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 ArcSec

 http://functions.wolfram.com/01.18.16.0206.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["a_", " ", RowBox[List["ArcSec", "[", "x_", "]"]]]], "+", RowBox[List["b_", " ", RowBox[List["ArcCoth", "[", "y_", "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "a"]], "2"], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["-", RowBox[List["Arg", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], "+", FractionBox["\[ImaginaryI]", "x"]]], ")"]], RowBox[List["\[ImaginaryI]", " ", "a"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", FractionBox["1", "y"]]], ")"]], RowBox[List["-", FractionBox["b", "2"]]]]]], "]"]]]], "-", RowBox[List["Arg", "[", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox["1", "y"]]], ")"]], RowBox[List["b", "/", "2"]]], "]"]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Im", "[", RowBox[List["Log", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], "+", FractionBox["\[ImaginaryI]", "x"]]], ")"]], RowBox[List["\[ImaginaryI]", " ", "a"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", FractionBox["1", "y"]]], ")"]], RowBox[List["-", FractionBox["b", "2"]]]]]], "]"]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Im", "[", RowBox[List["b", " ", RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox["1", "y"]]], "]"]]]], "]"]]]]]], 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"]]], ")"]], RowBox[List["\[ImaginaryI]", " ", "a"]]], "]"]]]], "-", RowBox[List["Arg", "[", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", FractionBox["1", "y"]]], ")"]], RowBox[List["-", FractionBox["b", "2"]]]], "]"]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Im", "[", RowBox[List["b", " ", RowBox[List["Log", "[", RowBox[List["1", "-", FractionBox["1", "y"]]], "]"]]]], "]"]]]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Re", "[", RowBox[List["a", " ", RowBox[List["Log", "[", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], "+", FractionBox["\[ImaginaryI]", "x"]]], "]"]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], ")"]]]], "+", RowBox[List["Log", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], "+", FractionBox["\[ImaginaryI]", "x"]]], ")"]], RowBox[List["\[ImaginaryI]", " ", "a"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", FractionBox["1", "y"]]], ")"]], RowBox[List["-", FractionBox["b", "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox["1", "y"]]], ")"]], RowBox[List["b", "/", "2"]]]]], "]"]]]]]]]]

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

 2007-05-02