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 ArcCsc

 http://functions.wolfram.com/01.17.16.0125.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ArcCsc", "[", "x_", "]"]], "+", RowBox[List["ArcSin", "[", "y_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["y", "x"]]], "+", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]], " ", RowBox[List["ArcSin", "[", RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", "y"]], "+", FractionBox[SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]], "x"]]], "]"]]]], SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["y", "x"]]], "+", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]], "2"]]], "+", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["-", FractionBox["y", "x"]]], "+", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["y", "x"]]], "+", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]], "2"]]]]], ")"]]]], "-", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["-", FractionBox["y", "x"]]], "+", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["y", "x"]]], "+", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]], "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[RowBox[List["\[ImaginaryI]", " ", "y"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["-", FractionBox["y", "x"]]], "+", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["y", "x"]]], "+", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]], "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[RowBox[List["\[ImaginaryI]", " ", "y"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]]]]]]

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

 2007-05-02