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 ArcTanh

 http://functions.wolfram.com/01.27.16.0220.01

 Input Form

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

 Standard Form

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

 MathML Form

 a tanh - 1 ( x ) + b csc - 1 ( y ) a π arg ( x + 1 ) - arg ( 1 - x ) + π 2 π - 2 π ( - arg ( ( 1 - x x + 1 ) - a 2 ) - arg ( ( 1 - 1 y 2 + y ) - b ) + π 2 π + 1 2 Im ( a log ( 1 - x x + 1 ) ) + π 2 π + Re ( b log ( 1 - 1 y 2 + y ) ) + π 2 π ) + π ( 1 - ( - 1 ) arg ( ( 1 - 1 y 2 + y ) - b ( 1 - x x + 1 ) - a 2 + 1 ) 2 π + 1 2 ) + 2 tanh - 1 ( ( 1 - x x + 1 ) - a 2 ( 1 - 1 y 2 + y ) - b - 1 ( 1 - 1 y 2 + y ) - b ( 1 - x x + 1 ) - a 2 + 1 ) 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 1 y 2 -1 1 2 y -1 -1 b 2 -1 1 2 a 1 -1 x x 1 -1 2 -1 b 1 -1 1 y 2 -1 1 2 y -1 2 -1 1 -1 -1 1 -1 1 y 2 -1 1 2 y -1 -1 b 1 -1 x x 1 -1 -1 a 2 -1 1 2 -1 1 2 2 1 -1 x x 1 -1 -1 a 2 -1 1 -1 1 y 2 -1 1 2 y -1 -1 b -1 1 -1 1 y 2 -1 1 2 y -1 -1 b 1 -1 x x 1 -1 -1 a 2 -1 1 -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["ArcCsc", "[", "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[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]], "+", FractionBox["\[ImaginaryI]", "y"]]], ")"]], RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]]], "]"]], "+", "\[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["Re", "[", RowBox[List["b", " ", RowBox[List["Log", "[", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]], "+", FractionBox["\[ImaginaryI]", "y"]]], "]"]]]], "]"]], "+", "\[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[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]], "+", FractionBox["\[ImaginaryI]", "y"]]], ")"]], RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "x"]], RowBox[List["x", "+", "1"]]], ")"]], RowBox[List["-", FractionBox["a", "2"]]]]]], "+", "1"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "+", FractionBox["1", "2"]]], "]"]]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["ArcTanh", "[", FractionBox[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "x"]], RowBox[List["x", "+", "1"]]], ")"]], RowBox[List["-", FractionBox["a", "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]], "+", FractionBox["\[ImaginaryI]", "y"]]], ")"]], RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]]]]], "-", "1"]], RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]], "+", FractionBox["\[ImaginaryI]", "y"]]], ")"]], RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "x"]], RowBox[List["x", "+", "1"]]], ")"]], RowBox[List["-", FractionBox["a", "2"]]]]]], "+", "1"]]], "]"]]]]]]]]]]

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

 2007-05-02