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ArcSinh






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.25.19.0003.01









  


  










Input Form





Im[ArcSinh[x + I y]] == Sign[Y] ArcCot[X/Abs[Y]] /; X == x + Sqrt[(1/2) (x^2 - y^2 + Sqrt[(x^2 + y^2 + 1)^2 - 4 y^2] + 1)] && Y == y + Sign[x y] Sqrt[(1/2) (-x^2 + y^2 + Sqrt[(x^2 + y^2 + 1)^2 - 4 y^2] - 1)] && !IntervalMemberQ[Interval[{-Infinity, -1}], I (x + I y)] && !IntervalMemberQ[Interval[{1, Infinity}], I (x + I y)]










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Im", "[", RowBox[List["ArcSinh", "[", RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]], "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Sign", "[", "Y", "]"]], " ", RowBox[List["ArcCot", "[", FractionBox["X", RowBox[List["Abs", "[", "Y", "]"]]], "]"]]]], "/;", RowBox[List[RowBox[List["X", "\[Equal]", RowBox[List["x", "+", SqrtBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox["x", "2"], "-", SuperscriptBox["y", "2"], "+", SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "+", "1"]], ")"]], "2"], "-", RowBox[List["4", " ", SuperscriptBox["y", "2"]]]]]], "+", "1"]], ")"]]]]]]]]], "&&", RowBox[List["Y", "\[Equal]", RowBox[List["y", "+", RowBox[List[RowBox[List["Sign", "[", RowBox[List["x", " ", "y"]], "]"]], " ", SqrtBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["x", "2"]]], "+", SuperscriptBox["y", "2"], "+", SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "+", "1"]], ")"]], "2"], "-", RowBox[List["4", " ", SuperscriptBox["y", "2"]]]]]], "-", "1"]], ")"]]]]]]]]]]], "&&", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", "1"]]]], "}"]], "]"]], ",", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], ")"]]]]]], "]"]]]], "&&", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["1", ",", "\[Infinity]"]], "}"]], "]"]], ",", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], ")"]]]]]], "]"]]]]]]]]]]]]










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





2001-10-29





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