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ArcTanh




Mathematica Notation

Traditional Notation









Elementary Functions > ArcTanh[z] > Representations through equivalent functions > With related functions > Involving cosh-1 > Involving tanh-1(z1/2) > Involving tanh-1(z1/2) and cosh-1(2 (-z)1/2/z-1)





http://functions.wolfram.com/01.27.27.2358.01









  


  








Input Form





ArcTanh[Sqrt[z]] == ((Pi z)/(4 Sqrt[-z^2])) (1 + ((2 (z + 1))/(z - 1)) Sqrt[((z - 1)/(z + 1))^2]) + (1/4) (1 - ((z + 1)/(z - 1)) Sqrt[((z - 1)/(z + 1))^2] + Sqrt[1/(1 + z)] Sqrt[1 + z] (1 + ((z + 1)/(z - 1)) Sqrt[((z - 1)/(z + 1))^2])) ArcCosh[(2 Sqrt[-z])/(-1 + z)] /; 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", "[", SqrtBox["z_"], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Pi]", " ", "z"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]]]], ")"]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "-", "1"]], RowBox[List["z", "+", "1"]]], ")"]], "2"]]]], RowBox[List["z", "-", "1"]]]]], ")"]]]], RowBox[List["4", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]]], "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "-", "1"]], RowBox[List["z", "+", "1"]]], ")"]], "2"]]]], RowBox[List["z", "-", "1"]]], "+", RowBox[List[SqrtBox[FractionBox["1", RowBox[List["1", "+", "z"]]]], " ", SqrtBox[RowBox[List["1", "+", "z"]]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "-", "1"]], RowBox[List["z", "+", "1"]]], ")"]], "2"]]]], RowBox[List["z", "-", "1"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["ArcCosh", "[", FractionBox[RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]], RowBox[List[RowBox[List["-", "1"]], "+", "z"]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[NotEqual]", "1"]]]]]]]]


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





2003-08-21