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variants of this functions
LegendreP




Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreP[nu,mu,2,z] > Representations through equivalent functions > With related functions > Involving Legendre functions





http://functions.wolfram.com/07.08.27.0013.01









  


  








Input Form





LegendreP[\[Nu], \[Mu], 2, z] == (-Csc[Pi \[Mu]]) (Sin[Pi \[Nu]] ((-z - 1)^(\[Mu]/2)/(z + 1)^(\[Mu]/2)) LegendreP[\[Nu], \[Mu], 3, -z] + (Pi/(Gamma[-\[Mu] - \[Nu]] Gamma[1 - \[Mu] + \[Nu]])) ((z + 1)^(\[Mu]/2)/(-z - 1)^(\[Mu]/2)) LegendreP[\[Nu], -\[Mu], 3, -z])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "2", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Mu]"]], "]"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "z"]], "-", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]]], RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", RowBox[List["-", "z"]]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["\[Pi]", " "]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], "]"]]]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "z"]], "-", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]]], RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", RowBox[List["-", "\[Mu]"]], ",", "3", ",", RowBox[List["-", "z"]]]], "]"]]]]]], ")"]]]]]]]]








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> &#957; </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> <mrow> <mo> - </mo> <mrow> <mi> csc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#956; </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#956; </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <semantics> <mi> &#120083; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[GothicCapitalP]&quot;, LegendreP] </annotation> </semantics> <mi> &#957; </mi> <mi> &#956; </mi> </msubsup> <mo> ( </mo> <semantics> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;-&quot;, &quot;z&quot;]], HoldComplete[LegendreP, 3]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mi> &#960; </mi> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#956; </mi> </mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> - </mo> <mi> &#956; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#956; </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#956; </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <semantics> <mi> &#120083; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[GothicCapitalP]&quot;, LegendreP] </annotation> </semantics> <mi> &#957; </mi> <mrow> <mo> - </mo> <mi> &#956; </mi> </mrow> </msubsup> <mo> ( </mo> <semantics> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;-&quot;, &quot;z&quot;]], HoldComplete[LegendreP, 3]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> LegendreP </ci> <ci> &#957; </ci> <ci> &#956; </ci> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <csc /> <apply> <times /> <ci> &#956; </ci> <pi /> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <sin /> <apply> <times /> <ci> &#957; </ci> <pi /> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreP </ci> <ci> &#120083; </ci> </apply> <ci> &#957; </ci> </apply> <ci> &#956; </ci> </apply> <apply> <apply> <ci> HoldComplete </ci> <ci> LegendreP </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreP </ci> <ci> &#120083; </ci> </apply> <ci> &#957; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> </apply> <apply> <apply> <ci> HoldComplete </ci> <ci> LegendreP </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Mu]"]], "]"]]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "z"]], "-", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", RowBox[List["-", "z"]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]]], "+", FractionBox[RowBox[List["\[Pi]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", RowBox[List["-", "\[Mu]"]], ",", "3", ",", RowBox[List["-", "z"]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "z"]], "-", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]]]]]]], ")"]]]]]]]]


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





2002-12-18