|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/07.12.17.0006.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
LegendreQ[\[Nu], \[Mu], 3, z] == (1/((2 \[Nu] + 1) z))
((\[Nu] - \[Mu] + 1) LegendreQ[\[Nu] + 1, \[Mu], 3, z] +
(\[Nu] + \[Mu]) LegendreQ[\[Nu] - 1, \[Mu], 3, z])
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]], " ", "z"]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "-", "\[Mu]", "+", "1"]], ")"]], " ", RowBox[List["LegendreQ", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", "\[Mu]", ",", "3", ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "+", "\[Mu]"]], ")"]], " ", RowBox[List["LegendreQ", "[", RowBox[List[RowBox[List["\[Nu]", "-", "1"]], ",", "\[Mu]", ",", "3", ",", "z"]], "]"]]]]]], ")"]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mtext> </mtext> <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'> <mi> z </mi> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> <mo> ⩵ </mo> <mfrac> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mtext> </mtext> <mrow> <mstyle scriptlevel='0'> <msubsup> <semantics> <mi> 𝔔 </mi> <annotation encoding='Mathematica'> TagBox["\[GothicCapitalQ]", LegendreQ] </annotation> </semantics> <mrow> <mi> ν </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> μ </mi> </msubsup> </mstyle> <mo> ( </mo> <mstyle scriptlevel='0'> <mi> z </mi> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <mi> μ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mtext> </mtext> <mrow> <mstyle scriptlevel='0'> <msubsup> <semantics> <mi> 𝔔 </mi> <annotation encoding='Mathematica'> TagBox["\[GothicCapitalQ]", LegendreQ] </annotation> </semantics> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> μ </mi> </msubsup> </mstyle> <mo> ( </mo> <mstyle scriptlevel='0'> <mi> z </mi> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <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> <ci> z </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> μ </ci> <ci> ν </ci> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreQ </ci> <ci> 𝔔 </ci> </apply> <apply> <plus /> <ci> ν </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> μ </ci> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreQ </ci> <ci> 𝔔 </ci> </apply> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> <ci> μ </ci> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "-", "\[Mu]", "+", "1"]], ")"]], " ", RowBox[List["LegendreQ", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", "\[Mu]", ",", "3", ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "+", "\[Mu]"]], ")"]], " ", RowBox[List["LegendreQ", "[", RowBox[List[RowBox[List["\[Nu]", "-", "1"]], ",", "\[Mu]", ",", "3", ",", "z"]], "]"]]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]], " ", "z"]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|