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 ArcCoth

 http://functions.wolfram.com/01.28.16.0190.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2007-05-02