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 ArcTanh

 http://functions.wolfram.com/01.27.16.0203.01

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

 Standard Form

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

 MathML Form

 tanh - 1 ( x ) - sinh - 1 ( y ) tanh - 1 ( ( - 1 ) - 1 2 - arg ( x y 2 + 1 - 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 ( x y 2 + 1 - y 1 - x 2 ) π ) arg ( y + y 2 + 1 ) + arg ( - x 1 - x 2 ) 2 π + ( - 1 ) 1 2 - arg ( x y 2 + 1 - y 1 - x 2 ) π + ( - 1 ) arg ( x y - y 2 + 1 ) π + 1 2 - arg ( 1 - x 2 ) 2 π + 1 2 - arg ( x y 2 + 1 - y 1 - x 2 ) π - 2 ( - 1 + ( - 1 ) 1 2 - arg ( x y 2 + 1 - y 1 - x 2 ) π ) 1 2 - arg ( y + y 2 + 1 ) + arg ( - x 1 - x 2 ) 2 π ) x -1 y -1 -1 1 2 -1 x y 2 1 1 2 -1 y 1 -1 x 2 1 2 -1 -1 1 -1 x 2 1 2 1 -1 x y -1 y 2 1 1 2 2 1 -1 x 2 -1 1 2 x y -1 y 2 1 1 2 -1 -1 1 2 2 1 -1 1 2 -1 x y 2 1 1 2 -1 y 1 -1 x 2 1 2 -1 -1 y y 2 1 1 2 -1 x 1 -1 x 2 1 2 -1 2 -1 -1 1 2 -1 x y 2 1 1 2 -1 y 1 -1 x 2 1 2 -1 -1 -1 x y -1 y 2 1 1 2 -1 1 2 -1 1 -1 x 2 2 -1 1 2 -1 x y 2 1 1 2 -1 y 1 -1 x 2 1 2 -1 -1 -1 2 -1 -1 1 2 -1 x y 2 1 1 2 -1 y 1 -1 x 2 1 2 -1 -1 1 2 -1 y y 2 1 1 2 -1 x 1 -1 x 2 1 2 -1 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["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]", " ", "x", " ", SqrtBox[RowBox[List[SuperscriptBox["y", "2"], "+", "1"]]]]], "-", 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["x", " ", "y"]], "-", SqrtBox[RowBox[List[SuperscriptBox["y", "2"], "+", "1"]]]]], ")"]], "2"], RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]]]]]], RowBox[List[RowBox[List["-", SqrtBox[RowBox[List[SuperscriptBox["y", "2"], "+", "1"]]]]], "+", RowBox[List["x", " ", "y"]]]]], "]"]], "-", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[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]", " ", "x", " ", SqrtBox[RowBox[List[SuperscriptBox["y", "2"], "+", "1"]]]]], "-", 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["y", "+", SqrtBox[RowBox[List[SuperscriptBox["y", "2"], "+", "1"]]]]], "]"]]]], 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]", " ", "x", " ", SqrtBox[RowBox[List[SuperscriptBox["y", "2"], "+", "1"]]]]], "-", RowBox[List["\[ImaginaryI]", " ", "y"]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]], "]"]], "\[Pi]"]]], "]"]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["Floor", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["Arg", "[", RowBox[List["1", "-", SuperscriptBox["x", "2"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "+", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["x", " ", "y"]], "-", SqrtBox[RowBox[List[SuperscriptBox["y", "2"], "+", "1"]]]]], "]"]], "\[Pi]"], "+", FractionBox["1", "2"]]], "]"]], "+", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x", " ", SqrtBox[RowBox[List[SuperscriptBox["y", "2"], "+", "1"]]]]], "-", 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]", " ", "x", " ", SqrtBox[RowBox[List[SuperscriptBox["y", "2"], "+", "1"]]]]], "-", 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["y", "+", SqrtBox[RowBox[List[SuperscriptBox["y", "2"], "+", "1"]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]]]], ")"]]]]]]]]]]

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

 2007-05-02