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variants of this functions
Fibonacci






Mathematica Notation

Traditional Notation









Hypergeometric Functions > Fibonacci[nu,z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving power function





http://functions.wolfram.com/07.06.21.0003.01









  


  










Input Form





Integrate[z^(\[Alpha] - 1) Fibonacci[\[Nu], z], z] == -((2^(\[Alpha] - \[Nu] - 1) z^\[Alpha] (z + Sqrt[4 + z^2])^(1 - \[Nu]) (4^\[Nu] (1 - \[Alpha] + \[Nu]) Cos[Pi \[Nu]] Hypergeometric2F1[ (1 - \[Alpha] - \[Nu])/2, 1 - \[Alpha], (3 - \[Alpha] - \[Nu])/2, (1/4) (z + Sqrt[4 + z^2])^2] + (z + Sqrt[4 + z^2])^(2 \[Nu]) (\[Alpha] + \[Nu] - 1) Hypergeometric2F1[(1 - \[Alpha] + \[Nu])/2, 1 - \[Alpha], (3 - \[Alpha] + \[Nu])/2, (1/4) (z + Sqrt[4 + z^2])^2]))/((-z) (z + Sqrt[4 + z^2]))^\[Alpha])/ ((1 - \[Alpha] + \[Nu]) (\[Alpha] + \[Nu] - 1))










Standard Form





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MathML Form







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</ci> </apply> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> &#945; </ci> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["\[Alpha]", "-", "\[Nu]", "-", "1"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]], RowBox[List["1", "-", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "z"]], " ", RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]]]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["4", "\[Nu]"], " ", RowBox[List["(", RowBox[List["1", "-", "\[Alpha]", "+", "\[Nu]"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "\[Alpha]", "-", "\[Nu]"]], ")"]]]], ",", RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["3", "-", "\[Alpha]", "-", "\[Nu]"]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]], "2"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]], RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Nu]", "-", "1"]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "\[Alpha]", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["3", "-", "\[Alpha]", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]], "2"]]]]], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[Alpha]", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Nu]", "-", "1"]], ")"]]]]]]]]]]]










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





2001-10-29





© 1998- Wolfram Research, Inc.