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 ArcCoth

 http://functions.wolfram.com/01.28.16.0196.01

 Input Form

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

 Standard Form

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

 MathML Form

 coth - 1 ( x ) - cos - 1 ( y ) coth - 1 ( ( - 1 ) 1 2 - arg ( y - 1 - y 2 x 1 - 1 x 2 ) π ( y x - 1 - y 2 ) 1 - 1 x 2 1 - ( - y x - 1 - y 2 ) 2 1 - 1 x 2 ) + 1 2 π ( - 1 + 2 ( 1 + ( - 1 ) 1 2 - arg ( y - 1 - y 2 x 1 - 1 x 2 ) π ) arg ( + x 1 - 1 x 2 ) + arg ( y + 1 - y 2 ) 2 π + ( - 1 ) - arg ( 1 - 1 x 2 ) 2 π + arg ( - y x - 1 - y 2 ) π + 1 2 + 1 2 - arg ( y - 1 - y 2 x 1 - 1 x 2 ) π + ( - 1 ) 1 2 - arg ( y - 1 - y 2 x 1 - 1 x 2 ) π - 2 ( - 1 + ( - 1 ) 1 2 - arg ( y - 1 - y 2 x 1 - 1 x 2 ) π ) 1 2 - arg ( + x 1 - 1 x 2 ) + arg ( y + 1 - y 2 ) 2 π ) x -1 y -1 1 2 -1 y -1 1 -1 y 2 1 2 x -1 1 -1 1 x 2 -1 1 2 -1 -1 y x -1 -1 1 -1 y 2 1 2 1 -1 1 x 2 -1 1 2 1 -1 -1 y x -1 -1 1 -1 y 2 1 2 2 1 -1 1 x 2 -1 -1 1 2 -1 1 2 -1 2 1 -1 1 2 -1 y -1 1 -1 y 2 1 2 x -1 1 -1 1 x 2 -1 1 2 -1 -1 x -1 1 -1 1 x 2 -1 1 2 -1 y 1 -1 y 2 1 2 2 -1 -1 -1 1 -1 1 x 2 -1 2 -1 -1 y x -1 -1 1 -1 y 2 1 2 -1 1 2 1 2 -1 y -1 1 -1 y 2 1 2 x -1 1 -1 1 x 2 -1 1 2 -1 -1 -1 1 2 -1 y -1 1 -1 y 2 1 2 x -1 1 -1 1 x 2 -1 1 2 -1 -1 -1 2 -1 -1 1 2 -1 y -1 1 -1 y 2 1 2 x -1 1 -1 1 x 2 -1 1 2 -1 -1 1 2 -1 x -1 1 -1 1 x 2 -1 1 2 -1 y 1 -1 y 2 1 2 2 -1 [/itex]

 Rule Form

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

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

 2007-05-02