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; }

 ArcCoth

 http://functions.wolfram.com/01.28.16.0191.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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