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variants of this functions
LegendreP




Mathematica Notation

Traditional Notation









Polynomials > LegendreP[n,mu,2,z] > Representations through equivalent functions > With related functions > Involving spherical harmonic functions





http://functions.wolfram.com/05.07.27.0012.01









  


  








Input Form





LegendreP[n, \[Mu], 2, z] == ((2 Sqrt[Pi] Sqrt[Gamma[1 + \[Mu] + n]])/ (Sqrt[1 + 2 n] Sqrt[Gamma[1 - \[Mu] + n]])) SphericalHarmonicY[n, \[Mu], ArcCos[z], 0]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["n", ",", "\[Mu]", ",", "2", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["2", " ", SqrtBox["\[Pi]"], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]", "+", "n"]], "]"]]]]], RowBox[List[SqrtBox[RowBox[List["1", "+", RowBox[List["2", " ", "n"]]]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "n"]], "]"]]]]]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", "\[Mu]", ",", RowBox[List["ArcCos", "[", "z", "]"]], ",", "0"]], "]"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> n </mi> <mi> &#956; </mi> </msubsup> <mo> ( </mo> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox[&quot;z&quot;, HoldComplete[LegendreP, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msqrt> </mrow> <mrow> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#956; </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mi> Y </mi> <mi> n </mi> <mi> &#956; </mi> </msubsup> <mo> ( </mo> <mrow> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> LegendreP </ci> <ci> n </ci> <ci> &#956; </ci> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#956; </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <ci> n </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> SphericalHarmonicY </ci> <ci> n </ci> <ci> &#956; </ci> <apply> <arccos /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreP", "[", RowBox[List["n_", ",", "\[Mu]_", ",", "2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", SqrtBox["\[Pi]"], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]", "+", "n"]], "]"]]]]], ")"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", "\[Mu]", ",", RowBox[List["ArcCos", "[", "z", "]"]], ",", "0"]], "]"]]]], RowBox[List[SqrtBox[RowBox[List["1", "+", RowBox[List["2", " ", "n"]]]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "n"]], "]"]]]]]]]]]]


Date Added to functions.wolfram.com (modification date)





2001-10-29