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/01.17.16.0152.01

 Input Form

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

 Standard Form

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

 MathML Form

 csc - 1 ( x ) + csch - 1 ( y ) 1 - 1 x 2 1 + 1 y 2 - x y ( 1 - 1 x 2 1 + 1 y 2 - x y ) 2 sin - 1 ( 1 + 1 y 2 x + 1 - 1 x 2 y ) + π arg ( 1 - 1 x 2 + x ) + arg ( 1 + 1 y 2 - 1 y ) 2 π ( 1 - 1 - 1 x 2 1 + 1 y 2 - x y ( 1 - 1 x 2 1 + 1 y 2 - x y ) 2 ) + 1 2 π ( 1 - 1 - 1 x 2 1 + 1 y 2 - x y ( 1 - 1 x 2 1 + 1 y 2 - x y ) 2 ) - π ( 1 - 1 x 2 1 + 1 y 2 - x y ( 1 - 1 x 2 1 + 1 y 2 - x y ) 2 + 1 ) - arg ( 1 - 1 x 2 + x ) - arg ( 1 + 1 y 2 - 1 y ) + π 2 π x y 1 -1 1 x 2 -1 1 2 1 1 y 2 -1 1 2 -1 x y -1 1 -1 1 x 2 -1 1 2 1 1 y 2 -1 1 2 -1 x y -1 2 1 2 -1 1 1 y 2 -1 1 2 x -1 1 -1 1 x 2 -1 1 2 y -1 1 -1 1 x 2 -1 1 2 x -1 1 1 y 2 -1 1 2 -1 1 y -1 2 -1 1 -1 1 -1 1 x 2 -1 1 2 1 1 y 2 -1 1 2 -1 x y -1 1 -1 1 x 2 -1 1 2 1 1 y 2 -1 1 2 -1 x y -1 2 1 2 -1 1 2 1 -1 1 -1 1 x 2 -1 1 2 1 1 y 2 -1 1 2 -1 x y -1 1 -1 1 x 2 -1 1 2 1 1 y 2 -1 1 2 -1 x y -1 2 1 2 -1 -1 1 -1 1 x 2 -1 1 2 1 1 y 2 -1 1 2 -1 x y -1 1 -1 1 x 2 -1 1 2 1 1 y 2 -1 1 2 -1 x y -1 2 1 2 -1 1 -1 1 -1 1 x 2 -1 1 2 x -1 -1 1 1 y 2 -1 1 2 -1 1 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["ArcCsch", "[", "y_", "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]]], "-", FractionBox["\[ImaginaryI]", RowBox[List["x", " ", "y"]]]]], ")"]], " ", RowBox[List["ArcSin", "[", RowBox[List[FractionBox[SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]], "x"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]]]], "y"]]], "]"]]]], SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]]], "-", FractionBox["\[ImaginaryI]", RowBox[List["x", " ", "y"]]]]], ")"]], "2"]]], "+", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]]], "-", FractionBox["\[ImaginaryI]", RowBox[List["x", " ", "y"]]]]], SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]]], "-", FractionBox["\[ImaginaryI]", RowBox[List["x", " ", "y"]]]]], ")"]], "2"]]]]], ")"]]]], "-", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]]], "-", FractionBox["\[ImaginaryI]", RowBox[List["x", " ", "y"]]]]], SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]]], "-", FractionBox["\[ImaginaryI]", RowBox[List["x", " ", "y"]]]]], ")"]], "2"]]]]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", 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["1", "y"]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]]], "-", FractionBox["\[ImaginaryI]", RowBox[List["x", " ", "y"]]]]], SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]]], "-", FractionBox["\[ImaginaryI]", RowBox[List["x", " ", "y"]]]]], ")"]], "2"]]]]], ")"]], " ", 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["1", "y"]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]]]]]]

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

 2007-05-02