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variants of this functions
GegenbauerC






Mathematica Notation

Traditional Notation









Hypergeometric Functions > GegenbauerC[nu,lambda,z] > Series representations > Generalized power series > Expansions at z==-1 > For the function itself > Generic formulas for main term





http://functions.wolfram.com/07.14.06.0068.01









  


  










Input Form





GegenbauerC[\[Nu], \[Lambda], z] \[Proportional] Piecewise[{{0, Element[-\[Nu], Integers] && -\[Nu] > 0}, {ComplexInfinity, Element[-\[Nu] - 2 \[Lambda], Integers] && -\[Nu] - 2 \[Lambda] >= 0}, {-((2^(1/2 - \[Lambda]) Sin[Pi \[Nu]] Gamma[-(1/2) + \[Lambda]] (1 + z)^(1/2 - \[Lambda]))/(Sqrt[Pi] Gamma[\[Lambda]])), Re[\[Lambda]] > 1/2 && !(Element[\[Nu], Integers] && \[Nu] >= 0)}, {(Sin[\[Nu] Pi]/Pi) Log[z + 1], \[Lambda] == 1/2 && !(Element[\[Nu], Integers] && \[Nu] >= 0)}, {(2^(1 - 2 \[Lambda]) Cos[Pi (\[Nu] + \[Lambda])] (-(1/2) - \[Lambda])! Gamma[\[Nu] + 2 \[Lambda]])/(Sqrt[Pi] Gamma[\[Lambda]] Gamma[\[Nu] + 1]), Re[\[Lambda]] < 1/2}}, -((2^(1/2 - \[Lambda]) Sin[Pi \[Nu]] Gamma[-(1/2) + \[Lambda]] (1 + z)^(1/2 - \[Lambda]))/(Sqrt[Pi] Gamma[\[Lambda]])) + (2^(1 - 2 \[Lambda]) Cos[Pi (\[Lambda] + \[Nu])] Gamma[1/2 - \[Lambda]] Gamma[2 \[Lambda] + \[Nu]])/(Sqrt[Pi] Gamma[\[Lambda]] Gamma[1 + \[Nu]])] /; (z -> -1)










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]_", ",", "\[Lambda]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[Piecewise]", GridBox[List[List["0", RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", "\[Nu]"]], ">", "0"]]]]], List["ComplexInfinity", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "-", RowBox[List["2", " ", "\[Lambda]"]]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "-", RowBox[List["2", " ", "\[Lambda]"]]]], "\[GreaterEqual]", "0"]]]]], List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "\[Lambda]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]]]], RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Lambda]", "]"]], ">", FractionBox["1", "2"]]], "&&", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Nu]", "\[GreaterEqual]", "0"]]]], ")"]]]]]]], List[FractionBox[RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["z", "+", "1"]], "]"]]]], "\[Pi]"], RowBox[List[RowBox[List["\[Lambda]", "\[Equal]", FractionBox["1", "2"]]], "&&", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Nu]", "\[GreaterEqual]", "0"]]]], ")"]]]]]]], List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["2", " ", "\[Lambda]"]]]]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Nu]", "+", "\[Lambda]"]], ")"]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Lambda]"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], "]"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]]]], RowBox[List[RowBox[List["Re", "[", "\[Lambda]", "]"]], "<", FractionBox["1", "2"]]]], List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "\[Lambda]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["2", " ", "\[Lambda]"]]]]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Lambda]", "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], "]"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]]]]], TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]]]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", RowBox[List["-", "1"]]]], ")"]]]]]]]]










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





2007-05-02