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GoldenRatio




Mathematica Notation

Traditional Notation









Constants > GoldenRatio > Representations through more general functions > Through other functions





http://functions.wolfram.com/02.02.26.0010.01









  


  








Input Form





GoldenRatio == (Sqrt[5] Fibonacci[\[Nu]] + Sqrt[4 Cos[Pi \[Nu]] + 5 Fibonacci[\[Nu]]^2])^(1/\[Nu])/2^\[Nu]^(-1) /; Element[\[Nu], Reals] && \[Nu] > 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["GoldenRatio", "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "/", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["5"], " ", RowBox[List["Fibonacci", "[", "\[Nu]", "]"]]]], "+", SqrtBox[RowBox[List[RowBox[List["4", " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], "+", RowBox[List["5", " ", SuperscriptBox[RowBox[List["Fibonacci", "[", "\[Nu]", "]"]], "2"]]]]]]]], ")"]], FractionBox["1", "\[Nu]"]]]]]], "/;", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Reals"]], "\[And]", RowBox[List["\[Nu]", ">", "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> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, Function[GoldenRatio]] </annotation> </semantics> <mo> &#10869; </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> / </mo> <mi> &#957; </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mn> 5 </mn> </msqrt> <mo> &#8290; </mo> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mi> &#957; </mi> </msub> </mrow> <mo> + </mo> <msqrt> <mrow> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <msubsup> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mi> &#957; </mi> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> / </mo> <mi> &#957; </mi> </mrow> </msup> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> &#957; </mi> <mo> &#8712; </mo> <mi> &#8477; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#957; </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <ci> GoldenRatio </ci> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Fibonacci </ci> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <power /> <apply> <ci> Fibonacci </ci> <ci> &#957; </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <cos /> <apply> <times /> <pi /> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> &#957; </ci> <ci> &#8477; </ci> </apply> <apply> <gt /> <ci> &#957; </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", "GoldenRatio", "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "/", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["5"], " ", RowBox[List["Fibonacci", "[", "\[Nu]", "]"]]]], "+", SqrtBox[RowBox[List[RowBox[List["4", " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], "+", RowBox[List["5", " ", SuperscriptBox[RowBox[List["Fibonacci", "[", "\[Nu]", "]"]], "2"]]]]]]]], ")"]], RowBox[List["1", "/", "\[Nu]"]]]]], "/;", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Reals"]], "&&", RowBox[List["\[Nu]", ">", "0"]]]]]]]]]]


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





2007-05-02