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http://functions.wolfram.com/07.06.06.0029.01
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Fibonacci[\[Nu], z] \[Proportional] Fibonacci[Subscript[\[Nu], 0], z] +
((1/Sqrt[4 + z^2])
((ArcSinh[z/2] ((z + Sqrt[4 + z^2])^(2 Subscript[\[Nu], 0]) +
4^Subscript[\[Nu], 0] Cos[Pi Subscript[\[Nu], 0]]) +
4^Subscript[\[Nu], 0] Pi Sin[Pi Subscript[\[Nu], 0]])/
(2^Subscript[\[Nu], 0] (z + Sqrt[4 + z^2])^Subscript[\[Nu], 0])))
(\[Nu] - Subscript[\[Nu], 0]) +
(1/2) (ArcSinh[z/2]^2 Fibonacci[Subscript[\[Nu], 0], z] +
(1/Sqrt[4 + z^2]) ((2^Subscript[\[Nu], 0] Pi
(Pi Cos[Pi Subscript[\[Nu], 0]] - 2 ArcSinh[z/2]
Sin[Pi Subscript[\[Nu], 0]]))/(z + Sqrt[4 + z^2])^
Subscript[\[Nu], 0])) (\[Nu] - Subscript[\[Nu], 0])^2 +
O[(\[Nu] - Subscript[\[Nu], 0])^3]
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Cell[BoxData[RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List[SubscriptBox["\[Nu]", "0"], ",", "z"]], "]"]], "+", RowBox[List[RowBox[List[FractionBox["1", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", SubscriptBox["\[Nu]", "0"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]], RowBox[List["-", SubscriptBox["\[Nu]", "0"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["ArcSinh", "[", FractionBox["z", "2"], "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]], RowBox[List["2", " ", SubscriptBox["\[Nu]", "0"]]]], "+", RowBox[List[SuperscriptBox["4", SubscriptBox["\[Nu]", "0"]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", SubscriptBox["\[Nu]", "0"]]], "]"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["4", SubscriptBox["\[Nu]", "0"]], " ", "\[Pi]", " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", SubscriptBox["\[Nu]", "0"]]], "]"]]]]]], ")"]]]], ")"]]]], " ", RowBox[List["(", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]", "0"]]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["ArcSinh", "[", FractionBox["z", "2"], "]"]], "2"], " ", RowBox[List["Fibonacci", "[", RowBox[List[SubscriptBox["\[Nu]", "0"], ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox["1", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["2", SubscriptBox["\[Nu]", "0"]], " ", "\[Pi]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]], RowBox[List["-", SubscriptBox["\[Nu]", "0"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", SubscriptBox["\[Nu]", "0"]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["ArcSinh", "[", FractionBox["z", "2"], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", SubscriptBox["\[Nu]", "0"]]], "]"]]]]]], ")"]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]", "0"]]], ")"]], "2"]]], "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]", "0"]]], ")"]], "3"], "]"]]]]]]]]
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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> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox["F", Fibonacci] </annotation> </semantics> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ∝ </mo> <mrow> <mrow> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox["F", Fibonacci] </annotation> </semantics> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </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> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> </msup> <mo> ⁢ </mo> <msup> <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> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <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> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> </msup> <mo> + </mo> <mrow> <msup> <mn> 4 </mn> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </msup> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mn> 4 </mn> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </msup> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </msup> <mo> ⁢ </mo> <mi> π </mi> </mrow> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 4 </mn> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <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> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox["F", Fibonacci] </annotation> </semantics> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Proportional </ci> <apply> <ci> Fibonacci </ci> <ci> ν </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <ci> Fibonacci </ci> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> <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> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <power /> <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> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <arcsinh /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <power /> <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> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 4 </cn> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <cos /> <apply> <times /> <pi /> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 4 </cn> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> <pi /> <apply> <sin /> <apply> <times /> <pi /> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> <pi /> <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 /> <pi /> <apply> <cos /> <apply> <times /> <pi /> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <arcsinh /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sin /> <apply> <times /> <pi /> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <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> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <arcsinh /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Fibonacci </ci> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List[SubscriptBox["\[Nu]\[Nu]", "0"], ",", "z"]], "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", SubscriptBox["\[Nu]\[Nu]", "0"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]], RowBox[List["-", SubscriptBox["\[Nu]\[Nu]", "0"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["ArcSinh", "[", FractionBox["z", "2"], "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]], RowBox[List["2", " ", SubscriptBox["\[Nu]\[Nu]", "0"]]]], "+", RowBox[List[SuperscriptBox["4", SubscriptBox["\[Nu]\[Nu]", "0"]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", SubscriptBox["\[Nu]\[Nu]", "0"]]], "]"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["4", SubscriptBox["\[Nu]\[Nu]", "0"]], " ", "\[Pi]", " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", SubscriptBox["\[Nu]\[Nu]", "0"]]], "]"]]]]]], ")"]]]], ")"]], " ", RowBox[List["(", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], ")"]]]], SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["ArcSinh", "[", FractionBox["z", "2"], "]"]], "2"], " ", RowBox[List["Fibonacci", "[", RowBox[List[SubscriptBox["\[Nu]\[Nu]", "0"], ",", "z"]], "]"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["2", SubscriptBox["\[Nu]\[Nu]", "0"]], " ", "\[Pi]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]], RowBox[List["-", SubscriptBox["\[Nu]\[Nu]", "0"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", SubscriptBox["\[Nu]\[Nu]", "0"]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["ArcSinh", "[", FractionBox["z", "2"], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", SubscriptBox["\[Nu]\[Nu]", "0"]]], "]"]]]]]], ")"]]]], SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], ")"]], "2"]]], "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], "]"]], "3"]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
