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 ArcTanh

 http://functions.wolfram.com/01.27.27.2010.01

 Input Form

 ArcTanh[Sqrt[z]] == ArcSinh[Sqrt[z/(1 - z)]] + (Sqrt[1 - z] Sqrt[1/(1 - z)] - 1) ((Pi I)/2)

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2003-08-21