[nu,mu,z]
LegendreQ[nu,mu,2,z]