![](/common/images/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
http://functions.wolfram.com/07.09.06.0012.01
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
LegendreP[\[Nu], m, 3, z] \[Proportional]
(-((Gamma[m - \[Nu]] Gamma[1 + m + \[Nu]] Sin[Pi \[Nu]])/(2^m (Pi m!))))
(-1)^m (z + 1)^(m/2) (z - 1)^(m/2) (1 + O[z - 1]) /;
Element[m, Integers] && m >= 0
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "m", ",", "3", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "m"]]], " ", RowBox[List["Gamma", "[", RowBox[List["m", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "m", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], RowBox[List["\[Pi]", " ", RowBox[List["m", "!"]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["m", "/", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["m", "/", "2"]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", RowBox[List["z", "-", "1"]], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <mi> 𝔓 </mi> <mi> ν </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox["z", HoldComplete[LegendreP, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> ∝ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <mfrac> <mi> m </mi> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </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> <mtext> </mtext> </mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> m </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> m </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> m </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> m </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> 𝔓 </ci> <ci> ν </ci> </apply> <ci> m </ci> </apply> <apply> <apply> <ci> HoldComplete </ci> <ci> LegendreP </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> m </ci> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <sin /> <apply> <times /> <pi /> <ci> ν </ci> </apply> </apply> <apply> <power /> <apply> <times /> <pi /> <apply> <factorial /> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <ci> m </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <ci> m </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> m </ci> <ci> ℕ </ci> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/clear.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| | ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", "m_", ",", "3", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "m"]]], " ", RowBox[List["Gamma", "[", RowBox[List["m", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "m", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["m", "/", "2"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["m", "/", "2"]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", RowBox[List["z", "-", "1"]], "]"]]]], ")"]]]], RowBox[List["\[Pi]", " ", RowBox[List["m", "!"]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]]] |
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
Date Added to functions.wolfram.com (modification date)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/common/images/spacer.gif) |
LegendreP[n,z] | LegendreP[nu,z] | LegendreP[nu,mu,z] | LegendreP[n,mu,2,z] | LegendreP[nu,mu,2,z] | |
|
|
|