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 ArcSec

 http://functions.wolfram.com/01.18.16.0139.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2007-05-02