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 GegenbauerC

 http://functions.wolfram.com/07.14.06.0047.01

 Input Form

 GegenbauerC[\[Nu], \[Lambda], z] \[Proportional] (-((2^(1 - 2 \[Lambda]) Cos[Pi (\[Lambda] + \[Nu])] Sin[Pi \[Nu]])/ (Pi^(3/2) Gamma[\[Lambda]]))) (I Pi^(3/2) 4^\[Lambda] E^(I Pi (1/2 - \[Lambda]) Floor[Arg[z - x]/(2 Pi)]) Csc[Pi \[Nu]] Floor[Arg[z - x]/(2 Pi)] Gamma[\[Lambda]] GegenbauerC[\[Nu], \[Lambda], -x] + Exp[Pi I (1 - 2 \[Lambda]) Floor[Arg[z - x]/(2 Pi)]] MeijerG[{{1 + \[Nu], 1 - 2 \[Lambda] - \[Nu]}, {}}, {{0, 1/2 - \[Lambda]}, {}}, (1/2) (1 + x)] + O[z - x]) /; Element[x, Reals] && x < -1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]", ",", "\[Lambda]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["-", 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["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], RowBox[List[SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox["4", "\[Lambda]"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]], RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]", ",", "\[Lambda]", ",", RowBox[List["-", "x"]]]], "]"]]]], "+", RowBox[List[RowBox[List["Exp", "[", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", "\[Lambda]"]]]], ")"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "]"]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "-", RowBox[List["2", " ", "\[Lambda]"]], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]]]]]], "]"]]]], "+", RowBox[List["O", "[", RowBox[List["z", "-", "x"]], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["x", "<", RowBox[List["-", "1"]]]]]]]]]]

 MathML Form

 C ν λ ( z ) - 2 1 - 2 λ cos ( π ( λ + ν ) ) sin ( π ν ) π 3 / 2 Γ ( λ ) ( π 3 / 2 4 λ π ( 1 2 - λ ) arg ( z - x ) 2 π csc ( π ν ) Γ ( λ ) arg ( z - x ) 2 π C ν λ ( - x ) + π ( 1 - 2 λ ) arg ( z - x ) 2 π G 2 , 2 2 , 2 ( x + 1 2 ν + 1 , - 2 λ - ν + 1 0 , 1 2 - λ ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "2"]], RowBox[List["2", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[FractionBox[RowBox[List["x", "+", "1"]], "2"], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["\[Nu]", "+", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Lambda]"]], "-", "\[Nu]", "+", "1"]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] + O ( z - x ) ) /; x TagBox["\[DoubleStruckCapitalR]", Function[Reals]] x < - 1 Condition Proportional Subscript C ν λ z -1 2 1 -1 2 λ λ ν ν 3 2 Gamma λ -1 3 2 4 λ 1 2 -1 λ z -1 x 2 -1 ν Gamma λ z -1 x 2 -1 Subscript C ν λ -1 x 1 -1 2 λ z -1 x 2 -1 MeijerG ν 1 -2 λ -1 ν 1 0 1 2 -1 λ x 1 2 -1 O z -1 x x x -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]_", ",", "\[Lambda]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", 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["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox["4", "\[Lambda]"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]], " ", RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]", ",", "\[Lambda]", ",", RowBox[List["-", "x"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[Lambda]"]]]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "-", RowBox[List["2", " ", "\[Lambda]"]], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", FractionBox[RowBox[List["1", "+", "x"]], "2"]]], "]"]]]], "+", RowBox[List["O", "[", RowBox[List["z", "-", "x"]], "]"]]]], ")"]]]], RowBox[List[SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["x", "<", RowBox[List["-", "1"]]]]]]]]]]]]

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

 2007-05-02