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 ArcCos

 http://functions.wolfram.com/01.13.16.0142.01

 Input Form

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

 Standard Form

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

 MathML Form

 cos - 1 ( x ) + sinh - 1 ( y ) cos - 1 ( ( - 1 ) 1 2 - arg ( x y + 1 - x 2 y 2 + 1 ) π ( x y 2 + 1 - 1 - x 2 y ) ) + 1 2 π ( 2 ( - 1 + ( - 1 ) 1 2 - arg ( x y + 1 - x 2 y 2 + 1 ) π ) arg ( x + 1 - x 2 ) + arg ( y + y 2 + 1 ) 2 π + ( - 1 ) 1 2 - arg ( x y + 1 - x 2 y 2 + 1 ) π + 2 ( 1 + ( - 1 ) 1 2 - arg ( x y + 1 - x 2 y 2 + 1 ) π ) 1 2 - arg ( x + 1 - x 2 ) + arg ( y + y 2 + 1 ) 2 π - 1 ) x y -1 1 2 -1 x y 1 -1 x 2 1 2 y 2 1 1 2 -1 x y 2 1 1 2 -1 1 -1 x 2 1 2 y 1 2 2 -1 -1 1 2 -1 x y 1 -1 x 2 1 2 y 2 1 1 2 -1 x 1 -1 x 2 1 2 y y 2 1 1 2 2 -1 -1 1 2 -1 x y 1 -1 x 2 1 2 y 2 1 1 2 -1 2 1 -1 1 2 -1 x y 1 -1 x 2 1 2 y 2 1 1 2 -1 1 2 -1 x 1 -1 x 2 1 2 y y 2 1 1 2 2 -1 -1 [/itex]

 Rule Form

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

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

 2007-05-02