|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/05.07.13.0010.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Wronskian[s^z LegendreP[n, \[Mu], 2, a r^z],
s^z LegendreQ[n, \[Mu], 2, a r^z], z] ==
(a r^z s^(2 z) Gamma[1 + \[Mu] + n] Log[r])/
((1 - a^2 r^(2 z)) Gamma[1 - \[Mu] + n])
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List["Wronskian", "[", RowBox[List[RowBox[List[SuperscriptBox["s", "z"], " ", RowBox[List["LegendreP", "[", RowBox[List["n", ",", "\[Mu]", ",", "2", ",", RowBox[List["a", " ", SuperscriptBox["r", "z"]]]]], "]"]]]], ",", RowBox[List[SuperscriptBox["s", "z"], RowBox[List["LegendreQ", "[", RowBox[List["n", ",", "\[Mu]", ",", "2", ",", RowBox[List["a", " ", SuperscriptBox["r", "z"]]]]], "]"]]]], ",", "z"]], "]"]], "\[Equal]", FractionBox[RowBox[List["a", " ", SuperscriptBox["r", "z"], " ", SuperscriptBox["s", RowBox[List["2", " ", "z"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]", "+", "n"]], "]"]], " ", RowBox[List["Log", "[", "r", "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["r", RowBox[List["2", " ", "z"]]]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "n"]], "]"]]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <mi> W </mi> <mi> z </mi> </msub> <mo> ( </mo> <mrow> <mrow> <msup> <mi> s </mi> <mi> z </mi> </msup> <mo> ⁢ </mo> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox["P", LegendreP] </annotation> </semantics> <mi> n </mi> <mi> μ </mi> </msubsup> <mo> ( </mo> <semantics> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> r </mi> <mi> z </mi> </msup> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["a", " ", SuperscriptBox["r", "z"]]], HoldComplete[LegendreP, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <msup> <mi> s </mi> <mi> z </mi> </msup> <mo> ⁢ </mo> <mrow> <msubsup> <semantics> <mi> Q </mi> <annotation encoding='Mathematica'> TagBox["Q", LegendreQ] </annotation> </semantics> <mi> n </mi> <mi> μ </mi> </msubsup> <mo> ( </mo> <semantics> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> r </mi> <mi> z </mi> </msup> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["a", " ", SuperscriptBox["r", "z"]]], HoldComplete[LegendreQ, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mfrac> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> r </mi> <mi> z </mi> </msup> <mo> ⁢ </mo> <msup> <mi> s </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> r </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> r </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> μ </mi> </mrow> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> W </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <ci> s </ci> <ci> z </ci> </apply> <apply> <ci> LegendreP </ci> <ci> n </ci> <ci> μ </ci> <cn type='integer'> 2 </cn> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> r </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> s </ci> <ci> z </ci> </apply> <apply> <ci> LegendreQ </ci> <ci> n </ci> <ci> μ </ci> <cn type='integer'> 2 </cn> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> r </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> r </ci> <ci> z </ci> </apply> <apply> <power /> <ci> s </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> μ </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ln /> <ci> r </ci> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> r </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Wronskian", "[", RowBox[List[RowBox[List[SuperscriptBox["s_", "z_"], " ", RowBox[List["LegendreP", "[", RowBox[List["n_", ",", "\[Mu]_", ",", "2", ",", RowBox[List["a_", " ", SuperscriptBox["r_", "z_"]]]]], "]"]]]], ",", RowBox[List[SuperscriptBox["s_", "z_"], " ", RowBox[List["LegendreQ", "[", RowBox[List["n_", ",", "\[Mu]_", ",", "2", ",", RowBox[List["a_", " ", SuperscriptBox["r_", "z_"]]]]], "]"]]]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["a", " ", SuperscriptBox["r", "z"], " ", SuperscriptBox["s", RowBox[List["2", " ", "z"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]", "+", "n"]], "]"]], " ", RowBox[List["Log", "[", "r", "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["r", RowBox[List["2", " ", "z"]]]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "n"]], "]"]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
LegendreP[n,z] | LegendreP[nu,z] | LegendreP[nu,mu,z] | LegendreP[nu,mu,2,z] | LegendreP[nu,mu,3,z] | |
|
|
|