Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











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





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Fibonacci", "[", "\[Nu]", "]"]], "\[Proportional]", RowBox[List["Piecewise", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", SqrtBox["5"]], SuperscriptBox["GoldenRatio", "\[Nu]"]]], ",", 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"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["2", SqrtBox["5"]]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "\[Nu]", " ", "\[Pi]"]], "-", RowBox[List["\[Nu]", " ", RowBox[List["ArcCsch", "[", "2", "]"]]]]]]]]], ",", 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"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["2", SqrtBox["5"]]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]", " ", "\[Nu]"]], "-", RowBox[List["\[Nu]", " ", RowBox[List["ArcCsch", "[", "2", "]"]]]]]]]]], ",", 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"]]]]]], "}"]]]], "}"]], ",", RowBox[List[FractionBox["1", SqrtBox["5"]], RowBox[List["(", RowBox[List[SuperscriptBox["GoldenRatio", "\[Nu]"], "-", RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", SuperscriptBox["GoldenRatio", RowBox[List["-", "\[Nu]"]]]]]]], ")"]]]]]], "]"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mi> &#957; </mi> </msub> <mo> &#8733; </mo> <mrow> <mo> &#62305; </mo> <mtable> <mtr> <mtd> <mfrac> <msup> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, Function[List[], GoldenRatio]] </annotation> </semantics> <mi> &#957; </mi> </msup> <msqrt> <mn> 5 </mn> </msqrt> </mfrac> </mtd> <mtd> <mrow> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> <mo> &lt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo> - </mo> <mfrac> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> - </mo> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> </mrow> </msup> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> <mo> &lt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> </mrow> <mo> &lt; </mo> <mn> 0 </mn> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo> - </mo> <mfrac> <msup> <mi> &#8519; </mi> <mrow> <mrow> <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> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </mrow> </msup> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &lt; </mo> <mn> 0 </mn> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <msup> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, Function[List[], GoldenRatio]] </annotation> </semantics> <mi> &#957; </mi> </msup> <mo> - </mo> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, Function[List[], GoldenRatio]] </annotation> </semantics> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> </msup> </mrow> </mrow> <msqrt> <mn> 5 </mn> </msqrt> </mfrac> </mtd> <mtd> <semantics> <mi> True </mi> <annotation encoding='Mathematica'> TagBox[&quot;True&quot;, &quot;PiecewiseDefault&quot;, Rule[AutoDelete, False], Rule[DeletionWarning, True]] </annotation> </semantics> </mtd> </mtr> </mtable> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> Fibonacci </ci> <ci> &#957; </ci> </apply> <piecewise> <piece> <apply> <times /> <apply> <power /> <ci> GoldenRatio </ci> <ci> &#957; </ci> </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> <apply> <and /> <apply> <lt /> <apply> <imaginary /> <ci> &#957; </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <apply> <plus /> <apply> <real /> <ci> &#957; </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





© 1998-2014 Wolfram Research, Inc.