|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/07.10.27.0001.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
LegendreQ[\[Nu] - 1/2, z]/LegendreP[\[Nu] - 1/2, z] -
LegendreQ[\[Nu] + 1/2, z]/LegendreP[\[Nu] + 1/2, z] ==
1/((\[Nu] + 1/2) LegendreP[\[Nu] - 1/2, z] LegendreP[\[Nu] + 1/2, z])
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List[FractionBox[RowBox[List[" ", RowBox[List["LegendreQ", "[", RowBox[List[RowBox[List["\[Nu]", "-", FractionBox["1", "2"]]], ",", "z"]], "]"]]]], RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["\[Nu]", "-", FractionBox["1", "2"]]], ",", "z"]], "]"]]], "-", FractionBox[RowBox[List["LegendreQ", "[", RowBox[List[RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ",", "z"]], "]"]], RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ",", "z"]], "]"]]]]], "\[Equal]", FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ")"]], " ", RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["\[Nu]", "-", FractionBox["1", "2"]]], ",", "z"]], "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ",", "z"]], "]"]]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mrow> <msub> <semantics> <mi> Q </mi> <annotation encoding='Mathematica'> TagBox["Q", LegendreQ] </annotation> </semantics> <mrow> <mi> ν </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mrow> <msub> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox["P", LegendreP] </annotation> </semantics> <mrow> <mi> ν </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <msub> <semantics> <mi> Q </mi> <annotation encoding='Mathematica'> TagBox["Q", LegendreQ] </annotation> </semantics> <mrow> <mi> ν </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mrow> <msub> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox["P", LegendreP] </annotation> </semantics> <mrow> <mi> ν </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> ⩵ </mo> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox["P", LegendreP] </annotation> </semantics> <mrow> <mi> ν </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox["P", LegendreP] </annotation> </semantics> <mrow> <mi> ν </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <times /> <apply> <ci> LegendreQ </ci> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <ci> z </ci> </apply> <apply> <power /> <apply> <ci> LegendreP </ci> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> LegendreQ </ci> <apply> <plus /> <ci> ν </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> <apply> <power /> <apply> <ci> LegendreP </ci> <apply> <plus /> <ci> ν </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> ν </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> LegendreP </ci> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <ci> z </ci> </apply> <apply> <ci> LegendreP </ci> <apply> <plus /> <ci> ν </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[FractionBox[RowBox[List["LegendreQ", "[", RowBox[List[RowBox[List["\[Nu]_", "-", FractionBox["1", "2"]]], ",", "z_"]], "]"]], RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["\[Nu]_", "-", FractionBox["1", "2"]]], ",", "z_"]], "]"]]], "-", FractionBox[RowBox[List["LegendreQ", "[", RowBox[List[RowBox[List["\[Nu]_", "+", FractionBox["1", "2"]]], ",", "z_"]], "]"]], RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["\[Nu]_", "+", FractionBox["1", "2"]]], ",", "z_"]], "]"]]]]], "]"]], "\[RuleDelayed]", FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ")"]], " ", RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["\[Nu]", "-", FractionBox["1", "2"]]], ",", "z"]], "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ",", "z"]], "]"]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|