Inverse sine : Continued fraction representations (formula 01.12.10.0001)

ArcSin

$Mathematica Notation$

Elementary Functions

Continued fraction representations

http://functions.wolfram.com/01.12.10.0001.01

Input Form

ArcSin[z] == (z Sqrt[1 - z^2])/ (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[{-Infinity, -1}], z] && !IntervalMemberQ[Interval[{1, Infinity}], z]

Standard Form

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

MathML Form

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Rule Form

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

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

2001-10-29