Inverse hyperbolic sine: Complex characteristics (formula 01.25.19.0003)

ArcSinh

$Mathematica Notation$

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

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Im", "[", RowBox[List["ArcSinh", "[", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], "]"]], "\[Equal]", 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"]], ")"]]]]]]]]], "\[And]", 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"]], ")"]]]]]]]]]]], "\[And]", RowBox[List["Not", "[", 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"]]]], ")"]]]]]], "]"]], "]"]], "\[And]", RowBox[List["Not", "[", 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"]]]], ")"]]]]]], "]"]], "]"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mi> sgn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> Y </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> cot </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> X </mi> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> Y </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> X </mi> <mo> ⩵ </mo> <mrow> <mi> x </mi> <mo> + </mo> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msqrt> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> Y </mi> <mo> ⩵ </mo> <mrow> <mi> y </mi> <mo> + </mo> <mrow> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msqrt> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mi> sgn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> ∞ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mi> sgn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> Y </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> cot </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> X </mi> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> Y </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; 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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