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 ArcCos

 http://functions.wolfram.com/01.13.16.0151.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2007-05-02