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 ArcSec

 http://functions.wolfram.com/01.18.16.0148.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2007-05-02