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variants of this functions
Fibonacci






Mathematica Notation

Traditional Notation









Integer Functions > Fibonacci[nu] > Differentiation > Symbolic differentiation





http://functions.wolfram.com/04.11.20.0003.02









  


  










Input Form





D[Fibonacci[\[Nu]], {\[Nu], n}] == (1/Sqrt[5]) (GoldenRatio^\[Nu] Log[GoldenRatio]^n - (((-1)^n/2) (E^(I Pi \[Nu]) (Log[GoldenRatio] - I Pi)^n + (Log[GoldenRatio] + I Pi)^n/E^(I Pi \[Nu])))/GoldenRatio^\[Nu]) /; Element[n, Integers] && n >= 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["\[Nu]", ",", "n"]], "}"]]], RowBox[List["Fibonacci", "[", "\[Nu]", "]"]]]], "\[Equal]", RowBox[List[FractionBox["1", SqrtBox["5"]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["GoldenRatio", "\[Nu]"], " ", SuperscriptBox[RowBox[List["Log", "[", "GoldenRatio", "]"]], "n"]]], "-", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], "2"], SuperscriptBox["GoldenRatio", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "GoldenRatio", "]"]], "-", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]]]], ")"]], "n"]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]", " ", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "GoldenRatio", "]"]], "+", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]]]], ")"]], "n"]]]]], ")"]]]]]], ")"]]]]]], "/;", " ", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]










MathML Form







<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> &#8706; </mo> <mi> n </mi> </msup> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mi> &#957; </mi> </msub> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> &#957; </mi> <mi> n </mi> </msup> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mn> 5 </mn> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, Function[GoldenRatio]] </annotation> </semantics> <mi> &#957; </mi> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mi> n </mi> </msup> <mo> ( </mo> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, Function[GoldenRatio]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> &#8290; </mo> <msup> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, Function[GoldenRatio]] </annotation> </semantics> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, Function[GoldenRatio]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, Function[GoldenRatio]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> &#957; </ci> <degree> <ci> n </ci> </degree> </bvar> <apply> <ci> Fibonacci </ci> <ci> &#957; </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <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> <ci> &#957; </ci> </apply> <apply> <power /> <apply> <ln /> <ci> GoldenRatio </ci> </apply> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <power /> <ci> GoldenRatio </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <pi /> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <ln /> <ci> GoldenRatio </ci> </apply> <apply> <times /> <imaginaryi /> <pi /> </apply> </apply> <ci> n </ci> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <ln /> <ci> GoldenRatio </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <pi /> </apply> </apply> </apply> <ci> n </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["\[Nu]_", ",", "n_"]], "}"]]]]], RowBox[List["Fibonacci", "[", "\[Nu]_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List[SuperscriptBox["GoldenRatio", "\[Nu]"], " ", SuperscriptBox[RowBox[List["Log", "[", "GoldenRatio", "]"]], "n"]]], "-", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["GoldenRatio", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "GoldenRatio", "]"]], "-", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]]]], ")"]], "n"]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]", " ", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "GoldenRatio", "]"]], "+", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]]]], ")"]], "n"]]]]], ")"]]]]]], SqrtBox["5"]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2001-10-29