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variants of this functions
Fibonacci






Mathematica Notation

Traditional Notation









Integer Functions > Fibonacci[nu] > Series representations > Asymptotic series expansions





http://functions.wolfram.com/04.11.06.0020.01









  


  










Input Form





Fibonacci[\[Nu]] \[Proportional] Piecewise[{{(1/Sqrt[5]) GoldenRatio^\[Nu], Im[\[Nu]] < 0 && Re[\[Nu]] - Pi Abs[Im[\[Nu]]] > 0}, {(-(1/(2 Sqrt[5]))) E^(I \[Nu] Pi - \[Nu] ArcCsch[2]), Im[\[Nu]] < 0 && Re[\[Nu]] + Pi Im[\[Nu]] < 0}, {(-(1/(2 Sqrt[5]))) E^((-I) Pi \[Nu] - \[Nu] ArcCsch[2]), Im[\[Nu]] > 0 && Re[\[Nu]] - Pi Im[\[Nu]] < 0}}, (1/Sqrt[5]) (GoldenRatio^\[Nu] - Cos[\[Nu] Pi]/GoldenRatio^\[Nu])] /; (Abs[\[Nu]] -> Infinity)










Standard Form





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MathML Form







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</ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <abs /> <apply> <imaginary /> <ci> &#957; </ci> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </piece> <piece> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> &#957; </ci> <pi /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#957; </ci> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </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> <apply> <and /> <apply> <lt /> <apply> <imaginary /> <ci> &#957; </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <lt /> <apply> <plus /> <apply> <times /> <pi /> <apply> <imaginary /> <ci> &#957; </ci> </apply> </apply> <apply> <real /> <ci> &#957; </ci> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </piece> <piece> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> </apply> <ci> &#957; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <pi /> <ci> &#957; </ci> </apply> </apply> </apply> </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> <apply> <and /> <apply> <gt /> <apply> <imaginary /> <ci> &#957; </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <lt /> <apply> <plus /> <apply> <real /> <ci> &#957; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <imaginary /> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </piece> <otherwise> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> GoldenRatio </ci> <ci> &#957; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <cos /> <apply> <times /> <ci> &#957; </ci> <pi /> </apply> </apply> <apply> <power /> <ci> GoldenRatio </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </otherwise> </piecewise> </apply> <apply> <ci> Rule </ci> <apply> <abs /> <ci> &#957; </ci> </apply> <infinity /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Fibonacci", "[", "\[Nu]_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[Piecewise]", GridBox[List[List[FractionBox[SuperscriptBox["GoldenRatio", "\[Nu]"], SqrtBox["5"]], RowBox[List[RowBox[List[RowBox[List["Im", "[", "\[Nu]", "]"]], "<", "0"]], "&&", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], "-", RowBox[List["\[Pi]", " ", RowBox[List["Abs", "[", RowBox[List["Im", "[", "\[Nu]", "]"]], "]"]]]]]], ">", "0"]]]]], List[RowBox[List["-", FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "\[Nu]", " ", "\[Pi]"]], "-", RowBox[List["\[Nu]", " ", RowBox[List["ArcCsch", "[", "2", "]"]]]]]]], RowBox[List["2", " ", SqrtBox["5"]]]]]], RowBox[List[RowBox[List[RowBox[List["Im", "[", "\[Nu]", "]"]], "<", "0"]], "&&", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], "+", RowBox[List["\[Pi]", " ", RowBox[List["Im", "[", "\[Nu]", "]"]]]]]], "<", "0"]]]]], List[RowBox[List["-", FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]", " ", "\[Nu]"]], "-", RowBox[List["\[Nu]", " ", RowBox[List["ArcCsch", "[", "2", "]"]]]]]]], RowBox[List["2", " ", SqrtBox["5"]]]]]], RowBox[List[RowBox[List[RowBox[List["Im", "[", "\[Nu]", "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], "-", RowBox[List["\[Pi]", " ", RowBox[List["Im", "[", "\[Nu]", "]"]]]]]], "<", "0"]]]]], List[FractionBox[RowBox[List[SuperscriptBox["GoldenRatio", "\[Nu]"], "-", RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", SuperscriptBox["GoldenRatio", RowBox[List["-", "\[Nu]"]]]]]]], SqrtBox["5"]], TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]]]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]










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





2007-05-02