html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 GegenbauerC

 http://functions.wolfram.com/07.14.06.0046.01

 Input Form

 GegenbauerC[\[Nu], \[Lambda], z] == (-((2^(1/2 - \[Lambda]) Sqrt[Pi] E^(I Pi (1/2 - \[Lambda]) Floor[Arg[-x + z]/(2 Pi)]) Sec[Pi \[Lambda]] Sin[Pi \[Nu]])/ Gamma[\[Lambda]])) Sum[((-1)^k/k!) (2 I (1 - x)^(1/2 - k - \[Lambda]) Cos[Pi (\[Lambda] + \[Nu])] Floor[Arg[-x + z]/(2 Pi)] Hypergeometric2F1Regularized[1/2 + \[Lambda] + \[Nu], 1/2 - \[Lambda] - \[Nu], 3/2 - k - \[Lambda], (1 - x)/2] + ((Pi (1 + x)^(1/2 - \[Lambda] - k))/(Gamma[1 - k - 2 \[Lambda] - \[Nu]] Gamma[1 - k + \[Nu]])) (2 I Cos[Pi (\[Lambda] + \[Nu])]^2 Csc[Pi \[Nu]] Csc[Pi (2 \[Lambda] + \[Nu])] Floor[Arg[-x + z]/(2 Pi)] + E^(I Pi (1/2 - \[Lambda]) Floor[Arg[-x + z]/(2 Pi)]) (-1 - Cos[Pi (\[Lambda] + \[Nu])]^2 Csc[Pi \[Nu]] Csc[Pi (2 \[Lambda] + \[Nu])]) Sec[Pi \[Lambda]]) Hypergeometric2F1Regularized[1/2 + \[Lambda] + \[Nu], 1/2 - \[Lambda] - \[Nu], 1/2 + k + \[Lambda], (1 - x)/2]) (z - x)^k, {k, 0, Infinity}] /; !Element[1/2 - \[Lambda], Integers] && Element[x, Reals] && x < -1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]", ",", "\[Lambda]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]]], " ", SqrtBox["\[Pi]"], SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["Sec", "[", RowBox[List["\[Pi]", " ", "\[Lambda]"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " "]], RowBox[List["k", "!"]]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "x"]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "k", "-", "\[Lambda]"]]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Lambda]", "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Lambda]", "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]", "-", "\[Nu]"]], ",", RowBox[List[FractionBox["3", "2"], "-", "k", "-", "\[Lambda]"]], ",", FractionBox[RowBox[List["1", "-", "x"]], "2"]]], "]"]]]], "+", " ", RowBox[List[FractionBox[RowBox[List["\[Pi]", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]", "-", "k"]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "k", "-", RowBox[List["2", " ", "\[Lambda]"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "k", "+", "\[Nu]"]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Lambda]", "+", "\[Nu]"]], ")"]]]], "]"]], "2"], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List[SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Lambda]", "+", "\[Nu]"]], ")"]]]], "]"]], "2"], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], ")"]]]], "]"]]]]]], ")"]], " ", RowBox[List["Sec", "[", RowBox[List["\[Pi]", " ", "\[Lambda]"]], "]"]]]]]], ")"]], RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Lambda]", "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]", "-", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "+", "k", "+", "\[Lambda]"]], ",", FractionBox[RowBox[List["1", "-", "x"]], "2"]]], "]"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "k"]]]]]]]]], "/;", RowBox[List[RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]], ",", "Integers"]], "]"]], "]"]], "\[And]", RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["x", "<", RowBox[List["-", "1"]]]]]]]]]]

