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variants of this functions
Fibonacci




Mathematica Notation

Traditional Notation









Hypergeometric Functions > Fibonacci[nu,z] > General characteristics > Branch cuts > With respect to z





http://functions.wolfram.com/07.06.04.0014.01









  


  








Input Form





BranchCuts[Fibonacci[\[Nu], z], z] == {{Interval[{(-I) Infinity, -2 I}], 1}, {Interval[{2 I, I Infinity}], -1}} /; !Element[\[Nu], Integers]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BranchCuts", "[", RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Infinity]"]], ",", RowBox[List[RowBox[List["-", "2"]], "\[ImaginaryI]"]]]], "}"]], "]"]], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["2", "\[ImaginaryI]"]], ",", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]], "}"]], "]"]], ",", RowBox[List["-", "1"]]]], "}"]]]], "}"]]]], "/;", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "Integers"]], "]"]], "]"]]]]]]








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> <mi> &#8492;&#119966; </mi> <mi> z </mi> </msub> <mo> ( </mo> <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> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </mrow> <mo> ] </mo> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mtext> </mtext> <mrow> <mo> { </mo> <mrow> <mrow> <mo> [ </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mo> , </mo> <mtext> </mtext> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> &#957; </mi> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <msub> <mi> &#8492;&#119966; </mi> <mi> z </mi> </msub> <mo> ( </mo> <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> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </mrow> <mo> ] </mo> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mtext> </mtext> <mrow> <mo> { </mo> <mrow> <mrow> <mo> [ </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mo> , </mo> <mtext> </mtext> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> &#957; </mi> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BranchCuts", "[", RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Infinity]"]], ",", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]"]]]], "}"]], "]"]], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]"]], ",", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]], "}"]], "]"]], ",", RowBox[List["-", "1"]]]], "}"]]]], "}"]], "/;", RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]]]


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





2001-10-29