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 GegenbauerC

 http://functions.wolfram.com/07.13.21.0002.01

 Input Form

 Integrate[z^(\[Alpha] - 1) GegenbauerC[\[Nu], z], z] == (1/(2^\[Alpha] \[Nu])) z^(\[Alpha] - 1) (ChebyshevT[\[Alpha] - 1, z] + I Sqrt[1 - z^2] ChebyshevU[\[Alpha] - 2, z]) (z (z + I Sqrt[1 - z^2]))^(1 - \[Alpha]) ((1/(\[Alpha] - \[Nu])) ((ChebyshevT[\[Nu] - \[Alpha], z] + I Sqrt[1 - z^2] ChebyshevU[\[Nu] - \[Alpha] - 1, z]) Hypergeometric2F1[(\[Nu] - \[Alpha])/2, 1 - \[Alpha], 1 + (\[Nu] - \[Alpha])/2, 1 - 2 z^2 - 2 I z Sqrt[1 - z^2]]) + (1/(2 + \[Nu] - \[Alpha])) ((ChebyshevT[2 + \[Nu] - \[Alpha], z] + I Sqrt[1 - z^2] ChebyshevU[1 + \[Nu] - \[Alpha], z]) Hypergeometric2F1[1 + (\[Nu] - \[Alpha])/2, 1 - \[Alpha], 2 + (\[Nu] - \[Alpha])/2, 1 - 2 z^2 - 2 I z Sqrt[1 - z^2]]) + (1/(\[Nu] + \[Alpha])) ((ChebyshevT[\[Nu] + \[Alpha], z] - I Sqrt[1 - z^2] ChebyshevU[\[Nu] + \[Alpha] - 1, z]) Hypergeometric2F1[-((\[Nu] + \[Alpha])/2), 1 - \[Alpha], 1 - (\[Nu] + \[Alpha])/2, 1 - 2 z^2 - 2 I z Sqrt[1 - z^2]]) - (1/(\[Nu] + \[Alpha] - 2)) ((ChebyshevT[2 - \[Nu] - \[Alpha], z] + I Sqrt[1 - z^2] ChebyshevU[1 - \[Nu] - \[Alpha], z]) Hypergeometric2F1[1 - (\[Nu] + \[Alpha])/2, 1 - \[Alpha], 2 - (\[Nu] + \[Alpha])/2, 1 - 2 z^2 - 2 I z Sqrt[1 - z^2]]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox["2", RowBox[List["-", "\[Alpha]"]]], "\[Nu]"], " ", SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List[RowBox[List["\[Alpha]", "-", "1"]], ",", "z"]], "]"]], "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["ChebyshevU", "[", RowBox[List[RowBox[List["\[Alpha]", "-", "2"]], ",", "z"]], "]"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", " ", RowBox[List["(", RowBox[List["z", "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]]], ")"]]]], ")"]], RowBox[List["1", "-", "\[Alpha]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["\[Alpha]", "-", "\[Nu]"]]], RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List[RowBox[List["\[Nu]", "-", "\[Alpha]"]], ",", "z"]], "]"]], "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["ChebyshevU", "[", RowBox[List[RowBox[List["\[Nu]", "-", "\[Alpha]", "-", "1"]], ",", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["\[Nu]", "-", "\[Alpha]"]], "2"], ",", RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List["1", "+", FractionBox[RowBox[List["\[Nu]", "-", "\[Alpha]"]], "2"]]], " ", ",", RowBox[List["1", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]]]]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["2", "+", "\[Nu]", "-", "\[Alpha]"]]], RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List[RowBox[List["2", "+", "\[Nu]", "-", "\[Alpha]"]], ",", "z"]], "]"]], "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["ChebyshevU", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]", "-", "\[Alpha]"]], ",", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", FractionBox[RowBox[List["\[Nu]", "-", "\[Alpha]"]], "2"]]], " ", ",", RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List["2", "+", FractionBox[RowBox[List["\[Nu]", "-", "\[Alpha]"]], "2"]]], ",", RowBox[List["1", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]]]]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["\[Nu]", "+", "\[Alpha]"]]], RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List[RowBox[List["\[Nu]", "+", "\[Alpha]"]], ",", "z"]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["ChebyshevU", "[", RowBox[List[RowBox[List["\[Nu]", "+", "\[Alpha]", "-", "1"]], ",", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