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 ArcCoth

 http://functions.wolfram.com/01.28.16.0157.01

 Input Form

 ArcCoth[x] + I ArcSin[y] == (-((I Sqrt[1 - 1/x^2] x Sqrt[(x y - I Sqrt[1 - y^2])^2/(-1 + x^2)])/ (x y - I Sqrt[1 - y^2]))) ArcSin[((I y)/x + Sqrt[1 - y^2])/ Sqrt[1 - 1/x^2]] + (I Pi Sqrt[1 - 1/x^2] x Sqrt[(x y - I Sqrt[1 - y^2])^2/(-1 + x^2)])/(2 (x y - I Sqrt[1 - y^2])) + I Pi (1 + (Sqrt[1 - 1/x^2] x Sqrt[(x y - I Sqrt[1 - y^2])^2/(-1 + x^2)])/ (x y - I Sqrt[1 - y^2])) Floor[(Arg[(I + I/x)/Sqrt[1 - 1/x^2]] + Arg[I y + Sqrt[1 - y^2]])/ (2 Pi)] - I Pi (-1 + (Sqrt[1 - 1/x^2] x Sqrt[(x y - I Sqrt[1 - y^2])^2/(-1 + x^2)])/ (x y - I Sqrt[1 - y^2])) Floor[-((-Pi + 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["ArcSin", "[", "y", "]"]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", "x", " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["x", " ", "y"]], "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]], "2"], RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["x", "2"]]]]], " "]], RowBox[List[RowBox[List["x", " ", "y"]], "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]]]]], RowBox[List["ArcSin", "[", FractionBox[RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "y"]], "x"], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]]], "]"]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", "x", " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["x", " ", "y"]], "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]], "2"], RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["x", "2"]]]]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["x", " ", "y"]], "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]]]]], "+", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", "x", " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["x", " ", "y"]], "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]], "2"], RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["x", "2"]]]]]]], RowBox[List[RowBox[List["x", " ", "y"]], "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]]]]], ")"]], " ", 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]"]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", "x", " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["x", " ", "y"]], "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]], "2"], RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["x", "2"]]]]]]], RowBox[List[RowBox[List["x", " ", "y"]], "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]]]]], ")"]], " ", 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[RowBox[List["\[ImaginaryI]", " ", "y"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]]]]]]]]

 MathML Form

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

 Rule Form

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

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