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http://functions.wolfram.com/04.11.19.0008.01
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Abs[Fibonacci[x + I y]] == (1/Sqrt[10])
Sqrt[-4 Cos[Pi x] Cos[2 y ArcCsch[2]] Cosh[Pi y] +
(2 (3 + Sqrt[5])^(2 x) + 4^x (Cos[2 Pi x] + Cosh[2 Pi y]))/
(1 + Sqrt[5])^(2 x) + 4 Sin[Pi x] Sin[2 y ArcCsch[2]] Sinh[Pi y]]
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Cell[BoxData[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Fibonacci", "[", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", SqrtBox["10"]], RowBox[List["(", RowBox[List["\[Sqrt]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "x"]], "]"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "y", " ", RowBox[List["ArcCsch", "[", "2", "]"]]]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["\[Pi]", " ", "y"]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["5"]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", "x"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["3", "+", SqrtBox["5"]]], ")"]], RowBox[List["2", " ", "x"]]]]], "+", RowBox[List[SuperscriptBox["4", "x"], " ", RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List["2", " ", "\[Pi]", " ", "x"]], "]"]], "+", RowBox[List["Cosh", "[", RowBox[List["2", " ", "\[Pi]", " ", "y"]], "]"]]]], ")"]]]]]], ")"]]]], "+", RowBox[List["4", " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "x"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "y", " ", RowBox[List["ArcCsch", "[", "2", "]"]]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["\[Pi]", " ", "y"]], "]"]]]]]], ")"]]]], ")"]]]]]]]]
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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> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <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> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo>  </mo> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mn> 10 </mn> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mo> √ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> + </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> x </mi> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mn> 4 </mn> <mi> x </mi> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <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> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> x </mi> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <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> <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> ⁢ </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> 4 </mn> <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> <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> ⁢ </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> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <abs /> <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'> 10 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <root /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 3 </cn> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> x </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 4 </cn> <ci> x </ci> </apply> <apply> <plus /> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <ci> x </ci> </apply> </apply> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <ci> y </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -2 </cn> <ci> x </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <cos /> <apply> <times /> <pi /> <ci> x </ci> </apply> </apply> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <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> <times /> <cn type='integer'> 4 </cn> <apply> <sin /> <apply> <times /> <pi /> <ci> x </ci> </apply> </apply> <apply> <sin /> <apply> <times /> <cn type='integer'> 2 </cn> <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> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Abs", "[", RowBox[List["Fibonacci", "[", RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]], "]"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "x"]], "]"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "y", " ", RowBox[List["ArcCsch", "[", "2", "]"]]]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["\[Pi]", " ", "y"]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["5"]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", "x"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["3", "+", SqrtBox["5"]]], ")"]], RowBox[List["2", " ", "x"]]]]], "+", RowBox[List[SuperscriptBox["4", "x"], " ", RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List["2", " ", "\[Pi]", " ", "x"]], "]"]], "+", RowBox[List["Cosh", "[", RowBox[List["2", " ", "\[Pi]", " ", "y"]], "]"]]]], ")"]]]]]], ")"]]]], "+", RowBox[List["4", " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "x"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "y", " ", RowBox[List["ArcCsch", "[", "2", "]"]]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["\[Pi]", " ", "y"]], "]"]]]]]]], SqrtBox["10"]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
