Lucas numbers: Complex characteristics (formula 04.22.19.0005)

LucasL

$Mathematica Notation$

Integer Functions

LucasL[nu]

Complex characteristics

Absolute value

http://functions.wolfram.com/04.22.19.0005.01

Input Form

Abs[LucasL[x + I y]] == (1/Sqrt[2]) Sqrt[(4^x Cos[2 Pi x] + 2 ((3 + Sqrt[5])^(2 x) + ((1 + Sqrt[5])^(2 (x - I y)) ((1 + Sqrt[5])^(4 I y) Cos[Pi (x - I y)] + 2^(4 I y) Cos[Pi (x + I y)]))/2^(2 I y)) + 4^x Cosh[2 Pi y])/(1 + Sqrt[5])^(2 x)]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["Abs", "[", RowBox[List["LucasL", "[", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", SqrtBox["2"]], RowBox[List["\[Sqrt]", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["5"]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", "x"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["4", "x"], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "\[Pi]", " ", "x"]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["3", "+", SqrtBox["5"]]], ")"]], RowBox[List["2", " ", "x"]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "y"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["5"]]], ")"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["x", "-", RowBox[List["\[ImaginaryI]", " ", "y"]]]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["5"]]], ")"]], RowBox[List["4", " ", "\[ImaginaryI]", " ", "y"]]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["x", "-", RowBox[List["\[ImaginaryI]", " ", "y"]]]], ")"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["4", " ", "\[ImaginaryI]", " ", "y"]]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], ")"]]]], "]"]]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["4", "x"], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "\[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> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <msub> <semantics> <mi> L </mi> <annotation encoding='Mathematica'> TagBox["L", LucasL] </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> 2 </mn> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mo> √ </mo> <mrow> <mo> ( </mo> <mrow> <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> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mn> 4 </mn> <mi> x </mi> </msup> <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> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <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> <mo> + </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </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> <mn> 4 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mn> 4 </mn> <mi> x </mi> </msup> <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> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <abs /> <apply> <ci> LucasL </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'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <root /> <apply> <times /> <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> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 4 </cn> <ci> x </ci> </apply> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <ci> x </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <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> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -2 </cn> <imaginaryi /> <ci> y </ci> </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> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <cos /> <apply> <times /> <pi /> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <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'> 4 </cn> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <imaginaryi /> <ci> y </ci> </apply> </apply> <apply> <cos /> <apply> <times /> <pi /> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 4 </cn> <ci> x </ci> </apply> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <ci> y </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Abs", "[", RowBox[List["LucasL", "[", RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]], "]"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["5"]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", "x"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["4", "x"], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "\[Pi]", " ", "x"]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["3", "+", SqrtBox["5"]]], ")"]], RowBox[List["2", " ", "x"]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "y"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["5"]]], ")"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["x", "-", RowBox[List["\[ImaginaryI]", " ", "y"]]]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["5"]]], ")"]], RowBox[List["4", " ", "\[ImaginaryI]", " ", "y"]]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["x", "-", RowBox[List["\[ImaginaryI]", " ", "y"]]]], ")"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["4", " ", "\[ImaginaryI]", " ", "y"]]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], ")"]]]], "]"]]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["4", "x"], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "\[Pi]", " ", "y"]], "]"]]]]]], ")"]]]]], SqrtBox["2"]]]]]]

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

2007-05-02