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 ArcCsc

 http://functions.wolfram.com/01.17.16.0172.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2007-05-02