[Nu]", "+", "\[Alpha]"]], "2"]]], ",", RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List["1", "-", FractionBox[RowBox[List["\[Nu]", "+", "\[Alpha]"]], "2"]]], ",", RowBox[List["1", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]]]]], "]"]]]], ")"]]]], "-", RowBox[List[FractionBox["1", RowBox[List["\[Nu]", "+", "\[Alpha]", "-", "2"]]], RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List[RowBox[List["2", "-", "\[Nu]", "-", "\[Alpha]"]], ",", "z"]], "]"]], "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["ChebyshevU", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]", "-", "\[Alpha]"]], ",", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "-", FractionBox[RowBox[List["\[Nu]", "+", "\[Alpha]"]], "2"]]], ",", RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List["2", "-", FractionBox[RowBox[List["\[Nu]", "+", "\[Alpha]"]], "2"]]], ",", RowBox[List["1", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]]]]], "]"]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 z α - 1 C ν ( 0 ) ( z ) z 2 - α ν z α - 1 ( T α - 1 ( z ) + 1 - z 2 U α - 2 ( z ) ) ( z ( z + 1 - z 2 ) ) 1 - α ( 1 α - ν ( T ν - α ( z ) + 1 - z 2 U - α + ν - 1 ( z ) ) 2 F 1 ( ν - α 2 , 1 - α ; ν - α 2 + 1 ; - 2 z 2 - 2 1 - z 2 z + 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["\[Nu]", "-", "\[Alpha]"]], "2"], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "\[Alpha]"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List["\[Nu]", "-", "\[Alpha]"]], "2"], "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], " ", "z"]], "+", "1"]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] + 1 α + ν ( T α + ν ( z ) - 1 - z 2 U α + ν - 1 ( z ) ) 2 F 1 ( - α + ν 2 , 1 - α ; 1 - α + ν 2 ; - 2 z 2 - 2 1 - z 2 z + 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox[RowBox[List["\[Alpha]", "+", "\[Nu]"]], "2"]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "\[Alpha]"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["1", "-", FractionBox[RowBox[List["\[Alpha]", "+", "\[Nu]"]], "2"]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], " ", "z"]], "+", "1"]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] + 1 ν - α + 2 ( T 2 - α + ν ( z ) + 1 - z 2 U 1 - α + ν ( z ) ) 2 F 1 ( ν - α 2 + 1 , 1 - α ; ν - α 2 + 2 ; - 2 z 2 - 2 1 - z 2 z + 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["\[Nu]", "-", "\[Alpha]"]], "2"], "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "\[Alpha]"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List["\[Nu]", "-", "\[Alpha]"]], "2"], "+", "2"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], " ", "z"]], "+", "1"]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] - 1 α + ν - 2 ( T 2 - α - ν ( z ) + 1 - z 2 U 1 - α - ν ( z ) ) 2 F 1 ( 1 - α + ν 2 , 1 - α ; 2 - α + ν 2 ; - 2 z 2 - 2 1 - z 2 z + 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", FractionBox[RowBox[List["\[Alpha]", "+", "\[Nu]"]], "2"]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "\[Alpha]"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["2", "-", FractionBox[RowBox[List["\[Alpha]", "+", "\[Nu]"]], "2"]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], " ", "z"]], "+", "1"]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] ) z z α -1 GegenbauerC ν 0 z 2 -1 α ν -1 z α -1 ChebyshevT α -1 z 1 -1 z 2 1 2 ChebyshevU α -2 z z z 1 -1 z 2 1 2 1 -1 α 1 α -1 ν -1 ChebyshevT ν -1 α z 1 -1 z 2 1 2 ChebyshevU -1 α ν -1 z Hypergeometric2F1 ν -1 α 2 -1 1 -1 α ν -1 α 2 -1 1 -2 z 2 -1 2 1 -1 z 2 1 2 z 1 1 α ν -1 ChebyshevT α ν z -1 1 -1 z 2 1 2 ChebyshevU α ν -1 z Hypergeometric2F1 -1 α ν 2 -1 1 -1 α 1 -1 α ν 2 -1 -2 z 2 -1 2 1 -1 z 2 1 2 z 1 1 ν -1 α 2 -1 ChebyshevT 2 -1 α ν z 1 -1 z 2 1 2 ChebyshevU 1 -1 α ν z Hypergeometric2F1 ν -1 α 2 -1 1 1 -1 α ν -1 α 2 -1 2 -2 z 2 -1 2 1 -1 z 2 1 2 z 1 -1 1 α ν -2 -1 ChebyshevT 2 -1 α -1 ν z 1 -1 z 2 1 2 ChebyshevU 1 -1 α -1 ν z Hypergeometric2F1 1 -1 α ν 2 -1 1 -1 α 2 -1 α ν 2 -1 -2 z 2 -1 2 1 -1 z 2 1 2 z 1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Alpha]"]]], " ", SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List[RowBox[List["\[Alpha]", "-", "1"]], ",", "z"]], "]"]], "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["ChebyshevU", "[", RowBox[List[RowBox[List["\[Alpha]", "-", "2"]], ",", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", " ", RowBox[List["(", RowBox[List["z", "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]]], ")"]]]], ")"]], RowBox[List["1", "-", "\[Alpha]"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List[RowBox[List["\[Nu]", "-", "\[Alpha]"]], ",", "z"]], "]"]], "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["ChebyshevU", "[", RowBox[List[RowBox[List["\[Nu]", "-", "\[Alpha]", "-", "1"]], ",", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["\[Nu]", "-", "\[Alpha]"]], "2"], ",", RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List["1", "+", FractionBox[RowBox[List["\[Nu]", "-", "\[Alpha]"]], "2"]]], ",", RowBox[List["1", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]]]]], "]"]]]], RowBox[List["\[Alpha]", "-", "\[Nu]"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List[RowBox[List["2", "+", "\[Nu]", "-", "\[Alpha]"]], ",", "z"]], "]"]], "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["ChebyshevU", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]", "-", "\[Alpha]"]], ",", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", FractionBox[RowBox[List["\[Nu]", "-", "\[Alpha]"]], "2"]]], ",", RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List["2", "+", FractionBox[RowBox[List["\[Nu]", "-", "\[Alpha]"]], "2"]]], ",", RowBox[List["1", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]]]]], "]"]]]], RowBox[List["2", "+", "\[Nu]", "-", "\[Alpha]"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List[RowBox[List["\[Nu]", "+", "\[Alpha]"]], ",", "z"]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["ChebyshevU", "[", RowBox[List[RowBox[List["\[Nu]", "+", "\[Alpha]", "-", "1"]], ",", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[Nu]", "+", "\[Alpha]"]], ")"]]]], ",", RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List["1", "-", FractionBox[RowBox[List["\[Nu]", "+", "\[Alpha]"]], "2"]]], ",", RowBox[List["1", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]]]]], "]"]]]], RowBox[List["\[Nu]", "+", "\[Alpha]"]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List[RowBox[List["2", "-", "\[Nu]", "-", "\[Alpha]"]], ",", "z"]], "]"]], "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["ChebyshevU", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]", "-", "\[Alpha]"]], ",", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "-", FractionBox[RowBox[List["\[Nu]", "+", "\[Alpha]"]], "2"]]], ",", RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List["2", "-", FractionBox[RowBox[List["\[Nu]", "+", "\[Alpha]"]], "2"]]], ",", RowBox[List["1", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]]]]], "]"]]]], RowBox[List["\[Nu]", "+", "\[Alpha]", "-", "2"]]]]], ")"]]]], "\[Nu]"]]]]]

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

 2001-10-29