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 ArcTanh

 http://functions.wolfram.com/01.27.16.0040.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["ArcTanh", "[", SqrtBox[SuperscriptBox["z", "2"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]], "+", RowBox[List[FractionBox[SqrtBox[SuperscriptBox["z", "2"]], "z"], RowBox[List["ArcTanh", "[", FractionBox["1", "z"], "]"]]]]]]]]]]

 MathML Form

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

 Rule Form

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

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

 2003-08-21