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variants of this functions
Fibonacci




Mathematica Notation

Traditional Notation









Integer Functions > Fibonacci[nu] > Representations through equivalent functions > With elementary functions





http://functions.wolfram.com/04.11.27.0002.01









  


  








Input Form





Fibonacci[\[Nu]] == (1/(2 Sqrt[5])) (2 GoldenRatio^\[Nu] - E^(\[Nu] ((-I) Pi - Log[GoldenRatio])) - E^(\[Nu] (I Pi - Log[GoldenRatio])))








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Fibonacci", "[", "\[Nu]", "]"]], "\[Equal]", FractionBox["1", RowBox[List["2", " ", SqrtBox["5"]]]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["GoldenRatio", "\[Nu]"]]], "-", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]"]], "-", RowBox[List["Log", "[", "GoldenRatio", "]"]]]], ")"]]]]], "-", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "-", RowBox[List["Log", "[", "GoldenRatio", "]"]]]], ")"]]]]]]], ")"]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mi> &#957; </mi> </msub> <mo> &#10869; </mo> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[TagBox[&quot;\[Phi]&quot;, Function[List[], GoldenRatio]], Function[GoldenRatio]] </annotation> </semantics> <mi> &#957; </mi> </msup> </mrow> <mo> - </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[TagBox[&quot;\[Phi]&quot;, Function[List[], GoldenRatio]], Function[GoldenRatio]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> - </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[TagBox[&quot;\[Phi]&quot;, Function[List[], GoldenRatio]], Function[GoldenRatio]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Fibonacci </ci> <ci> &#957; </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> GoldenRatio </ci> <ci> &#957; </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <ci> &#957; </ci> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <pi /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <ci> GoldenRatio </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <ci> &#957; </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <pi /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <ci> GoldenRatio </ci> </apply> </apply> </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> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Fibonacci", "[", "\[Nu]_", "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["2", " ", SuperscriptBox["GoldenRatio", "\[Nu]"]]], "-", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]"]], "-", RowBox[List["Log", "[", "GoldenRatio", "]"]]]], ")"]]]]], "-", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "-", RowBox[List["Log", "[", "GoldenRatio", "]"]]]], ")"]]]]]]], RowBox[List["2", " ", SqrtBox["5"]]]]]]]]


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





2001-10-29