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/01.29.16.0221.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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