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variants of this functions
ArcTan






Mathematica Notation

Traditional Notation









Elementary Functions > ArcTan[z] > Continued fraction representations





http://functions.wolfram.com/01.14.10.0006.01









  


  










Input Form





ArcTan[z] == z/(1 + z^2 + ContinueFraction[ {2 z^2 Floor[(1 + k)/2] (1 - 2 Floor[(1 + k)/2]), (1 + 2 k) (1 + (1/2) (1 + (-1)^k) z^2)}, {k, 1, Infinity}]) /; !IntervalMemberQ[Interval[{-Infinity, -1}], I z] && !IntervalMemberQ[Interval[{1, Infinity}], I z]










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcTan", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["z", RowBox[List["1", "+", SuperscriptBox["z", "2"], "+", RowBox[List["ContinueFraction", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["2", " ", SuperscriptBox["z", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "k"]], "2"], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "k"]], "2"], "]"]]]]]], ")"]]]], ",", RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"]]], ")"]], " ", SuperscriptBox["z", "2"]]]]], ")"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", "1", ",", "\[Infinity]"]], "}"]]]], "]"]]]]], "/;", RowBox[List[RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", "1"]]]], "}"]], "]"]], ",", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]], "&&", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["1", ",", "\[Infinity]"]], "}"]], "]"]], ",", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]]]]]]]]]]










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





2001-10-29