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ArcSin






Mathematica Notation

Traditional Notation









Elementary Functions > ArcSin[z] > Complex characteristics > Imaginary part





http://functions.wolfram.com/01.12.19.0003.01









  


  










Input Form





Im[ArcSin[x + I y]] == Sign[y] Log[X + Sqrt[X^2 - 1]] /; X == (1/2) Sqrt[(x - 1)^2 + y^2] + (1/2) Sqrt[(x + 1)^2 + y^2] && !IntervalMemberQ[Interval[{-Infinity, -1}], x + I y] && !IntervalMemberQ[Interval[{1, Infinity}], x + I y]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Im", "[", RowBox[List["ArcSin", "[", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["Sign", "[", "y", "]"]], " ", RowBox[List["Log", "[", RowBox[List["X", "+", SqrtBox[RowBox[List[SuperscriptBox["X", "2"], "-", "1"]]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["X", "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["x", "-", "1"]], ")"]], "2"], "+", SuperscriptBox["y", "2"]]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["x", "+", "1"]], ")"]], "2"], "+", SuperscriptBox["y", "2"]]]]]]]]]], "\[And]", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", "1"]]]], "}"]], "]"]], ",", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]]]], "]"]], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["1", ",", "\[Infinity]"]], "}"]], "]"]], ",", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]]]], "]"]], "]"]]]]]]]]










MathML Form







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Rule Form





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





2001-10-29