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/01.17.16.0185.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2007-05-02