Fibonacci[\[Nu], z] \[Proportional] (1/Sqrt[4 + z^2]) (((z + Sqrt[4 + z^2])/2)^\[Nu] - Cos[\[Nu] Pi]/((z + Sqrt[4 + z^2])/2)^ \[Nu]) /; (Abs[\[Nu]] -> Infinity)

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox["1", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], "2"], ")"]], "\[Nu]"], "-", RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], "2"], ")"]], RowBox[List["-", "\[Nu]"]]]]]]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]

<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> <msup> <mrow> <mo> ( </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> <mo> ) </mo> </mrow> <mi> &#957; </mi> </msup> <mo> - </mo> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </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> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </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> <power /> <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> <ci> &#957; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <cos /> <apply> <times /> <ci> &#957; </ci> <pi /> </apply> </apply> <apply> <power /> <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> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <apply> <abs /> <ci> &#957; </ci> </apply> <infinity /> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]]]], ")"]], "\[Nu]"], "-", RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]]]], ")"]], RowBox[List["-", "\[Nu]"]]]]]]], SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]

2001-10-29