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variants of this functions
GegenbauerC




Mathematica Notation

Traditional Notation









Hypergeometric Functions > GegenbauerC[nu,z] > Series representations > Generalized power series > Expansions at z==-1 > For the function itself > General case





http://functions.wolfram.com/07.13.06.0016.02









  


  








Input Form





GegenbauerC[\[Nu], z] \[Proportional] (2/\[Nu]) Cos[Pi \[Nu]] (1 - \[Nu]^2 (z + 1) - (((1 - \[Nu]) \[Nu]^2 (1 + \[Nu]))/6) (z + 1)^2 - \[Ellipsis]) + 2 Sqrt[2] Sqrt[z + 1] Sin[Pi \[Nu]] (1 + (1/3) (1/2 - \[Nu]) (1/2 + \[Nu]) (z + 1) + (1/30) (1/2 - \[Nu]) (3/2 - \[Nu]) (1/2 + \[Nu]) (3/2 + \[Nu]) (z + 1)^2 + \[Ellipsis]) /; (z -> -1)








Standard Form





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MathML Form







<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> C </mi> <mi> &#957; </mi> <mrow> <mo> ( </mo> <mn> 0 </mn> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mrow> <mfrac> <mn> 2 </mn> <mi> &#957; </mi> </mfrac> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <msup> <mi> &#957; </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#957; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> &#957; 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</ci> </apply> </apply> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <ci> &#957; </ci> </apply> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 30 </cn> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <plus /> <cn type='rational'> 3 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <ci> &#957; </ci> </apply> <apply> <plus /> <cn type='rational'> 3 <sep /> 2 </cn> <ci> &#957; </ci> </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> <apply> <ci> Rule </ci> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List[SuperscriptBox["\[Nu]", "2"], " ", RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]]]], "-", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[Nu]"]], ")"]], " ", SuperscriptBox["\[Nu]", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "2"]]], "-", "\[Ellipsis]"]], ")"]]]], "\[Nu]"], "+", RowBox[List["2", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List["z", "+", "1"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[FractionBox["1", "3"], " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]]]], "+", RowBox[List[FractionBox["1", "30"], " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox["3", "2"], "-", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", RowBox[List["-", "1"]]]], ")"]]]]]]]]


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





2001-10-29