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variants of this functions
Fibonacci






Mathematica Notation

Traditional Notation









Integer Functions > Fibonacci[nu] > Complex characteristics > Real part





http://functions.wolfram.com/04.11.19.0006.01









  


  










Input Form





Re[Fibonacci[x + I y]] == (1/Sqrt[5]) (((1/2) (1 + Sqrt[5]))^x Cos[y ArcCsch[2]] + ((1/2) (-1 + Sqrt[5]))^x ((-Cos[Pi x]) Cos[y ArcCsch[2]] Cosh[Pi y] + Sin[Pi x] Sin[y ArcCsch[2]] Sinh[Pi y]))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Re", "[", RowBox[List["Fibonacci", "[", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", SqrtBox["5"]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["5"]]], ")"]]]], ")"]], "x"], " ", RowBox[List["Cos", "[", RowBox[List["y", " ", RowBox[List["ArcCsch", "[", "2", "]"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox["5"]]], ")"]]]], ")"]], "x"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "x"]], "]"]]]], " ", RowBox[List["Cos", "[", RowBox[List["y", " ", RowBox[List["ArcCsch", "[", "2", "]"]]]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["\[Pi]", " ", "y"]], "]"]]]], "+", RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "x"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["y", " ", RowBox[List["ArcCsch", "[", "2", "]"]]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["\[Pi]", " ", "y"]], "]"]]]]]], ")"]]]]]], ")"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mrow> </msub> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mn> 5 </mn> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> y </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> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> y </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> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <msqrt> <mn> 5 </mn> </msqrt> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> x </mi> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <msqrt> <mn> 5 </mn> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> x </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> y </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> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <real /> <apply> <ci> Fibonacci </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <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> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <sin /> <apply> <times /> <pi /> <ci> x </ci> </apply> </apply> <apply> <sin /> <apply> <times /> <ci> y </ci> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <sinh /> <apply> <times /> <pi /> <ci> y </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <cos /> <apply> <times /> <pi /> <ci> x </ci> </apply> </apply> <apply> <cos /> <apply> <times /> <ci> y </ci> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <cosh /> <apply> <times /> <pi /> <ci> y </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> x </ci> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> x </ci> </apply> <apply> <cos /> <apply> <times /> <ci> y </ci> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Re", "[", RowBox[List["Fibonacci", "[", RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]], "]"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["5"]]], ")"]]]], ")"]], "x"], " ", RowBox[List["Cos", "[", RowBox[List["y", " ", RowBox[List["ArcCsch", "[", "2", "]"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox["5"]]], ")"]]]], ")"]], "x"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "x"]], "]"]]]], " ", RowBox[List["Cos", "[", RowBox[List["y", " ", RowBox[List["ArcCsch", "[", "2", "]"]]]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["\[Pi]", " ", "y"]], "]"]]]], "+", RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "x"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["y", " ", RowBox[List["ArcCsch", "[", "2", "]"]]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["\[Pi]", " ", "y"]], "]"]]]]]], ")"]]]]]], SqrtBox["5"]]]]]]










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





2007-05-02