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ArcTanh




Mathematica Notation

Traditional Notation









Elementary Functions > ArcTanh[z] > Representations through equivalent functions > With related functions > Involving sec-1 > Involving tanh-1(z) > Involving tanh-1(z) and sec-1(i (z2-1)/2z)





http://functions.wolfram.com/01.27.27.1533.01









  


  








Input Form





ArcTanh[z] == ((I (1 - z))/(2 (1 + z))) Sqrt[((1 + z)/(-1 + z))^2] ArcSec[(I (z^2 - 1))/(2 z)] + (Pi/4) (-(Sqrt[-z^2]/z) + (Sqrt[-z^2]/z - I) ((1 - z)/(1 + z)) Sqrt[((1 + z)/(-1 + z))^2]) /; Abs[z] != 1








Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcTanh", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]]]], ")"]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "+", "z"]], RowBox[List[RowBox[List["-", "1"]], "+", "z"]]], ")"]], "2"]], " ", RowBox[List["ArcSec", "[", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[SuperscriptBox["z", "2"], "-", "1"]], ")"]]]], RowBox[List["2", " ", "z"]]], "]"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]]]]], "+", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], "z"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], "z"], "-", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "+", "z"]], RowBox[List[RowBox[List["-", "1"]], "+", "z"]]], ")"]], "2"]]]], RowBox[List["1", "+", "z"]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[NotEqual]", "1"]]]]]]]]


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





2003-08-21