html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 ArcTanh

 http://functions.wolfram.com/01.27.16.0196.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2007-05-02