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 GegenbauerC

 http://functions.wolfram.com/07.14.06.0042.01

 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)] + (1/2) (2 Pi I E^(I Pi (1/2 - \[Lambda]) Floor[Arg[z - x]/(2 Pi)]) Floor[Arg[z - x]/(2 Pi)] Gamma[1 - \[Nu]] Gamma[1 + 2 \[Lambda] + \[Nu]] Hypergeometric2F1Regularized[1 - \[Nu], 1 + 2 \[Lambda] + \[Nu], 3/2 + \[Lambda], (1/2) (1 + x)] + Exp[Pi I (1 - 2 \[Lambda]) Floor[Arg[z - x]/(2 Pi)]] MeijerG[{{\[Nu], -2 \[Lambda] - \[Nu]}, {}}, {{0, -(1/2) - \[Lambda]}, {}}, (1/2) (1 + x)]) (z - x) + (1/8) (2 Pi I E^(I Pi (1/2 - \[Lambda]) Floor[Arg[z - x]/(2 Pi)]) Floor[Arg[z - x]/(2 Pi)] Gamma[2 - \[Nu]] Gamma[2 + 2 \[Lambda] + \[Nu]] Hypergeometric2F1Regularized[2 - \[Nu], 2 + 2 \[Lambda] + \[Nu], 5/2 + \[Lambda], (1/2) (1 + x)] - Exp[Pi I (1 - 2 \[Lambda]) Floor[Arg[z - x]/(2 Pi)]] MeijerG[{{-1 + \[Nu], -1 - 2 \[Lambda] - \[Nu]}, {}}, {{0, -(3/2) - \[Lambda]}, {}}, (1/2) (1 + x)]) (z - x)^2 + \[Ellipsis]) /; (z -> x) && Element[x, Reals] && x < -1

 Standard Form

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 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]] + 1 2 ( 2 π ( 1 2 - λ ) arg ( z - x ) 2 π π arg ( z - x ) 2 π Γ ( 1 - ν ) Γ ( 2 λ + ν + 1 ) 2 F ~ 1 ( 1 - ν , 2 λ + ν + 1 ; λ + 3 2 ; x + 1 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", "\[Nu]"]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]", "+", "1"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["\[Lambda]", "+", FractionBox["3", "2"]]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["x", "+", "1"]], "2"], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] + π ( 1 - 2 λ ) arg ( z - x ) 2 π G 2 , 2 2 , 2 ( x + 1 2 ν , - 2 λ - ν 0 , - λ - 1 2 ) 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["\[Nu]", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Lambda]"]], "-", "\[Nu]"]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "\[Lambda]"]], "-", FractionBox["1", "2"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] ) ( z - x ) + 1 8 ( 2 π π ( 1 2 - λ ) arg ( z - x ) 2 π arg ( z - x ) 2 π Γ ( 2 - ν ) Γ ( 2 λ + ν + 2 ) 2 F ~ 1 ( 2 - ν , 2 λ + ν + 2 ; λ + 5 2 ; x + 1 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["2", "-", "\[Nu]"]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]", "+", "2"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["\[Lambda]", "+", FractionBox["5", "2"]]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["x", "+", "1"]], "2"], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] - π ( 1 - 2 λ ) arg ( z - x ) 2 π G 2 , 2 2 , 2 ( x + 1 2 ν - 1 , - 2 λ - ν - 1 0 , - λ - 3 2 ) 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[RowBox[List["-", "\[Lambda]"]], "-", FractionBox["3", "2"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] ) ( z - x ) 2 + ) /; ( z "\[Rule]" x ) x TagBox["\[DoubleStruckCapitalR]", Function[Reals]] x < - 1 Condition Proportional Subscript C ν λ z 2 1 -1 2 λ λ ν ν 3 2 Gamma λ -1 -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 -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 1 2 2 1 2 -1 λ z -1 x 2 -1 z -1 x 2 -1 Gamma 1 -1 ν Gamma 2 λ ν 1 Hypergeometric2F1Regularized 1 -1 ν 2 λ ν 1 λ 3 2 x 1 2 -1 1 -1 2 λ z -1 x 2 -1 MeijerG ν -2 λ -1 ν 0 -1 λ -1 1 2 x 1 2 -1 z -1 x 1 8 2 1 2 -1 λ z -1 x 2 -1 z -1 x 2 -1 Gamma 2 -1 ν Gamma 2 λ ν 2 Hypergeometric2F1Regularized 2 -1 ν 2 λ ν 2 λ 5 2 x 1 2 -1 -1 1 -1 2 λ z -1 x 2 -1 MeijerG ν -1 -2 λ -1 ν -1 0 -1 λ -1 3 2 x 1 2 -1 z -1 x 2 Rule z x x x -1 [/itex]

 Rule Form

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

 2007-05-02