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 ArcSinh

 http://functions.wolfram.com/01.25.27.1839.01

 Input Form

 ArcSinh[Sqrt[(Sqrt[1 + z^2] - 1)/2]] == (1/2) ArcCsch[1/z] /; Inequality[-(Pi/2), Less, Arg[z], LessEqual, Pi/2]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcSinh", "[", SqrtBox[FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]], "-", "1"]], "2"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", "2"], RowBox[List["ArcCsch", "[", FractionBox["1", "z"], "]"]]]]]], "/;", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", FractionBox["\[Pi]", "2"]]]]]]]

 MathML Form

 sinh - 1 ( 1 + z 2 - 1 2 ) 1 2 csch - 1 ( 1 z ) /; - π 2 < arg ( z ) π 2 Condition 1 z 2 1 2 -1 2 -1 1 2 1 2 1 z -1 Inequality -1 2 -1 z 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSinh", "[", SqrtBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["z_", "2"]]]], "-", "1"]], ")"]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["ArcCsch", "[", FractionBox["1", "z"], "]"]]]], "/;", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", FractionBox["\[Pi]", "2"]]]]]]]]]

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

 2003-08-21