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 ArcCsch

 http://functions.wolfram.com/01.29.16.0152.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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