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http://functions.wolfram.com/07.05.21.0007.01
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Integrate[\[Nu]^(\[Alpha] - 1) ChebyshevU[\[Nu], z], \[Nu]] ==
((-(\[Nu]^\[Alpha]/(2 Sqrt[1 - z^2])))
(((-I) z + Sqrt[1 - z^2]) Gamma[\[Alpha],
\[Nu] (Im[ArcCos[z]] - I Re[ArcCos[z]])]
((-\[Nu]) (Im[ArcCos[z]] - I Re[ArcCos[z]]))^\[Alpha] +
(I z + Sqrt[1 - z^2]) Gamma[\[Alpha], (-\[Nu]) (Im[ArcCos[z]] -
I Re[ArcCos[z]])] (\[Nu] (Im[ArcCos[z]] - I Re[ArcCos[z]]))^
\[Alpha]))/((-\[Nu]^2) (Im[ArcCos[z]] - I Re[ArcCos[z]])^2)^\[Alpha]
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["\[Nu]", RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List["ChebyshevU", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], RowBox[List["\[DifferentialD]", "\[Nu]"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[Nu]", "\[Alpha]"], " "]], RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["\[Nu]", "2"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Im", "[", RowBox[List["ArcCos", "[", "z", "]"]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Re", "[", RowBox[List["ArcCos", "[", "z", "]"]], "]"]]]]]], ")"]], "2"]]], ")"]], RowBox[List["-", "\[Alpha]"]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List[RowBox[List["Im", "[", RowBox[List["ArcCos", "[", "z", "]"]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Re", "[", RowBox[List["ArcCos", "[", "z", "]"]], "]"]]]]]], ")"]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[Nu]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Im", "[", RowBox[List["ArcCos", "[", "z", "]"]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Re", "[", RowBox[List["ArcCos", "[", "z", "]"]], "]"]]]]]], ")"]]]], ")"]], "\[Alpha]"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "z"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List[RowBox[List["-", "\[Nu]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Im", "[", RowBox[List["ArcCos", "[", "z", "]"]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Re", "[", RowBox[List["ArcCos", "[", "z", "]"]], "]"]]]]]], ")"]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List[RowBox[List["Im", "[", RowBox[List["ArcCos", "[", "z", "]"]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Re", "[", RowBox[List["ArcCos", "[", "z", "]"]], "]"]]]]]], ")"]]]], ")"]], "\[Alpha]"]]]]], ")"]], " "]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> ∫ </mo> <mrow> <mrow> <msup> <mi> ν </mi> <mrow> <mi> α </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msub> <mi> U </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> ν </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <msup> <mi> ν </mi> <mi> α </mi> </msup> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <msup> <mi> ν </mi> <mn> 2 </mn> </msup> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> α </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> α </mi> <mo> , </mo> <mrow> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> α </mi> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> α </mi> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> α </mi> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> ν </ci> </bvar> <apply> <times /> <apply> <power /> <ci> ν </ci> <apply> <plus /> <ci> α </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> ChebyshevU </ci> <ci> ν </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> ν </ci> <ci> α </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <imaginary /> <apply> <arccos /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <real /> <apply> <arccos /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <ci> α </ci> <apply> <times /> <ci> ν </ci> <apply> <plus /> <apply> <imaginary /> <apply> <arccos /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <real /> <apply> <arccos /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <apply> <plus /> <apply> <imaginary /> <apply> <arccos /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <real /> <apply> <arccos /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <ci> α </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> <apply> <ci> Gamma </ci> <ci> α </ci> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <apply> <plus /> <apply> <imaginary /> <apply> <arccos /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <real /> <apply> <arccos /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> ν </ci> <apply> <plus /> <apply> <imaginary /> <apply> <arccos /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <real /> <apply> <arccos /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <ci> α </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["\[Nu]_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["ChebyshevU", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "\[Nu]_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[Nu]", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["\[Nu]", "2"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Im", "[", RowBox[List["ArcCos", "[", "z", "]"]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Re", "[", RowBox[List["ArcCos", "[", "z", "]"]], "]"]]]]]], ")"]], "2"]]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List[RowBox[List["Im", "[", RowBox[List["ArcCos", "[", "z", "]"]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Re", "[", RowBox[List["ArcCos", "[", "z", "]"]], "]"]]]]]], ")"]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[Nu]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Im", "[", RowBox[List["ArcCos", "[", "z", "]"]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Re", "[", RowBox[List["ArcCos", "[", "z", "]"]], "]"]]]]]], ")"]]]], ")"]], "\[Alpha]"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "z"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List[RowBox[List["-", "\[Nu]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Im", "[", RowBox[List["ArcCos", "[", "z", "]"]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Re", "[", RowBox[List["ArcCos", "[", "z", "]"]], "]"]]]]]], ")"]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List[RowBox[List["Im", "[", RowBox[List["ArcCos", "[", "z", "]"]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Re", "[", RowBox[List["ArcCos", "[", "z", "]"]], "]"]]]]]], ")"]]]], ")"]], "\[Alpha]"]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
