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 ArcCoth

 http://functions.wolfram.com/01.28.16.0173.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2007-05-02

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