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ArcSinh






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.25.27.1229.01









  


  










Input Form





ArcSinh[Sqrt[(Sqrt[z^2 - 1] - z)/(2 z)]] == (-((Sqrt[-1 + z] z^(3/2))/(2 Sqrt[(-z) (1 + z)]))) Sqrt[(1 + z)/(-1 + z)] Sqrt[-(1/z^2)] ArcCosh[1/z] + (Pi/(4 z)) (Sqrt[-(1/z)] Sqrt[I/z] Sqrt[(-I) z] z^(3/2) - Sqrt[-z^2] (1 + I Sqrt[I/z] Sqrt[I z]) + 3 I z (-1 + Sqrt[1 + 1/z] Sqrt[z/(1 + z)]))










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSinh", "[", SqrtBox[FractionBox[RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["z_", "2"], "-", "1"]]], "-", "z_"]], RowBox[List["2", " ", "z_"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "z"]]], " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["1", "+", "z"]], RowBox[List[RowBox[List["-", "1"]], "+", "z"]]]], " ", SqrtBox[RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]]], " ", RowBox[List["ArcCosh", "[", FractionBox["1", "z"], "]"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "z"]], " ", RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]]]]]]]]]], "+", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", FractionBox["1", "z"]]]], " ", SqrtBox[FractionBox["\[ImaginaryI]", "z"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "-", RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[FractionBox["\[ImaginaryI]", "z"]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "z"]]]]]]], ")"]]]], "+", RowBox[List["3", " ", "\[ImaginaryI]", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox["1", "z"]]]], " ", SqrtBox[FractionBox["z", RowBox[List["1", "+", "z"]]]]]]]], ")"]]]]]], ")"]]]], RowBox[List["4", " ", "z"]]]]]]]]]










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





2003-08-21