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

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
Fibonacci






Mathematica Notation

Traditional Notation









Hypergeometric Functions > Fibonacci[nu,z] > Representations through equivalent functions > With elementary functions





http://functions.wolfram.com/07.06.27.0007.01









  


  










Input Form





Fibonacci[\[Nu], z] == ((((-1)^(1/4) (2 Cos[Pi \[Nu]] + I Sin[Pi \[Nu]]))/ (Sqrt[2 - I z] Sqrt[z - 2 I])) Sin[2 \[Nu] ArcCsc[(2 (-1)^(3/4))/Sqrt[z - 2 I]]])/E^((I Pi \[Nu])/2) + (((I Sin[Pi \[Nu]])/Sqrt[4 + z^2]) Cos[(\[Nu]/2) (Pi + 2 I ArcSinh[z/2])])/ E^((I Pi \[Nu])/2)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]]]], ")"]]]], RowBox[List[SqrtBox[RowBox[List["2", "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]]], " ", SqrtBox[RowBox[List["z", "-", RowBox[List["2", " ", "\[ImaginaryI]"]]]]]]]], FormBox[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List[" ", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]], "2"]]]], TraditionalForm], RowBox[List["Sin", "[", RowBox[List["2", " ", "\[Nu]", " ", RowBox[List["ArcCsc", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]]]], SqrtBox[RowBox[List["z", "-", RowBox[List["2", " ", "\[ImaginaryI]"]]]]]], "]"]]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]], " ", FormBox[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List[" ", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]], "2"]]]], TraditionalForm], RowBox[List["Cos", "[", RowBox[List[FractionBox["\[Nu]", "2"], " ", RowBox[List["(", RowBox[List["\[Pi]", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", FractionBox["z", "2"], "]"]]]]]], ")"]]]], "]"]]]]]]]]]]










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> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 4 </mn> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mfrac> <mrow> <mtext> </mtext> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </mrow> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mtext> </mtext> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mroot> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mroot> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msqrt> <mrow> <mn> 2 </mn> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> z </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </mrow> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mfrac> <mrow> <mtext> </mtext> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </mrow> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> </mrow> <msqrt> <mrow> <mi> z </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Fibonacci </ci> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <sin /> <apply> <times /> <pi /> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <pi /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <cos /> <apply> <times /> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <arcsinh /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <pi /> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <cos /> <apply> <times /> <pi /> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <times /> <imaginaryi /> <apply> <sin /> <apply> <times /> <pi /> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <pi /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <sin /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> <apply> <arccsc /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <cn type='rational'> 3 <sep /> 4 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]]]], ")"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], ")"]]]]], " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "\[Nu]", " ", RowBox[List["ArcCsc", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]]]], SqrtBox[RowBox[List["z", "-", RowBox[List["2", " ", "\[ImaginaryI]"]]]]]], "]"]]]], "]"]]]], RowBox[List[SqrtBox[RowBox[List["2", "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]]], " ", SqrtBox[RowBox[List["z", "-", RowBox[List["2", " ", "\[ImaginaryI]"]]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], ")"]]]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "2"], " ", "\[Nu]", " ", RowBox[List["(", RowBox[List["\[Pi]", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", FractionBox["z", "2"], "]"]]]]]], ")"]]]], "]"]]]], SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]]]]]]]










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





2001-10-29





© 1998- Wolfram Research, Inc.