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variants of this functions
Fibonacci




Mathematica Notation

Traditional Notation









Hypergeometric Functions > Fibonacci[nu,z] > Integral representations > Contour integral representations





http://functions.wolfram.com/07.06.07.0001.01









  


  








Input Form





Fibonacci[\[Nu], z] == (Sin[Pi \[Nu]]/(2 Sqrt[Pi])) (Sin[(Pi \[Nu])/2] (1/(2 Pi I)) ContourIntegrate[ (Gamma[s] Gamma[(1 - \[Nu])/2 - s] Gamma[(1 + \[Nu])/2 - s])/ Gamma[1/2 - s]/(z^2/4)^s, {s, \[ScriptCapitalL]}] + (Sqrt[z^2]/z) Cos[(Pi \[Nu])/2] (1/(2 Pi I)) ContourIntegrate[ (Gamma[s + 1/2] Gamma[(1 - \[Nu])/2 - s] Gamma[(1 + \[Nu])/2 - s])/ Gamma[1 - s]/(z^2/4)^s, {s, \[ScriptCapitalL]}]) /; !Element[\[Nu], Integers]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " "]], RowBox[List["2", " ", SqrtBox["\[Pi]"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], FractionBox[RowBox[List["1", " "]], RowBox[List["2", "\[Pi]", " ", "\[ImaginaryI]"]]], RowBox[List["ContourIntegrate", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "-", "s"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "-", "s"]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s"]], "]"]]], SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "2"], "4"], ")"]], RowBox[List["-", "s"]]]]], ",", RowBox[List["{", RowBox[List["s", ",", "\[ScriptCapitalL]"]], "}"]]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[" ", SqrtBox[SuperscriptBox["z", "2"]]]], RowBox[List["z", " "]]], RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], FractionBox[RowBox[List["1", " "]], RowBox[List["2", "\[Pi]", " ", "\[ImaginaryI]"]]], RowBox[List["ContourIntegrate", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["1", "2"]]], "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "-", "s"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "-", "s"]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "s"]], "]"]]], SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "2"], "4"], ")"]], RowBox[List["-", "s"]]]]], ",", RowBox[List["{", RowBox[List["s", ",", "\[ScriptCapitalL]"]], "}"]]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "Integers"]], "]"]], "]"]]]]]]








MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], " ", RowBox[List["ContourIntegrate", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "-", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "-", "s"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "2"], "4"], ")"]], RowBox[List["-", "s"]]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s"]], "]"]]], ",", RowBox[List["{", RowBox[List["s", ",", "\[ScriptCapitalL]"]], "}"]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]]], "+", FractionBox[RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], " ", RowBox[List["ContourIntegrate", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["1", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "-", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "-", "s"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "2"], "4"], ")"]], RowBox[List["-", "s"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "s"]], "]"]]], ",", RowBox[List["{", RowBox[List["s", ",", "\[ScriptCapitalL]"]], "}"]]]], "]"]]]], RowBox[List["z", " ", RowBox[List["(", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]], ")"]]]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox["\[Pi]"]]]], "/;", RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]]]


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





2001-10-29