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ArcSinh




Mathematica Notation

Traditional Notation









Elementary Functions > ArcSinh[z] > Representations through equivalent functions > With related functions > Involving cos-1 > Involving sinh-1(z) > Involving sinh-1(z) and cos-1(2 z (-z2-1)1/2)





http://functions.wolfram.com/01.25.27.0184.01









  


  








Input Form





ArcSinh[z] == (Pi/4) (-(Sqrt[-z^2]/z) - I Sqrt[I/z] Sqrt[(-I) z] Sqrt[1/(1 - I Sqrt[2] z)] Sqrt[1 - I Sqrt[2] z] + I Sqrt[-(I/z)] Sqrt[I z] Sqrt[1/(1 + I Sqrt[2] z)] Sqrt[1 + I Sqrt[2] z] - Sqrt[z^2 + z^4]/(z Sqrt[-1 - z^2])) - (Sqrt[-1 - 2 z^2]/(2 z^2 Sqrt[1 + 2 z^2])) Sqrt[-(z^2/(-1 - z^2))] Sqrt[(-z^2) (-1 - z^2)] (Pi/2 - ArcCos[2 z Sqrt[-1 - z^2]])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ArcSinh", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["\[Pi]", "4"], RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], "z"]]], "-", RowBox[List["\[ImaginaryI]", SqrtBox[FractionBox["\[ImaginaryI]", "z"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", SqrtBox[FractionBox["1", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"], " ", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"], " ", "z"]]]]]]], "+", RowBox[List["\[ImaginaryI]", SqrtBox[RowBox[List["-", FractionBox["\[ImaginaryI]", "z"]]]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "z"]]], " ", SqrtBox[FractionBox["1", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"], " ", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"], " ", "z"]]]]]]], "-", FractionBox[SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", SuperscriptBox["z", "4"]]]], RowBox[List["z", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", SuperscriptBox["z", "2"]]]]]]]]], ")"]]]], "-", RowBox[List[FractionBox[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]]], RowBox[List["2", " ", SuperscriptBox["z", "2"], " ", SqrtBox[RowBox[List["1", "+", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]]]]]], " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], RowBox[List[RowBox[List["-", "1"]], "-", SuperscriptBox["z", "2"]]]]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", SuperscriptBox["z", "2"]]], ")"]]]]], " ", RowBox[List["(", RowBox[List[FractionBox["\[Pi]", "2"], "-", RowBox[List["ArcCos", "[", RowBox[List["2", " ", "z", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", SuperscriptBox["z", "2"]]]]]], "]"]]]], ")"]]]]]]]]]]








MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSinh", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], "z"]]], "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox[FractionBox["\[ImaginaryI]", "z"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", SqrtBox[FractionBox["1", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"], " ", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"], " ", "z"]]]]]]], "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["-", FractionBox["\[ImaginaryI]", "z"]]]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "z"]]], " ", SqrtBox[FractionBox["1", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"], " ", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"], " ", "z"]]]]]]], "-", FractionBox[SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", SuperscriptBox["z", "4"]]]], RowBox[List["z", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", SuperscriptBox["z", "2"]]]]]]]]], ")"]]]], "-", FractionBox[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]]], " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], RowBox[List[RowBox[List["-", "1"]], "-", SuperscriptBox["z", "2"]]]]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", SuperscriptBox["z", "2"]]], ")"]]]]], " ", RowBox[List["(", RowBox[List[FractionBox["\[Pi]", "2"], "-", RowBox[List["ArcCos", "[", RowBox[List["2", " ", "z", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", SuperscriptBox["z", "2"]]]]]], "]"]]]], ")"]]]], RowBox[List["2", " ", SuperscriptBox["z", "2"], " ", SqrtBox[RowBox[List["1", "+", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]]]]]]]]]]]]


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





2003-08-21