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 ArcTanh

 http://functions.wolfram.com/01.27.16.0142.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2007-05-02