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 ArcTanh

 http://functions.wolfram.com/01.27.16.0155.01

 Input Form

 ArcTanh[x] + ArcTan[y] == I Pi Floor[(-Arg[1 - x] + Arg[x + 1] + Pi)/(2 Pi)] - 2 I Pi (Floor[((1/2) Arg[(1 - x)/(x + 1)] - Arg[(1 - I y)^(I/2)] + Pi)/ (2 Pi)] + Floor[((1/2) Im[Log[(1 - x)/(x + 1)]] + Pi)/(2 Pi)] + Floor[(Pi - (1/2) Re[Log[1 - I y]])/(2 Pi)]) - 2 I Pi (Floor[(-Arg[(1 - I y)^(I/2)/Sqrt[(1 - x)/(x + 1)]] - Arg[(I y + 1)^(-(I/2))] + Pi)/(2 Pi)] + Floor[(Pi - Im[Log[(1 - I y)^(I/2)/Sqrt[(1 - x)/(x + 1)]]])/(2 Pi)] + Floor[((1/2) Re[Log[I y + 1]] + Pi)/(2 Pi)]) + I Pi (1 - (-1)^Floor[Arg[(1 - I y)^(I/2)/((I y + 1)^(I/2) Sqrt[(1 - x)/(x + 1)]) + 1]/(2 Pi) + 1/2]) + 2 ArcTanh[((1 - I y)^(I/2)/((I y + 1)^(I/2) Sqrt[(1 - x)/(x + 1)]) - 1)/ ((1 - I y)^(I/2)/((I y + 1)^(I/2) Sqrt[(1 - x)/(x + 1)]) + 1)]

 Standard Form

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

 MathML Form

 tanh - 1 ( x ) + tan - 1 ( y ) π arg ( x + 1 ) - arg ( 1 - x ) + π 2 π - 2 π ( - arg ( ( 1 - y ) / 2 1 - x x + 1 ) - arg ( ( y + 1 ) - 2 ) + π 2 π + 1 2 Re ( log ( y + 1 ) ) + π 2 π + π - Im ( log ( ( 1 - y ) / 2 1 - x x + 1 ) ) 2 π ) - 2 π ( 1 2 arg ( 1 - x x + 1 ) - arg ( ( 1 - y ) / 2 ) + π 2 π + 1 2 Im ( log ( 1 - x x + 1 ) ) + π 2 π + π - 1 2 Re ( log ( 1 - y ) ) 2 π ) + π ( 1 - ( - 1 ) arg ( ( y + 1 ) - 2 ( 1 - y ) / 2 1 - x x + 1 + 1 ) 2 π + 1 2 ) + 2 tanh - 1 ( ( 1 - y ) / 2 ( y + 1 ) - 2 1 - x x + 1 - 1 ( y + 1 ) - 2 ( 1 - y ) / 2 1 - x x + 1 + 1 ) x y x 1 -1 1 -1 x 2 -1 -1 2 -1 1 -1 y 2 -1 1 -1 x x 1 -1 1 2 -1 -1 y 1 -1 2 -1 2 -1 1 2 y 1 2 -1 -1 1 -1 y 2 -1 1 -1 x x 1 -1 1 2 -1 2 -1 -1 2 1 2 1 -1 x x 1 -1 -1 1 -1 y 2 -1 2 -1 1 2 1 -1 x x 1 -1 2 -1 -1 1 2 1 -1 y 2 -1 1 -1 -1 y 1 -1 2 -1 1 -1 y 2 -1 1 -1 x x 1 -1 1 2 -1 1 2 -1 1 2 2 1 -1 y 2 -1 y 1 -1 2 -1 1 -1 x x 1 -1 1 2 -1 -1 y 1 -1 2 -1 1 -1 y 2 -1 1 -1 x x 1 -1 1 2 -1 1 -1 [/itex]

 Rule Form

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

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

 2007-05-02