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ArcTanh




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.27.27.0557.01









  


  








Input Form





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








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["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]], ")"]], " ", SqrtBox[RowBox[List["1", "-", "z"]]], " ", SqrtBox[FractionBox["1", RowBox[List["1", "-", "z"]]]], " ", RowBox[List["ArcCos", "[", SqrtBox[FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "-", "z"]]], "-", SqrtBox[RowBox[List["-", "z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "z"]]]]]]], "]"]]]], "z"]]], "+", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]], ")"]], " ", SqrtBox[RowBox[List["1", "-", "z"]]], " ", SqrtBox[FractionBox["1", RowBox[List["1", "-", "z"]]]]]], "z"], "-", FractionBox[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], "z"]]], ")"]]]]]]]]]]


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





2003-08-21