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 ArcCos

 http://functions.wolfram.com/01.13.16.0199.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ArcCos", "[", "x_", "]"]], "-", RowBox[List["ArcSech", "[", "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", "[", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]]], ")"]], "\[ImaginaryI]"], "]"]]]], "-", RowBox[List["Arg", "[", FractionBox["1", RowBox[List[RowBox[List[SqrtBox[RowBox[List[FractionBox["1", "y"], "-", "1"]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", "y"]]]]]], "+", FractionBox["1", "y"]]]], "]"]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["Im", "[", RowBox[List["Log", "[", 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RowBox[List[RowBox[List[SqrtBox[RowBox[List[FractionBox["1", "y"], "-", "1"]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", "y"]]]]]], "+", FractionBox["1", "y"]]]], "+", "1"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]], "-", RowBox[List["Floor", "[", RowBox[List["-", FractionBox[RowBox[List["Arg", "[", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]]], ")"]], "\[ImaginaryI]"], RowBox[List[RowBox[List[SqrtBox[RowBox[List[FractionBox["1", "y"], "-", "1"]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", "y"]]]]]], "+", 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", "[", 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FractionBox[RowBox[List["Arg", "[", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]]], ")"]], "\[ImaginaryI]"], RowBox[List[RowBox[List[SqrtBox[RowBox[List[FractionBox["1", "y"], "-", "1"]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", "y"]]]]]], "+", FractionBox["1", "y"]]]], "]"]], "\[Pi]"]]], "]"]]]]], " ", RowBox[List["ArcCos", "[", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]]], ")"]], RowBox[List["-", "\[ImaginaryI]"]]], " ", RowBox[List["(", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]]], ")"]], RowBox[List["2", " ", "\[ImaginaryI]"]]], SuperscriptBox[RowBox[List["(", 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02