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/01.27.16.0171.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2007-05-02