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/01.29.16.0170.01

 Input Form

 ArcCsch[x] + I ArcTan[y] == (-((I x Sqrt[-((-1 - I Sqrt[1 + 1/x^2] x y)^2/(x^2 (1 + y^2)))] Sqrt[1 + y^2])/(1 + I Sqrt[1 + 1/x^2] x y))) ArcSinh[(I Sqrt[1 + 1/x^2] x - y)/(x Sqrt[1 + y^2])] + (Pi (1 + I Sqrt[1 + 1/x^2] x y))/ (2 x Sqrt[-((1 + I Sqrt[1 + 1/x^2] x y)^2/(x^2 (1 + y^2)))] Sqrt[1 + y^2]) + Pi I (1 - (I (1 + I Sqrt[1 + 1/x^2] x y))/ (x Sqrt[-((1 + I 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 - y)/Sqrt[1 + y^2]])/(2 Pi)] - Pi I (1 - (I x Sqrt[-((1 + I Sqrt[1 + 1/x^2] x y)^2/(x^2 (1 + y^2)))] Sqrt[1 + y^2])/(1 + I Sqrt[1 + 1/x^2] x y)) Floor[-((-Pi + 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["\[ImaginaryI]", " ", RowBox[List["ArcTan", "[", "y", "]"]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "x", " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["\[ImaginaryI]", " ", 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["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", "x", " ", "y"]]]]]]], RowBox[List["ArcSinh", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", "x"]], "-", "y"]], RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]]], "]"]]]], "+", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", 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["\[ImaginaryI]", " ", 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["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", 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["\[ImaginaryI]", " ", 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]", "-", "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["\[ImaginaryI]", " ", 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["\[ImaginaryI]", " ", 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]", "-", "y"]], SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]]]]]]]]

 MathML Form

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

 Rule Form

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

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

 2007-05-02