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; }

 ArcSec

 http://functions.wolfram.com/01.18.16.0145.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2007-05-02