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 ArcSin

 http://functions.wolfram.com/01.12.27.0180.02

 Input Form

 ArcSin[Sqrt[1 - z]] == (-I) ArcCosh[Sqrt[z]] /; Inequality[0, Less, Arg[z], LessEqual, Pi] || 0 < z < 1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcSin", "[", SqrtBox[RowBox[List["1", "-", "z"]]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["ArcCosh", "[", SqrtBox["z"], "]"]]]]]], "/;", RowBox[List[RowBox[List["0", "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "\[Pi]"]], "\[Or]", RowBox[List["0", "<", "z", "<", "1"]]]]]]]]

 MathML Form

 sin - 1 ( 1 - z ) - cosh - 1 ( z ) /; 0 < arg ( z ) π 0 < z < 1 Condition 1 -1 z 1 2 -1 z 1 2 Inequality 0 z 0 z 1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSin", "[", SqrtBox[RowBox[List["1", "-", "z_"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["ArcCosh", "[", SqrtBox["z"], "]"]]]], "/;", RowBox[List[RowBox[List["0", "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "\[Pi]"]], "||", RowBox[List["0", "<", "z", "<", "1"]]]]]]]]]]

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

 2001-10-29