 MathML Form

 C ν λ ( z ) - 2 1 2 - λ π π ( 1 2 - λ ) arg ( z - x ) 2 π sec ( π λ ) sin ( π ν ) Γ ( λ ) k = 0 ( - 1 ) k k ! ( 2 ( 1 - x ) 1 2 - k - λ cos ( π ( λ + ν ) ) arg ( z - x ) 2 π 2 F ~ 1 ( λ + ν + 1 2 , 1 2 - λ - ν ; 3 2 - k - λ ; 1 - x 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["\[Lambda]", "+", "\[Nu]", "+", FractionBox["1", "2"]]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]", "-", "\[Nu]"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["3", "2"], "-", "k", "-", "\[Lambda]"]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["1", "-", "x"]], "2"], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] + π ( x + 1 ) 1 2 - k - λ Γ ( 1 - k - 2 λ - ν ) Γ ( ν - k + 1 ) ( 2 csc ( π ν ) csc ( π ( 2 λ + ν ) ) arg ( z - x ) 2 π cos 2 ( π ( λ + ν ) ) + π ( 1 2 - λ ) arg ( z - x ) 2 π ( - csc ( π ν ) csc ( π ( 2 λ + ν ) ) cos 2 ( π ( λ + ν ) ) - 1 ) sec ( π λ ) ) 2 F ~ 1 ( λ + ν + 1 2 , 1 2 - λ - ν ; k + λ + 1 2 ; 1 - x 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["\[Lambda]", "+", "\[Nu]", "+", FractionBox["1", "2"]]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]", "-", "\[Nu]"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["k", "+", "\[Lambda]", "+", FractionBox["1", "2"]]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["1", "-", "x"]], "2"], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] ) ( z - x ) k /; 1 2 - λ TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] x TagBox["\[DoubleStruckCapitalR]", Function[Reals]] x < - 1 Condition Subscript C ν λ z -1 2 1 2 -1 λ 1 2 1 2 -1 λ z -1 x 2 -1 λ ν Gamma λ -1 k 0 -1 k k -1 2 1 -1 x 1 2 -1 k -1 λ λ ν z -1 x 2 -1 Hypergeometric2F1Regularized λ ν 1 2 1 2 -1 λ -1 ν 3 2 -1 k -1 λ 1 -1 x 2 -1 x 1 1 2 -1 k -1 λ Gamma 1 -1 k -1 2 λ -1 ν Gamma ν -1 k 1 -1 2 ν 2 λ ν z -1 x 2 -1 λ ν 2 1 2 -1 λ z -1 x 2 -1 -1 ν 2 λ ν λ ν 2 -1 λ Hypergeometric2F1Regularized λ ν 1 2 1 2 -1 λ -1 ν k λ 1 2 1 -1 x 2 -1 z -1 x k 1 2 -1 λ 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[FractionBox["1", "2"], "-", "\[Lambda]"]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["Sec", "[", RowBox[List["\[Pi]", " ", "\[Lambda]"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "x"]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "k", "-", "\[Lambda]"]]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Lambda]", "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Lambda]", "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]", "-", "\[Nu]"]], ",", RowBox[List[FractionBox["3", "2"], "-", "k", "-", "\[Lambda]"]], ",", FractionBox[RowBox[List["1", "-", "x"]], "2"]]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Pi]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]", "-", "k"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Lambda]", "+", "\[Nu]"]], ")"]]]], "]"]], "2"], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List[SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Lambda]", "+", "\[Nu]"]], ")"]]]], "]"]], "2"], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], ")"]]]], "]"]]]]]], ")"]], " ", RowBox[List["Sec", "[", RowBox[List["\[Pi]", " ", "\[Lambda]"]], "]"]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Lambda]", "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]", "-", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "+", "k", "+", "\[Lambda]"]], ",", FractionBox[RowBox[List["1", "-", "x"]], "2"]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "k", "-", RowBox[List["2", " ", "\[Lambda]"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "k", "+", "\[Nu]"]], "]"]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "k"]]], RowBox[List["k", "!"]]]]]]], RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]], "/;", RowBox[List[RowBox[List["!", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]], "\[Element]", "Integers"]]]], "&&", RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["x", "<", RowBox[List["-", "1"]]]]]]]]]]]]

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

 2007-05-02