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http://functions.wolfram.com/04.11.19.0005.01
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Conjugate[Fibonacci[x + I y]] == (1/(GoldenRatio^x Sqrt[5]))
(GoldenRatio^(2 x) (Cos[y Log[GoldenRatio]] - I Sin[y Log[GoldenRatio]]) -
Cos[Pi x] Cosh[Pi y] (Cos[y Log[GoldenRatio]] +
I Sin[y Log[GoldenRatio]]) + Sin[Pi x] Sinh[Pi y]
((-I) Cos[y Log[GoldenRatio]] + Sin[y Log[GoldenRatio]]))
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Cell[BoxData[RowBox[List[RowBox[List["Conjugate", "[", RowBox[List["Fibonacci", "[", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], "]"]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox["GoldenRatio", RowBox[List["-", "x"]]], SqrtBox["5"]], " ", RowBox[List["(", RowBox[List[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["Cos", "[", RowBox[List["\[Pi]", " ", "x"]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["\[Pi]", " ", "y"]], "]"]], " ", 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["Sinh", "[", RowBox[List["\[Pi]", " ", "y"]], "]"]], RowBox[List["(", 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", "]"]]]], "]"]]]], ")"]]]]]], ")"]]]]]]]]
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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> <mover> <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> </mover> <mo> ⩵ </mo> <mrow> <mfrac> <msup> <semantics> <mi> ϕ </mi> <annotation encoding='Mathematica'> TagBox["\[Phi]", Function[GoldenRatio]] </annotation> </semantics> <mrow> <mo> - </mo> <mi> x </mi> </mrow> </msup> <msqrt> <mn> 5 </mn> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <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> <mi> sinh </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> 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> <mo> - </mo> <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> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <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> 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> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> OverBar </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 /> <ci> GoldenRatio </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> <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> <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 /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <sin /> <apply> <times /> <ci> y </ci> <apply> <ln /> <ci> GoldenRatio </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <sin /> <apply> <times /> <pi /> <ci> x </ci> </apply> </apply> <apply> <sinh /> <apply> <times /> <pi /> <ci> y </ci> </apply> </apply> <apply> <plus /> <apply> <sin /> <apply> <times /> <ci> y </ci> <apply> <ln /> <ci> GoldenRatio </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <cos /> <apply> <times /> <ci> y </ci> <apply> <ln /> <ci> GoldenRatio </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <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> <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> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Conjugate", "[", RowBox[List["Fibonacci", "[", RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]], "]"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["GoldenRatio", RowBox[List["-", "x"]]], " ", RowBox[List["(", RowBox[List[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["Cos", "[", RowBox[List["\[Pi]", " ", "x"]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["\[Pi]", " ", "y"]], "]"]], " ", 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["Sinh", "[", RowBox[List["\[Pi]", " ", "y"]], "]"]], " ", RowBox[List["(", 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", "]"]]]], "]"]]]], ")"]]]]]], ")"]]]], SqrtBox["5"]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
