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 ChebyshevU

 http://functions.wolfram.com/07.05.21.0003.01

 Input Form

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

 Standard Form

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

 MathML Form

 z α - 1 U ν ( z ) z 1 ( 2 - α + ν ) ( α + ν ) ( 2 - α z α ( z ( z + 1 - z 2 ) ) - α ( T ν ( z ) - 1 - z 2 U ν - 1 ( z ) ) ( ( 2 - α + ν ) 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] + ( α + ν ) ( T 2 ν + 2 ( z ) + 1 - z 2 U 2 ν + 1 ( 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] ) ) z z α -1 ChebyshevU ν z 1 2 -1 α ν α ν -1 2 -1 α z α z z 1 -1 z 2 1 2 -1 α ChebyshevT ν z -1 1 -1 z 2 1 2 ChebyshevU ν -1 z 2 -1 α ν 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 α ν ChebyshevT 2 ν 2 z 1 -1 z 2 1 2 ChebyshevU 2 ν 1 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 [/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["ChebyshevU", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Alpha]"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", " ", RowBox[List["(", RowBox[List["z", "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]]], ")"]]]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["ChebyshevU", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "+", "\[Alpha]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List[RowBox[List["2", "+", RowBox[List["2", " ", "\[Nu]"]]]], ",", "z"]], "]"]], "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["ChebyshevU", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], ",", "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[RowBox[List["(", RowBox[List["2", "+", "\[Nu]", "-", "\[Alpha]"]], ")"]], " ", 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[RowBox[List["(", RowBox[List["\[Nu]", "-", "\[Alpha]", "+", "2"]], ")"]], " ", RowBox[List["(", RowBox[List["\[Nu]", "+", "\[Alpha]"]], ")"]]]]]]]]]

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

 2001-10-29