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variants of this functions
Fibonacci






Mathematica Notation

Traditional Notation









Integer Functions > Fibonacci[nu] > Complex characteristics > Signum value





http://functions.wolfram.com/04.11.19.0011.01









  


  










Input Form





Sign[Fibonacci[x + I y]] == (Sqrt[2] (Cos[Pi x] Cosh[Pi y] (-Cos[y Log[GoldenRatio]] + I Sin[y Log[GoldenRatio]]) + GoldenRatio^(2 x) (Cos[y Log[GoldenRatio]] + I Sin[y Log[GoldenRatio]]) + Sin[Pi x] (I Cos[y Log[GoldenRatio]] + Sin[y Log[GoldenRatio]]) Sinh[Pi y]))/GoldenRatio^x/ Sqrt[(2 GoldenRatio^(4 x) + Cos[2 Pi x] - 4 GoldenRatio^(2 x) Cos[Pi x] Cos[2 y Log[GoldenRatio]] Cosh[Pi y] + Cosh[Pi y]^2 + 4 GoldenRatio^(2 x) Sin[Pi x] Sin[2 y Log[GoldenRatio]] Sinh[Pi y] + Sinh[Pi y]^2)/GoldenRatio^(2 x)]










Standard Form





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MathML Form







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</mi> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, Function[GoldenRatio]] </annotation> </semantics> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> x </mi> </mrow> </msup> <mo> &#8290; </mo> <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> <mn> 2 </mn> <mo> &#8290; </mo> <mi> y </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, Function[GoldenRatio]] </annotation> </semantics> <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> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Sign </ci> <apply> <ci> Fibonacci </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> GoldenRatio </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <cos /> <apply> <times /> <pi /> <ci> x </ci> </apply> </apply> <apply> <cosh /> <apply> <times /> <pi /> <ci> y </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <sin /> <apply> <times /> <ci> y </ci> <apply> <ln /> <ci> GoldenRatio </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cos /> <apply> <times /> <ci> y </ci> <apply> <ln /> <ci> GoldenRatio </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> GoldenRatio </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> x </ci> </apply> </apply> <apply> <plus /> <apply> <cos /> <apply> <times /> <ci> y </ci> <apply> <ln /> <ci> GoldenRatio </ci> </apply> </apply> </apply> <apply> <times /> <imaginaryi /> <apply> <sin /> <apply> <times /> <ci> y </ci> <apply> <ln /> <ci> GoldenRatio </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <sin /> <apply> <times /> <pi /> <ci> x </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <cos /> <apply> <times /> <ci> y </ci> <apply> <ln /> <ci> GoldenRatio </ci> </apply> </apply> </apply> </apply> <apply> <sin /> <apply> <times /> <ci> y </ci> <apply> <ln /> <ci> GoldenRatio </ci> </apply> </apply> </apply> </apply> <apply> <sinh /> <apply> <times /> <pi /> <ci> y </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <root /> <apply> <times /> <apply> <power /> <ci> GoldenRatio </ci> <apply> <times /> <cn type='integer'> -2 </cn> <ci> x </ci> </apply> </apply> <apply> <plus /> <apply> <power /> <apply> <cosh /> <apply> <times /> <pi /> <ci> y </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> GoldenRatio </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> x </ci> </apply> </apply> <apply> <cos /> <apply> <times /> <pi /> <ci> x </ci> </apply> </apply> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> y </ci> <apply> <ln /> <ci> GoldenRatio </ci> </apply> </apply> </apply> <apply> <cosh /> <apply> <times /> <pi /> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> GoldenRatio </ci> <apply> <times /> <cn type='integer'> 4 </cn> <ci> x </ci> </apply> </apply> </apply> <apply> <power /> <apply> <sinh /> <apply> <times /> <pi /> <ci> y </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <ci> x </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> GoldenRatio </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> x </ci> </apply> </apply> <apply> <sin /> <apply> <times /> <pi /> <ci> x </ci> </apply> </apply> <apply> <sin /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> y </ci> <apply> <ln /> <ci> GoldenRatio </ci> </apply> </apply> </apply> <apply> <sinh /> <apply> <times /> <pi /> <ci> y </ci> </apply> </apply> </apply> </apply> </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["Sign", "[", RowBox[List["Fibonacci", "[", RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]], "]"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SqrtBox["2"], " ", SuperscriptBox["GoldenRatio", RowBox[List["-", "x"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "x"]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["\[Pi]", " ", "y"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Cos", "[", RowBox[List["y", " ", RowBox[List["Log", "[", "GoldenRatio", "]"]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", RowBox[List["y", " ", RowBox[List["Log", "[", "GoldenRatio", "]"]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["GoldenRatio", RowBox[List["2", " ", "x"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List["y", " ", RowBox[List["Log", "[", "GoldenRatio", "]"]]]], "]"]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", RowBox[List["y", " ", RowBox[List["Log", "[", "GoldenRatio", "]"]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "x"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Cos", "[", RowBox[List["y", " ", RowBox[List["Log", "[", "GoldenRatio", "]"]]]], "]"]]]], "+", RowBox[List["Sin", "[", RowBox[List["y", " ", RowBox[List["Log", "[", "GoldenRatio", "]"]]]], "]"]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["\[Pi]", " ", "y"]], "]"]]]]]], ")"]]]], SqrtBox[RowBox[List[SuperscriptBox["GoldenRatio", RowBox[List[RowBox[List["-", "2"]], " ", "x"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["GoldenRatio", RowBox[List["4", " ", "x"]]]]], "+", RowBox[List["Cos", "[", RowBox[List["2", " ", "\[Pi]", " ", "x"]], "]"]], "-", RowBox[List["4", " ", SuperscriptBox["GoldenRatio", RowBox[List["2", " ", "x"]]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "x"]], "]"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "y", " ", RowBox[List["Log", "[", "GoldenRatio", "]"]]]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["\[Pi]", " ", "y"]], "]"]]]], "+", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["\[Pi]", " ", "y"]], "]"]], "2"], "+", RowBox[List["4", " ", SuperscriptBox["GoldenRatio", RowBox[List["2", " ", "x"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "x"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "y", " ", RowBox[List["Log", "[", "GoldenRatio", "]"]]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["\[Pi]", " ", "y"]], "]"]]]], "+", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["\[Pi]", " ", "y"]], "]"]], "2"]]], ")"]]]]]]]]]]










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





2007-05-02





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