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 ArcCos

 http://functions.wolfram.com/01.13.16.0148.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2007-05-02