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http://functions.wolfram.com/04.11.21.0004.01
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Integrate[\[Nu]^(\[Alpha] - 1) Fibonacci[\[Nu]], \[Nu]] ==
(\[Nu]^\[Alpha]/(2 Sqrt[5])) ((-2 Gamma[\[Alpha], (-\[Nu]) ArcCsch[2]])/
((-\[Nu])^\[Alpha] ArcCsch[2]^\[Alpha]) +
Gamma[\[Alpha], \[Nu] ((-I) Pi + ArcCsch[2])]/
(\[Nu] ((-I) Pi + ArcCsch[2]))^\[Alpha] +
Gamma[\[Alpha], \[Nu] (I Pi + ArcCsch[2])]/(\[Nu] (I Pi + ArcCsch[2]))^
\[Alpha])
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["\[Nu]", RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List["Fibonacci", "[", "\[Nu]", "]"]], RowBox[List["\[DifferentialD]", "\[Nu]"]]]]]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox["\[Nu]", "\[Alpha]"], RowBox[List["2", " ", SqrtBox["5"]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", SuperscriptBox[RowBox[List["ArcCsch", "[", "2", "]"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List[RowBox[List["-", "\[Nu]"]], " ", RowBox[List["ArcCsch", "[", "2", "]"]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]"]], "+", RowBox[List["ArcCsch", "[", "2", "]"]]]], ")"]]]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]"]], "+", RowBox[List["ArcCsch", "[", "2", "]"]]]], ")"]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "+", RowBox[List["ArcCsch", "[", "2", "]"]]]], ")"]]]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "+", RowBox[List["ArcCsch", "[", "2", "]"]]]], ")"]]]]]], "]"]]]]]], ")"]]]]]]]]
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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> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox["F", Fibonacci] </annotation> </semantics> <mi> ν </mi> </msub> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> ν </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <msup> <mi> ν </mi> <mi> α </mi> </msup> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> α </mi> </mrow> </msup> <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> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> α </mi> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> α </mi> </mrow> </msup> <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> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> α </mi> </mrow> </msup> <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> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </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> Fibonacci </ci> <ci> ν </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <ci> ν </ci> <ci> α </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> <apply> <ci> Gamma </ci> <ci> α </ci> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <ci> ν </ci> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <pi /> </apply> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> <apply> <ci> Gamma </ci> <ci> α </ci> <apply> <times /> <ci> ν </ci> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <pi /> </apply> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <ci> ν </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <pi /> </apply> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> <apply> <ci> Gamma </ci> <ci> α </ci> <apply> <times /> <ci> ν </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <pi /> </apply> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </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["Fibonacci", "[", "\[Nu]_", "]"]]]], RowBox[List["\[DifferentialD]", "\[Nu]_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["\[Nu]", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", SuperscriptBox[RowBox[List["ArcCsch", "[", "2", "]"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List[RowBox[List["-", "\[Nu]"]], " ", RowBox[List["ArcCsch", "[", "2", "]"]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]"]], "+", RowBox[List["ArcCsch", "[", "2", "]"]]]], ")"]]]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]"]], "+", RowBox[List["ArcCsch", "[", "2", "]"]]]], ")"]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "+", RowBox[List["ArcCsch", "[", "2", "]"]]]], ")"]]]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "+", RowBox[List["ArcCsch", "[", "2", "]"]]]], ")"]]]]]], "]"]]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox["5"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
