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 GegenbauerC

 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

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]", ",", "\[Lambda]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List["Piecewise", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["-", "\[Nu]"]], ">", "0"]]]]]], "}"]], ",", "\[IndentingNewLine]", RowBox[List["{", RowBox[List["ComplexInfinity", " ", ",", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "-", RowBox[List["2", "\[Lambda]"]]]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "-", RowBox[List["2", "\[Lambda]"]]]], "\[GreaterEqual]", "0"]]]]]], "}"]], ",", RowBox[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]", "]"]]]]]]], ",", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Lambda]", "]"]], ">", FractionBox["1", "2"]]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "\[And]", RowBox[List["\[Nu]", "\[GreaterEqual]", "0"]]]], "]"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox[RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], "\[Pi]"], RowBox[List["Log", "[", RowBox[List["z", "+", "1"]], "]"]]]], ",", RowBox[List[RowBox[List["\[Lambda]", "\[Equal]", FractionBox["1", "2"]]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "\[And]", RowBox[List["\[Nu]", "\[GreaterEqual]", "0"]]]], "]"]]]]]], "}"]], ",", 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["\[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"]]]]], "}"]]]], "}"]], ",", 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]"]], "]"]]]]]]]]], "]"]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", RowBox[List["-", "1"]]]], ")"]]]]]]

 MathML Form

 C ν λ ( z ) 0 - ν + ~ - 2 λ - ν - 2 1 2 - λ sin ( π ν ) Γ ( λ - 1 2 ) ( z + 1 ) 1 2 - λ π Γ ( λ ) Re ( λ ) > 1 2 ν sin ( ν π ) log ( z + 1 ) π λ 1 2 ν 2 1 - 2 λ cos ( π ( λ + ν ) ) ( - λ - 1 2 ) ! Γ ( 2 λ + ν ) π Γ ( λ ) Γ ( ν + 1 ) Re ( λ ) < 1 2 2 1 - 2 λ cos ( π ( λ + ν ) ) Γ ( 1 2 - λ ) Γ ( 2 λ + ν ) π Γ ( λ ) Γ ( ν + 1 ) - 2 1 2 - λ sin ( π ν ) Γ ( λ - 1 2 ) ( z + 1 ) 1 2 - λ π Γ ( λ ) True TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]] /; ( z "\[Rule]" - 1 ) Condition Proportional Subscript C ν λ z 0 -1 ν SuperPlus OverTilde -2 λ -1 ν -1 2 1 2 -1 λ ν Gamma λ -1 1 2 z 1 1 2 -1 λ 1 2 Gamma λ -1 λ 1 2 ν ν z 1 -1 λ 1 2 ν 2 1 -1 2 λ λ ν -1 λ -1 1 2 Gamma 2 λ ν 1 2 Gamma λ Gamma ν 1 -1 λ 1 2 2 1 -1 2 λ λ ν Gamma 1 2 -1 λ Gamma 2 λ ν 1 2 Gamma λ Gamma ν 1 -1 -1 2 1 2 -1 λ ν Gamma λ -1 1 2 z 1 1 2 -1 λ 1 2 Gamma λ -1 Rule z -1 [/itex]

 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