Inverse hyperbolic cosecant: Continued fraction representations (formula 01.29.10.0001)

ArcCsch

$Mathematica Notation$

Elementary Functions

Continued fraction representations

http://functions.wolfram.com/01.29.10.0001.01

Input Form

ArcCsch[z] == Sqrt[1 + z^(-2)]/ (z (1 + (1 2)/ (z^2 (3 + (1 2)/(z^2 (5 + (3 4)/(z^2 (7 + (3 4)/(z^2 (9 + (5 6)/(z^2 (11 + \[Ellipsis])))))))))))) /; !IntervalMemberQ[Interval[{-1, 1}], I z]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcCsch", "[", "z", "]"]], "\[Equal]", " ", FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["-", "1"]]], SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", RowBox[List["-", "2"]]]]]]]], RowBox[List["1", "+", FractionBox[RowBox[List["1", "\[Times]", "2", SuperscriptBox["z", RowBox[List["-", "2"]]]]], RowBox[List["3", "+", FractionBox[RowBox[List["1", "\[Times]", "2", SuperscriptBox["z", RowBox[List["-", "2"]]]]], RowBox[List["5", "+", FractionBox[RowBox[List["3", " ", "\[Times]", "4", SuperscriptBox["z", RowBox[List["-", "2"]]]]], RowBox[List["7", "+", FractionBox[RowBox[List["3", " ", "\[Times]", "4", SuperscriptBox["z", RowBox[List["-", "2"]]]]], RowBox[List["9", "+", FractionBox[RowBox[List["5", " ", "\[Times]", "6", SuperscriptBox["z", RowBox[List["-", "2"]]]]], RowBox[List["11", "+", "\[Ellipsis]"]]]]]]]]]]]]]]]]]]]], "/;", "\[InvisibleSpace]", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "1"]], "}"]], "]"]], ",", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]], "]"]], "\[InvisibleSpace]"]]]]

MathML Form

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Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCsch", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["z", "2"]]]]], RowBox[List["z", " ", RowBox[List["(", RowBox[List["1", "+", FractionBox["2", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["3", "+", FractionBox["2", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["5", "+", FractionBox[RowBox[List["3", " ", "4"]], RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["7", "+", FractionBox[RowBox[List["3", " ", "4"]], RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["9", "+", FractionBox[RowBox[List["5", " ", "6"]], RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["11", "+", "\[Ellipsis]"]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]], ")"]]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "1"]], "}"]], "]"]], ",", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]]]]]]]]

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

2001-10-29