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variants of this functions
Fibonacci




Mathematica Notation

Traditional Notation









Hypergeometric Functions > Fibonacci[nu,z] > Series representations > Generalized power series > Expansions at nu==0





http://functions.wolfram.com/07.06.06.0001.01









  


  








Input Form





Fibonacci[\[Nu], z] \[Proportional] (1/Sqrt[z^2 + 4]) (2 Log[w] \[Nu] + (Pi^2/2) \[Nu]^2 + ((Log[w] (-3 Pi^2 + 2 Log[w]^2))/6) \[Nu]^3 + \[Ellipsis]) /; w == (z + Sqrt[z^2 + 4])/2 && (\[Nu] -> 0)








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox["1", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "4"]]]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Log", "[", "w", "]"]], "\[Nu]"]], "+", RowBox[List[FractionBox[SuperscriptBox["\[Pi]", "2"], "2"], SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["Log", "[", "w", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", SuperscriptBox["\[Pi]", "2"]]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["Log", "[", "w", "]"]], "2"]]]]], ")"]]]], "6"], " ", SuperscriptBox["\[Nu]", "3"]]], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List[RowBox[List["w", "\[Equal]", FractionBox[RowBox[List["z", "+", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "4"]]]]], "2"]]], "\[And]", RowBox[List["(", RowBox[List["\[Nu]", "\[Rule]", "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> <mrow> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 4 </mn> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> + </mo> <mfrac> <mrow> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> &#957; </mi> <mn> 2 </mn> </msup> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mrow> <mfrac> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 6 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#957; </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> w </mi> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 4 </mn> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#957; </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> Fibonacci </ci> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ln /> <ci> w </ci> </apply> <ci> &#957; </ci> </apply> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <ln /> <ci> w </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <ln /> <ci> w </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 6 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 3 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <ci> w </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> z </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> &#957; </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", RowBox[List["Log", "[", "w", "]"]], " ", "\[Nu]"]], "+", FractionBox[RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["\[Nu]", "2"]]], "2"], "+", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "w", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", SuperscriptBox["\[Pi]", "2"]]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["Log", "[", "w", "]"]], "2"]]]]], ")"]]]], ")"]], " ", SuperscriptBox["\[Nu]", "3"]]], "+", "\[Ellipsis]"]], SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "4"]]]], "/;", RowBox[List[RowBox[List["w", "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "4"]]]]], ")"]]]]]], "&&", RowBox[List["(", RowBox[List["\[Nu]", "\[Rule]", "0"]], ")"]]]]]]]]]]


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





2001-10-29