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/07.14.06.0045.01

 Input Form

 GegenbauerC[\[Nu], \[Lambda], z] == ((2^(1 - 2 \[Lambda]) Sin[Pi \[Nu]])/(Sqrt[Pi] Gamma[\[Lambda]])) Sum[(1/(2^k k!)) (Gamma[k - \[Nu]] Gamma[k + 2 \[Lambda] + \[Nu]] Cos[Pi (\[Lambda] + \[Nu])] (2 I E^(I Pi (1/2 - \[Lambda]) Floor[Arg[z - x]/(2 Pi)]) Floor[Arg[z - x]/(2 Pi)] - Sec[Pi \[Lambda]] Exp[Pi I (1 - 2 \[Lambda]) Floor[Arg[z - x]/(2 Pi)]]) Hypergeometric2F1Regularized[k - \[Nu], k + 2 \[Lambda] + \[Nu], 1/2 + k + \[Lambda], (1 + x)/2] + 2^(-(1/2) + k + \[Lambda]) Pi Sec[Pi \[Lambda]] (1 + x)^(1/2 - k - \[Lambda]) Exp[Pi I (1 - 2 \[Lambda]) Floor[Arg[z - x]/(2 Pi)]] Hypergeometric2F1Regularized[1/2 + \[Lambda] + \[Nu], 1/2 - \[Lambda] - \[Nu], 3/2 - 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[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["2", " ", "\[Lambda]"]]]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[SuperscriptBox["2", RowBox[List["-", "k"]]], RowBox[List["k", "!"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["k", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "+", RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], "]"]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Lambda]", "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", 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["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "-", RowBox[List[RowBox[List["Sec", "[", RowBox[List["\[Pi]", " ", "\[Lambda]"]], "]"]], " ", 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["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List["k", "-", "\[Nu]"]], ",", RowBox[List["k", "+", RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "+", "k", "+", "\[Lambda]"]], ",", FractionBox[RowBox[List["1", "+", "x"]], "2"]]], "]"]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "k", "+", "\[Lambda]"]]], " ", "\[Pi]", " ", RowBox[List["Sec", "[", RowBox[List["\[Pi]", " ", "\[Lambda]"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "k", "-", "\[Lambda]"]]], 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["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"]]], "]"]]]]]], " ", ")"]], " ", 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 sin ( π ν ) π Γ ( λ ) k = 0 2 - k k ! ( 2 k + λ - 1 2 π sec ( π λ ) ( x + 1 ) - k - λ + 1 2 π ( 1 - 2 λ ) arg ( z - x ) 2 π 2 F ~ 1 ( λ + ν + 1 2 , - λ - ν + 1 2 ; - k - 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["\[Lambda]", "+", "\[Nu]", "+", FractionBox["1", "2"]]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "\[Lambda]"]], "-", "\[Nu]", "+", FractionBox["1", "2"]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", "k"]], "-", "\[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] + cos ( π ( λ + ν ) ) Γ ( k - ν ) Γ ( k + 2 λ + ν ) ( 2 π ( 1 2 - λ ) arg ( z - x ) 2 π arg ( z - x ) 2 π - sec ( π λ ) π ( 1 - 2 λ ) arg ( z - x ) 2 π ) 2 F ~ 1 ( k - ν , k + 2 λ + ν ; k + λ + 1 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["k", "-", "\[Nu]"]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List["k", "+", RowBox[List["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["x", "+", "1"]], "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 2 1 -1 2 λ ν 1 2 Gamma λ -1 k 0 2 -1 k k -1 2 k λ -1 1 2 sec λ x 1 -1 k -1 λ 1 2 1 -1 2 λ z -1 x 2 -1 Hypergeometric2F1Regularized λ ν 1 2 -1 λ -1 ν 1 2 -1 k -1 λ 3 2 x 1 2 -1 λ ν Γ k -1 ν Gamma k 2 λ ν 2 1 2 -1 λ z -1 x 2 -1 z -1 x 2 -1 -1 λ 1 -1 2 λ z -1 x 2 -1 Hypergeometric2F1Regularized k -1 ν k 2 λ ν k λ 1 2 x 1 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[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["2", " ", "\[Lambda]"]]]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "k"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["k", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "+", RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], "]"]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Lambda]", "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", 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["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "-", RowBox[List[RowBox[List["Sec", "[", RowBox[List["\[Pi]", " ", "\[Lambda]"]], "]"]], " ", 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["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List["k", "-", "\[Nu]"]], ",", RowBox[List["k", "+", RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "+", "k", "+", "\[Lambda]"]], ",", FractionBox[RowBox[List["1", "+", "x"]], "2"]]], "]"]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "k", "+", "\[Lambda]"]]], " ", "\[Pi]", " ", RowBox[List["Sec", "[", RowBox[List["\[Pi]", " ", "\[Lambda]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "k", "-", "\[Lambda]"]]], " ", 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["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"]]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "k"]]], RowBox[List["k", "!"]]]]]]], RowBox[List[SqrtBox["\[Pi]"], " ", 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