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; }

 ArcCsch

 http://functions.wolfram.com/01.29.16.0186.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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