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http://functions.wolfram.com/07.07.17.0005.01
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z p[\[Nu], z] == p[\[Nu] + 1, z] + (\[Nu]^2/(4 \[Nu]^2 - 1))
p[\[Nu] - 1, z] /; p[\[Nu], z] ==
((Sqrt[Pi] \[Nu]!)/(2^\[Nu] Gamma[1/2 + \[Nu]])) LegendreP[\[Nu], z]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["z", " ", RowBox[List["p", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List["p", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", "z"]], "]"]], "+", RowBox[List[FractionBox[SuperscriptBox["\[Nu]", "2"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", SuperscriptBox["\[Nu]", "2"]]], "-", "1"]], ")"]], " "]]], RowBox[List["p", "[", RowBox[List[RowBox[List["\[Nu]", "-", "1"]], ",", "z"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["p", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["\[Nu]", "!"]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]], RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mi> p </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <msup> <mi> ν </mi> <mn> 2 </mn> </msup> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> p </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> ν </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> p </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> p </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> </msup> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mi> ν </mi> <mo> ! </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox["P", LegendreP] </annotation> </semantics> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <ci> z </ci> <apply> <ci> p </ci> <ci> ν </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> p </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> -1 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <ci> p </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> p </ci> <ci> ν </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <factorial /> <ci> ν </ci> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> ν </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> LegendreP </ci> <ci> ν </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["z_", " ", RowBox[List["p", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["p", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", "z"]], "]"]], "+", FractionBox[RowBox[List[SuperscriptBox["\[Nu]", "2"], " ", RowBox[List["p", "[", RowBox[List[RowBox[List["\[Nu]", "-", "1"]], ",", "z"]], "]"]]]], RowBox[List[RowBox[List["4", " ", SuperscriptBox["\[Nu]", "2"]]], "-", "1"]]]]], "/;", RowBox[List[RowBox[List["p", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["\[Nu]", "!"]]]], ")"]], " ", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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| LegendreP[n,z] | | LegendreP[n,mu,2,z] | | LegendreP[nu,mu,z] | | LegendreP[nu,mu,2,z] | | LegendreP[nu,mu,3,z] | | |
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){kind=small}
