html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 ArcCsc

 http://functions.wolfram.com/01.17.16.0190.01

 Input Form

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

 Standard Form

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

 MathML Form

 csc - 1 ( x ) - sech - 1 ( y ) - csc - 1 ( ( - 1 ) - 1 2 - arg ( 1 - 1 x 2 1 y - 1 1 + 1 y - 1 x y ) π - 1 y - 1 1 + 1 y x - 1 - 1 x 2 y ) - 1 2 π ( 1 2 ( 1 - ( - 1 ) - - arg ( 1 - 1 y ) 2 π ) ( 2 ( - 1 + ( - 1 ) 1 2 - arg ( 1 - 1 x 2 1 - 1 y 2 - 1 x y ) π ) arg ( 1 - 1 x 2 + x ) + arg ( 1 - 1 y 2 + y ) 2 π + ( - 1 ) 1 2 - arg ( 1 - 1 x 2 1 - 1 y 2 - 1 x y ) π + 2 ( 1 + ( - 1 ) 1 2 - arg ( 1 - 1 x 2 1 - 1 y 2 - 1 x y ) π ) 1 2 - arg ( 1 - 1 x 2 + x ) + arg ( 1 - 1 y 2 + y ) 2 π ) - 1 2 ( 1 + ( - 1 ) - - arg ( 1 - 1 y ) 2 π ) ( 2 ( - 1 + ( - 1 ) 1 2 - arg ( 1 - 1 x 2 1 - 1 y 2 + 1 x y ) π ) arg ( 1 - 1 x 2 - x ) + arg ( 1 - 1 y 2 + y ) 2 π + ( - 1 ) 1 2 - arg ( 1 - 1 x 2 1 - 1 y 2 + 1 x y ) π + 2 ( 1 + ( - 1 ) 1 2 - arg ( 1 - 1 x 2 1 - 1 y 2 + 1 x y ) π ) 1 2 - arg ( 1 - 1 x 2 - x ) + arg ( 1 - 1 y 2 + y ) 2 π ) ) x -1 y -1 -1 -1 1 2 -1 1 -1 1 x 2 -1 1 2 1 y -1 -1 1 2 1 1 y -1 1 2 -1 1 x y -1 -1 -1 1 y -1 -1 1 2 1 1 y -1 1 2 x -1 -1 1 -1 1 x 2 -1 1 2 y -1 -1 -1 1 2 1 2 1 -1 -1 -1 -1 1 -1 1 y -1 2 -1 2 -1 -1 1 2 -1 1 -1 1 x 2 -1 1 2 1 -1 1 y 2 -1 1 2 -1 1 x y -1 -1 1 -1 1 x 2 -1 1 2 x -1 1 -1 1 y 2 -1 1 2 y -1 2 -1 -1 1 2 -1 1 -1 1 x 2 -1 1 2 1 -1 1 y 2 -1 1 2 -1 1 x y -1 -1 2 1 -1 1 2 -1 1 -1 1 x 2 -1 1 2 1 -1 1 y 2 -1 1 2 -1 1 x y -1 -1 1 2 -1 1 -1 1 x 2 -1 1 2 x -1 1 -1 1 y 2 -1 1 2 y -1 2 -1 -1 1 2 1 -1 -1 -1 1 -1 1 y -1 2 -1 2 -1 -1 1 2 -1 1 -1 1 x 2 -1 1 2 1 -1 1 y 2 -1 1 2 1 x y -1 -1 1 -1 1 x 2 -1 1 2 -1 x -1 1 -1 1 y 2 -1 1 2 y -1 2 -1 -1 1 2 -1 1 -1 1 x 2 -1 1 2 1 -1 1 y 2 -1 1 2 1 x y -1 -1 2 1 -1 1 2 -1 1 -1 1 x 2 -1 1 2 1 -1 1 y 2 -1 1 2 1 x y -1 -1 1 2 -1 1 -1 1 x 2 -1 1 2 -1 x -1 1 -1 1 y 2 -1 1 2 y -1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ArcCsc", "[", "x_", "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSech", "[", "y_", "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", RowBox[List["ArcCsc", "[", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["-", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", FractionBox["1", "y"]]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", "y"]]]]]], "-", FractionBox["1", RowBox[List["x", " ", "y"]]]]], "]"]], "\[Pi]"]]], "]"]]]]], RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", FractionBox["1", "y"]]]], " ", SqrtBox[RowBox[List["1", 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FractionBox["1", RowBox[List["x", " ", "y"]]]]], "]"]], "\[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[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]]]], "+", FractionBox["1", RowBox[List["x", " ", "y"]]]]], "]"]], "\[Pi]"]]], "]"]]]]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["Arg", "[", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], "-", FractionBox["\[ImaginaryI]", "x"]]], "]"]], "+", RowBox[List["Arg", "[", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]], "+", FractionBox["\[ImaginaryI]", "y"]]], "]"]]]], 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02