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http://functions.wolfram.com/04.22.20.0003.01
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D[LucasL[\[Nu]], {\[Nu], n}] == GoldenRatio^\[Nu] Log[GoldenRatio]^n +
(1/(GoldenRatio^\[Nu] 2)) (E^(Pi I \[Nu]) (Pi I - Log[GoldenRatio])^n +
((-1)^n (Pi I + Log[GoldenRatio])^n)/E^(Pi I \[Nu])) /;
Element[n, Integers] && n >= 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["\[Nu]", ",", "n"]], "}"]]], RowBox[List["LucasL", "[", "\[Nu]", "]"]]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["GoldenRatio", "\[Nu]"], " ", SuperscriptBox[RowBox[List["Log", "[", "GoldenRatio", "]"]], "n"]]], "+", RowBox[List[FractionBox[SuperscriptBox["GoldenRatio", RowBox[List["-", "\[Nu]"]]], "2"], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Nu]"]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "-", RowBox[List["Log", "[", "GoldenRatio", "]"]]]], ")"]], "n"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", "\[Nu]"]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "+", RowBox[List["Log", "[", "GoldenRatio", "]"]]]], ")"]], "n"]]]]], ")"]]]]]]]], "/;", " ", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]
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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> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mi> n </mi> </msup> <msub> <semantics> <mi> L </mi> <annotation encoding='Mathematica'> TagBox["L", LucasL] </annotation> </semantics> <mi> ν </mi> </msub> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> ν </mi> <mi> n </mi> </msup> </mrow> </mfrac> <mo>  </mo> <mrow> <mrow> <msup> <semantics> <mi> ϕ </mi> <annotation encoding='Mathematica'> TagBox["\[Phi]", Function[List[], GoldenRatio]] </annotation> </semantics> <mi> ν </mi> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> log </mi> <mi> n </mi> </msup> <mo> ( </mo> <semantics> <mi> ϕ </mi> <annotation encoding='Mathematica'> TagBox["\[Phi]", Function[List[], GoldenRatio]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <msup> <semantics> <mi> ϕ </mi> <annotation encoding='Mathematica'> TagBox["\[Phi]", Function[List[], GoldenRatio]] </annotation> </semantics> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <semantics> <mi> ϕ </mi> <annotation encoding='Mathematica'> TagBox["\[Phi]", Function[List[], GoldenRatio]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> π </mi> </mrow> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <mo> + </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <semantics> <mi> ϕ </mi> <annotation encoding='Mathematica'> TagBox["\[Phi]", Function[List[], GoldenRatio]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> ν </ci> <degree> <ci> n </ci> </degree> </bvar> <apply> <ci> LucasL </ci> <ci> ν </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> GoldenRatio </ci> <ci> ν </ci> </apply> <apply> <power /> <apply> <ln /> <ci> GoldenRatio </ci> </apply> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <ci> GoldenRatio </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <pi /> <imaginaryi /> <ci> ν </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <pi /> <imaginaryi /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <ci> GoldenRatio </ci> </apply> </apply> </apply> <ci> n </ci> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> <imaginaryi /> <ci> ν </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <pi /> <imaginaryi /> </apply> <apply> <ln /> <ci> GoldenRatio </ci> </apply> </apply> <ci> n </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["\[Nu]_", ",", "n_"]], "}"]]]]], RowBox[List["LucasL", "[", "\[Nu]_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["GoldenRatio", "\[Nu]"], " ", SuperscriptBox[RowBox[List["Log", "[", "GoldenRatio", "]"]], "n"]]], "+", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["GoldenRatio", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "-", RowBox[List["Log", "[", "GoldenRatio", "]"]]]], ")"]], "n"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "+", RowBox[List["Log", "[", "GoldenRatio", "]"]]]], ")"]], "n"]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
