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 ArcSinh

 http://functions.wolfram.com/01.25.06.0012.02

 Input Form

 ArcSinh[z] \[Proportional] (Pi I)/2 - Sqrt[2] I Sqrt[I (z - I)] (1 + O[z - I])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["ArcSinh", "[", "z", "]"]], "\[Proportional]", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"], "-", RowBox[List[SqrtBox["2"], " ", "\[ImaginaryI]", SqrtBox[RowBox[List["\[ImaginaryI]", RowBox[List["(", RowBox[List["z", "-", "\[ImaginaryI]"]], ")"]]]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", RowBox[List["z", "-", "\[ImaginaryI]"]], "]"]]]], ")"]]]]]]]]]]

 MathML Form

 sinh - 1 ( z ) π 2 - 2 ( z - ) ( 1 + O ( z - ) ) Proportional z 2 -1 -1 2 1 2 z -1 1 2 1 O z -1 [/itex]

 Rule Form

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

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

 2001-10-29