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 ArcCoth

 http://functions.wolfram.com/01.28.16.0240.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2007-05-02