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 ArcTanh

 http://functions.wolfram.com/01.27.16.0181.01

 Input Form

 ArcTanh[x] - I ArcSin[y] == ((I Sqrt[1 - x^2] Sqrt[(-y - I x Sqrt[1 - y^2])^2/(1 - x^2)])/ (y + I x Sqrt[1 - y^2])) ArcSin[((-I) x y + Sqrt[1 - y^2])/ Sqrt[1 - x^2]] + (I Pi Sqrt[1 - x^2] Sqrt[(-y - I x Sqrt[1 - y^2])^2/(1 - x^2)])/ (2 (-y - I x Sqrt[1 - y^2])) + I Pi (1 + (Sqrt[1 - x^2] Sqrt[(-y - I x Sqrt[1 - y^2])^2/(1 - x^2)])/ (-y - I x Sqrt[1 - y^2])) Floor[(Arg[(I + I x)/Sqrt[1 - x^2]] + Arg[(-I) y + Sqrt[1 - y^2]])/ (2 Pi)] - I Pi (-1 + (Sqrt[1 - x^2] Sqrt[(-y - I x Sqrt[1 - y^2])^2/(1 - x^2)])/ (-y - I x Sqrt[1 - y^2])) Floor[-((-Pi + Arg[(I + I x)/Sqrt[1 - x^2]] + Arg[(-I) y + Sqrt[1 - y^2]])/ (2 Pi))]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcTanh", "[", "x", "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSin", "[", "y", "]"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "y"]], "-", RowBox[List["\[ImaginaryI]", " ", "x", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]], "2"], RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]]]], RowBox[List["y", "+", RowBox[List["\[ImaginaryI]", " ", "x", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]]], " ", RowBox[List["ArcSin", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "x", " ", "y"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]], "]"]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "y"]], "-", RowBox[List["\[ImaginaryI]", " ", "x", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]], "2"], RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "y"]], "-", RowBox[List["\[ImaginaryI]", " ", "x", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]]]]], "+", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "y"]], "-", RowBox[List["\[ImaginaryI]", " ", "x", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]], "2"], RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]]]], RowBox[List[RowBox[List["-", "y"]], "-", RowBox[List["\[ImaginaryI]", " ", "x", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]]]]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["Arg", "[", FractionBox[RowBox[List["\[ImaginaryI]", "+", RowBox[List["\[ImaginaryI]", " ", "x"]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]], "]"]], "+", RowBox[List["Arg", "[", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "y"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "y"]], "-", RowBox[List["\[ImaginaryI]", " ", "x", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]], "2"], RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]]]], RowBox[List[RowBox[List["-", "y"]], "-", RowBox[List["\[ImaginaryI]", " ", "x", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]]]]], ")"]], " ", RowBox[List["Floor", "[", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["-", "\[Pi]"]], "+", RowBox[List["Arg", "[", FractionBox[RowBox[List["\[ImaginaryI]", "+", RowBox[List["\[ImaginaryI]", " ", "x"]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]], "]"]], "+", RowBox[List["Arg", "[", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "y"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]]]]]]]]

 MathML Form

 tanh - 1 ( x ) - sin - 1 ( y ) 1 - x 2 ( - 1 - y 2 x - y ) 2 1 - x 2 1 - y 2 x + y sin - 1 ( 1 - y 2 - x y 1 - x 2 ) + π 1 - x 2 ( - 1 - y 2 x - y ) 2 1 - x 2 2 ( - 1 - y 2 x - y ) + π ( ( - 1 - y 2 x - y ) 2 1 - x 2 1 - x 2 - 1 - y 2 x - y + 1 ) arg ( + x 1 - x 2 ) + arg ( 1 - y 2 - y ) 2 π - π ( 1 - x 2 ( - 1 - y 2 x - y ) 2 1 - x 2 - 1 - y 2 x - y - 1 ) - arg ( + x 1 - x 2 ) + arg ( 1 - y 2 - y ) - π 2 π x -1 y 1 -1 x 2 1 2 -1 1 -1 y 2 1 2 x -1 y 2 1 -1 x 2 -1 1 2 1 -1 y 2 1 2 x y -1 1 -1 y 2 1 2 -1 x y 1 -1 x 2 1 2 -1 1 -1 x 2 1 2 -1 1 -1 y 2 1 2 x -1 y 2 1 -1 x 2 -1 1 2 2 -1 1 -1 y 2 1 2 x -1 y -1 -1 1 -1 y 2 1 2 x -1 y 2 1 -1 x 2 -1 1 2 1 -1 x 2 1 2 -1 1 -1 y 2 1 2 x -1 y -1 1 x 1 -1 x 2 1 2 -1 1 -1 y 2 1 2 -1 y 2 -1 -1 1 -1 x 2 1 2 -1 1 -1 y 2 1 2 x -1 y 2 1 -1 x 2 -1 1 2 -1 1 -1 y 2 1 2 x -1 y -1 -1 -1 x 1 -1 x 2 1 2 -1 1 -1 y 2 1 2 -1 y -1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ArcTanh", "[", "x_", "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSin", "[", "y_", "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "y"]], "-", RowBox[List["\[ImaginaryI]", " ", "x", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]], "2"], RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]]]], ")"]], " ", RowBox[List["ArcSin", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "x", " ", "y"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]], "]"]]]], RowBox[List["y", "+", RowBox[List["\[ImaginaryI]", " ", "x", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "y"]], "-", RowBox[List["\[ImaginaryI]", " ", "x", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]], "2"], RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "y"]], "-", RowBox[List["\[ImaginaryI]", " ", "x", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]]]]], "+", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "y"]], "-", RowBox[List["\[ImaginaryI]", " ", "x", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]], "2"], RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]]]], RowBox[List[RowBox[List["-", "y"]], "-", RowBox[List["\[ImaginaryI]", " ", "x", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]]]]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["Arg", "[", FractionBox[RowBox[List["\[ImaginaryI]", "+", RowBox[List["\[ImaginaryI]", " ", "x"]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]], "]"]], "+", RowBox[List["Arg", "[", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "y"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "y"]], "-", RowBox[List["\[ImaginaryI]", " ", "x", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]], "2"], RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]]]], RowBox[List[RowBox[List["-", "y"]], "-", RowBox[List["\[ImaginaryI]", " ", "x", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]]]]], ")"]], " ", RowBox[List["Floor", "[", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["-", "\[Pi]"]], "+", RowBox[List["Arg", "[", FractionBox[RowBox[List["\[ImaginaryI]", "+", RowBox[List["\[ImaginaryI]", " ", "x"]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]], "]"]], "+", RowBox[List["Arg", "[", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "y"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]]]]]]]]

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

 2007-05-02