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 ArcCsc

 http://functions.wolfram.com/01.17.16.0165.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2007-05-02

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