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variants of this functions
Fibonacci






Mathematica Notation

Traditional Notation









Integer Functions > Fibonacci[nu] > Complex characteristics > Signum value





http://functions.wolfram.com/04.11.19.0012.01









  


  










Input Form





Sign[Fibonacci[x + I y]] == (Sqrt[2] ((-((1/2) (-1 + Sqrt[5]))^x) Cos[Pi x] Cosh[Pi y] (Cos[y ArcCsch[2]] - I Sin[y ArcCsch[2]]) + ((1/2) (1 + Sqrt[5]))^x (Cos[y ArcCsch[2]] + I Sin[y ArcCsch[2]]) + ((1/2) (-1 + Sqrt[5]))^x Sin[Pi x] (I Cos[y ArcCsch[2]] + Sin[y ArcCsch[2]]) Sinh[Pi y]))/ Sqrt[-4 Cos[Pi x] Cos[2 y ArcCsch[2]] Cosh[Pi y] + (2 (3 + Sqrt[5])^(2 x) + 4^x (Cos[2 Pi x] + Cosh[2 Pi y]))/ (1 + Sqrt[5])^(2 x) + 4 Sin[Pi x] Sin[2 y ArcCsch[2]] Sinh[Pi y]]










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Sign", "[", RowBox[List["Fibonacci", "[", RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]], "]"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SqrtBox["2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox["5"]]], ")"]]]], ")"]], "x"]]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "x"]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["\[Pi]", " ", "y"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List["y", " ", RowBox[List["ArcCsch", "[", "2", "]"]]]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", RowBox[List["y", " ", RowBox[List["ArcCsch", "[", "2", "]"]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["5"]]], ")"]]]], ")"]], "x"], " ", RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List["y", " ", RowBox[List["ArcCsch", "[", "2", "]"]]]], "]"]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", RowBox[List["y", " ", RowBox[List["ArcCsch", "[", "2", "]"]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox["5"]]], ")"]]]], ")"]], "x"], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "x"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Cos", "[", RowBox[List["y", " ", RowBox[List["ArcCsch", "[", "2", "]"]]]], "]"]]]], "+", RowBox[List["Sin", "[", RowBox[List["y", " ", RowBox[List["ArcCsch", "[", "2", "]"]]]], "]"]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["\[Pi]", " ", "y"]], "]"]]]]]], ")"]]]], SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "x"]], "]"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "y", " ", RowBox[List["ArcCsch", "[", "2", "]"]]]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["\[Pi]", " ", "y"]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["5"]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", "x"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["3", "+", SqrtBox["5"]]], ")"]], RowBox[List["2", " ", "x"]]]]], "+", RowBox[List[SuperscriptBox["4", "x"], " ", RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List["2", " ", "\[Pi]", " ", "x"]], "]"]], "+", RowBox[List["Cosh", "[", RowBox[List["2", " ", "\[Pi]", " ", "y"]], "]"]]]], ")"]]]]]], ")"]]]], "+", RowBox[List["4", " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "x"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "y", " ", RowBox[List["ArcCsch", "[", "2", "]"]]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["\[Pi]", " ", "y"]], "]"]]]]]]]]]]]]










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





2007-05-02





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