|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/07.07.06.0022.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
LegendreP[\[Nu], z] \[Proportional]
(z^(-1 - \[Nu])/(2^\[Nu] (Sqrt[Pi] Gamma[-\[Nu]] Gamma[3/2 + \[Nu]])))
(Csc[Pi \[Nu]] (EulerGamma + PolyGamma[-\[Nu]] + PolyGamma[1 + \[Nu]] -
PolyGamma[2 + 2 \[Nu]]) + (Log[2] - Log[z - 1]) Sin[Pi \[Nu]])
(1 + O[1/z]) + ((2 \[Nu])!/(2^\[Nu] Gamma[1 + \[Nu]]^2)) z^\[Nu]
(1 + O[1/z]) /; (Abs[z] -> Infinity) && Element[\[Nu] - 1/2, Integers] &&
\[Nu] - 1/2 >= 0
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List["EulerGamma", "+", RowBox[List["PolyGamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["2", "+", RowBox[List["2", " ", "\[Nu]"]]]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "2", "]"]], "-", RowBox[List["Log", "[", RowBox[List["z", "-", "1"]], "]"]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]]]], ")"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["2", " ", "\[Nu]"]], ")"]], "!"]]]], SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], "2"]], SuperscriptBox["z", "\[Nu]"], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List[RowBox[List["\[Nu]", "-", FractionBox["1", "2"]]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["\[Nu]", "-", FractionBox["1", "2"]]], "\[GreaterEqual]", "0"]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <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> <mo> ∝ </mo> <mrow> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> ν </mi> </msup> </mrow> <msup> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mrow> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> csc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[EulerGamma]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> z </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> ν </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> LegendreP </ci> <ci> ν </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <factorial /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> ν </ci> </apply> <apply> <power /> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </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 /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> ν </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <csc /> <apply> <times /> <pi /> <ci> ν </ci> </apply> </apply> <apply> <plus /> <apply> <ci> PolyGamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <eulergamma /> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <sin /> <apply> <times /> <pi /> <ci> ν </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <apply> <abs /> <ci> z </ci> </apply> <infinity /> </apply> <apply> <in /> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <ci> ℕ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List["EulerGamma", "+", RowBox[List["PolyGamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["2", "+", RowBox[List["2", " ", "\[Nu]"]]]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "2", "]"]], "-", RowBox[List["Log", "[", RowBox[List["z", "-", "1"]], "]"]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], "]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["2", " ", "\[Nu]"]], ")"]], "!"]]]], ")"]], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], "2"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List[RowBox[List["\[Nu]", "-", FractionBox["1", "2"]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["\[Nu]", "-", FractionBox["1", "2"]]], "\[GreaterEqual]", "0"]]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| LegendreP[n,z] | | LegendreP[n,mu,2,z] | | LegendreP[nu,mu,z] | | LegendreP[nu,mu,2,z] | | LegendreP[nu,mu,3,z] | | |
|
|
|
){kind=small}
