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 | | http://functions.wolfram.com/04.11.19.0011.01 | 
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 | | 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)] | 
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 | | Cell[BoxData[RowBox[List[RowBox[List["Sign", "[", RowBox[List["Fibonacci", "[", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["(", 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"]], "]"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["\[Sqrt]", RowBox[List["(", 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"]]], ")"]]]], ")"]]]], ")"]]]]]]]] | 
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   <math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'>  <semantics>  <mrow>  <mrow>  <mi> sgn </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <msub>  <semantics>  <mi> F </mi>  <annotation encoding='Mathematica'> TagBox["F", Fibonacci] </annotation>  </semantics>  <mrow>  <mi> x </mi>  <mo> + </mo>  <mrow>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <mi> y </mi>  </mrow>  </mrow>  </msub>  <mo> ) </mo>  </mrow>  <mo> ⩵ </mo>  <mrow>  <mrow>  <mo> ( </mo>  <mrow>  <msqrt>  <mn> 2 </mn>  </msqrt>  <mo> ⁢ </mo>  <msup>  <semantics>  <mi> ϕ </mi>  <annotation encoding='Mathematica'> TagBox["\[Phi]", Function[GoldenRatio]] </annotation>  </semantics>  <mrow>  <mo> - </mo>  <mi> x </mi>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mrow>  <mi> cos </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> π </mi>  <mo> ⁢ </mo>  <mi> x </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mi> cosh </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> π </mi>  <mo> ⁢ </mo>  <mi> y </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <mrow>  <mi> sin </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> y </mi>  <mo> ⁢ </mo>  <mrow>  <mi> log </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <semantics>  <mi> ϕ </mi>  <annotation encoding='Mathematica'> TagBox["\[Phi]", Function[GoldenRatio]] </annotation>  </semantics>  <mo> ) </mo>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> - </mo>  <mrow>  <mi> cos </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> y </mi>  <mo> ⁢ </mo>  <mrow>  <mi> log </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <semantics>  <mi> ϕ </mi>  <annotation encoding='Mathematica'> TagBox["\[Phi]", Function[GoldenRatio]] </annotation>  </semantics>  <mo> ) </mo>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> + </mo>  <mrow>  <msup>  <semantics>  <mi> ϕ </mi>  <annotation encoding='Mathematica'> TagBox["\[Phi]", Function[GoldenRatio]] </annotation>  </semantics>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> x </mi>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mi> cos </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> y </mi>  <mo> ⁢ </mo>  <mrow>  <mi> log </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <semantics>  <mi> ϕ </mi>  <annotation encoding='Mathematica'> TagBox["\[Phi]", Function[GoldenRatio]] </annotation>  </semantics>  <mo> ) </mo>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  <mo> + </mo>  <mrow>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <mrow>  <mi> sin </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> y </mi>  <mo> ⁢ </mo>  <mrow>  <mi> log </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <semantics>  <mi> ϕ </mi>  <annotation encoding='Mathematica'> TagBox["\[Phi]", Function[GoldenRatio]] </annotation>  </semantics>  <mo> ) </mo>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> + </mo>  <mrow>  <mrow>  <mi> sin </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> π </mi>  <mo> ⁢ </mo>  <mi> x </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <mrow>  <mi> cos </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> y </mi>  <mo> ⁢ </mo>  <mrow>  <mi> log </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <semantics>  <mi> ϕ </mi>  <annotation encoding='Mathematica'> TagBox["\[Phi]", Function[GoldenRatio]] </annotation>  </semantics>  <mo> ) </mo>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> + </mo>  <mrow>  <mi> sin </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> y </mi>  <mo> ⁢ </mo>  <mrow>  <mi> log </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <semantics>  <mi> ϕ </mi>  <annotation encoding='Mathematica'> TagBox["\[Phi]", Function[GoldenRatio]] </annotation>  </semantics>  <mo> ) </mo>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mi> sinh </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> π </mi>  <mo> ⁢ </mo>  <mi> y </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  <mo> / </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mo> √ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <msup>  <semantics>  <mi> ϕ </mi>  <annotation encoding='Mathematica'> TagBox["\[Phi]", Function[GoldenRatio]] </annotation>  </semantics>  <mrow>  <mrow>  <mo> - </mo>  <mn> 2 </mn>  </mrow>  <mo> ⁢ </mo>  <mi> x </mi>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <msup>  <mi> cosh </mi>  <mn> 2 </mn>  </msup>  <mo> ( </mo>  <mrow>  <mi> π </mi>  <mo> ⁢ </mo>  <mi> y </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mo> - </mo>  <mrow>  <mn> 4 </mn>  <mo> ⁢ </mo>  <msup>  <semantics>  <mi> ϕ </mi>  <annotation encoding='Mathematica'> TagBox["\[Phi]", Function[GoldenRatio]] </annotation>  </semantics>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> x </mi>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <mi> cos </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> π </mi>  <mo> ⁢ </mo>  <mi> x </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mi> cos </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> y </mi>  <mo> ⁢ </mo>  <mrow>  <mi> log </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <semantics>  <mi> ϕ </mi>  <annotation encoding='Mathematica'> TagBox["\[Phi]", Function[GoldenRatio]] </annotation>  </semantics>  <mo> ) </mo>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mi> cosh </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> π </mi>  <mo> ⁢ </mo>  <mi> y </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> + </mo>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <msup>  <semantics>  <mi> ϕ </mi>  <annotation encoding='Mathematica'> TagBox["\[Phi]", Function[GoldenRatio]] </annotation>  </semantics>  <mrow>  <mn> 4 </mn>  <mo> ⁢ </mo>  <mi> x </mi>  </mrow>  </msup>  </mrow>  <mo> + </mo>  <mrow>  <msup>  <mi> sinh </mi>  <mn> 2 </mn>  </msup>  <mo> ( </mo>  <mrow>  <mi> π </mi>  <mo> ⁢ </mo>  <mi> y </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mo> + </mo>  <mrow>  <mi> cos </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> π </mi>  <mo> ⁢ </mo>  <mi> x </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mo> + </mo>  <mrow>  <mn> 4 </mn>  <mo> ⁢ </mo>  <msup>  <semantics>  <mi> ϕ </mi>  <annotation encoding='Mathematica'> TagBox["\[Phi]", Function[GoldenRatio]] </annotation>  </semantics>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> x </mi>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <mi> sin </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> π </mi>  <mo> ⁢ </mo>  <mi> x </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mi> sin </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> y </mi>  <mo> ⁢ </mo>  <mrow>  <mi> log </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <semantics>  <mi> ϕ </mi>  <annotation encoding='Mathematica'> TagBox["\[Phi]", Function[GoldenRatio]] </annotation>  </semantics>  <mo> ) </mo>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mi> sinh </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> π </mi>  <mo> ⁢ </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> 
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