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variants of this functions
LegendreP






Mathematica Notation

Traditional Notation









Polynomials > LegendreP[n,mu,2,z] > Series representations > Generalized power series > Expansions at z==-1





http://functions.wolfram.com/05.07.06.0014.01









  


  










Input Form





LegendreP[n, \[Mu], 2, z] == (Gamma[-\[Mu]]/(2^(\[Mu]/2) (Gamma[-\[Mu] - n] Gamma[1 - \[Mu] + n]))) (z + 1)^(\[Mu]/2) (1 + ((\[Mu] (1 + \[Mu]) - 2 n (1 + n))/(4 (1 + \[Mu]))) (z + 1) + ((\[Mu] (1 + \[Mu]) (2 + \[Mu])^2 - 4 (2 + \[Mu] (2 + \[Mu])) n - 4 (1 + \[Mu])^2 n^2 + 8 n^3 + 4 n^4)/(32 (1 + \[Mu]) (2 + \[Mu]))) (z + 1)^2 + \[Ellipsis]) /; !Element[\[Mu], Integers]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["n", ",", "\[Mu]", ",", "2", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "\[Mu]"]], "/", "2"]]], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Mu]"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "n"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "n"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["\[Mu]", RowBox[List["(", RowBox[List["1", "+", "\[Mu]"]], ")"]]]], "-", RowBox[List["2", " ", "n", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]]], RowBox[List["4", RowBox[List["(", RowBox[List["1", "+", "\[Mu]"]], ")"]]]]], RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["\[Mu]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Mu]"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", "+", "\[Mu]"]], ")"]], "2"]]], "-", RowBox[List["4", " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List["\[Mu]", " ", RowBox[List["(", RowBox[List["2", "+", "\[Mu]"]], ")"]]]]]], ")"]], " ", "n"]], "-", RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "\[Mu]"]], ")"]], "2"], " ", SuperscriptBox["n", "2"]]], "+", RowBox[List["8", " ", SuperscriptBox["n", "3"]]], "+", RowBox[List["4", " ", SuperscriptBox["n", "4"]]]]], RowBox[List["32", RowBox[List["(", RowBox[List["1", "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "\[Mu]"]], ")"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "2"]]], " ", "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Mu]", ",", "Integers"]], "]"]], "]"]]]]]]










MathML Form







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</mo> <msup> <mi> n </mi> <mn> 4 </mn> </msup> </mrow> </mrow> <mrow> <mn> 32 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> &#956; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> + </mo> <mi> &#956; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mtext> </mtext> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> &#956; </mi> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <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 /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> </apply> <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> <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> n </ci> </apply> </apply> </apply> <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> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> &#956; 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</ci> <apply> <plus /> <cn type='integer'> 2 </cn> <ci> &#956; </ci> </apply> </apply> </apply> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> &#956; </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 32 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> &#956; </ci> </apply> <apply> <plus /> <cn type='integer'> 2 </cn> <ci> &#956; </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> </apply> <apply> <notin /> <ci> &#956; </ci> <integers /> </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]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", FractionBox["\[Mu]", "2"]]]], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Mu]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[Mu]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Mu]"]], ")"]]]], "-", RowBox[List["2", " ", "n", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List["1", "+", "\[Mu]"]], ")"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[Mu]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Mu]"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", "+", "\[Mu]"]], ")"]], "2"]]], "-", RowBox[List["4", " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List["\[Mu]", " ", RowBox[List["(", RowBox[List["2", "+", "\[Mu]"]], ")"]]]]]], ")"]], " ", "n"]], "-", RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "\[Mu]"]], ")"]], "2"], " ", SuperscriptBox["n", "2"]]], "+", RowBox[List["8", " ", SuperscriptBox["n", "3"]]], "+", RowBox[List["4", " ", SuperscriptBox["n", "4"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "2"]]], RowBox[List["32", " ", RowBox[List["(", RowBox[List["1", "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "\[Mu]"]], ")"]]]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "n"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "n"]], "]"]]]]], "/;", RowBox[List["!", RowBox[List["\[Mu]", "\[Element]", "Integers"]]]]]]]]]]










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





2001-10-29





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